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# Entries from ElementaryMath

TwentyFirstCenturySkills 17 Jul 2005 - 21:02 CatherineJohnson

### update

In fact, teaching young children to build the next set of math facts on the math facts they already know is a good idea.

I'm pretty sure Parker & Baldridge recommend this approach (I'll check).

for more on 21st century skills, see MoreSingaporeMath

WickelgrenOnIntroducingAlgebra 08 Jul 2005 - 17:19 CarolynJohnston

I've been looking again at one of Catherine's favorite books, Math Coach (by Wayne and Ingrid Wickelgren).

Wayne and Ingrid have a lot to say about what they consider the most difficult aspects of elementary math -- long division and fraction manipulation. But it's what comes after that that interests me now: their discussion of the importance of teaching algebra early. Wayne suggests that the most important thing you can show your kid, what should motivate them most to want to continue in math, is the power of algebra to solve hard problems.

Most problems in prealgebra and early algebra start out something like this:

John is 27 years old. If his age is 3 times Pete's age, how old is Pete?

If you have a kid like Christopher or Ben, you know he's going to spit out the answer on the spot and tell you not to waste his time with this stupid letter stuff.

That's why Wayne Wickelgren suggests that, when you're ready to introduce your kid to the notion of algebra, the first thing you should do is sit down with him and let him watch you do a problem like this one:

In two years, Jean will be twice as old as Chris will be. In six years, Jean will be four times as old as Chris was last year. How old is Chris now?

In short, start with a demonstration of how algebra-at-your-fingertips gives you mindblowing powers. I was reading this last night and thinking: if I tell him that this problem is what algebra is all about, Ben will be blown away. Why scare him off? Maybe start with something simpler...

But the hard thing about this sort of problem isn't going to be doing the algebra: it's going to be setting up the equations, given the word problem. And that's going to be hard no matter how I try to teach it. Doing the mindless rote stuff required to crank out the answer, once you have the equations, is the easiest part of the problem. And I know Ben: he'll think that's the cool part.

Given that, I can't see a reason to hold off introducing algebra. Once a kid is at the sixth or seventh grade level in math, the heck with guess-and-check and pan-balance problems; the heck even with bar models. The most general tool that we currently have for solving word problems, and the only one that we have that isn't stymied by some word problem or other, is algebra. He may as well be motivated to go full speed ahead with the letters and symbols. Wickelgren says that algebra is the key to the castle; it's the most effective means for solving tricky math problems that's ever been devised. As such, you want it to be the tool that kids reach for instinctively when they have a tricky math problem to solve.

Here's a quote from a great article by Ethan Akin, "In Defense of Mindless Rote":

On the other hand, mathematics is cumulative and there are a great many skills that you have be unthinkingly familiar with. Every grumpy calculus teacher will tell you that most of the problems his students have come from weaknesses in algebra. For the students who say "I really understand it but...." the but is that for them algebra is not easy background knowledge. They are trying to build on a foundation of dust. A lot of college majors need a bit of calculus or statistics which are simply walled off to students who don't have sufficient skills in algebra. These are basically not hard subjects but they appear unnecessarily terrifying to such students.

Conversely, a practiced facility with algebra can provide its own positive reinforcement. Not only is the mathematics built on the algebra, but facility in algebra gives the student confidence in the face of new mathematical challenges. As the above discussion makes clear, such confidence is entirely justified.

I am motivated now to try to introduce real algebra by the end of the summer. No more pussyfooting around!

Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing

VisualLearningKThru2WikiPage 17 Jul 2005 - 16:51 CatherineJohnson

The comments thread of this post will be the page Becky C requested on the use of direct, explicit visual aids in teaching K-2 math.

Everyone can comment, edit & revise, so please share your experience & thoughts.

PartitiveAndQuotitivePedagogy 11 Jul 2005 - 17:45 CarolynJohnston

Catherine mentioned in one of her comments that she always finds it amusing when a mathematician encounters the notion of partitive vs. quotitive division:

I absolutely think there's all kinds of elementary math knowledge real mathematicians don't have, or did have but forgot, etc.

I always crack up when i see or read real mathematicians reacting to the 'partitive'-'quotitive' distinction in division.

They think it's ridiculous!

(And btw, I STILL can't explain the difference, so I'm not even going to bother to try....)

She's absolutely right. When I first encountered the notion of partitive vs. quotitive division (Liping Ma goes into a lot of detail about it in her book) I thought it was unnecessary obfuscation.

I know I never learned it myself. I don't know if my teachers knew it, but I know they never taught it to me (although Liping Ma says they didn't need to). And I don't know whether I need to know it in order to teach young children the full meaning of division, although Liping Ma says I do.

But as it happens, I do know what the difference is: my husband explained it to me in brilliantly simple terms (having learned it at the same time I did, and distilled its meaning more efficiently than I did). Here it is:

Partitive problems ask you to divide number of objects by number of groups, and get number of objects as an answer.

the partitive type of word problem asks this question: if I have x objects, and I want to split them into y groups, how many objects will be in each group?

Examples of partitive problems:

I have a board of length 16 inches, and I need to make 10 shorter boards of equal length out of it. How long can each board be? (16 objects, 10 groups)

I have a batch of 128 cookies. I need to split it into 8 equal bags of cookies. How many cookies will there be in each bag? (128 objects, 8 groups)

I have 12 cans of pears, and I need to serve 24 kids at lunch. How many cans of pears will each kid get? (12 objects, 24 groups)

It is somewhat difficult to frame word problems involving division by fractions as partitive problems, because you are dividing by the number of groups you want. Generally, you don't want a fractional number of groups. Note that in the problems I gave as examples of partitive division, the divisors are always whole numbers.

But here is a partitive word problem that uses a fractional divisor:

I have two cans of dog food that I need to split into 1-1/2 servings for my big and small dog. How many cans will be in a single serving? (2 objects, 1-1/2 groups -- awkward!)

Quotitive problems ask you to divide number of objects by number of objects, and get number of groups as an answer.

the quotitive word problem asks: If I have x objects, how many groups of y objects can I make from them?

Examples of quotitive problems:

I have a board of length 16 inches, and I need boards of length 1-3/4 inches. How many short boards can I cut from the longer board?(16 objects, 1-3/4 objects)

I have a batch of 128 cookies. I need to split it into bags of 12 cookies to give to children at school. How many such bags can I give away? (128 objects, 12 objects)

I have 12 cans of pears, and I need to serve a half can of pears to every kid at lunch. How many kids can I serve? (12 objects, 1/2 objects)

Problems involving division by fractions are easier to frame as quotitive word problems. Note that in the first and third sample problem, the divisor is a fraction; I didn't have to gin up an awkward problem involving big and small dogs in order to give you an example of quotitive division by a fraction.

Liping Ma's only point vis a vis quotitive and partitive division is that teachers should know the difference. It doesn't have to be explicitly laid out for the kids. But teachers need to know about it because they need to give a mix of types of word problems. She says that it may be obvious to us that numerically they are the same problems (in fact it is SO obvious that we miss the distinction!), but to the kids it may not be.

I'm not sure that's true, but I'm willing to give her the benefit of the doubt.

Liping Ma actually gave a set of US and a set of Chinese elementary school teachers the following problem: frame a word problem for 1-3/4 divided by 1/2.

The best of the Chinese teachers gave examples of both partitive and quotitive word problems; they were all able to give at least one word problem for the division. But some of the US teachers couldn't do the calculation.

The difference: in China, elementary math teachers are respected for what they do, and given time to consult with each other in order to improve their pedagogical knowledge. Elementary Chinese math teachers are specialists in math education.

Catherine has studied the Liping Ma book very carefully. I think Catherine concluded that the fundamental problem in the US is that teachers need release time to consult with each other and improve their knowledge.

I believe that the fundamental problem is that teaching is not a respected profession in the U.S., and that the other problems -- lack of release time, and mathematical weakness in the teachers themselves -- all follow from this.

MorganOnLearningModalities 18 Jul 2005 - 00:30 CarolynJohnston

This is a comment by Carolyn Morgan on the WillinghamOnLearningModalities thread: it's a beauty, so I'm posting it.

Things I note about her teaching approach:

• It's actually multimodal! She's talking, she's getting the kid to write AND draw his own pictures, she's doing whatever it takes to get the kid on track.
• It's feedback-driven: she switches gears in midstream if the kid isn't getting it.

Carolyn's post:

SteveH is correct. Whatever you do, you must bring all students to UNDERSTANDING. And this is what students really do want. "Understanding" or "not understanding" are the reasons students "hate" or "love" math.

A good math teacher learns how to "approach" a student having difficulty. The teacher has all of these ideas (hopefully) stored away back there some place, ready to be pulled out when needed. But a teacher's most important job is determining where the student's understanding fell apart, identifying where there might be gaps in reasoning, and knowing how to bridge those gaps. This is where choosing the right approach comes in. It might involve reteaching, reviewing a step that is being omitted, or helping a student reason through a difficult story problem.

So a hand goes up, and a student says, "I need help."

(Those are my favorite times of the math hour because it means I get to find the puzzle piece that is needed to make this all fit together in his/her head and give understanding to what I've just taught or to what's needed to solve this problem.)

So I have some choices, but I always look to see what the student has already done or tried. That tells me what to do next.

I then start by having the student read the problem to me (if it is a word problem).

Then I make a choice:

I might say, "OK, draw Bill's house. Now write 'B' on it for Bill. Now, draw the schoolhouse; now write 'S' on it. Now, draw the road from the house to the school. Now, look at the problem again to see how far it is to the school (and the student answers outloud 4 1/2 miles). OK, write that number on the map you've just drawn."

I could have drawn that little map for the student, and might do it under certain conditions, but having a student draw the map involves his sight and his movement (and mouth from speaking and ears from hearing his own voice) and it involves more importantly his BRAIN. (I've got to make sure his BRAIN is working and focused on the problem so he can "understand".)

So I would continue, "Start at Bill's house with your pencil; now walk to school. OK how far did Bill just walk? ('4 1/2 miles') OK, write that down. Now, he's at school, but he wants to come home, so have Bill walk back home. How far did Bill walk to get home? OK, write that down under the first number. How would you know how much he walked to school and back on that one day? ('add the two numbers together.') Good, do that. OK, but that is just one day. Now, let's read the problem one more time and let's see what the questiion was. (How far does Bill walk in a week going to school and back?) OK, now how could we figure that out?"

Many times the problem just works itself out in the student's brain as they begin to draw out a picture of the problem.

Or, if I've checked over his work and seen that he's added 4 1/2 miles 5 times for the 5 days of the week, I can see that he's overlooked a part of the question. So I have the student reread the question. Many times, the student will catch his own mistake when he hears his voice repeat the words "to school AND BACK". If not, I have him read just that part again.

Something really important: for some students it's just a matter of not knowing "how or where to get started". There are gaps in processing the information and gaps in understanding.

Not only does he need to know where to start, but he needs to know that where he is starting will get him going in the right direction and will help him get the right answer. This is very important to a student's confidence. If a student doesn't know "where to start" or isn't sure "if he's going about solving it properly", a teacher's trying to find the right modality isn't necessarily the answer.

This is where an "constructivist" approach is so devastating to the student. That student wants to be able to KNOW what to do to get the right answer.

It's terribly upsetting and deflating to a student not to know "where to start" and "if they're taking proper steps to solve the problem correctly." To leave this student to come up with his own idea isn't helping him.

Hopefully, though, a teacher will NOT just repeat the instructions that were given initially (if any were given). If a student didn't get it the first time, at least try a different approach.

One of my former students said of another teacher, "Why should I ask her for help? She always just repeats the same instructions that didn't make sense the first time." This student, smart as a cookie, just wanted to understand the entire process and to know how to work to get the right answer.

Carolyn Morgan On Conceptual Gaps
Congratulations Carolyn Morgan

MathAndTextPrototypeLesson 21 Jul 2005 - 13:56 CatherineJohnson

When I was in graduate school (DID I MENTION THAT I HAVE A PHD IN FILM STUDIES?) one of my professors told me that the definition of a reader is a person who owns more books than he can read before he dies.

I have now updated that definition for the impending ERA OF THE BLOOKI.

The definition of a reader is a person who owns so many books she can't even get her own web site read before she dies.

Now that's out of the way, I have managed to make a circuit of my favorite blogs this afternoon--and have discovered that J.D. has his prototype lesson up at Math and Text!

It looks wonderful.

I'm going to read it now.

### update

It is wonderful.

I love clean, lots-of-white-space invitations to maths...and there was something about the final lesson on figuring out which number is larger that made me happy.

That sensation is so reinforcing, that I think it ought to be an item on textbook write's & editor's lists: Does the student feel a click?

I was confused by just one part of the lesson, which was the first visual display. A middle school teacher has left a detailed comment explaining why she stumbled over it, too.

Take a look.

### update 2: more on the click

I'm realizing I've had many, many conversations in which people who like math bring up the click--that moment of knowing you've got it.

Either you've got the right answer, or you've got the concept.

That's what my cousin was talking about when she said it's incredibly boring never to know whether you got the right answer or not:

It’s boring when you don’t have the light bulb go off in your mind because, ‘Oh! I got it right!’

The best you could think was, ‘Well, maybe I got it right.

Our friends Fred & Wendy were here a couple of weekends ago, and Fred said exactly the same thing about maths.

He loved maths (I may have to give up on 'maths'....) and he wanted to study it at Yale, as an undergraduate. What he especially loved was the click.

He quickly realized that college-level maths was a different animal, and he shifted to statistics, eventually earning a Ph.D. in experimental psychology (and then a law degree after that).

Fred is a seriously smart guy (clerked for one of the Supremes, etc.).....and what's he talking about when he remembers math?

The click.

FirstPerson (interview with my cousin about Everyday Math)

CarolynMorganOnConceptualGaps 18 Jul 2005 - 19:27 CarolynJohnston

CarolynMorgan, who wrote the material in MorganOnLearningModalities, has written some more on conceptual gaps in students. She asked me to include it in her earlier post -- but that one was just perfect; just the right message and length. So I'm going to post the new piece here.

This highlights a teaching strategy that we used to use a lot in teaching at the college level, and on ourselves when learning new and difficult research material -- if a kid is stuck, have him work through a much simpler but still analogous example. Then work your way back up to the original problem.

Conceptual gaps

Often no matter what modality you try, a student just won't get it because there are conceptual gaps in his learning.

A precious student comes to mind. He had trouble with fractions. He just didn't understand the concept. He could recognize fractions, write fractions, read fractions properly. He could even (through tough perseverance) add, subtract, multiply, divide fractions. But the concept of a fraction was fuzzy. Very fuzzy.

His learning center teacher and I and other previous teachers, I'm sure, had him cut pies in half, divide groups into equal parts, but he continued to have those gaps. Something was just not clicking and none of us knew what it was, or wasn't.

I say all of this because that "conceptual gap" showed itself, not in computation of fractions, which he became very good at. It showed it's ugly head in the middle of a word problem, that most 5th graders could handle with very little trouble. A specific example comes to mind, which I will share now.

There is a problem in Saxon 6/5 something like this one:

Joe walked 288 feet, to the end of the pier and back. How long was the pier?

This student had no idea how to go about solving this problem. When he asked for help, I realized that it was because he really didn't understand what a fraction was, and how finding a half of 288 feet would solve his problem. I think he knew that there were two distances, but he didn't see them as halves.

To begin with the number was too huge for his mind to grasp. So I picked up his pencil and drew a line on his paper and said, "OK, here is another pier, but it's a very short pier. And when Joe walked on this pier to the end and back, he had walked 10 feet. How long was the pier?

He immediately, said, "Five feet."

I said, "Good for you. How did you know that?"

His answer was, "because 5 + 5 = 10". Notice how his mind was working. He still didn't see it as half of 10. That was why he couldn't solve the 288 feet problem. He didn't know two numbers that equaled 288. But he did know 2 numbers that equaled 10. And he picked 5 because he knew the 2 numbers had to be equal. But he didn't see it as halves.

So I knew we were only a part of the way there.

So I said to him, "OK, now, let's think about how we could work that problem so we could start with the 10 feet and know that he had walked 5 feet one way? Let's see if you can do another one and maybe that will help. Let's make another pier and make it shorter. (I'm drawing the pier.) OK, Joe walked to the end of the pier and back and he had walked 8 feet. How long is the pier?"

He immediately said "4 feet".

And so I said something like, "OK, when Joe walked 10 feet to the end and back, the pier was 5 feet long. (And we wrote that information down by the pier I had drawn.) And when Joe walked 8 feet to the end and back, the pier was 4 feet long (and we labeled that pier also)."

Now, my question: "OK, how could we work that problem to figure out that answer?"

And bless his heart. He said, "2 divided into 10". (Now I would have preferred that he say, "ten divided by two" but I was not going to quibble at this juncture.)

"Good for you," I said. "Now let's try that on another one. (drawing another pier) If Joe walked 20 feet going to the end of this pier and back, how long is the pier? How could we work that problem?"

And he understood the answer, and he smiled and wrote it.

"Now," I said. Let's look at our problem in the book. Joe walked 288 feet to the end of a LOOONNGG pier and back. How can we figure out the length of this long pier?

A HUGE, HUGE GRIN burst all over his face, as he said, "2 divided into 288".

It took getting down into smaller numbers of which he had some concept. He knew 8's and 10's. He could grasp numbers that size. And from there, he was able to know "how" to do the problem. He didn't understand fractions any better. He was just so happy that he knew how to get the right answer. And he felt successful.

That story happened two years ago. I don't know when he will fully grasp fractions. In a new story problem, in a new setting, he may have to be led all over again, step by step, from something small and "graspable" to something larger. And he may need that help for many years. I just hope he has teachers who understand him.

MorganOnLearningModalities
Congratulations Carolyn Morgan

CognitiveHoles 19 Jul 2005 - 16:27 CarolynJohnston

Bernie and I were talking tonight, and he told me a story that worried me a bit.

Ben came to visit us at work the other day, and wanted to get a snack from the vending machine. So he went into his dad's office and asked for some money. Bernie gave him a few coins, and Ben went into the snack room, picked out what he wanted, and put his money into the machine; but he didn't have enough. So he came in and asked for more; but he couldn't tell Bernie how much more he needed. He didn't seem to have much sense of how much more he needed, either.

Well, it wouldn't be the first time we came across this sort of gap in his understanding. We have a sort of a family byword for these things, very much like Catherine and Ed's no-common-sense-y; we call Ben's gaps his Cognitive Holes. They are located in unexpected places -- they're generally about something, like handling coins, that you think is very easy by comparison with other things he can do, like long division. And they tend to be very big gaping holes in his knowledge, and at first they were very frightening. But we come across them less often now than we used to, and we've found that once we know they are there, we can remediate them pretty quickly.

So I thought this was another run-of-the-mill Cognitive Hole.

Well, you tackle these by filling them in. Ben and I were ready for a change from what we've been doing lately, anyway (introductory equations, solved by adding and subtracting). We've been doing them all week, and struggling, and we finally got a 'click' a couple of nights ago (those babies are practically audible, aren't they?), and last night when he took his section test he got a 100. So tonight, when it was math time, instead of doing algebra, I got out some coins.

I had 3 quarters, and a dime. "OK, you're at our work, and you want a snack, and these are the coins I have", I told him. "The snack you want costs 60 cents. Which coins do you take?"

He went for two of the three quarters, and the dime. Good. "How much do I get back from the machine?" I asked. Nothing: good.

"OK, your snack costs 40 cents". He goes for the two quarters: he tells me the machine returns a dime.

"The snack costs 80 cents." He takes all the coins, and tells me the machine returns 5 cents.

In short, he passed my common sense test with flying colors, and Math Time was fun and a breeze for once. So what the heck was happening the other day? In short, what part of this Cognitive Hole we think we've uncovered am I not mapping correctly?

Tomorrow, we try it a little differently; we'll simulate the precise problem we had the other day with the snack machine at work. I'll give him too little money, tell him the snack costs a certain amount, and get him to tell me how much more he needs.

There may in fact be no Cognitive Hole, this time, just some situational rigidity. This is the deal with smart people on the autism spectrum; sometimes they know what they need to know, they just stiffen up when it comes time to apply it in the real world.

TitlesOfConstructivistMathCurricula 19 Jul 2005 - 01:46 CatherineJohnson

Jo Anne Cobasko has taken the time to construct a complete list of NCTM standards based math programs.

### update: Department of Corrections

This list is David Klein's handiwork, not Jo Anne's.

Thank you, David! (For everything you do.)

All of us should keep this handy, because none of these programs ever calls itself constructivist, and schools don't seem to advertise this piece of information, either.

When I first raised the issue of TRAILBLAZERS being a constructivist curriculum with a teacher on the textbook selection committee, she looked at me blankly. I got a number of those blank looks before I discovered that everyone in the school knows what the word constructivism means, and knows what a constructivist curriculum is.

The reason I know this is that I finally read the original committee report, which states explicitly that the new curricula must have a constructivist approach with modeling. I was a little behind the curve there.

### Elementary school

Everyday Mathematics (K-6)
TERC's Investigations in Number, Data, and Space (K-5)
Math Trailblazers (TIMS) (K-5)

### Middle school

Connected Mathematics (6-8)
Mathematics in Context (5-8)
MathScape: Seeing and Thinking Mathematically (6-8)
MATHThematics (STEM) (6-8)
Pathways to Algebra and Geometry (MMAP) (6-7, or 7-8)

### High school

Contemporary Mathematics in Context (Core-Plus Mathematics Project) (9-12)
Interactive Mathematics Program (9-12)
MATH Connections: A Secondary Mathematics Core Curriculum (9-11)
Mathematics: Modeling Our World (ARISE) (9-12)
SIMMS Integrated Mathematics: A Modeling Approach Using Technology (9-12)

### Programs explicitly denounced by over 220 Mathematicians and Scientists:

Cognitive Tutor Algebra
College Preparatory Mathematics (CPM)
Connected Mathematics Program (CMP)
Core-Plus Mathematics Project
Interactive Mathematics Program (IMP)
Everyday Mathematics
MathLand
Middle-school Mathematics through Applications Project (MMAP)
Number Power
The University of Chicago School Mathematics Project (UCSMP)

printable page

Thanks, Jo Anne, for taking the time to do this!

key words:
DavidKlein
listofconstructivisttextbooks
constructivist textbooktitles
NSFfundedcurricula

WhyIsSubtractionHarder 18 Jan 2006 - 14:23 CatherineJohnson

Christopher is sitting here doing his mixed practice, and he just asked me, "Why is subtraction harder than addition?"

He was doing the problem:

\$20 - e = \$3.47

I have no idea why subraction-with-borrowing is harder than addition-with-borrowing, or even if it is harder.

I'm asking all of you because I've noticed that sometimes the answer to incredibly simple-seeming questions tell you a huge amount that you didn't know before. Can't think of any examples offhand, but I'm going to start keeping track.

### update

Oh!

It's probably the left-to-right issue, yes?

QuestionAboutReciprocals 31 Jul 2005 - 18:04 CatherineJohnson

My 'benchmark' for the moment when I understand elementary mathematics well enough to move on is:

reciprocals

I find reciprocals utterly mysterious.

They're not quite in the magic category anymore, which is my benchmark for complete and total lack of comprehension. If a maths concept seems like magic, that means I know nothing.

Of late, inside the expanding math section of my brain, reciprocals have put a toe outside the magic category. But not much more than a toe.

### Danica McKellar

Tuesday's SCIENCE TIMES has a profile of Danica McKellar, who played Winnie on Wonder Days. It turns out she's a UCLA math major who published a proof (pdf file) now known as the Chayes-McKellar-Winn theorem.

### car wash problem

McKellar's web site has a mathematics section where she answers reader questions. She's a natural born teacher:

Q: Hi Danica, I heard a question from Mr. Feenie on a "Boy Meets World" episode which he claimed to be unanswerable. After hearing that, I decided to figure it out. If it takes Sam 6 minutes to wash a car by himself, and it takes Brian 8 minutes to wash a car by himself, how long will it take them to wash a car together?

Let's do it: This is a "rates" problem. The key is to think about each of their "car washing rates" and not the "time" it takes them. Alot of people would want to say "it takes them 7 minutes together" but that's obviously not right, after you realize that it must take them LESS time to wash the car together than either one of them would take.

So, what is Sam's rate? How much of a car can he wash in one minute? Well, if he can wash one car in six minutes, then he can wash 1/6 of a car in one minute, right? (think about that until it makes sense, then keep reading). Similarly, Brian can wash 1/8 of a car in one minute. So just add their two rates together to find out how much of a car they can do together, in one minute, as they work side by side on the same car: 1/6 + 1/8 = 7/24 of a car in one minute. That's their combined RATE. (Note: that's a little bit less than 1/3 of a car in one minute). From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done! Yes, I think they should work together, it gets done much more quickly that way. :)

By the way, you said when you watched the TV show you decided that YOU would figure it out, right? How did you do?

I love this. McKellar is teaching two things here:

• how to solve a rates problem
• how to read, study, & learn maths (that's metacognition)

First of all, she knows that math novices transfer their normal reading habits to maths books. By normal reading habits I mean that most of us, when we read a book of prose, read straight through at a fast clip, pausing only to underline or make notes in the margin.

You can't read a math book that way; in fact, I've come to feel you can't really read a math book at all. You have to do a math book, or work a math book. McKellar explicitly instructs her reader not to read the solution straight through, but to stop at key points and ponder until he or she gets the point, and is ready to go on to the next point.

She doesn't stop there, either. She also knows the precise spots in her explanation where most novices will need to stop and mull, and she tells them where those spots are.

She's giving novices direct instruction in metacognition.

As to the problem itself, she addresses the most common error novices will make confronting this particular rates problem, which is:

• figure that it takes the 2 boys 14 minutes to wash 2 cars
• so logically it must take them 7 minutes to wash 1 car

Amazing! And all in the space of a few short paragraphs.

I think McKellar's teaching skill here is connected to her acting. There's a large element of 'performance' in teaching, at least in my experience, and to be a good performer you have to know where your audience is, what they want to hear & what they need to hear.

She does.

### back to reciprocals

Here's my reciprocal question.

From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done!

I don't understand why you would use the reciprocal to solve this problem.

I understand perfectly well (let's hope) why you would divide 24 by 7.

I didn't even know you could use the reciprocal to find the answer.

### 7 fact families

I haven't had time to sit down and think this through, but I suspect the reciprocal answer to a rates problem is the same concept as the 7 fact family I put together after teaching the Primary Mathematics lesson on ratio & proportion (Primary Mathematics 6A Textbook, p. 21-46):

7 fact families

Dan K and Carolyn Morgan have given me some incredibly helpful advice on teaching fractions 'in a hurry.' I'll get it pulled up front as soon as I can, but in the meantime take a look at the Comments thread.

TwoMathEdBlogs 27 Jul 2005 - 19:26 CatherineJohnson

Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC."

And btw, these are not prerequisites for a serious college math course:

A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.

This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)

2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.

3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)

4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.

5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”).

6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)

7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

also added to the list by commenters:

I would add estimation and verification to that list. Students should know the difference between a sensible and nonsense answer.

Another blog by a college calculus professor: Learning Curves

NewStudyOnManipulatives 29 Jul 2005 - 17:27 CatherineJohnson

I want to follow up on Carolyn's post on Congressional math incentives, but before I do that here's a reminder:

Anne Dwyer has posted new notes on her summer math class.

And...quickly checking her page just now, noticed this comment:

So, what have I learned so far?

• they like games where they compete with one another
• they prefer pencil and paper exercises
• they like to figure out puzzles

This is fascinating, because Kevin Killion, of Illinois Loop, just pointed me to a new article in Scientific American showing that manipulatives are less effective than pencil and paper with young children. I'll write a bit more on this later (bike ride time!), but here's the critical passage:

Teachers in preschool and elementary school classrooms around the world use "manipulatives"--blocks, rods and other objects designed to represent numerical quantity. The idea is that these concrete objects help children appreciate abstract mathematical principles. But if children do not understand the relation between the objects and what they represent, the use of manipulatives could be counterproductive. And some research does suggest that children often have problems understanding and using manipulatives.

Meredith Amaya of Northwestern University, Uttal and I are now testing the effect of experience with symbolic objects on young children's learning about letters and numbers. Using blocks designed to help teach math to young children, we taught six- and seven-year-olds to do subtraction problems that require borrowing (a form of problem that often gives young children difficulty). We taught a comparison group to do the same but using pencil and paper. Both groups learned to solve the problems equally well--but the group using the blocks took three times as long to do so. A girl who used the blocks offered us some advice after the study: "Have you ever thought of teaching kids to do these with paper and pencil? It's a lot easier."

Dual representation also comes into play in many books for young children. A very popular style of book contains a variety of manipulative features designed to encourage children to interact directly with the book itself--flaps that can be lifted to reveal pictures, levers that can be pulled to animate images, and so forth.

Graduate student Cynthia Chiong and I reasoned that these manipulative features might distract children from information presented in the book. Accordingly, we recently used different types of books to teach letters to 30-month-old children. One was a simple, old-fashioned alphabet book, with each letter clearly printed in simple black type accompanied by an appropriate picture--the traditional "A is for apple, B is for boy" type of book. Another book had a variety of manipulative features. The children who had been taught with the plain book subsequently recognized more letters than did those taught with the more complicated book. Presumably, the children could more readily focus their attention with the plain 2-D book, whereas with the other one their attention was drawn to the 3-D activities. Less may be more when it comes to educational books for young children.

This perfectly supports the study Carolyn mentioned way back when, showing that fraction manipulatives are good for middle schoolers.

CA state study on manipulatives
Fraction Manipulatives
Fraction Manipulatives Part 2
New Study on Manipulatives Part 2

WichitaBoyOnMath 31 Jul 2005 - 22:15 CatherineJohnson

We have an embarrassment of riches! At least 2 great comments from WichitaBoy, and Ed sat down and wrote out his constructivism-as-psychoanalysis thoughts, too.

Here's one of WichitaBoy's observations:

"Writing is organizing." Now there's a great thought I can take to the bank.

Here's one back for you: professional mathematics is organizing. You have vague thoughts, you notice a vague pattern, and you try to organize your thoughts, to nail down the pattern, to really clarify what's going on beneath the hood. When you've nailed it completely, when you understand with perfect perspicacity the essence of the pattern, then you've got a proof of a new theorem. If you've really organized it, you've got a theorem that goes in the "Book of God".

There's this, too, in response to my saying that what is brilliant about Saxon Math--the structure--is largely invisible:
Read Confucius or Socrates. The ideal teacher should be able to fade into the background like the Cheshire cat. And so with the ideal textbook.

BernieOnVisualVersusSymbolicUnderstanding 03 Aug 2005 - 00:40 CatherineJohnson

Bernie Johnston (you may recognize the last name) left this comment in response to my post about visual versus symbolic understanding of math:

Not all mathematical understanding is visual. If it were, all mathematics would be geometry. That's essentially the view the ancient Greeks had, but the development of symbolic computation and an efficient number system have rendered that view obsolete by allowing us to approach mathematical understanding from other quite different directions.

Various mathematicians think in various ways. I've known strong algebraists who literally cannot draw a 3-dimensional cube and understand it. On the other hand, I've known topologists who could see 4-dimensional space in their mind's eye the way I can see a cube.

I prefer to think of different mathematical approaches as analogous to taking pictures of a house. You can take a picture from the front or from the sides or from the back, but none of these ipso facto captures the entire house. And even after viewing the pictures you might want to do a walk-through before buying.

So, while I don't know the definitive answer to the question of exactly what conceptual mathematics is, I'm certain that it need not be a visual representation alone.

I was relieved to learn this.

I think one problem for me, in trying to learn & re-learn math, is that I really have no idea what it looks like when a person has learned math.

All I really know is....they're good at math, majored in math, took graduate degrees in math, and now do or do not work in jobs requiring all kinds of math I don't understand. Makes it difficult to have any sense at all of where I'm headed, and how far I've come.

It's in my cart!

BasicCollegeMathematics 02 Aug 2005 - 01:14 CatherineJohnson

A round-about path from Vlorbik to Tall, Dark, and Mysterious to Basic College Mathematics at mathnotes.com.

Scroll down.

MathAndTextPrototypeLessonRevision 03 Aug 2005 - 17:05 CatherineJohnson

I've just noticed that J.D. has posted his revision of his prototype lesson at MathandText.

I can't wait to read it.

### update

OK, I have NOT read J.D.'s revision, because my copy of Adobe Reader has completely and totally gummed up my Mac.

It never ends.

MathForumArchivedNewsletters 14 Aug 2005 - 01:37 CatherineJohnson

I've just been alerted to a terrific resource, the Math Forum Newsletter.

They have an article about Kitchen Table Math in the latest issue! (Although so far I haven't been able to find it.....I don't think....)

Sigh.

However, I have managed to attach and display the logo they sent me!

BestPerformingStudentsPartThree 14 Nov 2005 - 02:32 CatherineJohnson

The question of how our top students compare to everyone else's top students has made me realize I need to be paying attention to this. My goal as a homeschooler-on-the-side is for Christopher to be able to major in a math-related subject in college if he chooses, which apparently means he should be able to score a 625 or higher on TIMSS.

So I'm going to start scouting information on all ranges of student achievement, and posting it here.

Here's my first:

Researchers determined which items students who achieved at the various levels on the total test were likely to get right. Then they placed the items on a scale from 200 to 750. So we have a pretty good idea of what the best students know that others have difficulty with.

Only the top 10 percent of 9-year-olds were likely to get this math item right. Students had to explain their answers verbally, symbolically or pictorially.

In the first part they had to indicate that 20 is twice as large as 10 or that 10 is half of 20. 10 percent of third graders and 21percent of fourth graders did this. A small number of students (less than 1 percent in any country ) received credit for satisfactory explanations even though they did not give a yes or no response to whether Julia was right.

U.S. percentages were 13 percent at third grade and 25 percent at fourth grade.

For the second part, only 6 percent of third graders and 15 percent of fourth graders responded correctly. 6 percent of U.S. third graders and 17 percent of U.S. 4th graders got credit. However, 30 percent or more got credit in Japan, Korea and Singapore.

I'm going to spring this one on Christopher tomorrow. I really can't tell whether he could have gotten this item right at age 9. If you showed him 10 girls and 20 boys he would have known instantly that boys and girls weren't half and half.

But I tend to think he would have been thrown by the sight of the numbers '10' and '20.'

As well, I'd say this problem imposes a high cognitive load. You have to keep Juanita and Amanda straight in your mind, unless you've developed seriously good informal chart-making skills, which Christopher has not done now and certainly had not done in 4th grade.

Christopher turned 11 yesterday (boo hoo).

His first impulse, as I feared, was to say 'yes,' Amanda is right.

He obviously had the 'environmental dependency' effect of seeing the numbers '10' and '20' and thinking: 1/2.

But then he corrected himself, and said, confidently, that Juanita is right and Amanda is wrong. (Nice to see that the Designated Stupid Person concept has spread to TIMSS, too.)

His explanation was a bit strangled, but it was right. He said, 'Well, if there's 1 girl for every 2 boys, then there's 1 girl and 2 boys, then 2 girls and 4 boys, then 3 girls and 6 boys...'

This is pretty interesting, because I think he had a 'number sense' or 'pattern' way of getting this answer. In other words, I think he got the answer without really knowing why or how he got it. He just knew it. Juanita's correct statement of the problem instantly became his statement of the problem; he didn't have to do any adding or subtracting or logical reasoning to test Juanita's statement.

Then, when I asked him to explain why Juanita was right, he explained how her answer would work as a kind of Fancy Skip Counting Mechanism. If you kept counting up by 2-to-1 ratios, eventually you'd hit 30 kids, and your ratio would be 10 girls, 20 boys.

After he gave this illustration I asked him, 'how many girls and how many boys would there be in the class' (forgetting that in fact THE PROBLEM TELLS YOU THIS UP FRONT) and Christopher said, instantly, '10 girls and 20 boys.'

When I asked him how he knew (TIMSS should just have 'Catherine' be the Designated Stupid Person) he said, 'I just knew it.'

Apparently he had forgotten the fact that we'd been given this information, too. Like mother like son.

In any case.....this is something I was talking to Carolyn about the other night: what is the relationship of implicit knowledge to expertise when you're talking about math?

Certainly in every other field (I think) implicit knowledge is a sign that you're getting good at what you do, because you don't have to think about it. You 'just know it.'

But math has been confusing for me in this realm.....our friend Fred was here a few weekends ago, and I asked him to take a look at a RUSSIAN MATH problem that was stumping me. Fred is a Big Brain; he went to Yale undergrad, then got a Ph.D. in experimental psychology at Stanford, I think it was; then got a law degree at Yale; then clerked for the Supreme Court.

So I hope you're impressed.

Anyway, Fred was keenly interested in math when he went to college, but pretty quickly found out that pure mathematics wasn't going to be for him.

### anti-constructivist digression

"I always loved finding the right answer," he said.

This is SO important; it's one of the core pleasures of math. Finding the right answer. Radical constructivists gleefully snatch this pleasure this pleasure away, the drips.

### back on topic

Anyway, once he realized that pure mathematics was beyond him, Fred moved to statistics. Looking at the Russian Math problem, he instantly knew how to do it. But he didn't know why He knew.

This was yet another Problem Involving Reciprocals, and Fred said, 'I don't know why I knew to use the reciprocal there.'

So......

This is where I get confused.

Fred is a super-smart person with, I would say, high expertise in elementary math & in applied math. On the other hand, he isn't doing a math-related job as a career, so maybe he's no longer in the 'expert' category after all these years. I don't know where to put him.

So I don't know what to think about the fact that he could instantly solve the RUSSIAN MATH problem, but didn't know why his solution worked. Is that a sign that he has advanced knowledge (because people with advanced knowledge often 'just know' things they can't explain), or a sign that he doesn't?

This brings me back to Christopher.

Watching and listening, I felt like the fact that he instantly knew Juanita was right was a sign he's developing expertise. It was as if math is starting to be 'in his bones.'

On the other hand, I don't think he could show me how to do the problem, if the problem were too advanced to do just by eyeballing it. (If the numbers weren't 'friendly.')

Actually, that's a good question. In the next day or two I'll find out what he would do with a more complicated version of this question.

How good are our best?
BestPerformingStudentsPartTwo
a word problem only the top 10% of 9 year olds solve
England vs America vs Singapore

LipingMa 24 Aug 2005 - 20:09 CatherineJohnson

Here's Liping Ma:

Multiple Approaches to a Computational Procedure: Flexibility Rooted in Conceptual Understanding

Although proofs and explanations should be rigorous, mathematics is not rigid.... Dowker (1992)asked 44 professional mathematicians to estimate mentally the results of products and quotients of 10 multiplication and division problems involving whole numbers and decimals. The most striking result of her investigation "was the number and variety of specific estimation strategies used by the mathematicians." "The mathematicians tended to use strategies involving the understanding of arithmetical properties and relationships" and "rarely the strategy of 'Proceeding algorithmically.'"

"To solve a problem in multiple ways" is also an attitude of Chinese teachers. For all four topics, they discussed alternative as well as standard approaches. For the topic of subtraction, they described at least three ways of regrouping, including the regrouping of subtrahends. [they also talked about which approach worked best in which situation] For the topic of multidigit multiplication, they mentioned at least two explanations of the algorithm. One teacher showed six ways of lining up the partial products. For the division with fractions topic the Chinese teachers demonstrated at least four ways to prove the standard algorithm and three alternative methods of computation.

For all the arithmetic topics, the Chinese teachers indicated that although a standard algorithm may be used in all cases, it may not be the best method for every case. Applying an algorithm flexibly allows one to get the best solution for a given case. For example, the Chinese teachers pointed out that there are several ways to compute 1 3/4 dividedby 1/2. Using decimals, the distributive law, or other mathematical ideas, all the alternatives were faster and easier than the standard algorithm. Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations--the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics.

ExtendedResponse 08 Nov 2005 - 22:52 CatherineJohnson

My sister-in-law, a fantastic teacher in central Illinois, says the Big New thing in math is extended response. She's going to fill me in when she finds out what it is.

In the meantime, I found this page of released extended response items on the ISAT.

### my extended response to extended response

OK, my initial reaction to extended response is: I'm against it.

Actually, make that mixed. My initial response is mixed.

Here's one of 2 released 2004 extended response gr5 items:

A company makes a wall calendar each year. The company sells ad space
around the calendar to local businesses. The cost of ad space is based on
the number of square units each ad contains. The company charges \$40.00
for Ad Space D. Using this information:

Draw an Ad Space that costs exactly \$60 in the gridded space on page 10 of

And here's the illustration:

I like this problem, although wiser heads here at ktm may give me reasons why I shouldn't, in which case I'll revise my opinion.

I like it because it's visual & spatial as well as 'numerical' (if that's the right word), and because I've found Christopher to be very challenged by any problem that asks him to combine numerical thinking or problem-solving with spatial 'thinking' or problem solving. And of course I love the Singapore bar models, and this problem reminds me of them.

I also like it because it has 2 steps: you have to figure out how much each square costs & then you have to figure out how many squares \$60 would buy.

I like the open-endedness of this particular problem, too. A child could simply count the number of squares in Ad Space D (40) and then divide 40 dollars by 40 squares to get \$1/square. Or he or she could notice that Ad Space D is a standard multiplication array, and multiply 4 by 10 to get 40. I'm sure a lot of kids would start out counting & then notice, mid-stream, that they could have arrived at their answer more efficiently by multiplying instead. Which is good. A little Math Object Lesson buried inside a story problem.

I like that!

Last but not least, I kind of like the fact that each square turns out to cost exactly one dollar. I don't know why. It reminds me of a genre of problems in Russian Math, in which you go through all kinds of elaborate, painstaking calculations only to end up with an answer of ONE. Or maybe TWO. Or, when things get really fancy, ONE HALF.

Interestingly, I'm finding, as I work my way through RUSSIAN MATH, that I'm becoming quite attached to the number one. Every time it crops up as an answer I think: I should have seen that coming. An answer of one always seems like a flag, a sign that there was an easier, more elegant way to do whatever it was I was doing.....but I missed it.

Russian Math has all kinds of 'surprise answers,' and I think a surprise answer in the middle of an ISAT could be slightly.....fun?

An answer of one is like a little joke.

### What I don't like...

...is the injunction to Explain in words how you got your answer and why you took the steps you did to solve the problem.

That is a terrible, terrible idea for a test.

It's a good thing to do on homework once in awhile, or in the classroom. RUSSIAN MATH asks students to write out explanations, although it doesn't ask students to explain how they did a problem. It asks them to restate the definitions & explanations given in the lesson.

Items like these can't possible be graded well on tests. They are far too time-consuming, and graders will end up scoring on length or number of explanations given. When you have items like these teachers are going to end up devoting all kinds of class time to writing extended responses, as Susan H says is already happening. We're looking at a massive waste of teachers' and students' time.

Last but not least, I'd bet the ranch you learn nothing from the verbal explanation that you didn't already learn by looking at the student's work.

Being able to produce a fluent, intelligible verbal explanation of a mathematical solution is almost certainly important for math teachers.

It's not important for the rest of us.

### I really don't like this one

The number of fifth-grade students going to the museum is greater than 30
but less than 50. Each student will have a partner on the bus. At the
museum, each tour group will have exactly 6 students.

How many students are going to the museum?

you took the steps you did to solve the problem.

Unless 5th graders in Illinois are doing a lot of prime factor problems, I don't see any reason to include an item like this one on a timed assessment.

First of all, no one should have to be doing discovery ON A TEST.

And second, this problem has two answers (36 & 42, right?), but the wording implies that it has just one answer, and that one answer is findable.

I am DISCOVERING the fact that I don't think red herrings belong in math classes. Certainly not in elementary school math classes.

What is the point? You are teaching children to distrust the English language at the precise moment they're learning grammar & composition. An unreliable narrator in a work of fiction can be a terrific device.

But an unreliable questioner in an examination is just wrong.

I'm against it.

### update: I forgot 48!

sigh

(thank you, Dan K)

### extended response in 8th grade

Here's the 2004 released 8th grade item:

Peter sold pumpkins from his farm. He sold jumbo pumpkins for \$9.00
each, and he sold regular pumpkins for \$4.00 each. Peter sold 80 pumpkins
and collected \$395.00.

How many jumbo pumpkins and regular pumpkins did he sell?

you took the steps you did to solve the problem.

The problem is fine, assuming these kids have actually been taught some algebra.

If they haven't, this is a discovery problem on a timed assessment, and I'm against it.

So, assuming they've learned how to set up & solve equations with unknowns, the problem is good. IMO.

The demand that the student explain each step in words is not.

### Russian Math rocks

Instead of writing about Russian Math, I should be downstairs (at the kitchen table!) actually doing some Russian Math.

So I think I'll sign off.

But tomorrow I'll give some examples of what a proper extended response item should be.

A proper extended response item should be a RUSSIAN MATH EXTENDED RESPONSE ITEM.

### update: scoring rubric for extended response

'Student Friendly' Mathematics Scoring Rubric

Assuming I'm reading this correctly (I feel a little distrustful), students must get all computations correct in order to earn the highest possible score of 4. They can earn a score of 3 with minor mistakes in computation, which I feel is fair, though others may disagree.

What I reject absolutely is the explanation section:

• I write what I did and why I did it.
• If I use a drawing, I can explain all of it in writing.

This is wrong. I don't believe a 4 should depend upon being able to supply an explanation in any case. But here you have a child who can explain why he or she did what she did in a drawing, which is no mean feat (and I'm in a position to know) and even that isn't enough.

Pace Anne, you'll notice that it's not OK for a child to explain what he/she has done by offering a mathematical demonstration, as the teachers in Liping Ma's book do. Anne's right about that; it struck me, too. Over and over again, when Liping Ma asks a Chinese teacher why he/she teaches an idea a certain way, the teacher responds by writing out a proof-like mathematical demonstration. That's what makes the book incredibly difficult (and incredibly valuable) to read for most of us; the teachers don't translate math into words, and neither does Ma.

For Chinese teachers, math is math.

This drops you to a 3:

• I write mostly about what I did.
• I write a little about why I did it.
• If I use a drawing, I can explain most of it in writing.

A couple of years ago the head of our school board sent out an email explaining the adoption of TRAILBLAZERS that included this line (from memory): In recent years math has become language-based.

WallStreetJournalSingaporeMath 12 Sep 2005 - 19:32 CatherineJohnson

I'm teaching my little Singapore Math class again this fall, in the Main Street School after-school program. Last year I had one blinding success, a boy who took to the Singapore bar models like a fish to water and decided, apparently as a direct result, that he liked math and wanted to do well in it. He was a Phase 3 kid, now boosted to Phase 4!

So I'm looking foward to it.

(The other kids all did great, too; I don't mean to draw negative comparisons. They just didn't experience major life epiphanies as a result of drawing bar models.)

I was revising my course writeup today, and had to go hunting for the WALL STREET JOURNAL article on Singapore math, which I apparently had neglected to post anywhere on the site. So here's the link.

Excerpts:

Singapore's curriculum was developed over the past few decades by math experts hired by the Ministry of Education, who continually interviewed math teachers to find out what works and where kids need help. The elementary textbooks cover only one-third of the topics typically found in U.S. textbooks, but the material is taught far more thoroughly. While rote learning plays a part, kids in Singapore also learn to use visual tools to understand abstract concepts.

Singapore math texts, for example, ask kids to draw bars and other diagrams to visualize problems -- a technique called "bar modeling." When this strategy is applied consistently over a number of years, children tend to be better able to break down complex problems and do rapid calculations in their head.

[snip]

The National Council of Teachers of Mathematics in the U.S. suggests that it might not be possible to copy what Singapore's done simply by importing its books. The success of its math program may have roots in Singapore's highly disciplined culture, where the entire community -- particularly parents -- expects kids to buckle down and work hard, argues the NCTM.

There's little doubt, though, that math teaching in America needs to be overhauled. Tuesday, Boston College will release a four-year global study that is expected to show the math gap with Asia remains. The college's last study, the 1999 Trends in International Mathematics and Science Study (TIMSS), ranked eighth-graders in Singapore the best in math, while U.S. kids came in 19th, just behind Latvia. American kids also fall further behind the longer they're in school; as fourth-graders, American kids ranked 7th on the 1995 study.

### today's horror factoids:

• Among U.S. freshmen who plan to major in science or engineering, one in
five requires remedial math courses

• Enrollment by U.S. citizens or permanent residents in graduate science and engineering programs, meantime,
dropped 10 percent between 1994 and 2001. Enrollment of foreign students grew 35 percent.

another link to the WSJ article: As math skills slip, U.S. schools seek answers from Asia

key words: decline in U.S. engineering math and science enrollment

EverydayMathLongDivision 13 Sep 2005 - 15:06 CatherineJohnson

Thanks to NYC HOLD I have a graphic of Everyday Math's substitute division algorithm. TRAILBLAZERS teaches the same approach, which it calls 'forgiving division.'

...instead of teaching long division, students are taught to divide numbers using the partial products method, a technique where children guess how many times a number goes into another and keep subtracting the guesses until they come up with the answer (see box). This method works, but it takes more time and doesn't allow the student to divide past the decimal point.

[snip]

Isaacs and others defend the alternative algorithms by explaining that they teach students how math works. The partial product method of division, for example, is a lot more transparent to students than the long division method.

I'm sure he's wrong about this. I found partial product division quite confusing myself when I used it.

otoh, I think partial product division might work as a teaching tool when used on simple demonstration problems. (I tried it on a complicated division problem and got completely lost mid-stream.) I might use a problem like 16 divided by 2 to show that division is repeated subtraction, analogous to multiplication being repeated addition.

I haven't tried it with any children just learning long division, but if I ever get a chance to, I'll take notes.

### the honeymoon

Some parents like the program as well. "It's sort of incredible," said Susan Pottinger, whose son Theo attends kindergarten at P.S. 261 in the Cobble Hill section of Brooklyn. "For him it's great fun. He's fascinated by numbers. He sees patterns everywhere," she said. "He'll put shoes away and alternate shoes with sneakers and say, 'See I'm making a pattern with my shoes.' "

We parents (well, some of us) spend those early elementary school years in a wonderland. Then the you-know-what hits the fan in 5th grade.

source:
Weighing the Factors Does the City's Standardized Math Curriculum Measure Up? By Amy Sara Clark

### update

Lone Ranger supplies this link to lattice multiplication, the method Everyday Math teaches children when they cover multiplication. Carolyn points out that lattice multiplication is distinctly opaque; it obscures rather than reveals the fact that multiplication depends on the distributive property.

Here's another link to lattice multiplication at Math Forum Carolyn posted awhile back.

why long division?

Everyday Math's alternative division algorithm
forgiving division
forgiving division, part 2
try this with forgiving division
who says long division is hard?
Everyday Math division algorithm fighting innumeracy at CO
conceptual understanding vs numbers

keywords: Columbiajournalismstudent EverdayMatharticle

KumonMathInDetroit 17 Nov 2005 - 13:28 CatherineJohnson

fyi:
KUMON math program

I've had an amazing email from an engineering professor who learned of Kitchen Table Math while she was in China (!)

(Apparently, not being listed on Google isn't a problem in China.)

She also sent me a copy of her paper on Kumon supplementation in Detroit schools (the results were incredible), and I'm waiting to see whether it's OK to post. In the meantime, she says it's fine to post her email:

I'm sure you must have come across Kumon mathematics? I'm a professor of engineering at Oakland University, and so mathematics is obviously very important to me. As a consequence, to make up for the problems with the American school system I've had my own daughters in the Kumon program for about ten years each--between the ages of three and thirteen. Their math skills are far better as a result. I was so impressed with the ideas behind Kumon (it is an outstanding supplement that provides the additional practice missing from K-12 math), that I started a program using the Kumon method in a local inner urban school district, Pontiac. The results are described in the attached paper.

Kumon provides the easiest, smartest way I've ever seen for a Mom to help her kids with math. I couldn't recommend it more highly.

One last thought. I've taught in China as well as the US. The US is definitely way ahead on the "creativity" side. But we are so far behind in math that it is ridiculous--and it is potentially crippling for our source of engineers and other professionals. There are many aspects involved in good engineering, for example, where a good math background is critical. Most of the engineering professors where I work now (Oakland University), are foreign born. Although I greatly respect my foreign-born colleagues, it's really an indictment of the American system that we can so rarely grow our own any more.

Thanks for your blooki, which I have bookmarked and will be following!

### Kumon for children with severe disabilities, too?

And, in a follow-up:

Actually, the woman who ran one of the Kumon centers I brought my children to originally got into Kumon because she saw how much it was helping a profoundly mentally disabled child who she was working with. So I suspect it may be surprisingly beneficial for Andrew. I couldn't have done the outreach in my local inner-urban outreach without the incredible help I got from Doreen Lawrence, the Vice President of Research for Kumon, North America. Her phone number is 248-755-2587, and her email is dlawrence@kumon.com. Doreen is a wonderful person who is deeply oriented towards helping children. I'm sure she'd be glad to answer any questions you might have about Kumon (she knows EVERYTHING about the program).

You can feel free to post anything from my letter that might help. I just apologize for the poor writing. I just got back from China and am still jet-lagged.

Over the next week or two I'll read through your website more carefully and get a better feel for what's going on (I just found out about your website while I was in China, but scarcely had any time available while I was there). I've a lot of thoughts and background information related to what you're doing, and have some interesting and relevent experience with national policy setters in academia on this topic, but am a little bogged down now working on a book, research papers, experiments, and grant proposals. You know, the usual academic stuff! So I will try posting some once I feel I understand more fully what you are doing and how you are doing it.

Thank you ever so much for providing a forum for something that is so important to our children!

Her name is Barbara Oakley & she has had an amazing life (e.g., she met her husband at the South Pole.....)

Plus--and I MUST post this--she's started a page of things she finds funny, which, thus far, has one link to a pdf file of what looks to be a PowerPoint presentation: Yours is a Very Bad Hotel.

All you World Traveling Kitchen Table Math denizens will relate.

### it's getting clearer now

Back when Carolyn and I started Kitchen Table Math, my one question was: Why?

Why exactly, in the middle of my life, am I spending 18 hours a day WRITING A MATH BLOG? Excuse me, a MATH BLOOKI.

This was my husband's question as well.

I'm just coming off a newyorktimesbestseller, the goal nonfiction writers spend their careers aspiring to reach.....shouldn't I be Following Up with another book? (I will follow up with another book; Temple and I are working up steam. But still. Kitchen Table Math is a detour.)

So what was I thinking?

Somehow, it seemed like I was supposed to be writing a math blooki.

That reason turns out to be, in large part, the people who write comments and set up pages and create dimensional dominoes and, now, send me an email out of the blue telling me I need to take Andrew to Kumon.

That is exactly what I need to do. I need to take Andrew to Kumon.

Andrew is my little locked-in boy; he's bright--so bright, it's there, you can see it--and I don't know how to reach him.

The folks at Kumon may not know how to reach him, either, but it's obvious to me I'm supposed to give it a shot. If they don't know, something there will give me a new idea. It's a lead.

I wasn't going to figure this out on my own.

I was telling my neighbor about this today, complaining that I can't think of these things myself. I have to have complete strangers tell me: take your severely autistic son to Kumon Math.

My neighbor said, 'You can never think what you're supposed to do about your own life.'

OakleyPapersOnline 19 Sep 2005 - 17:20 CatherineJohnson

Chris Adams found all of Barbara Oakley's research papers posted at her web site (something I probably could have done if I hadn't gotten sidelined by the humor page.....)

This is why it's a bad idea for me to try to learn math from textbooks with pictures of diving penguins.

Thank you, Chris!

### update

Oh, boy.

I'm gonna be reading all of her stuff.

Check out this title: IT TAKES TWO TO TANGO: HOW ‘GOOD’ STUDENTS ENABLE PROBLEMATIC BEHAVIOR IN TEAMS

This paper was written to describe a successful program developed to forestall non-cooperative behavior in team-related activities, and to provide an explicit guide for students on how to handle such problematic behavior if it does arise. The program involves creating self-awareness of the deleterious effects of typical, seemingly ‘nice’ behavior in a dysfunctional team situation. Indeed, it has proven to be a revelation to many students to find that their ethical, industrious, and well-meaning responses to non-cooperative behavior can often enable such unacceptable behavior to continue and even escalate.

I myself have Personally Experienced the deleterious effects of seemingly nice behavior in a Dysfunctional Team Situation, and I've never had the first clue how to deal with it.

Mostly I just fume and glare and fire off furiously angry body language in all directions, & end up looking like a lunatic.

I once did this on cable TV, trying to speak my piece at a school board discussion of TRAILBLAZERS.

### update update

OK, this paper is not going to solve my looks-like-a-lunatic-at-school-board-meetings problem.

It's about dealing with Hitchhikers & Couch Potatoes.

More t/k.....

MathLessonsPage 21 Sep 2005 - 15:48 CatherineJohnson

I've started to get the Math Lessons page pulled together. I'm sure I've forgotten posts that should be indexed there, so if you know of any, let me know. (Any lessons you especially like from other people's sites, like MathandText, for instance, should also be added.)

There's a link to 'Math Lessons' on the sidebar.

TeachnologyFreeWorksheets 21 Sep 2005 - 20:07 CatherineJohnson

Teachnology seems like a useful site.

Here are free online word problem worksheets.

And here are lots of free math worksheets.

I like this addition and subtract equations worksheet.

BestMadMinutesBook 22 Sep 2005 - 04:06 CatherineJohnson

I'm teaching the Singapore Math after-school class again, and I don't want to use Saxon's 5-minute sheets.

I need a 1-minute sheet (or online source).

Thanks--

OnlineMathResources 22 Sep 2005 - 22:30 CatherineJohnson

I came across all kinds of interesting-looking math web sites last night while looking for:

• integers worksheets

I didn't find either of the things I wanted (and almost spent \$29.95 to join some teacher site linked to by FunBrain just to be able to printout their number line sheet...).

But I found all of these:

• AAA Math (resources listed by grade thru gr8)
also has a potentially interesting page called World Education Levels. Unfortunately, I can't tell what 'world education levels' are without spending a lot more time on the site than I want to spend. LOTS of online quizzes that are corrected by the site, and they seem to be selling a software program on arithmetic.

• the aforementioned FunBrain Math Baseball is a classic.

• FunBrain's teacher site, the page that almost sold me a \$30 sheet of number lines. Has articles on behaviormanagement in the classroom that look good.

• Harcourt School Publishers' number line express Blecch. But maybe little kids would enjoy it. There's a talking lion railroad engineer.

• Math Cats how-to for teachers Definitely worth looking at.

• math clip art! possibly for autistic kids (I was on a major clip art tear a few years ago, when Andrew was in his PECS genius phase...)

• Mathsurf teacher's site word problems from Pearson Scott Foresman. If you're looking for story problems with multiple answers, this is the spot. Possibly (probably?) a good site to visit for problems your child may encounter in constructivist math courses -- worthwhile problems, as far as I can tell on cursory inspection.

• Mathsurf telling time worksheet (to print)

• Room 108 Looks decent. You can create online Mad Minute pages (must be answered & graded online)

• odd & even numbers possibly good for autistic kids? this site speaks the directions, although I don't think the directions are also written out in words. But any time an autistic child can hear the same words spoken by the same recorded voice it's a good thing, I believe. Site is simple and graphically compelling. Has a HUGE cursor (also great for autistic kids.)

• Primary Games good for autism? I have a feeling this might work with Andrew at some point in the near future. Very simple, has ONE moving image--'Squigly,' a little worm inside one of 10 apples who pops out of his apple and then disappears back inside every couple of seconds. The child has to tell which apple Squigly is in (first, third, fifth, and so on). The only bad part is that there's a lot of advertising crud at the top and the bottom of the page.

• Primary Games fishy counting game good for autism? terrific. Very, very simple counting game (as nice as the counting game they used to have on the Barney web site....

• Primary Games Tetris bubbles Great! I've been meaning to post a TIME MAGAZINE article saying girls improve their spatial-visualization skills when they play Tetris. This is, I think, a somewhat slower version of a Tetris game. (Slower is always good for me....) Stupid music, though.

• Primary Games time clock Terrific! Very simple & cute. You have to be able to use a mouse (Andrew & Jimmy both have huge MOUSE difficulties, unfortunately.)

### eureka

I will never, ever speak ill of the NCTM again.

They have FREE NUMBER LINES, 8 to a page!

Unfortunately, all 8 number lines start at 0 and contain only positive numbers....

### update

I take it back.

They do not appear to have posted a single number line on their web site that includes negative numbers as well as positive numbers and 0.

keywords: online interactive math resources tools nets manipulatives

WickelgrenOnYoungChildrenAndMath 17 Sep 2006 - 01:14 CatherineJohnson

back story:

My neighbor, the statistician, showed me her copy of Math Coach: A Parent's Guide to Helping Children Succeed in Math quite awhile back, before either of our kids had had any trouble in math class. I ordered a copy just because I order lots of copies of books I'd like to read but then don't.

So the book was sitting there on my shelf when Christopher came home with his 39 on the Unit 6 test & I subsequently failed to teach him fractions using SRA Math. I needed help.

It was the right book at the right time. A page-turner.

Most of what I believed to be true of math ed & math achievement, I discovered, was wrong. Severely wrong. I had been operating on the basis of sheer ignorance, naivete, and boneheaded cliche.

This is the observation that probably shocked me the most. It appears in Wickelgren's chapter on finding a school for your child:

There are schools with even less structure than Eastside. Take the Sudbury Valley School, a private K-12 school in a Boston suburb. This school gives each child complete freedom to choose how they spend their time at school. There are no classes except those specifically requested by a group of students. Children learn largely on their own, reading books, talking to each other and to teachers or outside experts, solving problems, playing games and sports, practicing musical instruments, doing arts and crafts, and anything else that can be done on the school grounds.

While you can read at length about the school's strengths on its web site, one of its biggest potential benefits is that every child can proceed at his or her own pace, in math and in other subjects as well.

There are also potential drawbacks. Since young children are not generally highly motivated to learn math, they may choose not to study much of it.

I was bowled over.

I had always thought kids want to learn things they're good at. Christopher is good at social studies, and he wants to learn it. At night he'll bug his dad to 'give me trivia questions.' (Give me superficial facts, Daddy!) Ed finally refused to do it anymore, because he ran out of trivia.

Christopher also has a collection of geography trivia books that he reads, and when he was 7 I read all of the first volume in the History of US series out loud to him as his bedtime story.

That was the book he wanted to hear.

So...I assumed kids wanted to learn subjects they had a talent for.

According to Wayne Wickelgren, this is not the case with math.

Or, at least, not generally. Math talent doesn't (necessarily) manifest itself in an obvious desire to learn the multiplication tables. (Or to write essays on My Special Number.)

### late bloomers

That one observation pretty much changed my life. I decided, then and there, that I didn't know whether Christopher had any talent for math or not, or what his eventual level of interest in the subject might be--or, more importantly--could be, given a decent education K-12.

I also knew he had good general intelligence, which meant he had the ability to learn a whole lot of math whether he was going to end up in a math-related career or not.

I decided right then and there that that was what was going to happen. Christopher was going to learn math, lots of it, and learn it well.

We were going to keep the doors open.

When Christopher reached college, he would be in a position to decide to pursue a math-related career or not. That decision would not have been made for him in 3rd grade, when he got sorted into Phase 3.

It wasn't too long after this that I met Carolyn and heard her story: flunked algebra in high school (right?), didn't decide to major in math until senior year in college, then got a Ph.D. In math. Another wake up call.

### more late bloomers

Two more stories.

One comes from Christopher's 4th grade teacher. Her daughter was reaching the end of high school, and it was time to do SAT prep.

So her mom hired a tutor, and within a couple of weeks the guy was reporting that her daughter had strong talent in math.

She had no idea. Neither she nor her daughter had the first clue that this kid had a knack for math. Now, working one-on-one with a tutor who, IIRC, had a Ph.D. in math (or engineering, possibly) she was flying.

I have no idea where that girl will end up, what she'll major in, or which job or career she'll pursue.

It doesn't matter. The point is: she's good at math, and she went through 11 years of formal education thinking she wasn't.

### you can't predict the future, or even the past

Story number two comes from a friend of ours. As a boy he had two or three chums who sat by each other in class & were bright kids. They were the kind of kids who could learn whatever you threw at them, and they got As in all their subjects & went to good colleges & universities. They got As in math, too, of course, but none of them was a whiz. Our friend became a lawyer.

One of the gang shocked everyone by growing up to become a world-famous econometrician.

No one can understand how this happened. This kid never showed any special talent for or interest in math. He was just a smart kid, like the rest of them. Our friend said that to this day, whenever any of them get together, they always ask each other how that friend could turn out to be not only an econometrician, but a world-famous one.

Go figure.

What I like about this story is the fact that not only could this boy's future as World Famous Econometrician not be predicted when he was 8, it can't be back-predicted now, when he's 40.

### Barbara Oakley's bio

I just remembered: Barbara Oakley is in the same category. Here's her bio:

I started studying engineering much later than many engineering students, because my original intention had been to become a linguist. I enlisted in the U.S. Army right after high school and spent a year studying Russian at the Defense Language Institute in Monterey California. The Army eventually sent me to the University of Washington, where I received my first degree–a B.A. in Slavic Languages and Literature. Eventually, I served four years in Germany as a Signal Officer, and rose to become a Captain. After my commitment ended, I decided to leave the Army and study engineering so that I could better understand the communications equipment I had been working with.

Barbara sent me an email that I won't quote without her permission (I'm WAY behind on email). But her story inside an email is more dramatic than her story here, though no different in outline. Barbara is a person who earned an entire B.A. degree in a humanties field and served a full stint in the Army before figuring out she wanted to major in engineering.

And the reason she decided to study engineering is pretty similar to the reason I've suddenly decided to study math; she got tired of not understanding the stuff she was working on. In her case, that was communications equipment; in my case it's K-12 math.

Obviously, Steve H is right, we simply cannoy be assigning grade school kids to our two Standing Committees: math whiz & math's not his thing.

### all English Language Arts all the time

from The Learning Gap by Harold Stevenson and James Stigler:

....American teachers like to teach reading; Asian teachers like to teach mathematics. When we asked teachers in Beijing, nearly all of whom were women, the subject they most liked to teach, 62 percent said mathematics, 29 percent said language arts. The reverse was found in Chicago: 33 percent mentioned mathematics and 47 percent mentioned language arts. There is more to the story than preference, however. Americans simply emphasize reading more than mathematics. Despite the large amount of time already spent in reading instruction, more than 40 percent of the suggestions made by Minneapolis mothers who wanted an increased emphasis on academic subjects said they thought that the subject should be reading. Fewer than 20 percent mentioned mathematics.

These data lead to the obvious conclusion that American children do less well in mathematics than do Chinese and japanese children partly because they spend less time studying mathematics....Conversely, American children may fare better in reading, relatively speaking, because they spend more time on this sujbect.

I mentioned yesterday: it's a commonplace for people to say, 'I was never any good at math.'

No one says, 'I was never any good at reading.'

### English Language Arts in Irvington

I've seen this here in Irvington.

My sense is that Irvington does a good job teaching reading. Not that I know what I'm talking about, but that's my sense. (fyi, after trying to teach out of the SRA Math book myself, I also think our grade school teachers are near-geniuses at teaching math, too.....& I'm not kidding about that. It was tough.)

• 2 periods of English language arts, one for reading & one for writing
• 1 period of social studies, taught by a teacher who told us, on back to school night, "I am an English language arts teacher at heart"
• 1 period of drama

That's 4 periods out of 8, half his day devoted to English language arts. He has 1 period for math, 1 period for science, and that's it. The other 2 periods are specials: study skills, music, art, drama, P.E., technology. Technology will mean creating an online 'portfolio' of his best work in 6th grade, not learning how to program. Study skills is about reading & taking notes, not doing problem sets.

And, on back to school night, the math teacher told us the kids would be keeping a math journal, because a lot of kids in accelerated math probably aren't as strong in ELA, so 'we try to help them with English language arts.'

Thus far she has done nothing of the sort, thank heavens, and she's stopped grading the kids' math tests on spelling, which she did last year. I gather she had a lot of complaints about it, and I made a point of asking her, in front of the other parents, whether she would be grading spelling this year, too. (This is what we call a warning shot.) So she told the kids she wouldn't, and she hasn't. otoh, Christopher is now spelling parenthesis parenthies, so be careful what you wish for.

### another story

This last story pretty much sums it up, I think.

I know I've mentioned the fact that we were clueless back when Christopher was in his early elementary years.

So, unbeknownst to us, he was placed in Phase 3 ELA as well as Phase 3 math. Actually, we're still clueless; I have no idea what kind of sorting & phasing they do with ELA. All I know is that in K-5 they divide the kids up into ability groups within the classroom, rather than separating them into different classes taught by different teachers, as they do with math.

In the hall outside Christopher's 4th grade class, after the year was over, I happened to run into his teacher and we fell into conversation, which led to the subject of Christopher's progress that year. I remember I was expressing gratitude for some especially good teaching she'd done, but I don't remember the details. It was probably about English language arts, since she taught him every subject but math.

One thing led to another, and suddenly I heard her saying, "Oh, I could see when he came into my class he wasn't a 3. He was much better than that. Sometimes you just have to ignore the tests."

So when was he a 3?

It took me a moment to recover, but I managed to keep her talking. "I pushed him," she said. "I knew he could do it." And, again: "You can't believe the tests."

Wow.

Here we have your dufus mom, completely out of the loop about tests, 3s, & 4s. And it doesn't matter; it doesn't hurt the kid. The teacher steps up to the plate, checks out the kid, decides for herself 'he's not a 3,' then sees to it he stops being a 3, and becomes a 4.

No extra reward, no extra praise, no extra payment or promotion. She just does it, because it's her job, and because she's good at it.

Perfect.

(And yes, I know; I'm tired of 3s and 4s, too. But 3s and 4s are a kind of shorthand, and a useful one.)

The point is: I have never heard this story told about a Phase 3 kid in math. Never.

Until this fall (that's another story), only a tiny handful of kids had ever moved from Phase 3 to 4. Maybe one 1 per year.

I've talked to the Chair of the middle school program about this issue, to one of the guidance counselors, to our 4-5 principal, and to numerous other teachers & parents.

Not one of them has mentioned the school or a teacher pushing a kid out of 3 and into 4. Whenever a move is made, the impetus has come from the parent, not the school. And the school resents it. (I've mentioned this before. We have a meta-narrative about pushy parents pressuring the school to put their kids in Phase 4 math when they don't belong there. Everyone subscribes to this narrative, including aides & other parents.)

The lesson I take away from this is that we really do have some major talent in some schools in this country, in the teaching of English Language Arts. I'm lucky to have my own kids in one such school district.

WickelgrenOnPrealgebra 16 Jul 2006 - 20:48 CatherineJohnson

Gulp.

A student can learn a year of pre-algebra math in three to six months studying three to ten hours per week, depending on the child's math aptitutde.

I'm gonna have to pick up the pace around here.

I've been working my way through Mathematics 6 since the beginning of June.

It is now the beginning of October.

RUSSIAN MATH has, estimating conservatively, 10,000 problems. At least 10,000. I have now worked 8000. In the process, I've learned a huge amount, although, sadly, even Enn Nurk & Aksel Telgmaa have not been able to dissuade me from the conviction that 7 x 6 = 43. If they can't do it, probably no one can.

I've just begun the last of RM's six chapters, and I was getting excited about starting algebra next. I can't wait.

So last night I took Saxon Math's placement test (pdf file) for algebra 1.

I got a 72.

### conclusion number one:

I am going to stop expressing reservations about the Saxon math series until I can actually take and pass a Saxon math test.

### conclusion number two:

wow

There are a boatload of topics I still don't know after doing 8000 complicated Russian computation, geometry, & word problems.

They are:

• using four 'unit multipliers' to convert 630 square yards to square inches: I have no idea what a unit multiplier is, or how to use it
• what a decimal part of a number is (I got the answer right, but only because I made a blind guess as to what a decimal part would be)
• negative exponents
• how to find the volume of a cylinder
• 'the method of cut and try' to find the square root of 20: to my knowledge, I have never heard the words 'cut and try' in my lifetime
• how to use a straightedge (what's a straightedge? I still don't know) and a compass to copy an angle
• how to find the area of a triangle (all I remember is: hypotenuse)
• how to find the probability that a die will first roll a 6 and then roll a 2, in that sequence
• base 2
• update 7-16-2006: I know all these things now, and will finish Lesson 81 (of 120) in Saxon Algebra 1 today.

So my first reaction, in Western polarizing fashion, was: I know nothing.

I know nothing, and I need to work through all 857 pages of Saxon Math 8/7 with Pre-Algebra before I can even think about setting foot inside a real algebra textbook.

I was depressed.

But then I calmed down a little and thought, mmmmm....maybe not.

Maybe I can just go through Saxon 8/7 and do every single lesson & every single problem related to these 9 topics.

Is that wrong?

update 7-16-2006: I ended up working through the entire book. Every lesson, every problem, every test. Then I took the Saxon placement test and placed into Algebra 2, but decided to start with Algebra 1. I'm glad I did.

Christopher began teaching himself Saxon Algebra 1/2 this summer (he starts 7th grade in th fall) so I'm reading through those lessons to make sure I didn't skip anything I need to practice - and just for the joy of encountering John Saxon's take on topics I already know.

Algebra 1 integrates algebra and geometry, though without proofs. I'll start Algebra 2 in September.

In one year I will have worked through:

• final chapter of Russian Math
• all of Saxon Math 8/7
• all of Saxon Algebra 1

That pace seems OK to me.

DougSundsethNumberLines 30 Sep 2005 - 21:37 CatherineJohnson

blank number lines (pdf file)

symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)

number lines: all positive numbers (pdf file)

number lines: all negative numbers (pdf file)

### update

If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.

addition & subtractions of integers review sheet

MathmanOnPractice 01 Oct 2005 - 15:03 CatherineJohnson

from mathman:

So how many exercises should I assign? I can't possibly grade them all. This is not an easy question to answer.

It's much easier to say how many exercises the student should do although most students won't care for what I have to say. The student should work as many exercises as it takes to be able to do them correctly most of the time as fast as he can physically write out a complete solution. When informed that he has made a mistake, he should be able to find and correct his error quickly. When it counts, given time to review his work carefully, he should be able come up with the correct solution every time.

This level of mastery opens the door to calculus, differential equations, linear algebra and the quantitative elements of any science.

I'm going to print this out, ask Christopher to read it out loud to me, and then post it above the dining room table. (We're still waiting on delivery of the Ikea desk I ordered a couple of week ago.)

### Willingham on overlearning

I re-read Practice Makes Perfect--But Only If You Practice Beyond the Point of Perfection every few months.

AharoniArticlePart1 09 Oct 2005 - 23:37 CarolynJohnston

This article by Ron Aharoni, which appeared in the Fall issue of American Educator, is brilliant. Catherine and I have both read it, and agree that there is enough in this article to chew over in multiple postings. So this one, I guess, will be the post that launches our discussion of it, and we'll tease out its excellent content gradually in more posts.

Aharoni, who is a math professor in Israel, got involved with a friend in a project to promote math education in elementary schools. His friends warned him off it, saying that elementary education is a whole different ballgame from professional mathematics. He went anyway.

He came in with some preconceived, rather idealistic notions of how elementary school math lessons should work, but wised up fast.

The banner I was carrying at that time was that of "experience". The children should experience abstract concepts concretely, I thought, after which the abstractions should occur by themselves. I took the kids out to the playground. We measured lengths of shadows and compared them to the lengths of the objects themselves, then used this information to calculate the height of trees according to their shadows. (This idea is borrowed from Thales, who was born in the 7th century B.C.) Then we measured the length and width of the classroom in various ways to find how many floor tiles fit into one square meter, and what the ratio was between the length of the classroom in meters and its length in tiles.

I learned the price of conceit the hard way: most of my lessons were a mess.

Aharoni discovered exactly what I did, when I started getting involved in my son's education; having a Ph.D. in math didn't mean I knew a darn thing about how to teach it to elementary school kids: I didn't. Not elementary school mathematics. It was different from any sort of teaching I'd done before. I was totally nonplussed, and Aharoni expresses it for both of us very well:

But what surprised me most was that I learned mathematics. Actually, a lot of it. This would not be the case had I gone to teach in a high school. The mathematical concepts there are known to a professional mathematician. In elementary school, it's the teaching of the most basic principles that counts; the nature of numbers, the meaning of the arithmetical operations, the principles of the decimal system. About these, it is rare for a mathematician to stop and think.

He addresses, in his article, the fact that the most common operations have multiple meanings. We adults get so used to moving between these meanings that we conflate them all in our minds, and when asked about the difference, can't even recall that there is a difference. Here's an example from Aharoni's article:

I was experienced enough to know that such confusion almost always originates from having skipped a stage. In this case the missing stage was the understanding that subtraction has more than one meaning. There is the meaning of diminution, where objects are removed: I had 5 balloons, 2 of them burst, how many do I have left? But there is also the meaning of comparison of quantities, where nothing disappears: There are 5 children in a group, 2 of them are boys. How many are girls? Or perhaps: How many more green apples than red apples are there? In these cases, too, the exercise is one of subtraction, but the meaning is different.

The various meanings of subtraction are an example of a fine point that has to be taught explicitly. Skipping this stage will result in later difficulties with word problems.

Another example of this conflation happened when Catherine asked me, and a couple of other math types, what the meaning of 'partitive' vs. 'quotitive' division was. She'd come across the concept in Liping Ma's book, which claimed that Chinese teachers could generate word problems involving fractions that were of either type, whereas most American teachers couldn't.

Well, mathematically there's no difference; division is division. Partitive vs. quotitive is just a pedagogical distinction that a teacher needs to know, in order to be sure that she can generate word problems that cover the whole set of possible problems, so that the kids will understand that the division operation is the same for all.

I didn't see the distinction at first; neither did any of the other mathematicians Catherine asked. We didn't need to in order to do our own work; but teaching elementary math is a whole different ballgame.

And here's an insight that I just love:

When I started teaching in elementary school, I was convinced that precise formulations and the explicit naming of principles was a matter for grownups. Children should learn things on an intuitive level, I thought. One of the greatest surprises that awaited me was to realize how wrong I was about that. Children need precise formulations. Such formulations consolidate their knowledge of the present layer and make it a safer basis on which higher layers can be built. Moreover, children love "adult" formulations and notations, and are proud of being able to use them. First-grade children who learn the notation "1/2" are happy to discover the notation for "1/3" by themselves.

It makes complete sense to me: grade schoolers realize that knowledge and skills are power, and grownups have the knowledge and skills. They don't want to learn dumbed-down, intuitive formulations of problems; they want to do what YOU do. It reminds me of how my Dad used to leave papers around with derivatives and integrals on them; I thought they looked like the coolest thing, and I wanted to grow up and do what he did.

More to come from both Catherine and me.

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

FractionPedagogyQuestion 13 Nov 2005 - 19:58 CarolynJohnston

Lone Ranger put this question on the Request Page:

Nuts and bolt help for my 8 year old daughter for all you math experts....So, we are working in book 4A in Singapore Math and are learning about fractions of a set. We move into problems such as 2/3 of 27. She can easily build this problem with beads or draw it so we move on to the the algorithm. My daughters asks me why 27 can be written as 27 over 1 and I am stumped. I cannot figure out how to "show " her like I can with improper fractions or mixed numbers. I told her it is a division problem but she wasn't ready to understand that. Any ideas?

This is one of those situations where I sputter a bit, trying to think up a good way to teach this to an 8 year-old. Here's my best shot at it (of course a good teacher has at least 3 ways to explain anything, so hopefully others out there will have other approaches):

You could say that the denominator represents the number of pieces that a unit is broken up into, and the numerator represents the number of such pieces that you have: she understands this intuitively when the denominator is a number greater than 1, I expect. Do comparative examples, perhaps, where you go from saying that 4/2 means you've split a unit into two pieces and you have four of them, to saying that 2/1 means you've split a unit into 1 piece and you have two of them.

So I guess that's one way to try to teach it; now we need two more.

NumberBondsVersusFourFactFamilies 13 Nov 2005 - 20:07 CatherineJohnson

From the Comment thread about Lone Ranger's approach to teaching an 8-year old why it's OK to write the number 5 as 5/1: I mentioned that Saxon Math uses four-fact families to teach the operations of arithmetic, while both constructivist curricula and Singapore math seem to use 'number bonds.'

Here's an example of a number bond flash card:

You can download these cards from DonnaYoung.org, a homeschooling resource that looks pretty good, and has a page of mostly terrific paper math manipulatives, including lots of circular fractions, terrific large-print math facts drill sheets, graph paper, play money, scale paper for household furniture arrangements, and some cool-looking empty worksheets with number lines on top.

It also has triangular addition and subtraction flash cards (pdf file).

from the directions:

To use the cards, hide one of the corner numbers with your thumb or finger and let the child tell you what the hidden number is.

### Saxon's fact families

Saxon Math does not use triangular flash cards.

Saxon uses four-fact families combined with Extreme Practice. If there is One Thing Christopher & I have overlearned from Saxon 6/5, it is FOUR FACT FAMILIES:

1, 2, 3

1 + 2 = 3
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2

Same deal with multiplication and division.

Here's a typical four-fact family problem from Lesson 2 in Saxon 7/6:

23.
Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.
12 + 24 = 36

Christopher can do that in his sleep.

So can I.

I probably have done it in my sleep.

I've been doing so much grade school math I sometimes dream about it.

### four weeks into Saxon 6/5

Quoting from a post I wrote on this subject awhile back:

About a month after Christopher and I began working with Saxon Math 6/5, he told me,

Multiplication and division are the big brothers, and addition and subtraction are the little brothers.

Then he said,

And multiplication and division are cousins.

This is a 9-year who, just 6 weeks earlier, had been flunking math.

You have to do a lot of four-fact fact families to come up with a thing like that.

### I vote for fact families

Triangular flash cards and number bonds are everywhere these days, but I don't like them. Here's why:

• First of all, the potential for confusion is huge. An addition & subtraction number card looks extremely similar to a multiplication & division number card, and separating factors from addends in a child's mind is a challenge under any circumstances.

• Second, triangular number bond cards aren't all that easy to 'read.' Kids don't naturally undestand visual displays of data; far from it. There's too much info on these cards, IMO.

• Third, number bonds are incredibly static, and I don't think math is static. Math is something you do, not something you look at. Four-fact families are action-packed; you get so good at them you can whip one of those babies out in a couple seconds flat. They're fun, and they absolutely (I'd bet money on it) prepare kids for the time when they're going to start solving problems like 2 + a = 5. When Christopher segued to 2 + a = 5 in Saxon 7/6 he didn't have a second's difficulty. He'd been inverse-operationing 2 + 3 for a year at that point, so 2 + a was just obvious.

• Last & certainly not least, I haven't had any luck with flash cards, period.

Not nearly as beautiful as Doug's number lines, but a good idea.

### oops

I've just noticed that Donna Young prefers sites not link to her printable forms, and in fact these links won't access the forms. Just go to her homepage, click on math, and then find what you're interested in. The math page is clear & easy to use.

Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes

NumicomDominoes 11 Oct 2005 - 19:50 CatherineJohnson

While I'm not keen on number bond flashcards, this is a variant on the number bond idea that I do like:

A domino activity using numicon plates rather than numbers. Can be used in a variety of ways i.e. as normal matching dominoes, two plates that make 5 to develop number bonds.

I'm thinking these could be terrific for Andrew, who has to learn math visually thus far, and who needs alternative visual representations so he doesn't get fixated on the felt-marker dot sheets he's been doing.

But I like these for any child. They're cool.

You can find these at a U.K. web site called Teaching Ideas, which has 4 pages of "numeracy" activities that look worth exploring.

### update: math games, too

Anne may want to check out this site. There are a bunch of math games, too. The site is a collaborative effort; lots of different teachers (I think) have contributed games & worksheets they created.

### this looks fun:

• Choose one child to start the game (Child A). This child should get up and stand behind the child who sits next to them (Child B).

• The teacher now asks a question which only Child A and Child B are allowed to answer. They should try to answer as quickly as possible, in order to beat each other. If either of them gets the answer wrong, they are allowed to answer again, until one of them gives the correct answer.

• If Child A answers correctly, he / she continues to stand and moves to the next person (Child C). Another question is then asked to Child A and Child C.

• If Child B answers correctly however, he / she gets up and stands behind the next person. Child A sits down. Another question is then asked to Child B and Child C.

You can continue asking questions for as long as you wish, and can even alter the level of difficulty of the questions to fit the children's ability. You can also ask questions based on a variety of topics (e.g. multiplication tables, number bonds).

A similar alternative to this game was contributed by Susan Aucoin:

##### The Game of Travel
To increase my students knowledge of Maths facts I play this game with my students.

Materials - flashcards of Math facts (add., sub., mult., or div.)

I start on one side of the classroom. The first student on the first row stands up next to the second person. I flash the card and the student who correctly answers first stands next to the following student on the row. The student who did not answer correctly or wasn't fast enough with the correct answer sits down in the desk of the student which he/she just competed with. We play for 15 minutes or so. The person who travelled around the classroom the most is the winner.

keywords: travel game travel flashcard game dominoes
Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes

RonAharoniOnTheFifthOperationOfArithmetic 14 Sep 2006 - 14:53 CatherineJohnson

Carolyn has kindly left my two favorite passages in Ron Aharoni's What I Learned in Elementary School for me to blooki.

Here's the first:

##### What Arithmetic Should Be Covered in Elementary School?

The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.

Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually five operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

I've thought about this observation every day since reading Aharoni's article. I probably can't explain why. At least, I can't at the moment. (Good thing I'm not taking the Regents, I guess.)

But it reminded me of a post Carolyn wrote early on:

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

These are the fraction tiles I like:

You can order extra tiles, too, which I have done. I've used these over and over again, with Christopher, and with at least two of his friends.

Worth their weight in gold.

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

AnneDwyerOnMathGames 10 Oct 2005 - 22:56 CatherineJohnson

I've tried many different methods for games. The main thing that I have noticed is that the weakest students tend to get the least practice with all these ideas. (For example, in the travel game, the strongest students answer the most questions and get the most practice.) Additionally, the weakest students get embarrassed.

So now, all my games are take turns. Even the weakest students get the same amount of practice. Additionally, I tend to pair weakness with weakness so that they both get pracitice in areas that need it. I may have three different games going on working on different weaknesses for different students.

MikeFeinbergKIPP 01 Dec 2006 - 21:25 CatherineJohnson

Here, via oldnewschoolteacher, is KIPP's Mike Feinberg on math ed:

### oh, snap

Everyone [in class] was decrying the fact that poor kids don't have the same things, and that they come into pre-K already behind. When they continue falling behind, middle school and high school teachers complain that "there just isn't enough time" to teach them, particularly with the mandated curriculum dictated by state exams.

I pointed out that, if what people were saying was correct, then that would mean that urban kids should have more time in the classroom, longer school days, and longer school years. This would allow them to catch up and give their teachers the chance to cover everything they wanted. I provided the KIPP schools as an example of a school system that does this, and gets amazing results. It works. More time in school and good instruction works.

My instructor was not pleased with this, though. He thought the idea was too "militaristic." He said, "I mean, what's the end goal?" I was flabbergasted, once again. Doesn't anyone get it? The goal is to give kids the skills and knowledge they need to choose the kind of lives they want to live. Period, end of story, I no longer want to talk to you, stupid idiot. But he has this whole notion of making people "good citizens" or getting them to "think critically" about the world. Ask yourself, what would you want for your child? Would you want her to get a great academic education and be able to do whatever she wanted, or would you want someone to teach her "how to be a good citizen" or "how to think critically"? I know, me too. And if the chips were down, my instructor would admit the same thing. The fact is that schools like KIPP are vaulting kids OUT OF POVERTY. They're giving them a fighting chance. And the concept of the schools is not that complex. Their motto is: Work hard. Be nice. And everything boils down to that in the end. There's no magic curriculum bullet. It's just hard work.

### what was it Orwell said about people being objectively something-or-other?

Oh, yes.

objectively pro-Fascist

oldnewschoolteacher again:

This guy, this instructor, he so decries poverty and "keeping poor kids poor" and "the pedagogy of poverty" but it is HIS reluctance to accept WHAT WORKS FOR KIDS that keeps them where they are.

I really don't understand. And I'm so angry about it.

I'm adding objectively pro-racist to the list.

[pause]

No. No, I'm not.

I'm going with functionally racist.

RonAharoniOnTeachingFractions 27 Oct 2005 - 01:49 CatherineJohnson

Interesting observations via email from Ron Aharoni.

But first, you might want to re-read this post on the fifth operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.”

### against pizza

I'd sent him the link to that post, which also included an earlier post of Carolyn's about rectangular fraction tiles being superior to circular pizza-pies:

I agree that sticking to the pie representation of fractions is harmful. I also prefer parts of rectangles.

But: I believe that it is important to take, from the very beginning, fractions of sets. What is a half of 6 apples? A quarter of a set of 8 pencils? And then, immediately, WHAT IS TWO QUARTERS, and three quarters, of that set? This conveys the meaning of "three quarters" better than the manipulatives.

And, an elaboration, from a second email:

I try to start with fractions of all kinds of objects - shapes as well as groups. In first grade, I start fractions with division. I give groups of kids all kinds of objects: one group gets a rectangle, another a circle, another group two rectangles, another group an apple, and another A GROUP OF 4 APPLES, and ask them to divide the objects they got into two parts. Later, each group tells what they did. We then discuss the notion of "a half of". Then each kid is asked to do work on his own - take halves of shapes, and a half of say 4 objects drawn on paper.

Then we can divide into three parts, and discuss what is a "third of something".

Then "two thirds" (just repeat twice the one third), then a quarter - all this can be easily done with second graders, even first graders.

### Carolyn says: 3/4 is 3 '1/4's'

This tracks with a point Carolyn made in an email last night:

A unit is rather like the denominator part of a fraction. Many of the rules regarding their manipulation are the same. I intuitively understand why that is, and I am going to try to write it up, but right now words elude me.

Here's a quick try to convey the idea by analogy, though -- the correct way to think of fractions is as a unit -- of the form 1/3, 1/4, 1/5, 1/8, etc. -- occurring some number of times, where that number is given by the numerator.

So you should think of 3/4 as being "the unit 1/4, occurring 3 times".

### on not using a child's pre-existing knowledge

One of the common-sense themes of 'metacognitively-aware' teaching, with which I normally agree, is that one should use what's already there, inside a child's head.

When it comes to fractions, the 'friendly fraction' 1/2 is probably more or less innate; children figure it out without having to be taught. (quoting from memory; not fact-checked)

I'm thinking the 'naturalness' of friendly fractions like 1/2, 1/4, 1/3 and so on -- all representing, for children just starting out, one obvious, natural whole divided into parts -- may be a problem as much as an opportunity.

All textbooks begin teaching fractions with the fraction 1/2.

Always, this is illustrated as 1/2 of a pizza.

I think that's probably a mistake. I'm thinking the idea of 1/2-of-a-pizza may be so deeply ingrained in children's (and grown-ups') minds that the jump to 1/2 of a group is that much harder to make.

### don't laugh

OK, I finally looked up the page in Christopher's 5th grade textbook that utterly threw me last year.

It was 'Lesson 58 Fractions of a Whole.'

The lesson began:

We've looked at a fraction of a whole unit. Now let's review fractions of numbers greater than 1.

Take 1/4 of three identical sandwiches.

There followed a page of drawings showing that 1/4 of 3 sandwiches is the same thing as 3/4 of 1 sandwich.

I didn't get it.

I could see it was true.

I could see that the drawing was 'true,' and I knew, of course that 1/4 x 3 = 3/4 x 1.

That wasn't the problem.

The problem was, I didn't get it.

I was having an especially hard time with the pizza pie chart image that kept popping into my mind:

My problem with this mental image, which was very strong & vivid, was that I simply could not stop seeing THREEFOURTHS.

THREEFOURTHS, to me, is a highly overlearned mental THING; if you say 3/4 to me, I'm going to start seeing visions of circles divided into fourths with 3 of the fourths shaded in.

Period.

I have no choice. It's like a song that's stuck in your head. Only it's not a song. It's a textbook illustration.

So there I was, trying to think about ONEFOURTH of 3, and forget it. It wasn't happening; it wasn't going to happen.

I just could not make that bright, vivid, 3/4-of-1-whole-circle turn into 1/4 of 3 circles.

I could imagine 3 circles, side by side, each divided into 4.

But after that my brain instantly jumped to the THREEFOURTHS clumps. I kept imagining, in sequence:

• first, the shaded 3/4 of Circle Number One (on the left)
• second, the shaded 3/4 Circle Number Three (on the right)

then

• the left-over 1/4 from Circle Number One added to the top half of Circle Number 2 (in the middle) AND the left-over 1/4 from Circle Number Three added to the bottom half of Circle number 2.....

...which I bet at this point nobody can even follow.

I certainly couldn't follow it. Not because it's hard, but because working memory wasn't put together to perform a sequential circle-dividing task of that magnitude.

### the magical number 5

I was thinking.

3 circles, 2 THREEFOURTHS chunks, 2 ONE-FOURTH chunks, and 2 TWO-FOURTH CHUNKS ought to come out to the magical number 7, plus or minus 2.

Apparently I'm down to the magical number 5.

### rescue

Finally my friend Debbie came to the rescue. (I bet I can't find her email....nope, can't find it). Paraphrasing:

The way I always think of this is as three 'one-fourths.' There are 3 sandwiches, and you take 1/4 from each sandwich. That gives you 3/4, or 3 separate one-fourths.

That one sentence clobbered my THREEFOURTHS image.

Suddenly I could 'see' separate little one-fourths pulled out of all 3 circles; I could see the individual one-fourths as.....units, I guess. Like Carolyn would say.

### in conclusion

This is what makes me wonder whether, in some cases, the 'natural math' a child (or adult) brings to class may not be the best hook.

In my case the problem wasn't just the probably-innate friendly fractions children & grownups understand without being taught. My problem was the image of the circle, which, as Carolyn points out, is not an easy thing to break into parts and then rearrange those parts in new configurations.

That's why circles represent things like 'eternal love' and the like, because we don't see circles as having beginnings, or ends—or pieces or parts.

Culturally speaking, at any rate, a circle is the Ultimate Whole.

All of this is a long way of saying that:

• I'm going to stick with rectangular fraction tiles
• Aharoni's idea of starting with 2/3 as well as 1/3 -- of teaching 2/3 in the same moment that you teach 1/3 -- is an excellent idea

### update

Here's a terrific example of why rectangular fraction tiles are superior to circles:

source:
Demonstrating division of fractions with pictures or manipulatives at Math Forum

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

DistributiveProperty 13 Oct 2005 - 17:57 CatherineJohnson

I keep forgetting to post this story.

A couple of nights ago I was doing my Russian Math on too-little sleep plus a glass of wine, and I found myself drawing a blank when the text asked me to multiply 24 by 7. I was sitting there complaining, '7 x 24, what's 7 x 24, oooohhhhhhh' (More Sleep, More Exercise, Less Wine coming right up) when I heard, from within my fog, Christopher calling out, "Distributive property! Distributive property!"

I was really tired.

So I kept moaning about What is 7 x 24, and Christopher kept calling out Distributive property! until finally I said, 'What are you talking about?"

Christopher said, 'It's 168! Use the distributive property! 7 x 20 is 140, 7 x 4 is 28!'

I've spent practically a whole year trying to teach Christopher the distributive property.

I had no idea he'd learned it.

GraphingCubesInSecondGrade 15 Oct 2005 - 00:20 CatherineJohnson

A morning in June at Thoreau Elementary School on Milwaukee's far north side, a school that uses the recommended math reform curriculum in Milwaukee Public Schools, known widely as Investigations:

In a second-grade class, two teachers, Blondell Currie and Monica Kelsey-Brown, lead two dozen students through an exercise in which hundreds of cubes in eight different colors are distributed to students in groups of three or four. The students make charts based on how many cubes of each color each group has, and then put all the cubes together to tally classwide totals.

This is part of learning how to collect data, Currie tells the class. In the course of an hour, actual numbers are rarely mentioned and there are only a few instances of actual addition. When the class pools all its data, it comes up with a total that is wrong. Currie and Kelsey-Brown praise the way the students did the project. Kelsey-Brown said later that the session is good because "it allows their critical-thinking skills to process information."

source:
Division flares up over math by Alan J. Borsuk SENTINEL JOURNAL 10-4-02

AskeyTeachesTeachers 15 Oct 2005 - 15:42 CatherineJohnson

Askey started teaching a course four years ago in Madison for prospective elementary school math teachers. One of the things he did was ask them a question from an eighth-grade math test that was used in an international study several years ago: Divide 25.56 by 0.004. Fewer than half got the right answer (which is 6,390).

### Liping Ma

[Liping Ma] asked teachers from each country to divide 1 3/4 by 1/2 and explain how they would teach that to students.

Only nine out of 21 U.S. teachers even got the right answer, and just one suggested a method of teaching how to divide a fraction by a fraction that Ma listed as "conceptually correct."

All 72 Chinese teachers gave correct answers, and 65 created more than 80 story problems for illustrating the process that were creative, easy to understand and appropriate.

(The correct answer is not 7/8, a common mistake. That's what you get when you multiply 1 3/4 by 1/2. When you divide by 1/2, the answer is 3 1/2. You are, in effect, figuring out how many 1/2's there are in 1 3/4.)

Ma asks in her book, "What kind of 'teaching for understanding' can we expect" from teachers who do not have a profound understanding of math themselves and who are not given much opportunity to work on improving how they do their work, including time to work on developing their teaching skills? Teachers in China work extensively, individually and in small groups, on ways to make sure they are doing their jobs well. American teachers have larger amounts of actual classroom time and very little time to work on improving their effectiveness.

"What U.S. teachers are expected to accomplish then is impossible," she says. "It is clear that they do not have enough time and appropriate support to think through thoroughly what they are to teach. And without a clear idea of what to teach, how can one determine how to teach it thoroughly?"

### the Math Summit

A math "summit meeting" held by the U.S. Department of Education in February ended with agreement on a three-point agenda for improving math education: Get the public more involved, launch more research on what works, and spur improved teacher knowledge. Two follow-up sessions have been held, and department officials are looking to the federal education law, No Child Left Behind, as a way to push for more math teachers who have strong backgrounds in math.

Get the public more involved.

So I guess we're doing our part!

### this is great

From the other side of the debate, Roger Shouse of Penn State told the same forum that "reformers' attacks on traditional practices appear to reflect a failure to distinguish between 'traditional math' and 'traditional math taught badly.' "

I'll say.

### lesson study

Liping Ma is associated now with the Carnegie Foundation in California and is beginning work on a project with a California school district on what is called "lesson study," an approach used in Japan and China in which teachers work together to improve their teaching methods.

I'm strongly in favor of this.

One of the main reasons I started working on Kitchen Table Math was that it was the only way I could rustle up colleagues, mentors, & master teachers for lesson study.

source:
Bottom line for math students: good teaching is what counts by Alan J. Borsuk SENTINEL JOURNAL 10-4-02

SuperficialKnowledge 19 Oct 2005 - 15:37 CatherineJohnson

from Steve H

I used to tell people that my son is a sponge for knowledge, but the school is feeding him with a teaspoon. When he was in Kindergarten, I used to leave out math worksheets on the table and he would walk by, sit down, and do them. He was thrilled. Adds and subtracts, less than, greater than, etc. I mentioned this to his Kindergarten teacher and she didn't look pleased.

He loves geography, but his first grade teacher said that "Yes, he has a lot of superficial knowledge."

Some parents would comment to him that perhaps he wasn't looking forward to school at the end of the summer. He didn't know what they were talking about. He loved school.

I'm curious. Did this teacher have children of his/her own?

Now I'm wondering whether I'm one of those parents who thinks her own school is fine; it's everyone else's school that's bad. (Can't remember at the moment which book I got that from, but it's standard. It's one of the things I love about Americans--or maybe about people in general--as a matter of fact. Normal people hold 'healthy illusions.' Politicians are lousy, but their own representatives are good. HMOs stink, but your own HMO is good. And so on.

My favorite all time statistic, from Positive Illusions by Shelley Taylor, a member of the psych department at UCLA, is this one:

### 95% of all people believe they drive 'better than average.'

Taylor's book has been out of print for awhile now, but it's well worth ordering a used copy from Amazon. It's one of those books that was (somewhat) life-altering for me. (You'll never see the issue of mercy-killing, euthanasia, and assisted suicide the same way again; that's for sure.)

So, back to my school, while it's true I did recently hear a teacher talk (incorrectly, I might add) about superficial knowledge, she was making this comment in a special ed meeting discussing a special needs kid's learning; it wasn't a crazy thing to say, given the context (though, as I say, I'm certain she was wrong in this particular case).

I just haven't heard teachers or administrators here make that kind of frankly anti-knowledge comment.

The one Big Sin committed here was the concerted effort to decrease the number of kids in accelerated math.

That's bad, but it was based on a premise 99% of parents seem to share: Math Brain.

Either you've got it, or you don't.

KumonPoll 17 Nov 2005 - 13:51 CatherineJohnson

Having hit the wall on afterschooling last night, I'm calling Kumon today.

There's a Kumon center at the Barnes & Noble mall, which is pretty close. I figured I could get their number from the Kumon web site, so I clicked on the Find a Center link.

Are there that many Kumon centers surrounding ktm readers?

Of course, what I'd really like to know is how many kids from Masters School are also attending Kumon.

Interesting.

There's a "Kumon Center" in walking distance of my house. I just called, and got a mom on the answering machine (obvious mom voice, that's how I know), saying, "Hi you've reached the X residence and Kumon Learning Center."

This reminds me of my best friend in high school, whose mom ran a beauty salon out of their basement.

### good grief

This person is a close neighbor of my friend Kris.

I wonder if she is my friend Kris.

Top secret mom-operated Kumon Centers in Irvington.

Strange.

KumonDay1 17 Nov 2005 - 14:22 CatherineJohnson

They weren't kidding about Kumon homework being easy.

I did mine this morning:
Total number problems: 115
Total number correct: 112 (apparently, in the parallel universe that is my brain, 7 x 57 sometimes equals 64)
Total time: 6 minutes, 10 seconds

##### Christopher:
Total number problems: 210
Total number correct: 210
Total time: 13 minutes

0 seconds

### blessed spill-over effect:

approximately 2 minutes spent fighting over Doing Spelling and/or Grammar

Normally the way fighting over Doing Spelling and/or Grammar works is this.

• Christopher demands a break 'first,' before getting down to work
• I protest, then cave
• I become distracted & lose track of time
• Christopher does not see fit to remind me his 15 minutes are up

That's part 1.

Part 2 begins when I come to and remember:

• SPELLING! GRAMMAR!
• I shout up the stairs: GET DOWN HERE RIGHT NOW! YOU HAVE SPELLING! YOU HAVE GRAMMAR!
• silence
• I shout up the stairs again
• silence again — or, sometimes, Christopher shouts WHAT???!!!
• I climb the stairs to our bedroom (where the PlayStation lives) stalk into the room, bark at my son:COME DOWNSTAIRS RIGHT NOW AND DO YOUR SPELLING
• Christopher, not taking eyes off screen: WAIT JUST 5 SECONDS!

etc.

It's too embarassing to go on.

Around here, 'break' means 'transition.' Christopher can't stand the idea of going directly from CHURCH to MOM'S HOMEWORK, or from SCHOOL to MOM's HOMEWORK, or from anything at all to MOM'S HOMEWORK. (He's conscientious about the school's homework, and seems often to enjoy doing it. He wants, and I think needs, 'a break' before doing his school homework, too. But he doesn't try to play out the clock.)

I sympathize with the transition business. But Christopher's Problem With Transitions long ago became a ploy, and I'm sick of it. Plus, it's rotten for my own frontal lobes not to mention my own productivity; as David Allen says, you need to get stuff OFF your mental list. David Allen is right. Constantly having to remember who's not doing what is eating up what little executive function I have left.

So on the way home from the KUMON Center yesterday I nipped the transition business in the bud. I said: I don't think you should take breaks before KUMON. You should just do your KUMON worksheets the instant you get home.

My timing was perfect. There's a little bit of Magic at the KUMON Center, and Christopher was still under its spell. 'OK,' he said, looking serious. Then, a little later, "I need to build up my speed on addition."

Today, after Christopher did his KUMON worksheets, I said, "You have spelling and grammar to do."

"NO!"

etc.

I did the Choose One routine ('you can choose which one you want to do'), which also elicited a big fat NO!

But within a couple of minutes, Christopher was calmly doing a page in Megawords.

Then he checked it himself.

This is gonna be good.

KumonMondayOctober24 17 Nov 2005 - 13:36 CatherineJohnson

I'm starting to see why, in the world according to KUMON, 4th grade math might be just about my speed.

Timed multiplication tests are hard. Surprisingly hard. Especially when you have:

a) fractured sleep
b) two large dogs whining & barking in your face

Sometimes I think it's a miracle my brain functions at all.

### today's score

sheets D6 a&b - D10 a&b
D6 a&b: 23 problems; 1:28; 0 errors
D7 a&b: 23 problems; 1:25; 1 errors
D8 a&b: 23 problems; 1:23; 1 errors
D9 a&b: 23 problems; 2:17; 1 errors
D10 a&b: 18 problems; 1:39; 0 errors
total problems: 110
total time: 8:12 minutes
total errors: 3

I hate errors.

### if you are a constructivist, please sit down before looking at this

The big red 0 here—and it has to be red—signifies a perfect score.

A big, fat, red zero!

That's not very friendly, I don't think.

And see Number 2 Pencil for another take on friendly numbers.

LinkingHighSchoolScoresToElementarySchool 31 Oct 2005 - 02:57 CatherineJohnson

I think this may be the first press release and/or news article (often one and the same thing, a little-known fact) to connect poor high school performance with what goes on in elementary school. Otoh, this article was published in 1998, so it's possible that the 'fourth-grade slump' meme has simply faded from view in the years since.

Penn State researchers think they know what is behind Johnny's and Janey's inability to do science and math, but Americans may not wish to make the changes that could improve performance.

"U.S. students, in general, show a drop in international rankings in math and science between the fourth and eight grades, which many educators and members of the press have called a slump," says Dr. Gerald K LeTendre, assistant professor of education. "Our studies indicate that this is not really a slump, but simply a continuation of low gains from year to year."

[snip]

"The initial reaction to our drop in ranking is to assume that our middle schools are at fault," says LeTendre. "But no one has looked at the overall trends," he told attendees today (Aug. 22) at the annual meeting of the American Sociological Association.

"Most countries do not move up or down in ranking from fourth to eighth to 12th grade," says Baker. "The U.S. is one of the few that does."

The United States starts above the mean in fourth grade science and is at the mean in eighth grade. In math, we are again above the mean in fourth grade but below the mean by eighth grade. The researchers agree that on the surface this has all the indications of a slump. However, the survey sampled third and fourth grades and a grade comparison shows that the U.S. is already losing ground in third grade.

"Low gains between third and fourth, indicate this is not a middle school problem and it is not a slump, but indicative of a system-wide low level of achievement," says LeTendre.

The researchers note that it is not high performance in other countries that pushes U.S. scores down, but something the United States is doing, or not doing, in our education systems to create this mediocrity.

Sociologists of education have observed that known since the early 1900s educational systems in countries have become extremely similar over time, but little is known about how this might influence achievement cross-nationally. Our performances in math and science should all be similar, however, they are not.

### do other countries have ed schools?

Apparently not.

The American system....employs teachers trained at universities in a wide variety of subjects besides teaching and their specialties. Other countries, however, have much tighter control over schools and teachers.

The American public is unlikely to accept a system like Singapore's, the number one country in the math and science rankings. There, teachers all receive exactly the same rigid training, school curriculums are uniform and the training institutes assign teachers to schools. Local and parental input to schools are nonexistent.

Agreed.

The American public is unlikely to accept a system like Singapore's.

The American public is likely, however, to accept a set of textbooks like Singapore's.

I'd bet the ranch.

One issue looked at by the researchers is the opportunity to learn—the students' access to material in the curriculum. In the U.S., subjects covered in one grade are often again covered in another grade, taking away time from new concepts. Other countries have much tighter upward spirals in learning, only repeating the minimum.

### so far, so good

Unfortunately, at this point the article goes off the rails:
Fixing what is wrong with the U.S. school system, however, could be problematic, say the researchers. The American system allows....a close parent teacher partnership....

I disagree.

The outlook is not totally grim. While U.S. 12th grade students were near the bottom in science, Minnesota fourth graders were the best in science worldwide.

Is this a joke?

source:U.S. Math And Science Scores Indicate Mediocrity

### middle schools are still worse

I'm not going to take the time to look it up right now, but I'm certain I've read, many times, that TIMSS data show no gain at all—zero—in math skills for U.S. students between the 7th and 8th grades.

I would be surprised to find that middle schools are simply as bad as elementary schools, but no worse.

Very surprised.

### I changed my mind

I decided to go look it up after all.

One of the most disappointing aspects of the TIMSS report as it described the United States was what a small amount of new learning actually occured during the eighth grade. Since both seventh- and eighth-graders took the same tests, researchers had the unique opportunity of creating a quasi-longitudinal study. Sadly, there was no significant difference between the scores of U.S. students at the end of seventh and eighth grades.

And see William Schmidt on U.S. middle schools.

### update

Here's Ken on Minnesota fourth graders holding the number one spot in science:

Most likely because hardly any science is taught anywhere at these early grade. I think Singapore doesn't even start teaching science until the third grade.

Summer Supplement Time
linking decline in high school scores to elementary school
research on summer regression
the time costs of not teaching to mastery
U.S. fourth graders not doing as well as thought
Phase 4 topic list, grade 6 class

KumonAverageStudentsBeyondGradeLevel 17 Nov 2005 - 13:25 CatherineJohnson

I've mentioned that one of the main differences between U.S. & Asian parents is that Americans see math as 'genetic.' You either have it or you don't.

Asian parents recognize innate talents as well, but are far more inclined to see high math achievement as a function of hard work, not genes.

Here's the KUMON company's take on the issue of advanced achievement for average kids:

Let's follow Mr. Kumon's thought a little further on this topic of self-study.

Our aim should be to educate our students so well through the Kumon Method that they don't have to depend solely on classroom activities to be able to deeply understand the course content. Students who develop this capacity will have a good chance to enter leading universities. To make this possible, we must help students acquire the ability of self-study from an early age and accelerate their level of study beyond their school grade. (Emphasis added)

This last line is the key. It follows that if Kumon teaches children to become independent learners, some of them will learn at a faster pace than their peers. This is certainly what happened to Mr. Kumon's own son, who was doing high school level work while still in the 5th grade. Multiply that times millions of students and you have a lot of children who are studying way beyond their "normal" school level.

Mr. Kumon recognized that self-learners are motivated by their own progress. It is only natural when climbing a mountain to look up and see what lies ahead. Students don't need to be pushed to scale these heights, but they do need to recognize that there are concrete goals and interesting challenges ahead. For that reason, it is important to encourage study of materials above the current grade level.

The biggest problem, in Mr. Kumon's view, was not getting children to want to tackle the challenges of advanced study, but getting parents and Instructors to believe that it was possible and desirable:

The most important and difficult feature for people to understand about the Kumon Method is having students advance beyond their actual school grade. The majority of people don't have the experience of studying material beyond their school grade. Consequently, they don't believe that children have that ability. Even Instructors find this fact hard to believe at first. The history of the Kumon Method can be called the history of our efforts to convince people of this fact.

That was written many years ago, but the same problem still exists. Many parents just don't believe that children can study beyond what is considered a "normal" level for them, or that it isn't "natural" for them to do so. If they see examples of children who are progressing at a faster-than-normal rate, they are inclined to say "The parents are pushing them too hard" or "That just isn't normal." In fact, advancing beyond grade level is a normal consequence of consistent, long-term Kumon study. Every year more and more young children in Kumon show that what we expect as "normal" for a certain age is more a reflection of the limits we have put on children than anything inherent in the child's ability to learn. More Kumon parents are spreading the word that what is really "normal" is for a child to be learning a wide variety of things, not under pressure or stress, but because children naturally enjoy learning about the world around them.

[snip]

Are most of the Kumon children who are studying one, two or three grade levels beyond their normal school level geniuses? I posed that question to Stan Laser, a former math and science teacher and later vice principal of Brooklyn High School in New York, where he was in charge of 1,200 students. Stan, who is now a Kumon Instructor, said that very few of the children who outperform later on are abnormally bright at an early age. "I sometimes ask the parents of children who turn out to be superstars in Kumon if those children were exceptionally bright when they were very young....In almost every case the parents say no, my child was just an average child." In other words, the great majority of the children who excel in elementary school or junior high are not geniuses. Their growth and success in attaining higher levels of ability has emerged through regular study.

All of this would have Toru Kumon nodding and smiling with a knowing look on his face. Isn't it obvious, he would say to us? Children are not only capable of advanced study, but need to be given the opportunity to advance. But....it took time for him to understand this process:

Initially, it was difficult to determine how children who advanced to higher levels would develop. Many Instructors worried that letting children advance so far would bring about other problems. But as children advance so far, they naturally develop self-motivaion and acquire self-esteem and self-confidence because of their abilities.

Everyone thinks it is perfectly natural when children who were exceptional when they started Kumon eventually move beyond their actual school grade. But if Instructors see that children who had average abilities at the beginning also advance by the same process, they will have a more profound understanding of the learning effect of the Kumon method.

I believe this account, because I've seen it myself, throughout my adult life: slow and steady wins the race.

Back when I was starting my dissertation, my advisor told me to write 500 words a day. Period.

If I wrote 500 words a day for 6 months, I'd have a dissertation.

I had already figured this out myself, but it was good to hear it from him.

### hard work versus steady work

I've noticed that hard work isn't always the key to success.

For a lot of undertakings, it's steady work that matters, or seems to.

Ed and I spar about this from time to time. He'll say, 'The problem with teaching math to children is that it requires a huge amount of drill, and drill is boring. So the question is, how do you make learning math less boring?'

This is a common sense view, but it's not what I see in my Singapore Math kids.

What I see is that they gain speed and accuracy—the KUMON goal—incredibly quickly.

I've mentioned my student who has a special needs classification. The first time he took a Saxon Fast Facts test, he needed 10 minutes to complete a 5-minute sheet.

He was so slow that I insisted on doing the writing for him the next time. That time he finished in 8 minutes.

The third time he took the test—the third time, with no practice in between—he did his own writing and came in under 5 minutes.

Kids learn fast, and they pick up speed fast, too.

I've also seen, in my own life, the effects of small amounts of effort put in on a daily basis. When KUMON says 'daily,' they mean daily. You do your KUMON worksheets every day of the week. No rest on Sunday.

But the worksheets aren't hard.

As far as I can tell, it's true that if you put in 10 minutes a day, day in and day out, you see shockingly high gains over time.

Judging by the experience of folks around here, of course, this is not the case in grad school.

And of course cognitive scientists have focused on the very high degrees of practice necessary to become, say, a virtuoso violinist.

But that research isn't really relevant.

Think about it. Is anyone trying to become a virtuoso of two-digit multiplication?

No.

What you're trying to do when you're learning elementary mathematics is to acquire speed and accuracy and then move on. That doesn't take 10 years of study. Nor does it take hours of drill.

It takes 10 to 20 minutes a day for 6 to 8 weeks.

I've come to feel that most people are missing how easy it can be to learn elementary mathematics when you have a coherent course of learning & practice.

keywords: drill and kill

MathBrainsInFourthGrade 01 Nov 2005 - 21:46 CatherineJohnson

OK, I'm a believer.

Math Brains!

They're real, alright.

Two kids in my class today were able to solve this problem:

Five workers take 5 hours to dig a ditch that is 5 yards long. How many workers are needed to dig 100 yards of ditch in 100 hours?

These kids are nine.

They're desperate for 'hard problems,' so if you've got 'em, I need 'em.

RequestForHelpWithBarModelProblems 13 Nov 2005 - 15:47 CatherineJohnson

My Singapore Math kids are in revolt because the bar model problems I've given them are too easy.

I had the same problem with Christopher. In his case, I handled things by giving him enough hard bar model problems that he finally realized he couldn't do them, at which point he agreed to back up & start at the beginning.

I can't do that with the Singapore Math kids because I only see them once a week for 8 weeks.

Plus, they're not my kids, and I'm not their (regular) teacher. My options are limited.

There are at least 3—maybe 4—kids in the class who are super-good at math.

They all love the Brain Maths problems, so I may just turn the class into the Brain Maths class. That's Plan B.

### Plan A

Plan A is to bring in some problems the kids can only do using bar models (and also turn the class into the BRAIN MATHS class. I figure, if they love doing BRAIN MATHS problems, go for it.)

I've just written up sheets for a Science News PuzzleZone problem that's going to be far more doable using bar models:

Seven apples must be shared equally among 12 children, but no apple is to be cut into more than four parts. How would you do it?

I also drew up bar models using the 'Hint' Science News gives kids to help them solve the problem. So they'll have a bunch of sheets to look at, think about, & play with while they're solving the problem. It suddenly occurred to me: these kids are nine. And: they are boys. Fine motor skills aren't their strength, except for the boy who's good at drawing.

As I did with Christopher, I'm taking the fine motor aspect of learning Singapore Math out of the 'equation' for now.

### anyone got any good problems?

If you have ideas for word problems around the level of a bright, math-savvy 9-year old that will be easier to solve with a bar model than without, I'd love to hear them.

And if you have story problems you know kids this age have enjoyed doing, I'd LOVE to see them.

Thanks!

### Brain Maths blurb

from the Singapore Math web site:
Brain Maths is a series of three books which contain a wide selection of mathematical and logical problems that help one to develop one's critical and logical thinking skills.

Recommended for readers of 9 years old and above, Brain Maths aims not only to increase one's IQ power, but also to develop one's mental flexibility.

Using puzzles and brain-teasers, Brain Maths has challenges catering for a wide audience - from the average math pupil to the gifted mathlete (one who competes in mathematical contests and competitions).

### one more favor

Can someone click on PuzzleZone and see if you get in?

I don't think it's a subscription-only site, but I'd like to know for sure.

Thanks.

IsSaxonPlusSingaporeTooMuch 07 Nov 2005 - 23:47 CarolynJohnston

We had a request today for some information about supplementing Saxon Math with Singapore Math...

First responders on the scene (with math tourniquets) were Susan and Dan...

Susan's response:

A homeschooler friend of mine once told me that many homeschoolers use both Singapore and Saxon at the same time. I'm presently using Saxon as the core supplement curriculum for my public school child, but I add Singapore problems to whatever chapter I'm on.

Singapore's word problems are better than any of the other books I've seen because they start with one and two steps and move up to 4+ steps by their level 5.

I don't know if you've seen The Well Trained Mind book, but it has an easy to follow schedule for homeschooling all subjects throughout the years of your child. You might get some ideas of how much to do from there. Since I'm an "after-schooler," as they call me, I haven't ever looked closely at the way they set up the teaching schedule, but it looks fairly thorough.

Dan's response:

I haven't homeschooled, so I feel a little uncomfortable commenting...but only a little.

I just wanted to ask if you were testing the multiplication (and, for that matter, addition) facts with timed tests. I'm pretty sure that timed fact tests are part of the Saxon school curriculum. It seems to be a consensus opinion here at KTM that these facts must be mastered to the point of automaticity. I certainly agree, and have found any lack of automaticity to be a major hindrance as students try to move forward.

And Diane replied..

DanK, Yes, I am using timed tests for addition and subtraction, and I use multiplication fact worksheets for drill, though I don't usually time them. We are just now moving into timed multiplication tests with Saxon.

SusanS, I have read "The Well Trained Mind" and I just revisited her suggestions for scheduling. An hour a day for math seems pretty typical for what most other homeschoolers I know are doing.

I am leaning towards getting Singapore and supplementing with it. Some of my friends who use Saxon with their kids just have the child work every other problem. I've been having my sons do every problem, and, as I commented earlier, it takes them about an hour. I don't want them to get overwhelmed by having an hour and a half of math every day, so I guess I would have to cut out some of the practice problems in Saxon.

So I'll weigh in now with a few thoughts...

I think an amalgam of Saxon and Singapore is a good choice for homeschooling. With Saxon, especially in the early grades, you can be sure that you're not missing out on any essential skills. I think Singapore has a good emphasis on word problems, and I like the way they get kids thinking algebraically very early.

I home-supplemented my son a lot the last two years (we had a constructivist curriculum in 4th and 5th grade—Everyday Math), and even though I'm knowledgeable about math, there were days when I felt up to the task of 'constructing his curriculum' (so to speak) and days when I just didn't. Saxon is a great support for homeschoolers who don't want to be carefully preparing their kids' lessons every day. Singapore takes a greater background knowledge of math, and is much harder for the kids to do independently than Saxon, so to do Singapore, you'll be making a commitment to get really involved with your kids' math. Not every homeschooler wants to do this.

I'd be reluctant to cut out every other Saxon problem on a regular basis, because I think those mixed practice problem sets are the genius of Saxon. They'll revisit a skill intermittently, and if your kids are only doing even problems, they'll miss getting the practice they need if the skill only appears in odd problems (it would be genius indeed if they had enough forethought to put a given skill alternately in even and odd problems!).

You could start by trying to add Singapore word problems to each math session, and see whether that worked; you might find the kids tolerate it pretty easily. If not, you might try switching off days. You wouldn't get through either curriculum as fast, but Saxon has a lot of repetition from one year to the next, so even if you didn't get all the way through a Saxon book you'd have little cause for worry.

Another thing you might consider doing is making Saxon your main text, and supplementing from one of the Singapore books that specializes in word problems, since that's where I think Singapore really has the most to offer. Singapore has a workbook series called Challenging Word Problems Books 1 - 6 (\$7.80 plus shipping; 129 pages), in a U.S. (as opposed to British English) edition. You can start at the workbook that's at the level your kids placed into; the problems are marked at a mixture of difficulty levels. This is definitely what I would do if I were constructing a homeschool program.

One more thought—my son, who has Asperger's Syndrome, got balky in second grade about doing math timed tests. He would basically refuse to deal with them, in class; although he knew the facts, he wouldn't do the timed tests because he was reluctant to deal with the time pressure. We ended up doing some heavy bribing to get him to move on those tests (once he did, he was fine). I think adding the time pressure factor is important to nudge the kids toward automaticity. Rewards in the form of treats or outings or privileges are good, I think. Competition can also be good, if it's friendly competition and not cutthroat (and if they're siblings that close in age, it could get ugly).

KDeRosasPageOnMathematiciansFindingCommonGroundWithConstructivists 04 Nov 2005 - 02:14 CatherineJohnson

What is important in mathematics?

### Direct Instruction math

Ken has also managed to find some sample pages from Connecting Math Concepts, which is a direct instruction curriculum (which I believe was designed by Engelmann??)

Ken will tell us...

### Wayne Bishop compares Saxon, CMC, Sadlier-Oxford, & Everyday Math

here

key words: SRA direct instruction sample lessons

BrianMickelthwaitOnKumon 17 Nov 2005 - 13:27 CatherineJohnson

I'm in the middle of reading Brian Micklethwait's terrific article about his experience as a KUMON instructor, but had to stop and post this passage:

There is also in Kumon what I think of as a very Japanese emphasis on the physical process of drawing the numbers and on physically handling the world generally. (Think of the Japanese fascination with hand-done graphics.) One of the ancillary games we get the children to play is simply placing numbers on a number board. This doesn’t just help them to understand numbers. It also helps them to get better at simply handling things, while thinking at the same time. As with so much of Kumon, doing the number board so that every number is where it should be is in principle very easy, so no child is humiliated by not being able to do it. But doing it fast isn’t so easy, so the cleverer ones are kept interested. (We also give the cleverer ones more complicated things, like “leave on the board only those numbers divisible by 3”.)

This emphasis on the physical handling of the world also explains, I think, why the Kumon people are so reluctant to get involved with computers. To me, an Anglo-Saxon techno-nerd, Kumon absolutely shouts computers. Each child doing an individually selected clutch of repetitive problems. Relentless and potentially very tedious marking. Even more tedious analysis to tell you what each child should be doing next. A huge apparatus of collective, centralised analysis to see which methods work best and to tell the rest of the world. This is surely the sort of stuff that computers — and their recent combined offspring, the Internet — were invented to supervise. But I sense that the Kumon people resist such notions. There’s so far been no mention of computers in any of the Kumon back-up or sales literature that I’ve seen. Computers, I hear them saying, would only complicate things.

I've come to believe that paper-and-pencil math is math—that there's something necessary, at least when you're learning,* about the experience of actually holding a pencil or a pen in your hand and solving problems.

Carolyn talks about the craft of math; Temple repeatedly & chronically encounters people who've learned to create scale drawings on computers and, as a direct result, cannot construct scale drawings. (Temple believes that the visusal processing and motor systems in the brain are connected. I won't be surprised to learn that she's right.)

I've been surprised at how unmoved Americans are by the Singapore bar models. I fell in love the instant I saw them, and wanted to draw them. With Sybilla Beckmann, I think the bar models are probably the reason for the Singapore curriculum's success.

I've mentioned several times that I've worked at least 300 bar model problems. I've said, too, that doing this changed my brain. I'd put money on it.

The thing is, I really don't know why this should be the case. I'd been thinking maybe they develop spatial reasoning, which is connected to mathematical ability.

It hadn't occurred to me that bar models might work simply because they involve lots more pencil-and-paper work than the traditional U.S. math curriculum.

But the explanation may be as simple as that.

When I first started drawing bar models, I badly wanted to paint one. I wanted to do a big, bold 'blow-up' of a Singapore bar model in oil, and hang it on the wall.

Maybe one day I will.

### what is the opposite of a fount of wisdom?

Here's Steve Leinwand:

Shouldn't we be as eager to end our obsessive love affair with pencil-and-paper computation as we were to move on from outhouses and sundials?

*Temple says that older people who learned to draw by hand & then switched to CAD have no problems at all. The problems turn up strictly in the work of younger employees, who've never done physical scale drawing using pencil and paper.

Swoop and Swoop
the craft of math

SingaporeStudentHelper 13 Nov 2005 - 14:51 CatherineJohnson

Parker & Baldridge on the student helpers:
The children pictured in the margins give the precise definitions and key ideas in very few words. These ‘student helpers’ often clearly convey an idea that might otherwise take an entire paragraph!

and remember N.S., also from TRAILBLAZERS, grade 5

Wow! That number is mind-boggling! Is it in the millions or in the billions? Reading and writing big numbers is not so easy. I've seen most of these words on the list before, but when I try to think about numbers in the millions, I get confused about what some of the words mean.

A Student Helper in the Singapore series would never be shown saying something like this.

There are pictures of children all through PRIMARY MATHEMATICS.

Not one is confused, bewildered, befuddled, or wrong.

FourthGradeMathEnrichment 17 Nov 2005 - 01:15 CatherineJohnson

The fourth grade math enrichment program is meant to complement and enhance the distict's new mathematics program, Math Trailblazers. It is a flexible, "push in" model that allows small groups of children the opportunity to explore the previously learned concepts in greater depth.

Most recently, all of the fourth graders constructed runways for various types of airplanes. The focus of the activity was to find the perimeter of different sized runways built with square tiles, noting patterns found during the process. As an enrichment activity, students built runways with shapes other than squares, such as triangles, pentagons and hexagons. Then, we discussed patterns and "function rules" that were devised from shape. With the discovery of each shape's function rule, we could calculate the perimeter of any number of blocks arranged in a line, without having to build the runway.

The math enrichment program can also extend any topic being taught in class. As fourth graders continue to explore our Hindu-Arabic number system with base ten blocks, enrichment activities will focus on other number systems such as the Roman numeral system. By studying the rules and symbols of other number systems, the students can compare and contrast the similarities, differences, pros and cons of our current number system.

As with Math Trailblazers, all enrichment activities align with NCTM and New York state mathematical standards appropriate for grade four. In addition, the children are often encouraged to show their work and explain their thought processes in complete sentences as they will be expected to do on the upcoming state test in March.

think and discuss

Our Hindu-Arabic number system, pros and cons

Hindu-Arabic number system
pros             cons

Roman numeral system
pros             cons

meanwhile, back in Singapore

11.
David spent 2/5 of his money on a story book.
The storybook cost \$20.
How much money did he have at first?

source:
Primary Mathematics 4A Textbook, p. 67

update
A lot of the Regulars (see Comments thread) think these activities are worthwhile.

This tells me I've done a poor job writing this post—and, in fact, thinking it through in the first place.

Thanks to all of you, I'm clearer now.

Here's where I am:

• First of all, it pains me that, apparently, it's only the 'enrichment kids' who are learning about the Roman numeral system and about irregularly-shaped figures. Saxon Math teaches these subjects to all children, as does Singapore Math.

• Second, having now spoken to the mother of a mathematically gifted child struggling with TRAILBLAZERS in 2nd grade, I'm finding myself drawn to the issue of what exactly my school district is doing for these kids. These are very brainy children; I know, because I have them in my Singapore Math class. I do not see or hear evidence of a commitment, in our district, to serve the needs of gifted children as we serve the needs of other children. The reason these children are having 'push-in' enrichment is that our district is philosophically opposed to allowing high-achieving and gifted children to move ahead in the curriculum. So now we're spending money on 'Math Differentiation' specialists to help teachers teach math to below-average, averge, high-achieving, and mathematically gifted and talented kids all in the same time and place. The decision to end tracking was made in a hierarchical, top-down fashion, and imposed upon protesting parents.

• Third, I'm highly skeptical of the use of manipulatives at this age. First of all, we do have some reasonably good research on the subject of manipulatives, which shows—counterintuitively—that manipulatives work for middle school kids, but not for elementary school children. Manipulatives may impede learning in earlier years. (see posts here and here) But leaving aside the handful of studies we have, what bothers me is that all of these kids are bright. Their math skills are strong; they're good readers; they have smart, dedicated parents to help with homework. These children, I believe, can be using the abstract symbols of mathmatics to reason and learn.

• Finally, I simply loathe the tone of this document. I know that's harsh, but that's the writer in me. The fourth grade math enrichment program is meant to complement and enhance the distict's new mathematics program, Math Trailblazers. This is spin, and, reading it, I feel my blood rise. When a document like this comes home in the backpack, I am not being treated with respect. I am being public relationed, in Carolyn's memorable phrase.

TheDivisionsOfMathematics 15 Nov 2005 - 13:03 CatherineJohnson

A ktm guest left a link to a terrific web site called The Divisions of Mathematics, and says that "You can follow the links there to find out what some of the fields in statistics are."

I'm posting the link in the 'book-style index.'

This is incredibly helpful for me. When I first started teaching Christopher I was constantly trying to figure out the various genres & subgenres of the field.

### NSF map of math

Here's the NSF's breakdown of the field:

• Algebra and Number Theory
• Topology and Foundations
• Geometric Analysis
• Analysis
• Statistics and Probability
• Computational Mathematics
• Applied Mathematics

hmm

I have to say, for me these categories raise as many questions as they answer, which I suppose was inevitable.

Good starting place, though.

### ah-hah

The perils of scanning.

If I'd read this first, I would have understood:

Another way to divide the portions of mathematics is by level of complexity. Elementary topics include arithmetic and measurement; intermediate topics include simple algebra and plane geometry. From there we may pass to somewhat more complex topics built upon these: trigonometry, "advanced" algebra, analytic geometry, and calculus.

This website is limited to topics more advanced than these; little mention will be made of topics which are typically not considered (except in their most elementary aspects) until a student has progressed through some University studies. Our intended audience at the site is the person who has already studied some mathematics courses beyond these at the university level, although in this tour we try to be more inclusive.

JamesMilgramOnLongDivisionAndTimeLagInMath 15 Nov 2005 - 21:42 CatherineJohnson

from James Milgram's talk:

In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.

First of all, I claim that taking—even asking to take [long division] out of the curriculum—shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

• Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.

• Long division is essential in learning to manipulate and factor polynomials.

• Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.

I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it's minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials.

So what, you ask? Again, this is a question that doesn't come up until the third year in college. At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory. It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory -- the chief underpinning of optimization and design in engineering, economics, and other areas.

[snip]

What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I'm referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.

But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop.

key words: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
NYU math major
overlearning
remediating Los Angeles algebra students
Inflexible Knowledge: The First Step to Expertise by Daniel Willingham
Matt Goff & Susan S on remediating gaps
Anne Dwyer on diagnosing gaps & request for 'gap' stories
formative assessment and Richard Nixon
Terminator

WhatToDoWithLinnkingCubes 17 Nov 2005 - 01:19 CatherineJohnson

I met a retired teacher at a bowling party the other day, who told me a story about math manipulatives.

He'd been a middle school teacher, and for some reason he was put in a 3rd grade class and given a crate full of math manipulatives. He had no idea what to do with them. He was especially stumped by the 'linking cubes,' which looked like Legos, but clearly were not Legos.

I had the same experience two summers ago when I orderd the Saxon Math manipulatives that go with the Saxon Kindergarten book. The Geoboard especially stumped me, but the linking cubes were a mystery, too.

Now I get it.

source:
Harcourt School Publishers Multimedia Math Glossary

I like Harcourt's glossary. The images are clean and sharp, and you can ask the website to speak the name of each concept or rule, which is fantastic for a child like Andrew who (probably) never hears the same word the same way twice.

A recorded word is much more likely to be a 'stable stimulus.'

On the other hand, their illustration of multiplication (in Grade 2) is visually wrong, I think. (Anyone else?)

MathPracticeSimplified 17 Nov 2005 - 17:17 CatherineJohnson

Searching for pre-algebra workbooks this morning, I found the Math Practice Simplified series at the Rainbow Resource Center. The books start in pre-school & run through 8th grade.

It's hard to get a look at any of the pages inside workbooks, but you can see one of the pages from the Pre-algebra book here.

And here's a list of all the Math Practice Simplified books.

For some reason, after staring at the computer screen for half an hour, it came to me that Math Practice Simplified is potentially better than all the other series whose covers I stared at this morning. I have no idea why. (Actually, I do. It's the clean graphic design.)

Rainbow Resource carries a number of workbook series, and their prices are terrific.

gold strike

Today's major find, though, was a website including the entire Glencoe Pre-Algebra Parent and Study Guide in pdf form.

Susan has mentioned before that Glencoe has a terrifically helpful web site, and this is fantastic. Every textbook series should have a Parent and Student Guide just like this one:

The Glencoe Parent and Student Study Guide is designed to help you support, monitor, and improve your child's math performance. These worksheets are written so that you do not have to be a mathematician to help your child.

I'm contemplating springing for this in print form; that's how good it is. Any concept Christopher is struggling with is instantly findable and directly expressed in the text. If he were using the Glencoe book in his class, I wouldn't hesitate.

I love the structural principal of these books.

Like KUMON, each page includes a brief explanation & illustration of the principal the student is practicing.

Beautiful!

Which reminds me. It's time for me to do my KUMON sheets.

Pre-Algebra, Parent and Student Study Guide Workbook (Paperback) by McGraw-Hill
(this is the print copy of the Guide, I believe)

Pre-Algebra, Skills Practice Workbook (no answer key?)
Glencoe Practice Workbook (no answer key?)

update

Illinois LOOP likes the Kelley Wingate workbook series.

from Teacher's Outlet:

Kelley Wingate: Math Practice
This curriculum-based series builds both math and test-taking skills. Practical problem-solving demonstrations and drill pages feature new skills and review. These reproducible resources are the perfect supplement to students' regular course of study. 96 flash cards and answer keys are included in every book.

She has an extensive series of workbooks covering phonics, math, 'math fun,' science, test-taking, study skills, and even writing. There may be more; these are the workbooks carried by Teacher's Outlet. Search the site, and you'll get the list.

I'm contemplating ordering the study skills book. Yet another topic I'm apparently going to be afterschooling.

Did I mention that Christopher's 'study skills' class is doing character education?

I think I did.

StevesSchoolDistrictSkillsGap 19 Nov 2005 - 19:32 CatherineJohnson

comment left by Steve

She seemed hesitant to criticize. She said that the biggest transition problems are the study skills, amount of homework, and attention to mathematical details. She said that they emphasize precision rather than "close enough". That is telling. I told her that I can also see a gap in content and skills and I wanted to get a list of textbooks/syllabi of the high school math courses so I could judge for myself.

I find that shocking.

Incredible.

This is the school district where 25% of the parents have pulled their kids out & sent them to private schools.

When I told Ed about it he said, 'The entire administration should be fired.'

No kidding.

JohnsHopkinsCenterForTalentedYouth 26 Nov 2005 - 23:18 CatherineJohnson

Scanning this list of standardized tests Lone Ranger left, I was led to the Johns Hopkins Center for Talented Youth.

Here's their list of criteria for eligible students:

Many students in grades K-1 participate in our math program. Since there is no talent search for students below grade two, students in grades K-1 can establish eligibility by submitting scores in the 97th percentile or higher on one of the following tests: CogAT, Woodcock Johnson Test of Achievement, or Keymath, OR a Full Scale score of 130 or higher with a minimum 127 performance score on the WISC or Stanford-Binet. Access to these tests usually requires a psychologist, counselor, or school-based testing specialist. See our Diagnostic and Counseling Center for help.

I taught a writing course for Johns Hopkins CTY years ago. College-level freshman rhetoric; 12 year old students.

It was great.

KeyMath

fyi: Our schools, and apparently a lot of others, use the KeyMath test. One of the special ed teachers at the middle school told me it's terrific.

If anyone knows if and how parents can administer KeyMath, let me know.

update

Oh, forget it.

I just looked up the price on AGS's Math-Level Indicator: Quick Group Placement Test, thinking it would be cheaper than KeyMath, because it's Quick.

Wrong.

I don't think I need to spend upwards of \$300 for a Quick Group Placement Test.

We'll just carry on as we have been until Christopher can pass the end-of-5th-grade Singapore Math test 2 or 3 years from now.

SubtractionDivision 29 Nov 2005 - 16:03 CatherineJohnson

24
÷2
12

My kids are doing division this way, and one of them asked how to divide 92 by 2. He thought you couldn't do it, because 2 doesn't go evenly into 9.

update

The TRAILBLAZERS scope and sequence chart shows that TRAILBLAZERS does not teach an algorithm for long division until the end of 4th grade.

HowCanCollegeFreshmenFillGaps 30 Nov 2005 - 18:34 CatherineJohnson

from Anne Dwyer:

Going back to the question of what to do with students who have large gaps in their background:

We (someone, I forget who) once asked this question on this site: once you have these large gaps in your knowledge, can you ever catch up and close all the gaps?

I think this is especially relevant at the college level. There is a basic math course at our community college, but it goes incredibly slowly. The prealgebra class gives basic lip service to large number problems, then goes straight into algebra. Towards the end of the course, the curriculum goes back to decimals and percentages and conversion factors. But, by then, many of the students are completely and totally lost.

Then, they break basic algebra into two classes: elementary algebra and intermediate algebra.

Even with a tutor, there isn't enough time to determine where the weaknesses are and to go back and correct while the student is taking the class. This would require them to work on parellel tracks: making up gaps and keeping up in class. Everything is geared towards students keeping up in class not preparing the student with the basics for the class.

This was a conundrum for me.

I don't know the structure of mathematics well enough to be able to tell where I have significant gaps and where I don't; if I did, I (probably) wouldn't have gaps.

This is why I decided to go back and re-learn everything from the beginning. That way, I figured, whatever gaps I didn't know I had would get taken care of.

I didn't end up being able to do that, mainly because I had to keep up with Christopher. So I started in 5th grade, where he was.

I'm wondering whether KUMON would be a good idea for students in this position.

I started in Level D—roughly 4th grade—and moved to Level E after two weeks.

Algebra begins in Level H.

Each level has 200 worksheets, and you do 5 worksheets a day, 6 days a week. (I think you're supposed to do 5 worksheets a day, 7 days a week, but Mr. Liu only gives me enough for 6 days.)

If you figure roughly 7 weeks per level, I'll move from Level E to Level H in 21 weeks. I can do that and easily do everything else; my KUMON worksheets are the least demanding part of my day. So I think an ill-prepared college student swamped with remedial work could do KUMON sheets and keep up with his classwork.

I gather there are some adults taking KUMON; I wonder if any of them have written about it.

CommentsFromKtmGuest 19 May 2006 - 16:26 CatherineJohnson

I was discussing this bliki last night with a friend, who is a former teacher with experience in elementary, middle, and high school, and with both IEP and non-IEP classes, and she says she also preferred teaching the IEP adaptive behavior students. Not only was there a well-defined plan with exactly specified goals for each student, but also she was dealing with the same classroom management problems as the regular ed teachers, except with only five students and an emergency button on the wall!

Absolutely. Christopher's brilliant 5th grade teacher told me she was asked to teach the Phase 4 class and she opted, instead, to teach Phase 2, which was children one year below grade level. Many (perhaps all) of them had IEPs, which meant the school was required, by law, to teach them to mastery.

She said a lot of them were terrified of math. Some would even start crying. Every single child in her class scored above 80% on her first big chapter test, using the same book the rest of the school was using.

Steve said one day that all students should have IEPs. I've often felt this way myself. Now that I've read Engelmann I formulate this slightly differently. I'd like to see the law changed to state that all children are entitled to be taught to mastery (leaving it to the Engelmann's of this world to figure out what that would mean as a matter of public law and policy).

As things stand, the entitlement to a public education does not mean an entitlement to learn the content being taught.

It means an entitlement to be exposed to that content.

I need an emergency button on my wall.

Another comment:

I don't recall either of my parents (1 Ph.D. in chemical engineering, 1 math major) helping me with my homework, ever. Well, okay, there was the one time in 10th grade where my mom helped me set up the electric typewriter so I could type up a 10-15 page term paper, but other than that, they had no idea what I was studying, what was assigned, or when it was due.

I did every single one of my shadow boxes and other projects by myself. (And the teachers could tell, I'm sure.)

This bliki has made me think about the elementary math education that I experienced in school, and I have come to realize that I don't remember a thing of the instruction -- because I wasn't paying attention at all. I don't think I ever had to do math homework at home until high school, because I was doing it in class while the teacher was instructing, or I did it the previous week by working ahead in class while the teacher was talking, or whatever.

I do, however, remember how to do fractions, decimals, long division, algebra, and calculus. I can even take square roots with a paper and pencil, something I taught myself out of an 1899 math book my mom found at a church yard sale. I am a little rusty at geometry proofs, but I can do geometry puzzles like the ones in the Singapore 6B entrance exam.

(Okay, okay, they encouraged and indulged my math mania by buying me math books and letting me read ahead in their high school and college texts. So sue me... that's not really helping with my homework. :) )

This comes up all the time.

Nobody I know had parents spending hours hauling them bodily through math and English language arts.

And yet most of us learned as much if not more than our own kids seem to be learning. I talked to Temple (Grandin) about this yesterday; she learned all fraction operations to mastery in the 6th grade, and she's used math all her life in her stock yard and meatpacking plant designs. This was a developmentally disabled child learning fractions to mastery in 6th grade. (I'll have to ask her how much time her mother spent filling in the gaps. I'll bet not much.)

What happened?

AleksIndividualizedLearningAssistant 03 Dec 2005 - 21:25 CatherineJohnson

Does anyone know anything about it?

All I know about it is that a blogger named Parent Pundit used it with her daughter with good results.

slipped my mind

hmm

I see that back in May I was planning to 'check out' ALEKS right away.

Obviously that didn't happen.

Time for me to read Getting Things Done again.

If I can find it.

David Allen has a blog

This could be interesting.

update 6-30-2006: David Allen doesn't have a blog.

good grief

Now here is a photo I would not publish on my blog if I were David Allen.

David Allen needs a blog consultant.

I think by now most of us here at ktm could set up shop as blog consultants.

if you're killing time?

Why is David Allen providing me with suggestions on how to kill time this weekend?

Wouldn't I be killing time reading David Allen's blog because I have a problem with killing time?

Think and discuss.

a parent's experience with ALEKS
ALEKS Graphic
formative assessment on wheels
ParentPundit uses ALEKS to fix Everyday Math
ALEKS question
ALEKS assessment coming right up

AleksAndIndividualizedProblemSets 04 Dec 2005 - 01:01 CatherineJohnson

This is the aspect of ALEKS that intrigues me:

• Adaptive, dynamically chosen small set of questions

• Details precisely what the student knows

• Constantly updated as work is completed

The idea of 'dynamically chosen' worksheets sounds good, but I wonder whether you gain anything you don't with a program like KUMON, where the worksheets aren't dynamically chosen. Saxon Math has students do the same worksheet many times during a school year, and I know from experience it works fine. You don't need a new mix of problems every time you practice.

On the other hand, even small gains in efficiency would add up over time.

formative assessment on wheels

Interesting.

Here's a link to the research/marketing paper ALEKS has posted on their web site:

ABSTRACT

This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a novel theory for the assessment of a student’s knowledge in a topic and gives details on a practical implementation in the form of a software system available on the Internet. The basic concept of the theory is the ‘knowledge state,’ which is the complete set of problems that an individual is capable of solving in a particular topic, such as Arithmetic or Elementary Algebra. The task of the assessor—which is always a computer—consists in uncovering the particular state of the student being assessed, among all the feasible states. Even though the number of knowledge states for a topic may exceed several hundred thousand, these large numbers are well within the capacity of current home or school computers. The result of an assessment consists in two short lists of problems which may be labelled: ‘What the student can do’ and ‘What the student is ready to learn.’ In the most important applications of the theory, these two lists specify the exact knowledge state of the individual being assessed. This work is presented against the contrasting background of common methods of assessing human competence through standardized tests providing numerical scores. The philosophy of these methods, and their scientific origin in nineteenth century physics, are briefly examined.

Of course now I'm super-intrigued.....

This is all I need, right now. One more high-concept math-learning scheme.

Curiosity doesn't seem to kill cats, but it's going to be the end of me.

a parent's experience with ALEKS
ALEKS Graphic
formative assessment on wheels
ParentPundit uses ALEKS to fix Everyday Math
ALEKS question
ALEKS assessment coming right up

AnneDwyerOnTutoring 16 Dec 2005 - 21:44 CatherineJohnson

What I've noticed with my tutoring students is this: if they don't understand something in math class, they try to find a procedure or "trick" that works everytime.

Since they don't really understand it, when they have to go back and do it on a test or later, they don't remember the "trick" exactly and their answers are consistent, but wrong.

For example, I was tutoring a student in basic math. He didn't really understand that a whole number has an implied decimal after the number (e.g. 3 is really 3. for a decimal problem)

When he first learned to divide decimals and he was following the teacher's examples, he was doing the problems right: So if he was dividing .045 into 15, he moved the decimal over three places for the .045 and three places for the 15. He even managed to get it right on the first test.

But he did them wrong on every test after that. When we were studying for the final, I was able to watch him do the problems.

Since he really didn't understand, he made up his own "trick". In the problem above, he would move the decimal over for the .045 correctly, but he put the decimal point in front of any number inside the divisor sign. So .045 into 15 became 45 into 150 instead of 15,000. And, because he had taught himself this trick, he ignored all decimal points inside the divisor sign. So even .045 into 1.5 became 45 into 150.

Needless to say, it took a while to find the problem and then to correct it.

IMO, with Christopher, because the class is going so fast and he doesn't always understand what he is doing, he will figure out his own rule and then apply it. You have to go back and see exactly what he is doing when he does the problems so you can identify the error he is making.

We are in fraction & decimal he** around here, which is annoying because I don't think we would be with Saxon or Primary Mathematics—and we weren't going into this course.

This is part of what I mean when I say Christopher is 'losing knowledge' he already had, or experiencing 'math regression,' or just......getting all jumbled up. I think he is becoming uncertain of procedures and knowledge he used to have fairly well nailed-down. (Though I don't know.)

Anyway, both of the ideas here strike me as excellent ideas.

First of all, I'm going to start writing whole numbers with a decimal point and some zeros to the right. I know that will help.

And second, I'm going to keep my eye open for 'invented shortcuts.'

One strategy I've begun, which I think is going to improve matters, is that I'm continually telling him that 'math shortcuts' come from the longer equations he's learned in the past. His teacher seems to be teaching only the shortcuts—either that, or he's only picking up the shortcuts, not the explanation for why they work. Either way, the result is the same: he's learning math tricks.

Last night, when I insisted on showing him why you could invert and multiply, he got his 'eureka' smile.

I'm sure he will have forgotten what I told him by today, but I'm going to keep hammering away at this.

I do think that the basic principle—that math shortcuts come from general principles he already knows—will stay with him, and will help.

source:
reciprocals

TheMathPage 21 Dec 2005 - 19:55 CatherineJohnson

I like The Math Page

What do you think?

AnneDwyerIsObsessed 19 Dec 2005 - 17:20 CatherineJohnson

from Anne Dwyer:

How do you know you're obsessed with mathematics education?

When you walk into a used book store and have to buy a Grammar School Arithmetic book published in 1892 because you want to see what math education was like before the progressive movement got involved.

Here are some cool things that I hadn't seen before:

The book teaches how to divide by a fraction (flip and multiply) but it also teaches this method for simplifying a fraction: Reduce 3/4/5/6 to a simple fraction (of course it was written as three fourths over five sixths) The answer: divide the top and bottom by 12 which is the lowest common multiple of 4 and 6 and it reduces to 9/10. I like this method because it works just like getting an equivalent fraction.

A number is divisible by 2 if the last or right hand digit is even.

A number is divisible by 4 if the number denoted by the last two digits is divisible by 4.

A number is divisible by 8 if the number denoted by the last three digits is divisible by 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisble by 9 if the sum of its digits is divisible by 9.

A number is divisible by 5 if its last digit is a 0 or 5.

A number is divisible by 25 if the number denoted by the last two digits is divisible by 25.

A number is divisible by 125 if the number denoted by the last three digits is divisible by 125.

A number is divisible by 6 if its last digit is even and the sum of its digits are divisible by 3.

A number is divisible by 11 if the difference between the sum of the digits in the odd places is either 0 or a multiple of 11.

Well, I have a roped-off pew in the church of my heart for the obsessed.

Edie: An American Biography by Jean Stein

key words: divisibility

IfTheStudentHasntLearned 23 Dec 2005 - 22:16 CatherineJohnson

revision

From Catherine:

Our new pretend-shirt specifically says "If the student hasn't learned, the school hasn't taught," not 'the teacher hasn't taught'.

No more thoughtless (and unintended) teacher-bashing.

Seriously. I'm the last person to want to make teachers feel blamed and bashed, seeing as how half my relatives have been or are currently teachers. I'm sure I'll be one again at some point, too.

The problem is that, when you talk about schools, it's the teachers who are visible. They're in the trenches, so they get the blame. (I realize I'm not telling teachers anything they don't know.) I know better than that, but I've been sounding like I don't.

Time for a course correction.

From Carolyn:

Hey, my entire family on my mother's side were also teachers, every man and woman Jack of them. I've been a teacher too; so has Catherine.

My observation is that policy flows downhill in a school, and the buck stops with the teachers. They get the responsibility, but not the authority; policy changes really have to start with upper management.

We're here to put the pressure on upper management, and support the teachers in doing what they know how to do.

TeachingMathToTeachers 19 May 2006 - 21:52 CatherineJohnson

Susan J left a link to Racial Equity Requires Teaching Elementary School Teachers More Mathematics (pdf file) by Patricia Clark Kenschaft.

I'm just beginning it, but so far it's right up my alley:

Seventy-five black people with at least one degree in mathematics responded to a variety of questions, including, “What can be done to bring more blacks into mathematics?”

[snip]

[the most common answer by far was] “Teach mathematics better to all American children. The way it is now, if children don’t learn mathematics at home, they don’t learn it at all, so any ethnic group that is underrepresented in mathematics will remain so until children are taught mathematics better in elementary school.”

[snip]

Like most Americans, I found it difficult to believe how poorly prepared mathematically they are.mathematically by our system. They need to be taught. I have found them eager and quick to learn—and appallingly ignorant of the most basic mathematics.

“Teach us math! Teach us math! Teach us math!” chanted dozens of elementary school teachers during one after-school workshop. There was an amazed silence while we all absorbed what had just happened. Then one of them said, “If you taught us math the way you did just now, we could teach it to the children.” They all nodded emphatically. This incident followed my statement that those of us who thrive mathematically have had some good mathematical experience early, typically at home. Someone had asked for an example out of my own childhood, and I had explained how my father had described the meaning of pi to me several months before I started kindergarten. Their response was the chanting, “Teach us math!”

The rest of the article is an account of Kenschaft's math classes for elementary school teachers.

I believe we need far less ed school and far more on-the-job training.

For me, that would include classes like Kenschaft's.

It's not reasonable to expect thousands of math majors to pour into K-8 education.

It is reasonable to expect that the dedicated and able people who've gone into K-8 education can continue to learn elementary school mathematics on the job, as Chinese teachers do. Chinese teachers typically have the equivalent of a high school education here, and their knowledge of math is not astonishing when they begin work. I imagine they start at a higher level than our teachers do—I'd have to check to see whether Liping Ma addresses this—but the fact is, Chinese teachers gain profound knowledge of elementary mathematics by studying the high-quality textbooks they must teach and meeting with colleagues to discuss those books.

If we think all kids can learn math, why don't we think all teachers can learn math?

The fact that they didn't learn math in their own schools & colleges is no reason to think they can't possibly learn math now, when they're employed and motivated to do their jobs well.

Ed ran summer institutes for high school history teachers. They were starved for real history and real colleagues, and they were smart.

That's the kind of professional development I'd like to see.

Let's have fewer Workshops on Differentiated Instruction, and more Summer Institutes in math, reading, writing, and history.

kids teaching kids

It has been my observation that the reason that scores are higher in white districts is that some parents teach their children mathematics at home, and these children teach many of the others. It has appeared to me that the teachers are no better prepared in the high-scoring districts.

I wouldn't be surprised to learn that elementary school teachers in high-scoring districts are no better prepared in mathematics than teachers in low-scoring districts—although I guess I'd been assuming that they were.

What did take me by surprise was Kenschaft's blunt statement that we parents are the entire reason high-scoring schools are high-scoring.

And I was gobsmacked by her assertion that kids like ours, who are being taught math at home, are in turn teaching math to other kids at school.

That possibility simply hadn't crossed my mind.

Which is funny, because Christopher taught his fourth grade partner-in-flunking how to do two-digit times two-digit multiplication.

Christopher. A kid who a couple of months before had been flunking math.

His friend hadn't gotten any remedial teaching at home, so Christopher taught him multiple-digit multiplication.

Our assistant superintendent told me that another kid in his school taught him algebra. A kid! The teacher was impossible, he said (and later on took credit for the Asst. Superintendent's progress.)

Of course, I was suitably scandalized by this story.

But it didn't occur to me to wonder how it was that the friend happened to know algebra.

You hear it said, often, that schools like Irvington's have high scores because their parents have high SES.

It's time to operationalize that statement.

How exactly does a high SES translate to my kid knows how to divide fractions?

Forget IQ differences, real or not; no one has an IQ so high he just naturally knows how to divide fractions. People have to learn how to divide fractions, which means someone has to teach them.

If Kenschaft is right, those people are the math brain parents and their kids.

it's always worse than you think

[The] principal invited me to consider that school “my school”. He and the teachers really wanted to help the students. Its students had a median achievement in mathematics of about the 25th percentile on the “Iowas”, one of the lowest levels in Newark. I am now convinced that its rank was due to the fact that the principal did not pressure the teachers to cheat in any way on standardized tests. When I told him this years later, his eyes widened. He was president of the principals’ union. “What? You are saying…” I nodded. Since then I have read numerous reports of systemic cheating on standardized tests and other forms of deception by school administrators...

A friend of mine was, I think, president of the PTSA in an affluent district when it was discovered that a teacher was cheating on the tests. She was walking around the room telling the kids the answers, IIRC. The principal put the teacher on leave, and the school blew up. The other teachers were bitterly upset; the parents went to war (many parents supported the teacher and attacked the parents who had complained as whistleblowers); many, many students left.

I lost contact with that friend not long after, so I have no idea whether the school even survived.

This was not a school in Newark.

communication skills for the 21st century

During my first class teaching elementary school children, a fifth grader raised his hand and asked, “What is that word you keep using instead of take away?” Enter “minus”—for fifth graders!

fast change

The best first-grade teacher told me she never bothered to teach subtraction during the first half of the year because the children couldn’t learn everything at once. I started visiting the school in October, and it seemed to me natural to teach addition and subtraction together. She told me she would not reinforce my teaching of subtraction between my weekly visits, and I said that was no problem.

One of the games I played with the children was holding five unifix blocks in front of me, putting them behind my back, and bringing forward three. “How many are behind my back?” I asked. The children could answer correctly. Then I told them that one way of writing this was “5 – 3 = 2”.

“Oh, no!” said the teacher.

“Because subtraction means “take away” and you took away two blocks. So it should be written ‘5 – 2 = 3.’” I explained that subtraction could mean “take away”, but it could also mean “missing addend”. It seemed to me that since the children could see three blocks, “5 – 3 = 2” was preferable, but “5 – 2 = 3” is not wrong. The next week we explored the “difference” meaning of subtraction and the “motion” meaning. (I walk five steps toward the window and three steps away. How many steps am I from where I began?)

She was startled when half the children passed the subtraction part of the November standardized test—without any reinforcement from her. She had never had a child pass it before. The crucial role of mathematical knowledge on the part of the teacher was becoming obvious to me.

white people can't jump (update 6-26-06: what does this heading mean?)

My first time in a fifth grade in one of New Jersey’s most affluent districts (white, of course), I asked where one-third was on the number line. After a moment of quiet, the teacher called out, “Near three, isn’t it?” The children, however, soon figured out the correct answer; they came from homes where such things were discussed. Flitting back and forth from the richest to the poorest districts in the state convinced me that the mathematical knowledge of the teachers was pathetic in both. It appears that the higher scores in the affluent districts are not due to superior teaching in school but to the supplementary informal “home schooling” of children.

The only thing wrong with this observation is: it's not so informal.

I'm working my tushie off here.

PhoneBookMath 11 Jan 2006 - 16:00 CatherineJohnson

direct from Duke University,

phone book math key

AnotherStudentMovesToPhase4 11 Jan 2006 - 21:17 CatherineJohnson

This is amazing.

I just got off the telephone with the mother of one of the kids in this fall's after-school Singapore Math class. She said my class gave him 'that extra inspiration' (something like that) he needed — and that he had been moved up to the accelerated class!

That makes 3 kids who've moved to the accelerated track after taking the class — 3 out of 5 or 6 children in all. (Some of the kids in the class were already in the accelerated track.)

I'm stunned.

First of all, I had no idea this child wasn't already in the accelerated track. He's a Math Guy.

Second.....whoa.

I've worked hard on the class, but it isn't much of a class (yet). I don't have good classroom management skills, I'm teaching kids after they've had a full day, I'm still feeling my way, etc.

Also: I'm using a curriculum designed to be used 5 days a week, not 1 day a week after school. That's a huge challenge.

I wonder what's going right?

These kids weren't crazy about the bar models; they liked Brain Maths.

On the other hand, by the end of the class we were doing two-variable algebra problems, and most of them were using bar models to figure out which operations to use.

My question is whether the main reason these kids jump forward has to do with motivation. As Nick's Mama said, I've fallen in love with math, and it shows. Some of the teachers at the Main Street School love math (maybe a lot of them do). Mrs. Woeckener, Christopher's Phase 4 teacher last year, sure did. She'd been an accountant for 15 years before becoming a teacher, and if you raised the subject of math with her she'd say, "I love math."

But they have to teach all day long, and they're on the hook, and so are the kids. In school, math is serious business; it's the children's job.

The other thing is: I'm just discovering math, and that shows, too. Normally I wouldn't think it's a good idea for a teacher to be an obvious amateur (and I think you could get killed taking such a stance in middle school).

But in an after-school class on Singapore Math, it seems to work.

Ms. Duque (now D'Arcy), Christopher's brilliant 5th grade teacher, told me last year she thought it was good for Christopher to see me learning math along with him. She said I was modeling how to learn math and how to tackle a problem and relish the challenge.

I wonder whether this is a case of 'infectious enthusiasm'?

Hard to tell how much the Singapore curriculum per se has to do with it. The kids in last year's class were using SRA Math; the kids this year are using TRAILBLAZERS. So I don't see this as a Singapore-versus-constructivism smackdown (wrestling terminology).The boy whose mother I just spoke with didn't even like doing the bar models. Some of the kids have loved them, and really taken off with them, but not him.

He was a BRAIN MATHS guy.

So I have no idea what's going on!

All I know is, it's very cool. I'm thrilled.

2 Singapore Math Class kids move to Phase 4
another student moves to Phase 4

WhatDoYouThinkOfThisProblem 14 Jan 2006 - 20:39 CatherineJohnson

from Houghton Mifflin Math, Grade 2, Chapter 6:

Cara, Nick, and Tammy had 96 cookies (8 dozen) to sell at a bake sale.

• Nick sold as many cookies as Tammy sold plus 5 more.

• Cara sold 10 fewer cookies than Tammy sold.

How many cookies did they have left?

I like it.

In fact, I like it a lot.

Am I wrong?

Here's the Hint:

Use the hundred chart or make a list. Then add or subtract.

And here's the Solution, which shows two ways to solve the problem.

Houghton Mifflin Education Place

PanBalanceInSaxonMath 20 Feb 2006 - 17:44 CatherineJohnson

I LOL'd when I read one of Carolyn's patented dry observations on the follies of 21st century math instruction:

Apparently you can understand the heck out of pan balances, and still have trouble with understanding and manipulating equations.

I distinctly recall being charmed the first time I saw a pan balance in Algebra to Go.

And of course I loved Carolyn's pan balance drawings:

I also had a lot of fun playing with the pan balance problems in the National Library of Virtual Manipulatives.

But I continued to experience a disconnect between pan balances and 'isolate the variable,' or 'do the same thing to both sides,' until I finally did Investigation 7 in Saxon 8/7: "Balanced Equations."

back from fun-filled Con Ed hiatus

We had an ice storm Saturday night, then a wind storm Wednesday morning, and there are so many trees down all over Westchester it's like Hurricane Katrina without the water.

Also without the trilliions of dollars in property damage, the loss of life, the breakdown of civil order, the helicopters, Wolf Blitzer, and the international expressions of shock and opprobrium.

Apart from that, it's exactly like Hurricane Katrina.

Anyway, the electricity went off at noon; the garage door is electric; the car was in the garage; the road to town was blocked; the side roads were blocked; when the electricity went back on the internet connection didn't; and so on.

All in all, about what you'd expect.

where was I?

Right.

I have no idea what I was planning to say about pan balances....apart from the fact that -- it's coming back to me now -- John Saxon can write a Pan Balance lesson like nobody's business.

The reason John Saxon can write a pan balance lesson like nobody's business is that he doesn't just slap down a drawing of a pan balance and expect the student to see the light.

Instead, he carefully develops his pan balance analogy, presenting the student with a sequence of 3 or 4 drawings of pan balances, one after the other, each one representing a step in the solution of an equation.

And he explains the whole thing in words. Words, pictures, numbers, and variables. Kit and caboodle.

Here he is:

Equations are sometimes called balanced equations [ed.: wonderful!] because the two sides of the equation "balance" each other. A balance scale can be used as a model of an equation. We replace the equal sign with a balanced scale. The left and right sides of the equation are placed on the left and right trays of the balance. For example, x +12 = 33 becomes

• (insert drawing of pan balance with x + 12 on the left side and 33 on the right)

Using a balance-scale model we think of how to get the unknown number, in this case the x, alone on one side of the cale. Using our example, we could remove 12 (subtract 12) from the left side of the scale. However, if we did that, the scale would no longer be balanced. So we make this rule for ourselves.

Whatever operation we perform on one side of an equation, we also perform on the other side of the equation to maintain a balanced equations.

We see that there are two steps to the process.

Step 1: Select the operation that will isolate the variable.

Step 2: Perform the selected operation on both sides of the equation.

Click.

This is perfect.

Instead of plopping a pan balance down in the middle of the page and expecting the student to discover its meaning, Saxon explains what the image means, and why it works.

Then he takes you through the steps which can only be implicit in a static drawing of a lone pan balance.

Then he has you draw your own pan balances.

I'm sick Christopher isn't using Saxon.

I'm so sick he isn't using Saxon, that I may try to squeeze Saxon back into our 'schedule.'

Saxon - Prentice-Hall smackdown Part 2

I've mentioned Christopher seems to be not only not gaining new knowledge, but to be losing the knowledge he already had.

Here's why.

The Saxon pan balance 'Investigation' opens with addition & subtraction equations.

Then the same Saxon Investigation proceeds to multiplication and division equations, reminding students in passing that multiplication and addition are related.

Prentice-Hall splits all of this up into separate lessons, and never the twain shall meet.

Multiplication and division go together.

Integers go together.

Decimals go together.

Fractions go together.

They're all in their separate lesson-boxes.

If the student doesn't make the connection, the connection doesn't get made.

I see why David Klein says all American textbooks are constructivist.

Technically, Prentice-Hall is a traditional book.

But nothing is explained, beyond the bare minimum. It's like a website with a lot of info to sort through (David has made that observation before), or a reference book with problem sets.

I don't know why they don't just buy these kids a Dictionary of Mathematics and let it go at that. There's a bunch of them out there.

-- CatherineJohnson - 20 Jan 2006

TeachingBinaryLikeSocrates 24 Jan 2006 - 23:30 CarolynJohnston

Rick Garlikov is a kind of modern-day Socrates, living in Birmingham, Alabama. He has a business mentoring students and their parents, and his philosophical writings are available online. Charlie Martin sent me a link to an article by Rick, entitled The Socratic Method: Teaching by Asking instead of Telling. The specific thing he is teaching is the notion of binary numbers, to a group of typical third graders.

Have a look: the whole transcript of the class session is there, and it's interesting. The guy can think on his feet. But it's more his interpretation of the situation that I wanted to dwell on here.

The experiment was to see whether I could teach these students binary arithmetic (arithmetic using only two numbers, 0 and 1) only by asking them questions. None of them had been introduced to binary arithmetic before. Though the ostensible subject matter was binary arithmetic, my primary interest was to give a demonstration to the teacher of the power and benefit of the Socratic method where it is applicable. That is my interest here as well. I chose binary arithmetic as the vehicle for that because it is something very difficult for children, or anyone, to understand when it is taught normally; and I believe that a demonstration of a method that can teach such a difficult subject easily to children and also capture their enthusiasm about that subject is a very convincing demonstration of the value of the method.

Many of the questions are decided before the class; but depending on what answers are given, some questions have to be thought up extemporaneously. Sometimes this is very difficult to do, depending on how far from what is anticipated or expected some of the students' answers are. This particular attempt went better than my best possible expectation, and I had much higher expectations than any of the teachers I discussed it with prior to doing it.

I like that he is not arguing that the Socratic method can replace other methods as the bread-and-butter teaching method in the classroom.

This method takes a lot of energy and concentration when you are doing it fast, the way I like to do it when beginning a new topic. A teacher cannot do this for every topic or all day long, at least not the first time one teaches particular topics this way. It takes a lot of preparation, and a lot of thought. When it goes well, as this did, it is so exciting for both the students and the teacher that it is difficult to stay at that peak and pace or to change gears or topics. When it does not go as well, it is very taxing trying to figure out what you need to modify or what you need to say. I practiced this particular sequence of questioning a little bit one time with a first grade teacher. I found a flaw in my sequence of questions. I had to figure out how to correct that. I had time to prepare this particular lesson; I am not a teacher but a volunteer; and I am not a mathematician. I came to the school just to do this topic that one period.

But I don't agree with everything he says:

The chief benefits of this method are that it excites students' curiosity and arouses their thinking, rather than stifling it. It also makes teaching more interesting, because most of the time, you learn more from the students -- or by what they make you think of -- than what you knew going into the class. Each group of students is just enough different, that it makes it stimulating. It is a very efficient teaching method, because the first time through tends to cover the topic very thoroughly, in terms of their understanding it. It is more efficient for their learning then lecturing to them is, though, of course, a teacher can lecture in less time.

Note the implicit assumption that lecturing to children 'stifles' their thinking. I doubt this very much; I expect that what really stifles their thinking is a constant diet of television.

Catch the reference to teaching being more interesting with the Socratic method. Teacher boredom is a real problem in the classroom; while a constantly changing group of kids is struggling to come up to speed on the material the teacher is teaching, so that they can move on, the teacher may never move on. It's up to the teacher to figure out ways to grow him or herself professionally, in such a way that it helps and doesn't harm the children's learning.

I disagree with the claim that the children learn the idea deeply the first time it is taught by the Socratic method. That's nonsense. We've all had the experience of listening to a really great teacher explain something, feeling we know it cold and are greatly enriched by the lecture, only to find we really remember none of it when it's time to sit down and do the homework -- i.e., to actively produce the knowledge ourselves. It's producing the knowledge yourself that leads to deep understanding. Leading questions are still just leading; what could be more 'sage on the stage'?

The other thing I disagree with is the notion that Socratic teaching serves as a type of in situ formative assessment:

It gives constant feed-back and thus allows monitoring of the students' understanding as you go. So you know what problems and misunderstandings or lack of understandings you need to address as you are presenting the material. You do not need to wait to give a quiz or exam; the whole thing is one big quiz as you go, though a quiz whose point is teaching, not grading. Though, to repeat, this is teaching by stimulating students' thinking in certain focused areas, in order to draw ideas out of them; it is not "teaching" by pushing ideas into students that they may or may not be able to absorb or assimilate. Further, by quizzing and monitoring their understanding as you go along, you have the time and opportunity to correct misunderstandings or someone's being lost at the immediate time, not at the end of six weeks when it is usually too late to try to "go back" over the material.

These claims may be true of the one or two children who are tracking the teacher all the way through the process. As usual, there will be some number of children who aren't following the discussion, or who develop some real misunderstanding of what's going on. Those kids are probably not going to speak up during a Socratic method class, any more than they would a normal class; and trying to correct those misapprehensions would derail the discussion, probably in a direction that Socrates wouldn't want to go. Good formative assessment has to assess everyone.

-- CarolynJohnston - 21 Jan 2006

SteveOnWhyKitchenTableMath 16 Sep 2006 - 20:28 CatherineJohnson

The only kids who are prepared to take a proper college prep math (esp. honors or AP courses) track in high school are those kids who are very smart or get help outside of the school. The current crop of fuzzy, low expectation, no mastery, discovery, spiraling math curricula are HARMFUL to kids. In the old days, traditional math may have been taught very poorly or inconsistently, but I don't think that was on purpose (perhaps incompetence and neglect played a part). Nowadays, perhaps there are more controls and teachers are more consistent (with the program), but the math curricula do not get students from point A (counting numbers in Kindergarten) to point B (a full course in algebra in eighth or ninth grade). This IS on purpose.

The problem of education is not some myopic teacher-perspective view of the problem. It is not "if only". If only we had more money. If only we had smaller class sizes. If only we didn't have to meet (trivial) state standards. If only the administration would get off my back. If only parents would get off my back. If only we had a better school culture. It is much more fundamental than that and it's not just about the teachers.

KTM exists because schools are not doing their jobs. Parents have to do it at home at the kitchen table. KTM is not ranting. It contains specific help for parents that they cannot get from the teachers, administration, school committee, or parent/teacher groups. Most of the regulars here have spent a whole lot of time working within their systems. It doesn't work.

After Christopher failed 2 of 6 units in 4th grade math, I had the Bayesian perception that unless I learned math myself, he would be out of the running for any career involving math in any way.

That perception may have been wrong. I'll never know how things might have turned out if I hadn't plunged into re-teaching Christopher his math, plunged into re-learning math myself, and ultimately plunged into writing and, more importantly, reading Kitchen Table Math.

Looking back, I think it's right to say that I myself was locked out of any career involving math in any way.

In my own school days, I was taught to mastery. That teaching stood me in good stead. I had 'shopkeeper's arithmetic' down cold, and I was able to start over again learning math in mid-life, and make quick progress.

But it wasn't enough to let me take math in college. And at that age, in college, I didn't know what I didn't know. I didn't know whether I liked math or not, whether I might be reasonably good at math or not, whether I should be doing something related to math or not....I didn't know anything. if I thought about it at all, I just figured I wasn't a 'math person.'

As one of Carolyn's old professors says, the last person you want making life decisions is a 19-year old.

When we were in Los Angeles over vacation, I spent time with the now-grown children of friends.

These kids have had fantastic educations, every one of them in private schools, including Catholic schools.

None of them is headed toward a math-related field at the moment (these kids are high school seniors & college freshmen) but each one of them could choose a math-related field if he or she wanted to do so. The door is open.

That's what I want for Christopher (and for Andrew, obviously, if I can get him there). I want the door to be open.

We've chosen to live in a high-tax suburban town with good schools. This was our version of choosing a private school. Talk about not knowing what you don't know.

The Irvington math track, thus far, isn't going to put Christopher in position to choose a math-related career.

Everyone says the high school is fantastic, and given the principal there I'm sure it is.

But when I talk to parents whose kids have taken AP calculus at IHS — and those kids are the only American kids who are competitive with their peers in other countries — what I hear is this:

His dad is really good at math, so he helped him all the way through.

In other words: my son made it through AP calculus because his dad knows calculus.

I have also heard this:

My son couldn't find a calculus tutor anywhere. He had to get through it on his own.

The woman who told me this has an advanced degree in math herself.

Carolyn says she finds it hard to believe that there could be no calculus tutors in all of Westchester County, and I agree.

But — and here's the point — I can't take the chance.

Maybe there'll be calculus tutors in Westchester when Christopher gets to Irvington High School, and maybe there won't.

Maybe Christopher would have gotten back on track without my turning into Math Mom, and maybe he wouldn't have.

I don't know.

I couldn't take the chance.

-- CatherineJohnson - 26 Jan 2006

MilgramStatementToCongress2000 02 Feb 2006 - 23:09 CatherineJohnson

I am honored to be here today and to be able to share my observations on the state of mathematics education in this country with the distinguished members of the Committee on Education and the Workforce.

The K - 12 teachers in this country are dedicated professionals, deeply committed to teaching our children. They persevere in the face of difficult conditions and low pay. I have the utmost respect for them. But all too often, their knowledge of mathematics is extremely superficial, and when this happens they are easily swayed by trendy and unproven programs which typically offer a superficial treatment of the subject, leading to weak backgrounds in their students.

Perhaps a local parent described this situation best when she wrote me recently that the curriculum was getting fuzzier and fuzzier, and she "concluded that by and large most teachers support it because it makes them feel OK about math - they understand language, not symbols." She continues, "I cannot tell you how many times I have heard from administrators and teachers, how, if they had had "this" math when they were in school, perhaps they, too, would have been perceived as a `math person'."

I am a research mathematician, and research in esoteric areas of mathematics is essentially all I did besides teaching graduate and undergraduate classes in mathematics at Stanford until four years ago.

Two things obligated me to spend much of my time for the last three years studying issues related to K - 12 mathematics.

The first was some courses I gave in New Mexico, where I had too many bright, very highly motivated students in my mathematics classes whose third rate K - 12 educations in mathematics could not be overcome no matter how hard these students were willing to work.

The second came from the Presidential Commission designing Clinton's proposed national eighth grade mathematics exam. The commission - including many of the foremost math education specialists in the country - distributed a list of 14 proposed problems. I and my colleagues at Stanford were amazed to find that 3 of the problems had serious errors.

One was so ill posed that it could not be solved. One had an incorrect solution included with it.

We later testified to the Clinton commission about these difficulties, and it became clear that the level of mathematical understanding on the part of the mathematics educators on this panel was unimpressive.

There is a distinction between math educators who are primarily interested in questions involving education, and mathematicians who know about mathematics. While educational issues are unquestionably important there has been a tendency recently to focus on educational questions at the expense of mathematics content. I was disturbed when I realized that it is these people who are determining the mathematics that our children learn in school. I was especially disturbed in view of the dramatic drop in content knowledge that we have been seeing in the students coming to the universities in recent years.

Since 1989 the percentage of entering students in the California State University System - the largest state system in the country - that were required to take remedial courses in mathematics have increased almost 2 1/2 times from 23% in 1989 to 55% today. And CSU admission is restricted to the top 30% of California high school graduates! This failure has important consequences for the nation. Although large numbers of US students entering the universities say they are interested in majoring in technical areas, very few actually get such degrees today.

The total number of technical degrees awarded to US citizens recently is approximately 28,000 yearly, while there are currently about 100,000 new jobs in these areas each year. Last year congress had to mandate an additional 142,000 new work visas for technically trained people, and these visas were used up by June 11, 1999, so great was the demand.

A large part of the blame rests with mathematics programs of the type recommended by the Department of Education recently as exemplary or promising.

All but possibly one of the programs in the list recommended by the Department of Education, represent a single point of view towards teaching mathematics, the constructivist philosophy that the teacher is simply a facilitator. Standard algorithms for operations like multiplication and division are not taught, but students are advised to construct their own algorithms. At all stages hand held calculators are used for arithmetic calculations. There are no means for students to develop mastery of basic arithmetic operations. Algebra is short-changed as well.

These programs all are designed to closely align with the 1989 NCTM Mathematics Standards: standards which explicitly embody all the principles above, and specifically require that skills in algebra be downplayed. Indeed, the co-chairman of the Department of Education Expert Panel on Mathematics, Steven Leinwand, recently stated that the curricula endorsed by the Department of Education "create a common core of math that all students can master." Not material that all students NEED to know or SHOULD master, imply material that HE believes all students can learn. (Incidently, Department of Education statistical analysis - C. Adelman, 1999 - show that success in algebra in high school is the single most important predictor of degree attainment in college.)

The high school programs, Core-Plus and IMP, both place heavy emphasis on topics such as discrete mathematics at the expense of basic algebra, and do not come near the level indicated in e.g., the new California Standards for most of the topics there.

However, programs such as these are completely consistent with the previous California Mathematics Standards. Consequently, at least three of them, CPM, Mathland and IMP, have been in wide use in California for up to 10 or more years. (MathLand and IMP were developed in the late 1980s at the same time that the 1989 NCTM Standards were being developed, and were introduced into California Schools by 1989.)

Recent studies of the SAT mathematics scores of high schools which use IMP showed a consistent and significant decline over the last ten years.

Moreover, high schools that use IMP in California scored below the state means, and those that expressed satisfaction with the program scored, on average, 10 points lower than those which were dropping the program or otherwise were dissatisfied with it.

It was the introduction of CMP and TERC (another NSF funded curriculum published by Dale Seymour -- designed for grades K - 5) in the Palo Alto school system that sparked the initial parental revolt which became the California Mathematics Wars.

It was the introduction of Everyday Mathematics in the Princeton Township School District, which led to the parental revolt in Princeton. This led to the involvement of a number of faculty members in both mathematics and physics at Princeton University and the Institute for Advanced Study in Princeton in trying to reform mathematics teaching in the district.

It was the use of TERC in one school system in Massachusetts, which prompted numerous members of the Harvard Mathematics Department to sign the open letter to Secretary Riley.

The support for these programs in the Department of Education is ultimately the responsibility of the Education and Human Resources Department, EHR, at the National Science Foundation. EHR funded the development of at least six of the "exemplary and promising" programs.

It is also probably worth noting that at the present time there is no valid research which shows that any of the programs of this type are effective.

At least equally important are the Systemic Initiatives funded by EHR, which have the objective of pushing the districts where these initiatives are awarded to adopt curricula in mathematics which align with the 1989 NCTM Mathematics Standards.

In California, there is one systemic initiative from EHR still functioning, a grant to Los Angeles Unified School District, LAUSD, the nations second largest district with 711,000 students. The people involved in this initiative resisted attempts to change the system in place there, while similar districts such as Sacramento Unified began to make major changes.

Two years ago, the two districts had equally bad scores - around the thirtieth percentile - on the California Statewide mathematics exams. This last year LAUSD had essentially the same score as previously while the Sacramento Unified scores jumped dramatically, particularly in the lower grades, due to their shift away from whole language and constructivist math.

Incidentally, I had been told two years ago that getting a grant from EHR in a mathematics related area required that one buy into the list of ideas discussed above. As a test of this I obtained all the (over 4000) abstracts for the last 9 years from EHR for awarded grants that involved mathematics.

I tested a random sample of about 200 for a few key phrases such as NCTM Standards, group learning, and discovery learning. All but four of them contained at least one of these phrases.

In conclusion, I believe that the sad state of mathematics education among high school graduates in this country is primarily the responsibility of two agencies: the Department of Education and Human Resources at the NSF, and the Department of Education. The programs they develop and push simply set too low a standard.

Written Testimony of R. James Milgram February 2, 2000

Written Testimony of R. James Milgram February 2, 2000, summary points

• "I cannot tell you how many times I have heard from administrators and teachers, how, if they had had "this" math when they were in school, perhaps they, too, would have been perceived as a `math person'."

• Two things obligated me to spend much of my time for the last three years studying issues related to K - 12 mathematics.

• The first was some courses I gave in New Mexico, where I had too many bright, very highly motivated students in my mathematics classes whose third rate K - 12 educations in mathematics could not be overcome no matter how hard these students were willing to work.

• The second came from the Presidential Commission designing Clinton's proposed national eighth grade mathematics exam. The commission - including many of the foremost math education specialists in the country - distributed a list of 14 proposed problems. I and my colleagues at Stanford were amazed to find that 3 of the problems had serious errors.

• There is a distinction between math educators who are primarily interested in questions involving education, and mathematicians who know about mathematics

• it is [math educators, not mathematicians] who are determining the mathematics that our children learn in school.

• I was especially disturbed in view of the dramatic drop in content knowledge that we have been seeing in the students coming to the universities in recent years.

• Since 1989 the percentage of entering students in the California State University System - the largest state system in the country - that were required to take remedial courses in mathematics have increased almost 2 1/2 times from 23% in 1989 to 55% today

• Although large numbers of US students entering the universities say they are interested in majoring in technical areas, very few actually get such degrees today.

• The total number of technical degrees awarded to US citizens recently is approximately 28,000 yearly, while there are currently about 100,000 new jobs in these areas each year. Last year congress had to mandate an additional 142,000 new work visas for technically trained people, and these visas were used up by June 11, 1999, so great was the demand.

• All but possibly one of the programs in the list recommended by the Department of Education, represent a single point of view towards teaching mathematics, the constructivist philosophy that the teacher is simply a facilitator

• There are no means for students to develop mastery of basic arithmetic operations. Algebra is short-changed as well.

• These programs all are designed to closely align with the 1989 NCTM Mathematics Standards: standards which explicitly embody all the principles above, and specifically require that skills in algebra be downplayed.

• the co-chairman of the Department of Education Expert Panel on Mathematics, Steven Leinwand, recently stated that the curricula endorsed by the Department of Education "create a common core of math that all students can master." Not material that all students NEED to know or SHOULD master, imply material that HE believes all students can learn

• The support for these programs in the Department of Education is ultimately the responsibility of the Education and Human Resources Department, EHR, at the National Science Foundation. EHR funded the development of at least six of the "exemplary and promising" programs

• Recent studies of the SAT mathematics scores of high schools which use IMP showed a consistent and significant decline over the last ten years.

• I had been told two years ago that getting a grant from EHR in a mathematics related area required that one buy into the list of ideas discussed above. As a test of this I obtained all the (over 4000) abstracts for the last 9 years from EHR for awarded grants that involved mathematics.

• I tested a random sample of about 200 for a few key phrases such as NCTM Standards, group learning, and discovery learning. All but four of them contained at least one of these phrases.

On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
National Research Council

Executive Summary, page 3

Under the auspices of the National Research Council, this committee’s charge was to evaluate the quality of the evaluations of the 13 mathematics curriculum materials supported by the National Science Foundation (NSF) (an estimated \$93 million) and 6 of the commercially generated mathematics curriculum materials (listing in Chapter 2).

The committee was charged to determine whether the currently available data are sufficient for evaluating the effectiveness of these materials and, if these data are not sufficiently robust, the committee was asked to develop recommendations about the design of a subsequent project that could result in the generation of more reliable and valid data for evaluating these materials.

[ellipsis]

These 19 curricular projects essentially have been experiments. We owe them a careful reading on their effectiveness. Demands for evaluation may be cast as a sign of failure, but we would rather stress that this examination is a sign of the success of these programs to engage a country in a scholarly debate on the question of curricular effectiveness and the essential underlying question, What is most important for our youth to learn in their studies in mathematics? To summarily blame national decline on a set of curricula whose use has a limited market share lacks credibility. At the same time, to find out if a major investment in an approach is successful and worthwhile is a prime example of responsible policy. In experimentation, success and worthiness are two different measures of experimental value. An experiment can fail and yet be worthy. The experiments that probably should not be run are those in which it is either impossible to determine if the experiment has failed or it is ensured at the start, by design, that the experiment will succeed. The contribution of the committee is intended to help us ascertain these distinctive outcomes.

[ellipsis]

The charge to the committee was “to assess the quality of studies about the effectiveness of 13 sets of mathematics curriculum materials developed through NSF support and six sets of commercially generated curriculum materials.”

[ellipsis]

In response to our charge, the committee finds that:

The corpus of evaluation studies as a whole across the 19 programs studied does not permit one to determine the effectiveness of individual programs with high degree of certainty, due to the restricted number of studies for any particular curriculum, limitations in the array of methods used, and the uneven quality of the studies.

source: On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations (2004)
Mathematical Sciences Education Board (MSEB)
Center for Education (CFE)
available online or purchase, pages 3 & 188

learning a year or more of math in 2 months
James Milgram on long division & lag time in math learning
NYU math major

-- CatherineJohnson - 02 Feb 2006

JohnnyAndKarin 06 Feb 2006 - 15:52 CatherineJohnson

This is so beautiful, I'm posting the whole essay.

From the Los Angeles Times
X = Karin (Johnny) {gt} 95%
What does it take to learn algebra? First you have to master the fundamentals.
By Karin Klein
KARIN KLEIN is an editorial writer for The Times.

February 4, 2006

JOHNNY PATRELLO was a greaser. I was a dork. And yet, despite our rigidly stratified school culture, we came together in the spring of 1968 at Walt Whitman Junior High School, where I tutored Johnny in algebra.

I thought about Johnny again as I read The Times' series this week on L.A.'s dropout problem. Algebra, the reporters found, is an insurmountable stumbling block for many high school students.

What struck me was that the reasons why Johnny can't do algebra in L.A. today are remarkably similar to why Johnny Patrello couldn't do algebra almost four decades ago in Yonkers, N.Y.

Johnny and I were brought together by Mrs. Elizabeth Bukanz, the algebra teacher. Mrs. Bukanz wore her sandy hair in a frizzy French twist and her glasses on a chain. But she was gentle and smiling, and she had passion — at least for what she called "the beauty of algebra." I, too, loved its perfect logic and tidy solutions, so unlike my messy teenage life.

But Johnny was deaf to algebra's siren song. He was flunking, and Mrs. Bukanz hoped that if I used my study halls to tutor him, he might score at least 65% on the New York State Regents exam. Passing the exam allowed even failing students to move on to high school, which started in 10th grade; otherwise, Johnny would be left behind.

Johnny wore his leather jacket in class despite the spring warmth, and he habitually tilted his face toward the floor so that when he looked up at me, he seemed embarrassed. Yet for such a cool guy, he was surprisingly friendly and committed to giving this a try.

Things looked pretty hopeless to both of us those first couple of sessions, as Johnny stumbled through algebra problems while I tried to figure out exactly what he didn't understand. Then, as we took it down to each step of each little calculation, the trouble became clear: Johnny somehow had reached ninth grade without learning the multiplication tables.

Because he was shaky on those, his long multiplication was error prone and his long division a mess. As Johnny tried to work algebraic equations, his arithmetic kept bringing up weird results. He'd figure he was on the wrong track and make up an answer.

This discovery should have made us feel worse. How could we possibly make up for a dearth of third-grade skills and cover algebra too?

But at least we knew where to start.

We spent about half of those early sessions on multiplication drills. Seven times eight, eight times seven — Johnny could never remember. As an adult, in memory of Johnny's struggles, I would rehearse my kids at an early age in that one math fact. Get that 56 down, I would tell them, and the rest of multiplication is a snap.

Today's failing high school students, though plagued by more poverty and upheaval than Johnny or I ever knew, bring the same scanty skills to algebra class, according to The Times' series. They never quite grasped multiplication tables, but still they moved on to more complicated math.

Who can focus on the step-by-step logic of peeling back an equation until "x" is bared when it involves arithmetic that comes slow and slippery, always giving a different answer to the same calculation?

Yet in all these decades, the same school structure that failed Johnny goes on, dragging kids through the grades even if they don't master the material from the year before. This especially makes no sense for math, which is almost entirely sequential.

Leaving children back isn't a solution; it simply makes them feel stupid. They learn, like Johnny, to look at the floor. The floor can't embarrass them.

What I learned from Johnny — aside from the fact that greasers could be sweet-natured and very, very smart — is that schools are structured to help administrators feel organized, not to help children learn.

Young children's skills are all over the map, yet we corral them into second grade, third grade and so forth, where everyone moves at one pace in all subjects. Better to group them according to their skills in each subject, without the "grade" labels, and let them move on to the next skill when they have mastered the one they were on. If they're not getting it, give them extra tutoring, but don't push them forward until they're ready. This way, there is no failure — only progress.

It requires a sea change in thinking, but it's not impossible or even all that hard. Back before standardized tests put classes in lockstep, some progressive schools already were using team teaching to do this in math as well as reading and writing.

Johnny finally nailed seven times eight, then with amazing quickness worked his way through basic "x" problems up to multiple variables and beyond. Still, I couldn't quite catch him up to a year's worth of work in a couple of months. And on a sweltering June day, with humidity that neared 100%, the regents exam came, faster than we felt ready for it.

A couple of weeks later, I saw Johnny in the hall. He shot me a dazed look and broke the news — 95%! That moment has wiped from memory my own regents score. But I won the algebra award at the graduation ceremony. Johnny cheered, apparently undaunted by the fear of appearing uncool.

We lost touch in high school. I was college-prep, he was voc-ed. We would pass occasionally in the halls, and he would glance up from the floor and say, "Hi, teach!"

I know he received his diploma because I see his picture in my old yearbook, wearing a suit and tie instead of his leather jacket. His eyes still look up cautiously from his slightly downcast face, as though he is a bit surprised to be there.

BEFORE I USED Johnny's full name in a story that would reach more than a million readers, it was only right to try to contact him for permission. Directory assistance found one John Patrello, not too far from Yonkers.

The phone was answered by his wife, Joann. It was the same Johnny, but he had died a year and a half ago of a massive stroke, leaving behind Joann and four children.

As she and I talked, both of us in tears at times, it was amazing how much of what I remembered about Johnny continued throughout his life — the tough outer look, the sweetness a millimeter underneath, the quick mind, the habit of tilting his face toward the floor. His eldest is a doctor; the second, a teacher. His teenage daughter wants to be a journalist, and I'll see what I can do to help her along the way.

Johnny became an auto mechanic. ("He loved math, and you know auto repair involves a lot of math," Joann said. Yes, it does.)

Another thing Joann told me about Johnny: He was incredibly fast at multiplication.

-- CatherineJohnson - 05 Feb 2006

CarolynOnMasteryLearning 07 Feb 2006 - 19:54 CatherineJohnson

I was just doing some Librarian work on ktm (linking like posts with like, dropping 'back doors' into existing posts, posting links in the book-style index) — and I discovered that Carolyn wrote a post on mastery learning back in May!

How good is mastery learning? Two of the review studies looked at mastery learning by itself and with combinations of other curricula, and found that mastery learning by itself produces better results than what was termed 'conventional instruction'. However, mastery learning got its best results when used with other teaching techniques. One study got decent results for "mastery learning with corrective feedback" (meaning -- electric shock? The review didn't say), but got its best results from mastery learning with 'enhanced cues' -- extremely detailed instructions to the students on how to do problems.

Another study found that mastery learning and cooperative learning strongly enhanced each other (note: cooperative learning is structured working-together among students, as opposed to simply being stuck in groups to do your homework together: see part two of this series).

It's interesting, reading this post now, not least because I recognize one of the author's names: Doug Carnine.

Report to the California State Board of Education

-- CatherineJohnson - 06 Feb 2006

SchmidtInLosAngelesTimes 08 Feb 2006 - 15:44 CatherineJohnson

The LA Times series on Los Angeles high schools includes a terrific interview with William Schmidt, U.S. TIMSS NATIONAL RESEARCH COORDINATOR 1998.

I love this line:

...we know that by the end of eighth grade, U.S. students are probably some two years behind their counterparts in most of the rest of the world…. Middle school in the rest of the world is about algebra, geometry, chemistry, physics. In this country it's still about a lot of arithmetic and what I call "rocks and body parts."

rocks and body parts

Never heard that one.

elementary school & fractions

A: Studies show that middle school is where we lose a great deal of ground, at least internationally. The middle school in these other countries in mathematics is much more demanding. And it's much more of a transition into what we in this country first do in high school. So when our kids come into high school, they're a couple of years behind already. And our high schools just can't make it up.

Basically, the middle school is not preparing kids adequately. But it actually goes all the way back even into primary school, where, again, the kids are not being prepared well, don't understand and aren't able to do the computations associated with simple arithmetic.

Q: So it starts in elementary school?

A: We need to really start a much more serious, clearly defined, coherent curriculum all the way back there, and then we'd have a better shot at doing better with our kids. A lot of mathematics in this country is not designed very coherently. It doesn't progress from the simple to the more complex in ways that are reflective of the mathematics discipline. There's a sequence of things that make the most sense. And very typically in American schools, these sequences are not very clearly laid out.

Q: What do you mean?

A: Fractions, for example, are very difficult for students. Instead of introducing the concept clearly enough so that they understand fractions as numbers on the number line, we oftentimes try to move too quickly to other parts of fractions, such as the operations, before they really have a clear understanding of what fractions are and how they fit into the broader number system.

So kids are trying to learn how to operate on these things, and at the same time they really don't understand what they are, so things get very muddled in their minds.

For me, this is very helpful.

I have yet to meet a mathematician, engineer, or applied-math professional who didn't tell me that fractions are the bottleneck.

I believe it, but I don't quite understand what it means. That's due, in part, to the fact that I haven't begun to re-learn algebra. (Algebra starts in April.)

But I'm also confused, at this stage of the game, over what exactly K-6 kids should be learning about fractions in order not to fall apart later on.

The idea that kids need to understand fractions as numbers on a number line before performing the four operations on them is extremely helpful.

I tend to think he's right about this, though not in quite as literal a sense as this passage implies. Based entirely in my experience of teaching math to Christopehr & re-teaching math to myself, I wouldn't say you need to nail down fractions are numbers before here's how fractions are added & subtracted.

I'd say that fractions-are-numbers can be illustrated and taught via fractions-are-numbers-that-can-be-added-and-subtracted.

My sense is that you want to spend a great deal of time using the number line and using rulers to show both that fractions are numbers and that fractions are numbers that can be added and subtracted.

Saxon Math & fractions

Saxon Math has a number of interesting approaches to fractions:

• kids are asked to count in fractions: 1/4; 1/2; 3/4; 2; 2 1/4; 2 1/2; 2 3/4; 3

• kids do mental math with fractions

• I believe kids are asked to do some skip-counting with fractions as well (not sure)

• fractions, especially fractions of groups, are taught via bar models

is there one perfect method?

Q: Are the nations throughout the world using a different curriculum? Do they have different teaching methods?

A: That's actually a point I want to make very clearly. There doesn't seem to be one perfect method for doing this across the world. Different countries have different methods, just like we do here in the United States.

The real issue is the what. What it is that they're studying, in what grade levels in what sequence and at what level of rigor. Those are the issues that become important, not the how. It's more the what.

Q: We're just not being hard enough.

A: Yes. That's it, in a certain sense. As we move through the grades, we keep repeating topics year after year. We try to do too many topics at each grade level. We coined the phrase the "mile-wide, inch-deep curriculum" as a characteristic of the U.S., which means they just keep repeating these topics and as a result they have so much every year that it's too much for the kids to try to learn.

In these other countries it's a more focused attention on a smaller number of topics that progress across the grades in a logical fashion, leading to higher levels of expectation as you get up in the grades, like in the middle grades. And that's what we need.

Here's where I suspect he's missed — or perhaps slighted — a key issue, which is teaching to mastery.

I'd put money on it that in fact there is one 'perfect method' of teaching math, which is to make sure students learn to mastery.

I also suspect that in some (or perhaps many?) countries parents, not schools, are responsible for seeing to it their children learn to mastery. I'd almost bet the ranch that's the case in Singapore.

If this is so, classrooms could in fact look very different to researchers. Once parents take on the job of formative assessment, schools gain a great deal of leeway, to put it mildly.

Wherever the learning-to-mastery is actually taking place, whether at school, at home, or in both locations, you're going to see the same things. You're going to see massed practice, you're going to see distributed practice, and you're going to see overlearning.

Assuming I understand the findings of cognitive science correctly, and I think I do, there is no other way.

mile-wide, inch-deep

....As we move through the grades, we keep repeating topics year after year. We try to do too many topics at each grade level. We coined the phrase the "mile-wide, inch-deep curriculum" as a characteristic of the U.S., which means they just keep repeating these topics and as a result they have so much every year that it's too much for the kids to try to learn.

In these other countries it's a more focused attention on a smaller number of topics that progress across the grades in a logical fashion, leading to higher levels of expectation as you get up in the grades, like in the middle grades. And that's what we need.

Again, why is it we 'keep repeating topics,' and why is it that as a result our kids have 'too much to learn'?

Kids in other countries end up learning far more than our kids do.

This is a perfect description of Christopher's accelerated math class, I must say. They're covering a zillion topics; it's far too much to learn in the time they have.

What he's not mentioning is the fact that when you cram too many topics into one school year, the kids end up learning nothing well.

Easy prediction: 'mile-wide, inch-deep' is going to be interpreted, in the next cycle of math reform, to mean we should teach fewer math topics, period. Teach fewer topics and continue not teaching to mastery.

teacher prep

Q: In Los Angeles, some educators say they have a hard time finding qualified teachers. Is that a problem for other nations?

A: For some. In the elementary grades, everybody struggles with this, because elementary teachers have to teach all the subjects. But once you get into about middle school, this is more of a problem in the United States, where our teachers are not as well prepared as the teachers in these other countries.

We are doing a study right now across six countries in which we very clearly find that U.S. teachers — U.S. teachers from middle school — are not being … required to take the same level of mathematics that is true in other countries. Teachers that are going to teach middle school mathematics have to have a stronger background.

IIRC, in Asian countries teachers begin to specialize as early as the 4th grade. I'd like to see that in our schools. Teaching both English language arts and math well in 5th grade is a huge undertaking.

By the numbers

A 2003 study found that U.S. 15-year-olds scored low among industrialized nations on the PISA* mathematics test.

 Rank Country Score 1. Hong Kong (region) 550 2. Netherlands 538 3. Japan 534 4. Belgium 529 5. Australia 524 6. New Zealand 523 7. Norway 495 8. Hungary 490 9. Latvia 483 (tie) United States 483 11. Russia 468 12. Italy 466

* Program for International Student Assessment -
Source: American Institutes for Research

Chapter 1 Why Schools Matter (pdf file)

-- CatherineJohnson - 07 Feb 2006

CollegePrep 09 Feb 2006 - 01:24 CatherineJohnson

via eduwonk, a link to an Ed Sector anaylsis, High Schools Failing to Prepare Many College-Bound Students for Science Careers.

factoids

• science, technology, engineering, and mathematics = STEM

• 82 percent of high school kids say they plan to go to college, but only 51% are in college prep [ed.: awhile back I read some material from Roy Ohrbach, of U.C. Riverside, showing that often Hispanic parents have no idea their kids aren't in college prep — Ohrbach's been traveling around CA, IIRC, giving parents papers in Spanish explaining what the college track is & how to find out if your child is in it]

• definition of college prep: 4 years of English, 3 years of math, science, and social studies, 2 years of foreign language, and 1 semester of computers — 31 percent of high school graduates complete this basic college preparatory curriculum

• 14 percent earn math or science credit in Advanced Placement (AP) or International Baccalaureate (IB) programs

• about 60 percent of students who take AP tests in Biology, Chemistry, and AB Calculus get a score of "3" or better

• 12 percent of h.s. kids take calculus

• 40 percent take trig (this includes the 12% of all h.s. kids who go on to take calculus)

I wouldn't think these figures were so bad, if it weren't for the charts below. Forty percent of all h.s. kids making it through trigonometry sounds OK to me (not that I would know...)

But when you look at how many of these kids take no math at all in college — around 70 percent — that seems pretty bad to me.

Because of poor middle school preparation, tracking, inadequate guidance counseling, low-quality instruction, or a simple absence of available courses, too many students are permanently knocked off the pathway to a STEM career early in high school or even before. This is particularly true for low-income and minority students. No one tells them or their parents that by failing to enroll in a rigorous, math-oriented college prep curriculum, they're effectively making a life decision to forgo the opportunity to pursue a career as a scientist or engineer.

This isn't just a problem for low-income & minority students. It's a problem for just about anyone who majored in the social sciences or humanities.

I had no idea, when Christopher was tracked into Phase 3 math in 3rd grade, that he'd been tracked out of calculus in high school. None. (spaced repetition, I know) I had no idea that a) there is a 'math track' and b) it starts young. People with jobs like mine naturally assume that math works like everything else. You go to high school, you graduate, you go to college, you choose a major — and the major can be anything you decide you're interested in. All doors are open.

update: Tracy & Matt Goff weigh in below

from Tracy

On the topic of the importance of doing maths, I know two girls who were tracked out of what they wanted to do by not doing maths.

One was told by her guidance councillor that she didn't need Maths With Calculus to get into engineering, only Maths With Statistics. (You could take two maths courses in the last year of high school).

Another was told by the Head of Chemistry that she didn't need to do another maths course at uni for her chemistry degree. Then she couldn't do an advanced organic chemistry course because she didn't have enough of a calculus background and had to change the topic of her PhD.

here's Matt G

I would not say that it is not possible to get a degree in a STEM field without having had calculus in High School. One of my math major classmates as an undergraduate had not had calculus in High School and he did fine starting in Calculus in college (which many students need to do anyway, even if they have already had Calculus in High School). I knew at least one person while I was at graduate school who had started in the basic algebra class and worked her way up through the math program (she was a non-traditional/adult student).

It is, however, my impression that if you have (barely) made it through algebra in High School, the chances are pretty decent that in some way for some reason you have been turned off to math (and likely science). At that point it seems very unlikely that you would choose to major in a STEM related field. That is to say, I think the barrier to students entering STEM fields is mostly a matter of perception and/or expectation, rather than something fundamental and insurmountable. It may take a year or more extra, and you probably won't get your degree form Cal Tech or MIT, but there are plenty of schools where the motivated student can work through the math/science curriculum (and whatever prerequisites might be necessary) and enter a STEM field.

That makes sense to me (based in extremely limited knowledge of what it takes to succeed in college math, obviously.)

It's never struck me as likely that not taking AP calculus would knock a kid out of any kind of math at all in college. And based in Ed's view of AP history (not especially positive) I assume most AP students are going to have to repeat calculus in college.

My AP calculus goal for Christopher is almost entirely pragmatic.

I'm assuming that if Christopher sets AP calculus as his goal (which, at this point, he has) he'll work hard in lower level courses, and learn more.

I also assume that taking calculus twice is a good thing. (Maybe it's not, but for me it's been good to do basic math twice.)

Rudbeckia Hirta on taking calculus twice

Bad calculus is worse than no calculus. I'd much rather have students in my class with a solid algebra background + no calculus than those who took a purely algorithmic high school calculus classes. Just this week one of my students (in Calculus 1) told me, "I already know calculus. It's when you take the number up top and put it down in front and lower it."

But perhaps I say this because this week I am teaching the limit definition of the derivative.

[snip]

I would say that a bad calculus course would be one that emphasized the easy, algorithmic calculations while minimizing the historical context, the applications, the technical details that make it all work, and the importance of mathematical precision in phrasing and justifying statements.

A crude analogy would be a history class that was only about dates and places and names (bad) and one that involved analysis of the issues involved and their context (in addition to the dates and places and names) (good).

You can probably teach a BIRD how to take the derivative of a polynomial function. Knowing when to do it, why you can, and what it means requires a person (who probably has taken a good calculus course).

The problem that I face is that my students (who are at the dualistic thinking stage of the Perry Model) believe that their high school teacher's point of view ("Calculus is about computing derivatives and integrals") is the right one and that mine ("Calculus is a subject in which mathematical techniques were developed to solve problems relating to areas and tangents.") is not. If they came to me thinking, "In my high school calculus course, I learned a little bit about part of calculus," then it would be OK. But instead they tend to think, "In my high school calculus course, I learned calculus. And my college is SO MEAN AND UNFAIR by making me take this so-called calculus course that ISN'T REALLY CALCULUS because it contains all sorts of stupid and unimportant stuff like proofs and limits and word problems!"

I had never heard of the Perry model - it's terrific.

Ed is constantly trying to talk college undergraduates out of stage 2.

-- CatherineJohnson - 07 Feb 2006

MathOnBroadway 11 Feb 2006 - 16:34 CatherineJohnson

Ed and I went to see Beauty & the Beast Thursday night (NYU had discount tickets on sale). The first act was so boring we almost left at intermission, but the second act was great. There were a zillion little kids in the theater having the time of their life, so that was fun, too.

Seeing as how I had my AlphaSmart with me, I was able to record, almost verbatim, the conversation behind me. The person speaking is female:

He said, Why are you going to be gone for 6 weeks?

I said, Well 6 weeks divided by 3.

He said, 5?

I said, Divided by 3.

He said, 3?

I said, 2. Six divided by 3 is 2. I’ll be gone 2 weeks. And what do you care anyway? You’re not even in my group.

True story.

-- CatherineJohnson - 11 Feb 2006

LindaSchrockTaylorOnMathAtSchool 16 Feb 2006 - 02:17 CatherineJohnson

I was just trying to de-code the mysteries of Saxon Physics, when I came across this observation from Linda Schrock Taylor:

Frequently we are asked, "When do you end the school year in homeschooling?" My answer is always, "When the last math lesson has been completed and the final exam passed with flying colors." I think it is important that students complete books, especially math books. Each year I would note that even the best math teachers in the public school where I taught were only completing about 42% of each math book prior to the start of summer vacation. The students then went home for eleven weeks, and returned to face the next book in the sequence—even though they were never taught the last 58% of the material in the prerequisite class! Still people wonder why American students fall ever further behind in math!

update: more on not finishing the book

I am thoroughly convinced that Accelerated Math can do things for students in math that are almost impossible to accomplish otherwise. The instant feedback and the emphasis on mastery ensure that students do not just coast through the program without truly learning the material. While the teacher (or someone) still has to do much of the teaching, students can be much more independent much of the time, and can cruise quickly through objectives that come easily to them. I have never made it through the end of the math book with any of my classes - I'm lucky to get past the halfway point with some of them. But with AM, motivated students can master EVERY SINGLE objective for the grade level library they work through, eliminating the gaps I see in the math skills of most of my students.

update: question

I remember reading somewhere — and posting — that math textbooks have approximately 23% new content each year....the rest being review of content taught in years before.

I have no idea where I read this, or where I posted it — and am now wondering whether I dreamed the whole thing up.

Does this factoid sound familiar to anyone?

Saxon Physics mystery

Charles found a site selling Saxon Physics, which a Saxon rep told Carolyn is out of print, at a nice price....but I can't figure out what comes with.

The same site also offers the Solutions Manual (\$27.99) and Saxon Physics, Answer Key Booklet & Test Forms (99 cents!)

Maybe I've gone blind or lost my capacity to read, but I simply cannot tell whether the Saxon Physics Home Study Kit — "Offering 100 physics lessons, tests, answers, periodic table, charts, and more: all you need to teach a complete physics course" — also includes a Solutions Manual and an Answer Key & Test Forms.

I'm guessing it does not include a Solutions Manual (but why would that be?) & does include an answer key & tests, seeing as how it says it includes an answer key & tests, & does not mention a Solutions Manual.

Nevertheless, I'm confused.

Megawords 14% off

The site offers Megawords at 14% off the regular price.

I love this

Linda Schrock Taylor...is a free-lance writer and the owner of "The Learning Clinic," where real reading, and real math, are taught effectively and efficiently.

I'm going to have to get in touch with her.

-- CatherineJohnson - 12 Feb 2006

WhyDoWeHavePerCent 14 Feb 2006 - 20:49 CatherineJohnson

I was teaching Christopher Lesson 77 in Saxon 8/7 today: "Percent of a Number, Part 2."

Saxon does something incredibly cool.

He gets kids started on writing equations to solve very simple fraction & percent problems by using WP, WN, WD, and WF to mean "What percent?" "What number?" "What decimal?" and "What fraction?"

Take the question:
What percent of 40 is 25?

The 'of' translates easily & directly to x; the 'is' translates easily and directly to =; the 'what number' translates to WN: WP x 40 = 25

What number is twenty-five percent of 80?
WN = .25 x 80

Fifteen percent of what number is 45?
.15 x WN = 45?

Seventy-five is what decimal part of 20?
75 = WD x 20

What fraction of 56 is 42?
WF x 56 = 42

This system allows Saxon to teach percent, decimal, and fraction problems close together, without students getting lost mid-stream. (At least, I assume this system works....it worked with Christopher today, so he's my 'n of 1.')

more Saxon subscripts

While I'm on the subject of Saxon's painstaking efforts to support the student, 8/7 also uses subscripts to set up proportions.

In the town of Centerville, 261 of the 300 working people do not carpool. What percent of the people carpool?

Saxon uses PC ('percent who carpool) and PN (percent who don't carpool) in the 'ratio box' he teaches students to construct.

PC/100 = 39/300

When I was a kid, everything was X.

I don't actually remember that being a problem for me.....but having everything be X today would sure be a problem now that I'm trying to teach this stuff.

so why do we have percents?

I wanted to tell Christopher why we're doing percent & proportion problems.....and I realized I don't necessarily know why.

I assume that percents were invented, or 'agreed upon,' to give everyone a common & efficient standard of comparison.

In other words, percents are another kind of 'math machine,' an invention that makes things go faster.

Is that right?

Are there other reasons?

And: when did people start using percents?

Saxon 8/7 a remedial book?

This is something I don't quite fathom.

Linda Schrock Taylor sees 8/7 as a remedial book.

But I love this book....and certainly don't experience it as remedial. (I finished Lesson 84 last night, out of 130. Saxon books have 120 lessons plus 12 'Investigations' plus an appendix or two. There are 133 lessons in Saxon 8/7 altogether; I've done 91.)

Any thoughts?

I'm thinking Saxon 8/7 must repeat a great deal of 7/6, which I haven't used beyond the first 20 lessons.....

But I don't know.

Percents, derived from per centum, or "per hundred", give you a common denominator with which to compare ratios.

So, instead of having to compare 3/4 with 7/9 with 2/3, you can compare 75% with 77.8% with 66.7%.

— and from Old Grouch:

And my Webster's Collegiate gives that derivation and first use of "percent" as French "per" + Latin "centum" in 1568.

— and from Steve:
Why percents? Because it's nicer to have some numbers to the left of the decimal point. Notice that scientific notation always puts one number to the left of the decimal point. Actually, I think I might prefer two digits to the left, rather than one.

Thanks!

Christopher learns how to do proportion/percent problem from Saxon Math

-- CatherineJohnson - 13 Feb 2006

NyuMathMajor 03 Oct 2006 - 01:13 CatherineJohnson

Ed talked to an undergraduate majoring in math today.

I guess the kid spontaneously told Ed that, "Calculators are the worst thing that ever happened to math students."

Ed said he almost burst out laughing, because next this student went on to say that nobody who used calculators as a kid can do fractions, and if you can't do fractions you can't do calculus.

Ed said this guy could have been channelling me.

The student also said that, in high school, his calculus teacher had told the students who were having trouble, "You're having trouble because you used calculators in grade school and you never learned to do fractions." It was obvious to her. He spent quite a lot of time describing automaticity to Ed, and how important it is.

Ed asked why he hadn't used calculators as a child, when everyone else was, and the answer was chilling: he hadn't used calculators because he 'was into' math, he liked it, and he wanted to do the calculations by hand.

What that tells me is that only the natural born Math Brains are going to make it through these days — natural born Math Brains who know they're natural born Math Brains.

Your basic kid is going to use the calculator if the teacher hands it to him.

Then he's going to regret it later on.

That's what happened to the other kids in his high school calculus class.

Ed asked him whether the kids who'd used calculators could catch up.

The kid didn't think so. At least, he hadn't seen it happen.

Math is hard, he said. It's hard, it takes a long time to learn, and he didn't think a high school student who'd lost that much time could make it up.

That's what James Milgram said, too.

no calculators in Irvington

I don't think any of the grade school kids here are using calculators.

One of main criterion for choosing a new math curriculum was (paraphrasing) 'constructivist approach.'

One of the other main criterion was emphasis on math facts & computation.

TRAILBLAZERS was the only constructivist curriculum they considered that stressed fluency in math facts.

(I assume they're teaching the traditional long division algorithm in spite of the fact that TRAILBLAZERS teaches 'forgiving division,' but I don't know. Nevertheless, nobody's passing out baskets of calculators.)

Good for them.

which reminds me

I had to buy Christopher an expensive graphing calculator (or something) last fall, for Middle School.

He never used it once, and then finally lost the thing.

Good riddance!

His teacher is letting them use calculators for the first time this year, to calculate circumference & area of circles. I'm not even sure that's such a good idea.

Since he's doing KUMON, though, I figure it's OK. He's incredibly fast & accurate on the KUMON sheets.

Of course, the two "Fraction Levels" - E & F - are killers.

-- CatherineJohnson - 14 Feb 2006

RaysArithmetic 22 Feb 2006 - 23:35 CatherineJohnson

I've just discovered a series of arithmetic textbooks from the 1800s while cruising geometry workbooks at christianbooks.com. (fyi, Charles put me on to christianbooks, which has the apparently-out-of-print Saxon Physics for a good price. I'm still mulling that one.)

According to the publisher, Ray's Arithmetic was the most popular arithmetic series in the 1800s, selling more than 120,000,000 copies.

Does anyone know anything about these books? Have you used them? Seen them? Read them?

The books have glowing reviews at Amazon. My ADD TO BOOKBAG finger is starting to twitch.

The 8-volume set is \$100, but you can buy individual titles as well.

Christianbooks has posted 14 pages of Ray's New Practical Arithmetic online.

titles
Ray's New Primary Arithmetic
Ray's New Intellectual Arithmetic
Ray's New Practical Arithmetic
Key to Ray's New Arithmetics (Primary, Intellectual)
Ray's New Test Examples in Arithmetic
Ray's New Higher Arithmetic
Key to Ray's New Higher Arithmetic
Ray's New Arithmetics-Parent Teacher Guide

uh-oh

I'm going to get myself in serious trouble.

Fortunately, the listing appears to be closed.

I've sent an email to the seller just to make sure.

sources:
Amazon
Biblical Worldview Learning Center
Farm Country General Store
Homeschoolingbooks.com
Mott Media

-- CatherineJohnson - 21 Feb 2006

SingaporeMathInWisconsin 01 Mar 2006 - 00:04 CatherineJohnson

Charles left a link to a story in Milwaukee's Journal Sentinel, via Education News, Less may be more when it comes to math.

Students in Singapore are introduced to roughly half the number of new math topics a year as students in the United States are. Experts and policy analysts say Singapore's emphasis on depth over breadth is a formula for success.

The thicker the textbooks and the greater the volume of math topics introduced a year, the less likely American students and teachers are to achieve similar results, says Alan Ginsburg, director of the policy and program studies service at the U.S. Department of Education.

"There's no way you can teach twice the amount of mathematics to the same depth that Singapore does," says Ginsburg, co-author of a 2005 report called "What the United States Can Learn From Singapore's World-Class Mathematics System," published by the American Institutes for Research.

He says the Singapore method of teaching math also puts a bigger emphasis on understanding instead of "mechanical" memory, and on visualization of the problems.

"I feel like the biggest difference is the visualization," says Julia Rothacher, 12, a sixth-grader at University School.

Previously, she says, she attended Cumberland Elementary School in Whitefish Bay, where her class used the mathematical reasoning-based curriculum known as Everyday Math.

To appreciate what visualization can do, consider a problem that Neuwirth gave her class. It was considered the hardest question on a Massachusetts state assessment for 10th-graders, based on data that showed that more than half of the 72,000 test-takers got the question wrong, according to The Boston Globe, which published the problem.

Of the people in attendance at a recent baseball game, one-third had grandstand tickets, one-fourth had bleacher tickets, and the remaining 11,250 people in attendance had other tickets. What was the total number of people in attendance at the game?

The four choices were: A) 27,000, B) 20,000, C) 16,000 or D) 18,000.

Neuwirth's sixth-graders - without using the calculators that Massachusetts' 10th-graders could use - went to work.

Alexis Block, 12, did the problem on the board.

She drew 12 boxes of the same size, because 12 is the lowest common denominator of the denominators 3 and 4 in one-third and one-fourth, respectively.

She wrote "GS" for grandstand tickets above four - or one-third - of the 12 boxes, and "B" for bleachers above three - or one-fourth - of the 12 boxes.

She wrote 11,250 below the remaining five boxes, then divided 11,250 by five to get the value for each box - 2,250.

She multiplied the value of each box by 12 and got the correct answer for the total number of people in attendance: 27,000.

[ed.: Unfortunately, the reporter didn't ask Julia whether her classmates at Cumberland could do this problem.]

• one-fourth of audience had bleacher tickets
• one-third of audience had grandstand tickets
• the remaining 11,250 people in attendance had other tickets
• QUESTION: how many people in the audience altogether?

spaced repetition: more than half of Massachusetts 10th graders missed this problem

Singapore's bar models are gold. Saxon Math uses them, too; students draw bar models in virtually every problem set in Saxon Math 8/7. I'm going to have Christopher doing them all summer.

using bar models to prep for the state test

I used a bar model to show Christopher how to do this problem from the Glencoe test prep booklet he brought home over break:

6.N.17 Multiply and divide fractions with unlike denominators

Pizza Pizzaz was running a special on their 1/2 pepperoni and 1/2 cheese pizza. Mary, Jorge, and Shaun wanted to share a pizza, but they only liked the cheese half. If they shared equally, what fraction of the total pizza would they each be able to eat?

I fault Christopher's teacher for assigning virtually no story problems all year long.

Word problems are the true manipulatives.

Every concept she's teaching should be illustrated & practiced through extremely simple word problems to start — word problems so simple the kids can do them in their heads.

Christopher had no idea — zero — that this problem called for division of a fraction.

If you tell him to divide 1/2 by 3 he can do it in 2 seconds flat.

But his procedural knowledge is completely divorced from any actual situation in which one would divide 1/2 by 3.

So I drew a bar model, and of course he saw right away that this problem requires you to divide 1/2 into 3 parts.

At the beginning of the year I was having him do 2 or 3 bar models a day. I'm going to have to get back to that.

what does the AIR study of Singapore Math find?

The conclusion of the story is quite misleading:

But it's not certain that Singapore Math is making a difference in U.S. test scores.

[snip]

In his study of Singapore's math system, Ginsburg, of the U.S. Department of Education, looked at four sites in the U.S. where the Singapore approach had been adopted. Only two of those sites achieved results superior to control groups, and those two sites got additional staff development.

"It's not magic," Ginsburg says. "You can't just give out textbooks."

I've read most of the AIR report (pdf file), and I'd say that the impression it leaves is that the Singapore textbooks are as close to magic as it gets.

Here is the actual conclusion of the report:

The two pilot sites (out of four) that had both a stable population of higher performing students and a clear staff commitment to support the introduction of the Singapore mathematics textbooks produced sizeable improvements in student outcomes.

As far as I'm concerned, that's two out of two.

In North Middlesex, Massachusetts, the school system of about 5,000 was selected by the state education agency to pilot the Singapore textbooks. Over two years, the percentage of those students who participated in the Singapore pilot and scored at the advanced level on the grade 4 Massachusetts assessment increased by 32 percent over two years. The pilot schools had strong district and staff support. Over two years, Baltimore’s Ingenuity Project increased the proportion of its students who scored at the 97th percentile or above by 17 percent. The Ingenuity Project serves gifted Baltimore students and can select highly skilled teachers capable of teaching the mathematical reasoning underlying the challenging Singapore problems.

The two other Singapore pilot sites, which in one case had uneven staff commitment to the project and in the other case had a more transient, lower income population, produced uneven or disappointing results.

• The Montgomery County outcomes were positively correlated with the amount of professional training the staff received. Two Singapore pilot schools availed themselves of extensive professional development and outperformed the controls; two other pilot schools had low staff commitment coupled with low exposure to professional training and were actually outperformed by the controls. Professional training is important in helping teachers understand and explain the nonroutine, multistep problems in the Singapore textbooks. Teachers also need preparation to explain solutions to Singapore problems, which often require students to draw on previously taught mathematics topics, which the Singapore textbook, in contrast to U.S. textbooks, does not reteach.

• The Paterson, New Jersey, school, with an annual student turnover of about 40 percent, fared no better on the New Jersey grade 4 test than the district average over two years. Having such a high student turnover meant that many 4th graders were exposed to the Singapore mathematics textbook for the first time - by definition, not a fair test of the cumulative effects of exposure to the textbook.

Offhand, I don't see where this is a triumph of professional development. None of these teachers went back to college or took additional courses in advanced mathematics. They spent a fair amount of time learning how to teach the Primary Mathematics series, and their commitment was high.

It seems extremely unlikely to me — again, having read the report — that the same degree of professional development focused on teaching EVERYDAY MATH would have produced results like these.

update: from the AIR report

This is funny:

The most serious mismatch occurred in Paterson, where grade 4 teachers supplemented the Singapore mathematics textbook with their U.S. textbook to cover a few topics, notably statistics and probability, that were on their grade 4 state assessment but not in the Singapore grade 4 textbook. Unfamiliar with the pedagogy laid out in Singaporean Teachers’ Guides, several sites were also concerned that the Singapore textbooks did not stress written communication skills by requiring students to explain their answers.

OK, that's not funny ha-ha.

Speaking of funny ha-ha, I'm going to have to find the email our school board president sent to parents explaining the adoption of TRAILBLAZERS by saying - and I think I'm quoting - 'math has become language-based.'

if you want to teach bar models to your child

I'm afraid the simplest and quickest - but not the least expensive - approach is to buy the four PRIMARY MATHEMATICS books for grade 3 & just work through all the story problems, start to finish:

• Primary Mathematics 3A Textbook (\$8.00)
• Primary Mathematics 3A Workbook (\$8.00)
• Primary Mathematics 3B Textbook (\$8.00)
• Primary Mathematics 3B Workbook (\$8.00)

I say this because last summer I tried to have Christopher do the problems in Challenging Word Problems Book 3 (3rd grade), and it was just too hard. [update 3-23-2006: I've misspoken. The only difficult problems in Book 3 are those in the "Challenging Problems" sections. The problems were too hard for Christopher to do while also learning to construct his first bar models.]

Both KUMON & DI advocate backing kids up to a point before their level of expertise, and that's what I needed to do with Christopher. He was annoyed that the bar model problems in 3A were too easy, but in fact he hadn't learned the 'core' bar models representing addition, subtraction, multiplication, and division & he kept getting tripped up.

Finally even Christopher agreed to go back and start from the beginning.

the principle: when you're learning a new skill, start at the beginning

You might be able to start at the beginning by purchasing the 2nd grade Challenging Word Problems book, but unfortunately I don't have a copy, so I can't say.

UPDATE - THERE ARE SAMPLE PAGES ONLINE!

Looking at these pages, Challenging Word Problems Book 2 (\$7.80) might be a good way to go.

Basically, you need to teach your child the 'core' bar models corresponding to the 4 operations. There are essentially 5 'core bar models' (I think):

• subtraction as diminution ('take away')
• subtraction as comparison ('difference')
• multiplication
• division

If you're more math-savvy than I was when I first started working with PRIMARY MATHEMATICS, you might just want to have your child practice these 5 forms using whatever very simple word problems you happen to have around - including 'number problems,' such as 'What number is the difference between 9 and 7?' (I'll get samples posted.)

Come to think of it, that's probably what I'll do this summer: massed practice on the 5 models.

I'll figure out the core fraction-percent-ratio-proportion bar models & teach those, too.

the bar model for subtraction as comparison

This is from Challenging Word Problems Book 2 (second grade):

bar models for parents

I like The Essential Parents' Guide to Primary Maths (\$10.50) quite a bit. As luck would have it, the 3 sample pages on bar models cover the 'Comparison Concept":

subtraction as the difference between 2 numbers
study sheet: subtracting integers & absolute value
notes on integer, subtraction, & absolute value study sheet
subtraction has two meanings - Word document

-- CatherineJohnson - 25 Feb 2006

DanKOnRightAnswersAndPleasureInMath 02 Mar 2006 - 03:07 CatherineJohnson

This is something I've been thinking about for awhile now, but hadn't gotten around to bringing up.

How do you get a kid to like math?

I didn't worry about it much when Christopher was in grade school, because I was boss; Christopher had to do what I said.

I'm still boss here in middle school - the boss of last resort at least - but it's a fight to the death, and the day when Christopher makes his own decisions about what he likes and doesn't like, and will and won't pursue, isn't far off.

Is he going to graduate high school with a superb grasp of K-12 mathematics?

If his father and I have anything to say about it, the answer is yes.

But I'm not sure how much we will have to say about it 4 years from now.

Another thing: as the British report on the UK's dwindling supply of mathematicians and applied math types points out, if we hope to increase the number of people who are good at math, we have to increase the number of students who want to be good at math. We don't have a math draft, after all.

from the report:

At this point we should perhaps comment on an apparent contradiction underlying our analysis.

(i) We know that many students find mathematics hard.
(ii) Yet our goal is to attract more students to the study of mathematics.

A crude “consumerist” model of education might lead one to conclude that one has no choice but to “drop the price” – that is, to concentrate on making mathematics “easier”. Yet we have repeatedly emphasised both (a) the need to strengthen basic technique and to expect more students to integrate one-step routines into multi-step wholes, and (b) the urgent need for a massive increase in the number of students taking A level Mathematics. How can such talk be realistic? And how can it be achieved?

These are serious questions – provided they are not merely rhetorical. Resolving the present crisis will not be easy; but, as we shall try to indicate, there is no essential contradiction in the analysis.

First, one has to understand that the long term challenge of ensuring a natural flow of home-grown mathematically competent graduates is quite different form the short term goal of selling off an unfashionable product simply by “dropping the price”.

Second, one has to recognise that a modern economy is mathematical in so many ways that we really have no choice but to find ways of producing a reliable flow of mathematically competent graduates – unless, that is, we are content to become a dependency of those countries that do appreciate the essentially “mathematical” character of a modern economy.

Third, we need to remember that the number taking A level Mathematics as recently as 1989 was more than 50% larger than at present, so there is no obvious logical reason why the goal is unrealistic.

So the question of motivating or inspiring kids to like math, and to want to pursue it, is important.

Constructivists seem to have given this question thought, and I've seen at least two real-live kids around here who love TRAILBLAZERS, and are having the time of their life with it. (I've seen more a few more who dislike it...)

Even so, if I had to bet, I'd bet that the answers constructivists have come up with are wrong, for the reason Dan K points out:

My daughter threw a (minor) fit today about having to do an extended response math homework problem. "I got the right answer," she wailed. "I did it in my head. Why do I have to write a paragraph about it?" I know I would have been as least as whiny back in my youth. I'm a mature adult now, so I enjoy learning stuff just to learn it. I can remember, though, wanting to get by with as little work as possible back in school (I still don't like working too much). To me, some of the best things about math, as opposed to other subjects, were 1) there's a right answer; 2) no term papers; and 3) no essay questions. Modern educators have eroded those benefits.

I agree.

I agree, because people who are good at math so often say that the thing they loved about math when they were kids was that there was a right answer.

For a particularly spectacular example, consider Lisa Randall (\$?)

There used to be an ice cream parlor in the student center at the Massachusetts Institute of Technology. And it was there, in the summer of 1998, that Lisa Randall, now a professor of physics at Harvard and a bit of a chocoholic, and Raman Sundrum, a professor at Johns Hopkins, took an imaginary trip right out of this earthly plane into a science fiction realm of parallel universes, warped space and otherworldly laws of physics.

They came back with a possible answer to a question that has tormented scientists for decades, namely why gravity is so weak compared with the other forces of nature: in effect, we are borrowing it from another universe. In so doing, Dr. Randall and Dr. Sundrum helped foment a revolution in the way scientists think about string theory - the vaunted "theory of everything" - raising a glimmer of hope that coming experiments may actually test some of its ineffable sounding concepts.

Their work undermined well-worn concepts like the idea that we can even know how many dimensions of space we live in, or the reality of gravity, space and time. The work has also made a star and an icon of Dr. Randall. The attention has been increased by the recent publication to laudatory reviews of her new book, "Warped Passages, Unraveling the Mysteries of the Universe's Hidden Dimensions," A debate broke out on the physics blog Cosmic Variance a few weeks ago about whether it was appropriate, as a commentator on NPR had said, to say she looked like Jodi Foster.

"How do we know we live in a four-dimensional universe?" she asked a crowd who filled the Hayden Planetarium on a stormy night last week.

"You think gravity is what you see. We're always just looking at the tail of things."

Although it is the unanswerable questions that most appeal to her now, it was the answerable ones that drew her to science, especially math, as a child, the middle of three daughters of a salesman for an engineering firm, and a teacher, in Fresh Meadows, Queens. "I really liked the fact that it had definite answers," Dr. Randall said.

source:
Scientist At Work | Lisa Randall: On Gravity, Oreos and a Theory of Everything by Dennis Overbye
NEW YORK TIMES
November 1, 2005, Tuesday

So here we have the Holy Grail, the object of millions of dollars of NSF-funded curriculum-building and conference-hosting, a Woman in Physics - at Harvard, no less!

What got her interested in math?

Lisa Randall

update: more from Dan

I'll also add that math is not the only realm in which there are right answers. Some kids of all ages enjoy playing games like Trivial Pursuit or the home version of Who Wants to be A Millioinaire? It's not because you're going to get rich winning the home edition. It's because it's fun to get the right answer. That's the whole point of the game.

And don't forget variable reinforcement!

maths in England
maths in England, part 2
more maths in England, part 2
top students in England, US, & Singapore
why do kids like math?
another brilliant person who liked getting right answers (scroll down)
Catherine's cousin talks about Everyday Math

Call for national debate on maths teaching GUARDIAN
Where will the next generation of UK mathematicians come from? (GOVT REPORT: pdf file)

The Beauty of Branes SCI AMER
On Gravity, Oreos and a Theory of Everything NYTIMES (possibly \$)
On Gravity, Oreos and a Theory of Everything NYTIMES (pdf file)

extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

-- CatherineJohnson - 27 Feb 2006

MathOlympiadProblems 02 Mar 2006 - 06:47 CatherineJohnson

source:
Math Olympiad Contest Problems for Elementary and Middle Schools
Dr. George Lenchner

the solutions

creative problem solving

Happy July Fourth (Moise & Downs)

-- CatherineJohnson - 28 Feb 2006

RatioProblemFromTheBbc 01 Mar 2006 - 01:27 CatherineJohnson

What is ratio?

Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example:

• Use 1 measure screen wash to 10 measures water
• Use 1 shovel of cement to 3 shovels of sand
• Use 3 parts blue paint to 1 part white

Ratio is the number of parts to a mix. The paint mix is 4 parts, with 3 parts blue and 1 part white.

The order in which a ratio is stated is important. For example, the ratio of screenwash to water is 1:10. This means for every 1 measure of screenwash there are 10 measures of water.

Mixing paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all.

3 parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white paint.

If the mix is in the right proportions, we can say that it is in the correct ratio.

This mission - improving the basic skills of adults - probably means Skillwise is going to be a good resource for math problems real people use in the real world: the kinds of problems you can show a child - or a Washington Post columnist - who's wondering out loud whether he'll ever 'use math' once he's out of school.

more real-world math

from the Delta College Teaching/Learning Center:

Ratio and Proportion in Nursing Math

Proportion is often used to calculate a dosage. Suppose a drug comes in tablets of 150mg. The dosage ordered is 375mg. How many tablets are needed? Here is the problem:

To Solve for x, we have to cross-multiply:

x = 2.5 tablets

I don't know about you, but I would like any nursing staff taking care of me & mine to know this stuff cold.

Ratio And Proportion in Nursing Math: Sample Problems & Answers

more Nursing Math

-- CatherineJohnson - 28 Feb 2006

PopQuiz 01 Mar 2006 - 20:34 CatherineJohnson

2x - 14 = 7 - x

source:
Can School Board Hopefuls Handle a Pop Quiz?
from Google Master: same story, no registration required
by Steve Hymon
February 27, 2006
LA TIMES

and get your timers out —

How long does it take you to solve this problem?

9x - 9 = x + 7

I'm guessing nobody needed 2 minutes & 45 seconds.

-- CatherineJohnson - 01 Mar 2006

HowDoYouTeachChildrenToSolveWordProblems 03 Apr 2006 - 03:15 CatherineJohnson

The New York State test is coming up on March 14 - 15.

The kids aren't doing well on the sample tests they've taken. Only 2 out of 19 in Christopher's class got a 4 - 'exceeds state standards' - on the one they did last week. Two 4s in an 'accelerated' math class. [update: turns out that's 2 out of all 3 Phase 4 classes, which is close to 60 kids. Two of sixty children in the Irvington Middle School accelerated math class exceed state standard on a practice test.]

It's a joke.

Christopher got a 3 on the sample test, and of course I'm determined that he earn a 4 on the real one; don't ask me why. Same reason people climb Mount Everest, probably. [update 4-23-2006: no, that's not why. Christopher's 4's on NY state tests to date are at odds with the grades he receives in his classes at Irvington Middle School. Part of our new data warehousing initiative involves comparing grades in school to scores on state tests.]

Mount Everest aside, this is a golden opportunity for Christopher finally to learn something about solving word problems. I've mentioned several times that they've done essentially no word problems this year; I'm thinking they must not have done many in 5th grade, either, though I don't recall.

Saxon 6/5, I do recall, does not stress word problems. Or, rather, Saxon teaches word problems very, very carefully, slowly, and deliberately. Kids learn different genres of problems, such as 'problems with equal groups' and practice one-step versions of those problems to mastery. I don't think they do two-step problems until Saxon 7/6 or maybe even 8/7 (though I could be wrong).

This always used to bother me about Saxon. Singapore Math has two-part problems starting in 3rd grade or possibly even earlier. However, now that I'm almost done with Saxon 8/7 myself, I can see the point.

Back when I wrote my dissertation (on 1950s film comedy, no less) I talked about the 'narrational presence' in movies, by which I meant the implied director or author hovering over the proceedings. The narrational presence in a Saxon book is a kind and intelligent person who really, really wants you to learn math - and doesn't expect your parents to hire a tutor or send you to cram school to see to it that you do.

So Saxon builds word problem solving skills slowly, incrementally, and logically. After awhile you're doing two-step and three-step word problems, you're doing them easily, and you're doing them without your parents ever having spent \$300 to attend a 30-hour weekend seminar on how to understand changes in math instruction.

Unfortunately that's not what we need here.

We need teach-to-crammery problem-solving strategies, and we need them today.

We need teach-to-crammery problelm-solving strategies today because the state test has an open-ended question section that's a killer. It's wall-to-wall story problems, none of which Christopher has ever seen or done. He got 20 out of 25 multiple choice questions right on the sample test. That's not great, but that it will improve easily with practice.

He got 13 out of 24 open questions right. Awful.

The smartest child in the class missed 5 of the open questions. This is a kid who, from where I sit, is unstoppable. And she's scoring 5 wrong out of 24.

'make a chart'

I spent this weekend teaching Christopher the fantastically helpful charts that are in Saxon, Dolciani, and Brown and Dolciani (Brown's book being a terrific basic algebra text, btw. In the past, inexpensive teacher's editions for Brown have been easy to find.)

How I wish I'd known about 'word problem charts' when I was a kid. They're incredible.

And how I wonder why Prentice Hall doesn't have them.

I'll post a couple of examples, but in the meantime, here's the simplest one:

I find this beautiful.

• It's simple, clean, and instructive. Every time Christopher fills out a Dolciani/Saxon/Brown-type chart he rehearses and 'sees' again the relationships among these numbers.

• Once a value has been entered in its correct place on the chart, the student doesn't have to hold it in memory. Nor does he have to re-read the problem to re-find whichever number he's forgotten while remembering whichever number he's (currently) remembering. When you're just learning to solve word problems, you're constantly forgetting one number while remembering some other number. People always say that the 'big problem' with word problems is they're hard to read, but I'm starting to think the big problem is they're impossible to remember. Which may amount to the same thing, of course.
These charts take such an enormous burden off of working memory that I wonder whether Temple might have been able to learn algebra if someone had taught her to construct them.)

• Finally, the fraction bar is already there, implicitly and almost explicitly, in the lines of the chart. When I pointed this out to Christopher he said, 'Oh, yeah' in his happy 'I get it' voice.

more charts

update

Here are the Prentice-Hall triangle charts.

Horrid.

So here's my question.

Last night, watching Christopher read word problems, I could see that he had no clue.

He wasn't even pulling out the numbers, especially; his approach seemed completely haphazard. He seems just to guess positions and operations.

The minute I showed him the charts, he started knowing what he was after & being able to find it in the problem.

He needs a strategy.

At the moment, I'm telling him to circle each 'math fact' and underline the question. I also suggested using yellow highlighter to highlight the math facts and blue to highlight the question. He likes that idea, but I'm not sure it's practical for the state test, which is timed.

But I'm wondering whether I also ought to make up some kind of 'teaching template' he would have to fill in for each problem he does.

Something like this:

question: _________________

what I know: _______________

what I know _______________

what I know _______________

what I need to find out (if needed) _____________

what I need to find out (if needed) _____________

I thought of this because I saw somebody on a website somewhere do something similar. Now, of course, I have no idea what or where that website was.

Any suggestions?

I gather Mildred and Tim Johnson's book, How to Solve Word Problems in Algebra, is the best of the lot, but I probably don't have time to pick up a copy before next week.

how do you teach your child word problems?
mini problems (important)

teachtocrammery

-- CatherineJohnson - 06 Mar 2006

MiniProblems 15 Jul 2006 - 16:33 CatherineJohnson

I've been complaining for months about the lack of word problems in Christopher's math class.

The kids memorize one procedure/rule/formula a day, do a few calculations, and march on. As a direct result, IMO, their knowledge really is rote as opposed to procedural. At least, Christopher's is. And I've had enough math talks with other kids in the class to know some of them are in the same boat.

Today I had a eureka moment reading a Comment left by Kathy Iggy:

The old math books I found (the same ones I used in grade school) have lots of what they call "mini problems" used to illustrate how a recently taught concept would be presented in a word problem. Megan likes these because of their brevity and she doesn't have to struggle with comprehension that much.

For example:

20 yards of ribbon. 1/4 used for dress. How much ribbon used?

That's IT!

mini problems

That's the concept, and the phrase, I've been looking for.

mini problems:word problems :: basic skills:higher order skills .

That's from Ken, and he's exactly right.

[update 4/23/2006: no! he's not right! Actually, he's write about using mini problems to teach word problems; I'm talking about mini problems to teach math - to teach the fundamental concept in a lesson. Awhile back I realized that word problems are the 'real manipulatives.' Now I know what I mean by that.

All concepts should be taught — illustrated — with mini problems. All concepts, every last one.

PRIMARY MATHEMATICS does this; SAXON MATH does it; KUMON does it. I'll post examples.

I've come to feel that the first word problems illustrating a new concept should be so simple children can do them in their heads.

For example, the very first ratio word problem a child does should be something like this:

Christopher bought one pencil for one dollar.
How many pencils can he buy for two dollars?

The question should be written this way, too: on two separate lines, so the child sees instantly that the first sentence is the set-up, and the second sentence is the question. Richard Brown's revision of Mary Dolciani's BASIC ALGEBRA, a book I like very much, does this for many of its word problems. I'll post some of those, too, as I get to it.

mini problems are applications

The problem with word problems is that, in the U.S., they're always hard.

Word problems are so hard people have apparently come to think that if a word problem isn't hard it isn't really a word problem.

I'm wondering if we ought to ditch the phrase 'word problem' (ditto for 'story problem') and adopt the word 'application.'

A better idea: we should think about the point of word problems.

Some word problems are written and assigned to give students practice.

Many word problems are written and assigned to assess whether students have developed flexible knowledge.

I'm talking about a third purpose, which is instruction. I'm talking about word problems designed to teach.

instructional word problems

A word problem is an application. A super-simple, starter word problem explains and demonstrates a mathematical concept by showing students how the concept is applied.

As a matter of fact, an instructional word problem shouldn't even be a 'problem.' It should just be a question, and the answer should be obvious.

A simple, instructional mini-problem should not test the child, should not challenge the child, and certainly should not trick the child.

It should teach.

examples to come

be sure to see Google Master's comment

how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 07 Mar 2006

StickingPointsInAreaAndPerimeter 09 Mar 2006 - 23:07 CatherineJohnson

I've finally found an image to explain one of the toughest concepts for kids:

Christopher has a dreadful time with figures like these, and so did my neighbor's son last year (no report on how he's faring this year).

Christopher simply can't 'see' that if all the angles are right angles, the right side equals:

8 ft + 5 ft

Nor can he see that the short horizontal line segment between the 8 & the 5 equals:

30 ft - 22 ft

I would like to have a few worksheets of figures like these.

taking a measure without starting at 0

Here's another category of problem that's incredibly hard for kids to do:

Doug, if you're around, and you feel like taking on another project, this is something I'll wager every grade schooler on the planet could use.

Christopher would be in much better shape today if he'd been given a bunch of 'simple' measurement problems in which the left side of the object being measured is placed somewhere other than 0 on the ruler.

For the life of me, I don't know why kids aren't bringing home such assignments as homework.

Some of you may remember that, last year, Christopher eked out a '4' on the TONYSS ('Test of New York State Standards,' a test schools in NY state can purchase from a private company to use in 'off' years). His score was one point above the cut-off.

The scale he flubbed was measurement!

I was shocked.

I'd been working around the clock with him (at an age when he was still willing to work with his mom) — and he flunks measurement! (Apparently, this was true of kids all over the state.)

Then we heard from teachers explaining that measurment is a difficult concept and skill to learn. Meanwhile the Singapore series takes measurement as one of its core subjects. They place huge emphasis on that topic.

Live and learn.

Now I see why measurement is a) difficult and b) incredibly valuable.

Think how much knowledge and skill goes into figuring out a problem like the line measurement above.

1
You can figure out the measure of the line either by adding or subtracting fractions.

2
The fact that you can figure it out by adding or subtracting reinforces the concept that addition and subtraction are inverse operations.

3
You can also figure out what the measurement is by counting-up using fractions instead of whole numbers. 'Counting by fractions' is an incredibly valuable activity. You almost can't not see that 'fractions are numbers' when you count by fractions. Saxon Math has numerous Mental Math tasks requiring students to count up (and, I think, back down) by fractions.

1/5, 2/5, 3/5, 4/5, 1, 1 1/5, 1 2/5, 1 3/5, 1 4/5, 2

btw, Schoolhouse Tech has a very nice sheet of fraction number lines available for download. (pdf file)

update from Doug

Doug recommends drawing simple perimeter problems on quadrille paper, like this one from Enchanted Learning (I think you have to be a member to download the sheet):

I'm going to print out the sheet and see if Christopher readily transfers from the quadrille problems to problems written on blank paper.

from last year

I just found a number of comments about kids and measurement that I'd forgotten:

My first year teaching high school freshman (I just finished my 3rd year at a urban neighborhood school) I was completely shocked that none, and I mean none, of the kids could measure using an inches ruler.

How can they get out of middle school, or even grade school, not knowing how to measure? I still have no clue. I doubt its the constructivists fault due to their fondess for hands-on, manipulatives, and project, which all lend themselves to measurement.

What I have observed:

• Metric OK, Inches Not -- While the kids can't (or won't) measure in inches, many (but not all) can measure using a centimeter ruler. Fractions rear their ugly head again.

• Estimation, Schmestimation -- The kids do not know when it is, or is not, appropriate to estimate. The kids have trouble estimating measurements between the lines of the ruler. But the kids are very willing to make bad estimates to avoid having to figure out what the little lines mean. 2 5/16 inevitably becomes 2 1/2.

• What is a protractor? -- The kids REALLY don't know how to use a protractor (except as a frisbee). Most don't even know that its purpose is to measure angles.

A side note related, I believe, to measurement. Each year I do a lesson where we compare the kids height in inches to their shoe size. The majority of the kids do not know how tall they are, let alone how to convert the height in inches.

So by all means get a ruler, protractor, some measuring cups and spoons, and a kitchen scale (or even better a pan balance) and start measuring everything around the house!

Barry's reaction:
Interesting observation and good advice. I just purchased the Saxon Math 76 book for 6th grade, and I notice that many of the problems have a scale on the page (in inches, sometimes divided into 8ths, 16ths, etc depending on the problem), with a line above it and students are asked to give the length of the line. I thought it strange to have such measurement practice but now I don't.

(Obviously that's part of our problem around here. We skipped Saxon 7/6 and went directly from 6/5 to 8/7.

from Interested Teacher:

Learning to read/measure from an 'inch' ruler has to be incremental. Younger students can't look at a ruler and automatically discern what all of those marks mean. They have to be taught to find the 'half' mark and measure using the 'half' marks. Then add the 'fourth' marks, (Don't be surprised that students don't automatically know that the 'half' mark also becomes a 'fourth' mark.) Then have students measure using the 'fouth' and half' marks. And so on, going into 'eighth' marks, etc. Practice between each incremental step.

Practice is necessary so students develop the skill of disregarding the smaller (16ths and 32nds) marks. For some students, with visual discrimination problems, this is horribly difficult.

Saxon 6/5 covers through 'fourths' and I add a little 'eighths' for more advanced students.

I was looking through Passport to Mathematics,Book 1, a text that I am previewing for personal reasons, and I see lots and lots of metric work, but little with feet and inches. On pg. 32, students measure to the nearest inch, and nothing else that I can see until pg. 318. With no review of 'half' and 'fourth' inches, it jumps to 'eights' -- there is one problem.

-- CatherineJohnson - 09 Mar 2006

GlencoeListsGrade6NewYorkStandards 12 Mar 2006 - 00:02 CatherineJohnson

In a fit of civic-mindedness I've decided to type up Glencoe's list of NY State standards to be assessed in Tuesday's test.

I'm coming to love Glencoe. I've mentioned their Parent and Student Study Guide, which they've made available to everyone free online, as well as Glencoe's Diagnose - Prescribe - Practice test prep booklet, which has been terrifically helpful. IMO it would have been helpful whether we'd had a state test coming up or not.

A couple of weeks ago my friend Kris said she wished she had a list — a simple list — of the procedures & concepts her son is learning this year.

That way she could keep track of what he knows and doesn't know, and quickly give him a few more problems to do when she sees he's weak on something.

I think every parent needs a List, and I imagine most teachers either need such a list or already have one.

Glencoe's list of 'Strands and Performance Indicators,' at the front of the booklet, is just the ticket. I spent quite a bit of time searching the NY Department of Education website looking for a list of Grade 6 skills and concepts that made sense — a list that specified math that could actually be done on a test.

What I wasn't looking for were standards like 'Understand that some ways of representing a problem are more efficient than others,' or 'Act out or model with manipulatives activities involving mathematical content from literature.'

If it's there on the NY website, and I have to assume it is, I didn't find it. [update: found it ]

Then the Glencoe test prep book came home and I had what I needed. Glencoe lays it all out in 4 pages.

Below are the 'Post-March' Grade 5 'Strands and Performance Indicators' that are tested in 6th grade.

When I type up the rest of the Strands and Performance Indicators I'll post them on a side page with a link here and elsewhere.

New York State Mathematics
Content Strands, Grade 5, Post-March Indicators

These indicators from Grade 5 are assessed on the Grade 6 Test.

STRAND ALGEBRA
Variables and Expressions
5.A.2 Translate simple verbal expressions into algebraic expressions
5.A.3 Substitute assigned values into variable expressions and evaluate using order of operations
Equations and Inequalities
5.A.4 Solve simple one-step equations using basic whole-number facts
5.A.5 Solve and explain simple one-step equations using inverse operations involving whole numbers

STRAND GEOMETRY
Coordinate Geometry
5.G.12 Identify and plot points in the first quadrant
5.G.13 Plot points to form basic geometric shapes (identify and classify)
5.G.14 Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes)

STRAND STATISTICS AND PROBABILITY
Probability
5.S.5 List the possible outcomes for a single-event experiment
5.S.6 Record experiment results using fractions/ratios
5.S.7 Create a sample space and determine the probability of a single event, given a simple experiment (e.g., rolling a number cube)

source:

another Glencoe Parent and Student Study Guide

Searching for the URL for the Pre-Algebra Parent and Student Study Guide, I found this Glencoe guide to their book MATHEMATICS: APPLICATIONS AND CONNECTIONS COURSE 2.

I haven't looked at it yet.

Pre-Algebra, Parent and Student Study Guide Workbook at Amazon

-- CatherineJohnson - 11 Mar 2006

OpenEndedProblemsInMathematicsEducation 16 Mar 2006 - 21:43 CatherineJohnson

While visiting Hung Hsi-Wu's website yesterday, I found an article he published last year on The Role of Open-Ended Problems in Mathematics Education.

I'm still not at the level where I can easily read his work, though I think I could muscle my way through. (I should put this proposition to the test, shouldn't I?)

That said, I assume Hung Hsi-Wu is talking about the kind of problem I think of as Problems The Kids Can't Do. We call them Extended Response here in Irvington; they have various names elsewhere. Possibly the most famous such problem is the haybaler problem from IMP, which Barry Garelick posted awhile back. Google "haybaler problem" and you get 619 hits.

Here is Hung Hsi-Wu:

Open-ended problems have become a popular educational tool in mathematics education in recent years. Since mathematical research is nothing but a daily confrontation with open-ended problems, the introduction of this type of problems to the classroom brings mathematical education one step closer to real mathematics. The appearance of these problems in secondary education is therefore a welcome sight from a mathematical standpoint. More than this is true, however. While these problems may represent something of a pedagogical innovation to the professional educators, the fact is that many mathematicians have made use of them in their teaching all along and do not regard their presence in the classroom as any kind of departure in educational philosophy. For example, I myself have often given such problems in my homework assignments and exams.2 Nevertheless, I have chosen to take up this topic for discussion here because, after having reviewed a limited amount of curricular materials for mathematics in the schools, I could not help but notice that they pose certains hazards in practice. These hazards include the possibility of misinforming the students about the very nature of mathematics itself.

Two things:

a) open-ended problems are not confined to high school mathematics.

and

b) I'm going to dive in and take it as a given that open-ended problems for 9-year olds and misinformation about the very nature of mathematics go hand-in-hand.

But in fact, I don't know.

What do you think of these problems?

Eggs for 9 year olds (pdf file)
multiples of 4 that end in 4 for 13 year olds (pdf file)
the Million Dollar Job a group problem for 8th graders (pdf file)

source:
Sources of Mathematics Open-Response Items
World Class Arena

it's always worse than you think

I've now skimmed enough of Hung Hsi Wu's article to see that I was right. His subject is Problems The Kids Can't Do.

And, yes, it's always worse than you think:

...in discussing these three problems with some teachers, I was astounded to be told by one and all that they considered the first part of Problem I (“WHAT MIGHT ITS AREA BE??”) to be a good problem because it allows the students to make up their own questions and answers, but that they thought the second part (“WHAT WOULD THE LARGEST AREA BE??”) was bad because it pins down the students to a single correct answer. Since a good part of mathematics, pure or applied, is pre-occupied precisely with such maximization problems, we have here an example of an educational philosophy that has distorted the way a group of teachers think about the subject they are supposed to teach.5 This should be a matter of grave concern.

ComputersInTheClassroomPartTwo 19 Mar 2006 - 19:20 CatherineJohnson

"No pilot project in educational technology has ever been declared a failure."

source:
High Tech Heretic
Clifford Stoll

computers in the classroom
ed technology never fails
"Computer Delusions"
another negative study

-- CatherineJohnson - 19 Mar 2006

SanFranciscoKippStudyFromKenDeRosa 29 Mar 2006 - 01:36 CatherineJohnson

SRI International has released a new study (pdf) of the new KIPP schools in San Francisco. It is close to 100 pages long but a good read. Not surprisingly, the KIPP kids are achieving better academic results although not yet stellar results since these schools are so new. In any event, the paper goes into detail as to what KIPP experience is like:

1. Long days (7:15 am to 5 pm)
2. Saturday classes
3. Mandatory summer sessions
4. Strict discipline
5. High academic expectations, usually using CA approved textbooks

I'd characterize the KIPP method as a brute force method of instruction that happens to work. However, I also happen to believe that similar results could be achieved with far less effort if:

1. KIPP started their program at K or 1 instead of grade 5 after these kids have had 5 years of failure in the public schools,

2. Used more praise, than punition (though the punition may be necessary for these kids at the stage they get them), and

3. Used a more efficient accelerated instructional Program. For example the DI programs achieve similar results using far less instructional time, even for low performers.

Nonetheless, KIPP shows what can be achieved with low performers with a little hard work and effective class room management, neither of which they get in the traditional classroom.

Catherine here. Every so often it crosses my mind that Ken and I may have been separated at birth; "brute force" is the exact term I've often used, in my own mind, to characterize KIPP's approach — and I say 'brute force' with a smile. I'm an enormous fan of the KIPP Academy, to the point where I've actually broached the possibility, with Christopher, of sending him there as an exchange student. (He says no.)

The KIPP people know what they're doing, and I'm not going to pick nits. But I do ask myself whether they absolutely need 6 days a week, schooldays lasting 'til 5, plus some of the summer to do what they're doing.

On the other hand, Christopher and I often put in some time on both weekend days as well as quite a few vacation days.....so I'm raising this question just to raise it, because I'm interested, and curious.

Rafe Asquith says, "There are no shortcuts."

But efficiencies and productivity gains are possible in most other realms (unless I'm overstating the case?)....why shouldn't there be efficiencies possible even in the realm of remediation and closing-of-gaps?

Or is more always more?

KIPP for all

from the U.S. News interview with Feinberg & Levin:

Finding qualified teachers to sign on to this cruise, however--even with the higher salaries KIPP pays--is a growing challenge, one that Feinberg and Levin say they can't solve without taking control of the training and certification process themselves. Already, KIPP runs a training program for principals at the Haas School of Business at the University of California-Berkeley. Extending that to teachers is an ambitious goal, one that would very likely require new legislation in individual states. But Levin, nothing if not persistent, insists that anything less is just tinkering around the edges. "Teaching has to become one of our society's most critical professions, rewarded and respected," he says. "And the cartels that control entry--the unions, the education schools--need to be addressed."

I'm in.

-- CatherineJohnson - 23 Mar 2006

UnderstandingReciprocals 27 Mar 2006 - 20:23 CatherineJohnson

[I've finally read the thread on teaching reciprocals - great stuff. I'll get it all pulled into this post so it shows in the archives tomorrow...]

I was bewildered by reciprocals when I started re-teaching myself arithmetic. I could divide fractions easily; I also knew when to divide fractions; I could even make up a proper word problem to illustrate the division problem 3 1/4 ÷ 3/4, something the American teachers in Liping Ma's study couldn't do.

But I had no idea why reciprocals worked.

Reciprocals seemed like magic.

Part of the problem, I now realize, is that I saw so many visual illustrations of fractions that I thought I was supposed to be able to visualize reciprocals, too. And I couldn't. Try as I might, I couldn't conjure up a Reciprocal Picture in my mind's eye.

So I was stumped.

Then one day, cruising math sites on the web, I saw this explanation of reciprocals and the whole thing became blindingly clear and obvious at once:

The minute I saw a fraction division problem rewritten as a "complex fraction" (is that the term?) I saw why you start with a division problem and you end up with a multiplication-by-the-reciprocal problem:

Et voila!

Division of a fraction turns into invert-and-multiply.

This is why I keep saying that 'Math Brains,' math teachers, and math textbooks alike all need to tell us rookies when a procedure or a formula is a 'shortcut' for a longer procedure.

That observation never seems to make much sense to people who are savvy at math, but I think it's important for novices. I know it's important for me, or was. I was 'stuck' in the idea that reciprocals ought to be 'obvious' the same way, say, adding 2 plus 2 and getting 4 is obvious. A reciprocal ought to be a 'thing' for me, in the way a 2 or a 3 is a 'thing.'

I think I'll stop now. I'm probably scandalizing all of you folks who Actually Know Math.

However, this is a good thing to know about students.

When you're learning, or relearning, arithmetic, you're constantly shown procedures and concepts that 'make sense' in a concrete, you-can-pick-it-up-and-hold-it-in-your-hand kind of way. Even place value makes sense that way, although I've come to feel that it probably shouldn't.

The addition and subtraction of fractions continues to make sense in this way — i.e. you can draw fraction addition & subtraction easily. You can 'see' it.

Fraction multiplication was, for me, the first operation & concept that didn't readily lend itself to visualization, in spite of the fact that everyone does produce drawings of fraction multiplication all the time in every textbook.

But fraction division was impossible. With the division of fractions, you hit the visual wall, or maybe the visual cliff. I suspect lots of arithmetic students have the same experience.

My feeling about teaching fraction division is this:

• For most students, you should delay teaching the 'why and how' of invert-and-multiply until students have mastered equivalent fractions and the multiplicative identity property of 1

• You should also tell students directly that invert-and-multiply is a shortcut, the same way cross-multiplying is a shortcut. You end up with invert-and-multiply after a couple of intermediate steps.

-- CatherineJohnson - 23 Mar 2006

SupportingChildrenWithGaps 29 Mar 2006 - 19:09 CatherineJohnson

Haven't had a chance to look at this yet —

From the UK Standards Site:

Supporting children with gaps in their mathematical understanding

also from the UK site:

Am I missing something here?

-- CatherineJohnson - 27 Mar 2006

FutureTeachers 31 Mar 2006 - 01:53 CatherineJohnson

Susan just emailed me this link to Rudbeckia's site!

Here's what Rudbeckia calls the 'honors version' of The Rule (two numbers: multiply, more than two numbers: add) —

Re: Item 8
A pretty good way to get at least a C when solving word problems:

If there are two 1- or 2-digit numbers: add them.
If there are two 2- or 3-digit numbers: subtract the smaller from the larger.
If there are 2 numbers, one single digit, the other 2 digit: multiply them.
If there are 2 numbers, one substantially larger than the other: divide the larger by the smaller.
If there are 3 numbers, each less than 30: find the least common multiple.
If there are 3 numbers, each less than 100: find the greatest common factor.

Clearly on item 8, you should follow the last rule, which is what the student did. And s/he even checked the work by finding the GCF two different ways. The rules don't always work (obviously), but they work often enough that you will get a reasonable number of questions correct. (And you will get others ridiculously wrong.) Of course, each student will have tweaked the rules according to the styles of their teachers and textbooks, so not every student will follow the same set of rules.

It's MUCH easier than actually READING the words and trying to puzzle out what's going on.

One of RH's commenters asked the same question I have: do we know why the first student has factored the numbers in order to arrive at an answer?

That looks familiar to me.

Is this something out of one of the fuzzy books?

-- CatherineJohnson - 29 Mar 2006

DowsLanePrincipalTalksAboutHandwriting 08 Oct 2006 - 22:18 CatherineJohnson

Just got a call from our old principal at Dows Lane (K-3), Joe Rodriguez.

Golly, we miss Dows Lane.

We miss Main Street School (4-5).

Homesick!

oops, out of time

Christopher's been sick for days, and is getting worse.....and our doctor is out of town.

So I'm off to Ossining to see the doctor who's filling in for her.

Back later -

Home again; Christopher will live.

Also, he will probably not end up in the emergency room suffering dehydration, as he did this time last year, when he had this same virus.

Good.

Anyway, back to Joe.

Long story short, I had asked our school board president what the proposed 'Math and Handwriting' books for Dows Lane were. Our board president had apparently forwarded my question to the assistant superintendent for curriculum, and the assistant superintendent had asked Joe to give me a call and fill me in. So he did.

Turns out they're not buying "Math and Handwriting" books, they're buying some math books and some handwriting books. They're two different things.

That's cool, because Joe said Andrea, the occupational therapist who works with Andrew, told him they must give the kids another year of practice with printing before starting them on cursive. They used to teach cursive in 2nd grade; now they'll teach it in 3rd grade.

GREAT!

I told him what a mess Christopher's printing is, and what a problem it is when it comes to math, and added that everyone over 70 has great handwriting because they were taught handwriting at school until they'd mastered it.

Joe disagreed. "My handwriting isn't any good," he said.

Joe is 50. 55, tops.

I said, "Joe, you're not 70."

Joe said he had 8 years of handwriting instruction and daily practice in Catholic school and it didn't work. That was depressing.

He said back when he was teaching, he had to concentrate to write legibly on the board. He'd start writing a sentence in the top lefthand corner writing a sentence, and end up down in the middle of the board. A lot of teachers, he said, can just blast their way across the board and it comes out looking great.

I told him I call that Teacher Handwriting.

Talking to Joe made me homesick. Back at Dows Lane we weren't having to fight constant skirmishes over bullying teachers and lousy computer-generated mid-term reports delivered to your mailbox on Christmas eve and 20-point deductions because the State Test made you do it.

At Dows Lane, and at Main Street School, you had conversations about things like How come Joe had 8 years of handwriting instruction from the nuns and he still can't write a straight line on the blackboard?

At Irvington Middle School, when you see Scott he tells you, "I'm very protective of my teachers."

Or, "I protect my staff."

One time he asked me, on the phone, if I thought he was protective of his teachers.

At the time I was in the full flush of gratitude that he'd rescued Christopher from Mrs. Roth's class, and I said, admiringly, Yes! I think you take good care of your teachers. Which is what he wanted to hear.

Of course that was a sign.

I was talking about that to Ed today.

He said, "If you listen, people always tell you who they are."

update: compare and contrast

Ed just ran into one of our closest friends from Dows Lane at the video store.

This mom is very on top of things, and has been extremely concerned about TRAILBLAZERS, to the point of enrolling her child in KUMON.

She told Ed she's resolved 'every' concern she had with the school. She's worked closely with her child's teacher, and the teacher has responded to every issue, and made changes where necessary.

Every one of her concerns has now been addressed and resolved.

Ed said, "Joe runs a tight ship."

a cordial email

Meanwhile, we are not working closely with our teacher.

We are not working with Ms. K at all.

Ms. K. has not responded to our emails.

Ed raised this issue with Scott Fried, who said something about cordial conversations. Our emails, he said, were not cordial.

True.

So, on Wednesday evening, I wrote a cordial email to Ms. Kahl:

Ms. K — we haven’t heard back from you about Christopher’s grade on the blueprint project.

And now this. This project was his one and only success in math this year. He spent four hours working on it. Ed had to supervise; he couldn’t do it alone. But he did all the work, and he figured out how to do all the work with guidance.

We’re working so hard to keep his motivation up. This is the age when boys check out. Some of his friends already are checked out (these are kids who moved from Phase 4 to Phase 3).

Before I started working with him he was completely turned off to math. I got him liking it again.

He’s very discouraged now.

We really need some help here.

Catherine

This doesn't happen at Dows Lane.

update 4-19-2006: 20 days and counting.....still no response...

-- CatherineJohnson - 31 Mar 2006

CollegeStudentWhoDoesntKnowMean 12 Apr 2006 - 19:19 CatherineJohnson

via Joanne Jacobs, a post at Right Wing Prof called I would like an answer.

After he spends 3 hours teaching a college freshman what an average is, Right Wing Prof wants to know why he's doing the job of K-12.

I know the answer to that.

In global terms, the answer is simple: K-12 is about inputs, not outputs. It's about inputs, not outputs, as a matter of law.

Students are legally entitled to receive a public education. They are not legally entitled to learn.

Legally, a public education is a set of inputs: teachers, administrators, textbooks, school buildings. Nowhere does the law state that a student is entitled to learn the content covered in school. You've probably noticed that no one ever sues a school district because his child graduated high school not knowing what an average is. We sue over everything else under the sun. We sue doctors for malpractice, we sue companies for selling tobacco, we sue McDonald's because the coffee was too hot — why has no parent ever sued a school for not teaching his child how to read or write or solve a mathematical problem?

The answer is that students have no legal entitlement to learn. Learning is an output; learning is the intended result of the inputs.

Students have a right to inputs. Students do not have a right to outputs.

UPDATE 11-8-2006: right answer, wrong question. Many parents have sued public schools in many states; the courts have universally ruled in favor of schools. If the child fails to learn, "it could be something about the kid." Therefore, courts have ruled that the school cannot be held accountable for learning. (I don't know the legal ins and outs of this. Could the school be held accountable for a broad class of students failing to learn? Could one file a class action lawsuit on behalf of non-classified kids who've failed to learn? I don't believe anyone's done it, which implies to me that someone probably tried and failed. But I don't know.)

IEPs for all

There is one exception to this rule.

Children with special needs have a federal entitlement actually to learn the material covered in school.

Each year the school must sit down with the parents and hash out a formal, legally binding document called an Individualized Education Plan, or IEP, stating exactly what knowledge and skills the child will be expected to learn in the coming year. Parents can convene a new IEP meeting any time they wish; schools must show that the student has actually learned what is listed on the IEP.

That's the law. Good luck enforcing it, of course. The reality of special ed is that it's an adversarial system, parents are called 'advocates,' and you spend huge quantities of your life fighting the school.* In Los Angeles we went to practically every meeting with our lawyer.

Nevertheless, the law exists. Your child is legally entitled to learn.

Christopher has no entitlement to learn. None, zip. Logically, then, if he doesn't learn, it's not obviously the school's fault. Legally speaking, the school is doing what it's supposed to be doing.

So whose fault is it? It's Christopher's fault. He's not organized, or he's not paying attention, or he doesn't realize that Middle School Is Hard while Fifth Grade Was Easy, or whatever. The school and the parents both seek an explanation within the child.

I'm sure this is why we've seen the explosion in numbers of special needs students. The instant a student goes from being 'typical' to having 'special needs,' he gains a legal entitlement to learn.

I'm not saying that parents 'play' the system, though I hope some do. A bad system should be played. I think an economist would analyze the huge increase in special ed population as a case of people & systems responding to bad incentives.

Bad incentives operate below the level of consciousness, for the most part. It's not that parents and teachers consciously think to themselves: If we get him classified, the school will have to teach him. Things just go that way.

I include teachers in this category, because many teachers are frustrated by school policies requiring them to march through content students haven't mastered. Christopher's brilliant teacher in 5th grade, Ms. Duque, told me that she'd been asked to teach the accelerated Phase 4 math class but had turned it down in favor of Phase 2, which had a number of students on IEPs. She always preferred to teach students with IEPs, she said, because the IEP gave her the legal right to teach to mastery.

There are plenty of administrators who feel this way, too. I mentioned that in Los Angeles we always went to meetings with our attorney in tow.

Well, that wasn't a problem! The administrators loved her. They all went way back. You could practically see them breathing sighs of relief once Valerie showed up. They were under pressure to withhold services; once Valerie was there they knew she was going to fight them all the way and win; so they had far more ability to do what they wanted to do, which was to do their level best to see to it that kids with disabilities learned everything they could.

Parents want their children to learn, teachers want their students to learn, administrators and school boards want their students to learn. But the system is set up to cover content in a spiralling sequence, not to teach to mastery.

That produces large numbers of kids who fall behind. When they've fallen two years behind - the formal definition of an LD - they can be classified as learning disabled, which triggers a legal entitlement to be taught to mastery.

When parents and teachers both want children taught to mastery, but the system blocks teaching to mastery, the incentive to move large numbers of kids into the sole category that will allow them to be taught to mastery is immense.

This process doesn't have to be conscious, and I don't think it is conscious 99% of the time. That's the tragedy. Everyone believes the categories. I was stunned to learn from my neighbor, a clinical psychologist, that from her perspective 'learning disability' isn't a diagnosis. "Learning disability" is a legal category used by schools to assign services. (This gets complicated. The law defines a learning disability as an actual brain-based disability. However, school districts often define a learning disability as 'normal intelligence, two years behind grade level.')

People think learning disabilites are real the way diabetes or cancer are real. In many school districts, once a child falls two years behind in a spiralling curriculm, he is 'referred' for testing, which invariably finds a problem in the child, not the teaching. At that point he 'qualifies'** for the label of 'learning disabled' and the label becomes real.

He is a learning disabled child.

It has crossed my mind that one answer to our problems with Irvington Middle School is to figure out a way to get Christopher classified.

That was one of the first questions asked by the principal at our Team Meeting. 'He doesn't have a learning disability, does he?' He sounded almost hopeful.

Life would be easier for everyone if Christopher had a classification. Come to think of it, if I hadn't done any reteaching at home, he just might qualify for a special needs classification in math by now. Dyscalculia anyone?

I'm not going to do it. I have no interest in working the system so as to have a third child 'classified.' I'm not even sure I could do it, though, knowing me, I probably could.

Nope. This is a battle for my one typical child as a typical child.

He should have the same rights his autistic brothers do.

* Last year, in Irvington, the school board openly announced an illegal policy at a board meeting. There was a shortfall in the budget, they said, so they had come up with a plan to save money by bringing all the special needs kids 'back to district.' When a school can't meet an IEP student's needs, the district has to pay to send him elsewhere. That's expensive. Balancing a budget on the backs of special ed kids by taking them out of programs that meet their needs and bringing them back to programs that don't is illegal. Everyone hired lawyers, the district hired a superb new interim director of special ed, and that initiative came to a screeching halt. He's done a superb job creating a 'transitional' program for Jimmy. We're hoping desperately we can keep this administrator for two more years.

** 'Qualifies' is the term used. A child must 'qualify' for 'services.' A typical child does not qualify for services.

key words: blame the student school psychologist
Pamela Darr Wright summary of Galen Alessi study
Evolving Functions for the School Psychologist
Whose Fault Is It?
educational rights of special need children versus typical children
Engelmann on Galen Alessi study
Pamela Darr Wright posted to ktm
"public school has never been about outputs..."

-- CatherineJohnson - 11 Apr 2006

KippGoesToKindergarten 04 Oct 2006 - 16:11 CatherineJohnson

Trying to track down a Jay Matthews column on St. Anne's school in Brooklyn, I came across this column saying KIPP has started an elementary school in Houston.

That's good news.

And check this out.

They're combining Saxon Math with Everyday Math:

At SHINE, Brenner says, he is blending the more modern Everyday Math with the more traditional Saxon Math for first-graders. The proponents of those two teaching programs have been at war for 20 years; can combining them really work? I'd predict that joining such radically different elements would cause an explosion, like when I used to toss manganese shavings into the surf to illuminate beach parties.

Brenner seemed unfazed by my doubts. "Our kids are off the charts in math," he says. I haven't surrendered my skepticism, but I will visit his school, and then watch what happens when Laura Bowen brings all this here, where Washington can get a really good look at it.

I'm not surprised.

My friend with the kids in the fantastic private school told me her school combines Everyday Math with traditional math. They seem to do nothing but EM for the first couple of years; then they shift.

I was shocked when she told me this, and assumed that her kids were getting shortchanged.

Then she faxed me her son's math homework.

WAY past anything kids are doing in public schools. This boy was doing long division with a gazillion digits; no forgiving division anywhere in sight. The word problems were serious and challenging - challenging at his level. My friend was shocked that we have to reteach math at night. She and her husband never reteach any subjects at all. The kids in her school are way up at the top of U.S. kids, and they're learning everything they know at school.

Barry has mentioned before that James Milgrim thinks Everyday Math would be a good supplemental program when used with a traditional math curriculum.

Looks like he's right.

-- CatherineJohnson - 12 Apr 2006

EducationJournalism 19 Apr 2006 - 17:54 CatherineJohnson

at D-Ed Reckoning

...the kids who need the most parental support are least likely to have parents who are able to provide effective support. That's why we send these kids to school in the first place, isn't it? Because their parents do not have the ability and/or are unwilling to teach them. The assumption going in should be that no parental support will be forthcoming. The instructional design should be premised on that.

Ed just wrapped up the college interview season (he interviews for Princeton).

His impression, interviewing a number of very high achieving kids, is that the kids who take & succeed in AP calculus all have 'Math Brain' parents. Most of them have parents who weren't just 'good at math' in college, but are actually working in math-related fields today. Some of these kids have two parents working in math-related fields.

That jibes with what I've heard from the few parents of high schoolers with whom I've discussed the issue so far.

Kids who 'go the distance' in high school math have parents who can reteach and/or tutor algebra, trig, and calculus at home. I'm developing an image of a hereditary 'Math Elite' here in America, a Special Caste. If your folks didn't take calculus, you're not taking it either! *

I just have to hope I can get to calculus and come close to mastering it before Christopher does....

....which brings me to —

I've just this moment finished Saxon 8/7!

Investigation 11 is behind me!

The one thing I did wrong, in case any of you are interested, was that I didn't do the 'Fast Facts' sheets students are supposed to complete at the beginning of each lesson.

That was a mistake. Towards the end of the book I realized I need a fair amount of simple memorization of things like metric conversions, area & volume formulas, terminology & the like.

So I'll be doing a Fast Fast sheet every day for awhile.

But I start Algebra 1 tomorrow! Yay!

I'm using Saxon Algebra and Dolciani together (Solution Key), with Foerster (THANK YOU, BOOK FAIRY!) on the side.

Can't wait.

Foerster algebra

wow

Check out this review of Foerster from Mathematically Correct.

The book provides an outstanding opportunity for student learning. Even achievement at the highest levels is supported, although sometimes only at good levels rather than outstanding levels.

Perhaps the greatest strength of this program lies in the abundance and quality of student exercises, especially application word problems. But, virtually all ratings of this program are outstanding. Simply put, it does a good job of the topic of introductory algebra.

I may have to re-think (as well as track down the supporting teacher materials for Foerster...)

Maybe I'll do all 3!

* The kids I know who went to private and/or Catholic School took calculus and succeeded in it regardless of their parents' careers or number of math courses taken when they were young, but this hasn't been true of the kids I know in public schools. I'll keep asking. My sense is that even very good public schools rely on parents to get their kids through math.

Greta recommends Foerster
more from Greta

-- CatherineJohnson - 14 Apr 2006

ArithmeticToAlgebra 15 Jul 2006 - 16:33 CatherineJohnson

source:
A rational approach to education: Integrating behavioral, cognitive, and brain science
John Bruer
Herbert Spencer lecture
Oxford University
October 18, 2002
PowerPoint presentation

lecture notes accompanying this slide

The notes accompanying this slide are confusing, so I'll give you my own translation.

In this study, students worked the problems listed above, and teachers predicted which problems would be harder & which would be easier.

Teachers correctly predicted that students would do better on the arithmetic problems - i.e., that arithmetic is easier than algebra.

They incorrectly predicted that students would do better on the symbol problems than on the word problems. In fact, these word problems were easier for students than the symbol problems.

Look at the 6 problems listed in the table. Which do you think are the easiest to solve for late elementary/early secondary students to solve? Rank them from 1 (easiest) to 6 (hardest).

Types of problems:
first 3 are arithmetic (start known), last 3 are algebra (start unknown)
first and fourth are story problems
second and fifth are word problems
third and sixth are symbol problems.

Here is how teachers ranked the problems versus how the students performed on solving problems of the various types.

Teacher rankings agreed with student performance on arithmetic versus algebra: The teachers [say that] algebra is more difficult and students find it so.

However, teachers erred within these two categories ranking story problems as being more difficult for students than symbol problems. Whereas in both these categories students performed best on story problems and worst on symbol problems.

Based on this and other research Ken Koedking and Mitch Nathan have shown that (American) teachers hold “symbol precedence view” of student mathematical development. They believe that symbolic problem solving and equation manipulation skills develop prior to the ability to execute verbal reasoning about number.

The “symbol precedence view” dominates textbooks for teaching algebra. As a result algebra instruction does not build on students’ prior strengths and understanding, but is orthogonal to them.

We can use students’ prior understanding of the verbal number system to help them acquire what appears to be most difficult for them, mastery of mathematical formalism and symbolism.

THERE ARE SIMILAR EXAMPLES FROM OTHER DOMAINS: PHYSICS, READING COMPREHENSION, WRITING AND COMPOSITION.

mini word problems

This year, relearning pre-algebra myself and trying to get Christopher through his pre-algebra class at school, I've discovered the importance of "mini word problems." By "mini problem" I mean super-simple word problems that illustrate the concept being taught in class and in the textbook.

Mini problems are a profound strength of the Singapore Math series. Children do word problems from Day One, and parents can buy a Challenging Word Problems work book for first grade. The word problems in Singapore Math teach the concept.

The idea of word-problems-that-teach has been a revelation to me. I'm used to 'killer' word problems, problems written to stump, trick, and otherwise mystify 90% of the children in any given class. Word problems, in my experience, have been used as the gatekeeper.

That's not strictly true, of course. All textbooks, and presumably all teachers, also use word problems to teach how to do word problems.

I'm talking about word problems used to teach math.

out loud word problems

I've come to think that, ideally, the first word problems a student does in a lesson should be what Russian Math calls 'Out loud problems,' or mental math.

For instance, the kids I know had a heck of a time with the chapter on ratio & proportion a couple of months back. That's because everything in Ms. K's class is taught by rote memorization, and once you start trying to memorize ratio & proportion things can get jumbled quickly.

I told Christopher & his friend M. that the first proportion problems they should have done was:

The stationery store was selling pencils 2 for a dollar. M. bought 5 pencils and paid how much?

The next problem should have been, The stationery store was selling pencils 2 for a dollar fifty. M. bought 5 pencils and paid how much?

The students would have done these problems in their heads, without writing the distraction of having to write things down.

Mini problems should be so simple that they are completely transparent. No trick language, no trick sequencing of verbs and numbers, etc. A mini problem should be a super-simple, see-through application of the concept being taught.

The mini word problem is the true 'manipulative' of K-8 math.

1st grade worked problem, Primary Mathematics

2nd grade worked bar model, Primary Mathematics

update from Mark Roulo

My son is working his way through Singapore Math (just finished K, starting on 1st grade), which is irrelevant except to set context. We are doing things like 3 + 4 and 8 - 2. Nothing (yet) with serious multiple digits.

Sometimes adding and subtracting doesn't "click" for him (strangely, this seems day to day ... he might have been fine the day before).

If my wife or I translate the problems into word problems with firetrucks, he almost always gets the problem. So 3 + 4 he usually gets (if he didn't, we'd still be working in the K booklets). When he doesn't get it, this almost always works: "If I have three firetrucks at the station and four more arrive, how many do I have?"

This only works with firetruck related stories.

I'm guessing that the firetrucks somehow make this more concrete for him on those days when he is having trouble not with the mathy part of the problem, but with the abstraction.

[Sidenote: I'm wondering how far this can be pushed: I've got six stations with three trucks each, how many trucks do I have? Pumper engines have three firefighters and ladders have four. If I have 10 trucks total and 36 firefighters, how many of each type of truck do I have? Etc.]

from Molly

I noticed the same thing yesterday when my 4 year old son was attempting to play a dice game. He would yell out 5 - 2 and have no idea what the answer might be. As soon as I rephrased the problem as "If I gave you 5 cookies and you ate 2, how many would be left?" he would respond immediately with the correct answer. Cookies, toys and frogs have worked at our house. I'll have to try firetrucks next.

from Anne

This is similar to how my son Daniel learned the times tables.

He is a huge football fan. If you phrased any problem in terms of football, he could always get the answer.

When it came time to teach the times tables, I started with the 7 times tables (touchdowns). He already knew up to 7 x 8. (It was rare that a team got more than 8 touchdowns.) I then proceeded to the three times tables (field goals). After that, everything else just seemed to come easily.

These stories of parents teaching symbolic problems to children jibe perfectly with Willingham's report that learning proceeds from the concrete and specific to the abstract:

[T]he mind tends to remember new concepts in terms that are concrete and superficial, not abstract or deep.

can neuroscience tell us how to teach?

conclusion of the Bruer's lecture:

I have tried to share with some thoughts on how behavioral, cognitive, and neural science might be related and how they might be integrated to support an applied science of teaching and learning.

Currently, seeking insights for education from basic cellular neuroscience and fine brain morphology is not viable. Attempts to do so overlook what cognitive and behavioral research tells us about learning and thus do not bring the best science to the problem. The popularity and appeal of such explanations are detrimental to rational applied science of learning.

Cognitive neuroscience and functional brain imaging studies, likewise have little to offer in themselves for an applied science of learning. These studies are dependent on pre-existing cognitive models.

The cognitive psychological level is of importance to educators. The most powerful and applicable educational research tends to characterize educational problems in terms of cognitive models and to develop interventions or insights for interventions based on these models. There is a long history of success with this research program, but it is difficult to motivate educators to enact them or the public to support them.

how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 23 Apr 2006

NationalMathematicsAdvisoryPanel 06 Oct 2006 - 16:53 CatherineJohnson

Barry Garelick alerted me to the formation of the National Mathematics Advisory Panel, which follows on President Bush's SOTU American Competitiveness Inititiative, announced in his January SOTU.

In His State Of The Union Address, President Bush Announced The American Competitiveness Initiative (ACI) To Encourage American Innovation And Strengthen Our Nation's Ability To Compete In The Global Economy. This ambitious strategy will increase Federal investment in critical research, ensure that the United States continues to lead the world in opportunity and innovation, and provide American children with a strong foundation in math and science. The American Competitiveness Initiative commits \$5.9 billion in FY 2007, and more than \$136 billion over 10 years....

[snip]

Education Is The Gateway To Opportunity And The Foundation Of A Knowledge-Based, Innovation-Driven Economy. To prepare our citizens to compete more effectively in the global marketplace, the American Competitiveness Initiative proposes \$380 million in new Federal support to improve the quality of math, science, and technological education in our K-12 schools and engage every child in rigorous courses that teach important analytical, technical, and problem-solving skills....providing grants for targeted interventions, and developing curricula based on proven methods of instruction, and developing curricula based on proven methods of instruction. The American Competitiveness Initiative includes a number of new and expanded programs, including:

• The Advanced Placement/International Baccalaureate (AP/IB) Program to expand access of low-income students to AP/IB coursework by training 70,000 additional teachers over five years to lead AP/IB math and science courses.

• An Adjunct Teacher Corps to encourage up to 30,000 math and science professionals over eight years to become adjunct high school teachers.

• Math Now for Elementary School Students and Math Now for Middle School Students to promote promising and research-based practices in math instruction, prepare students for more rigorous math courses, and diagnose and remedy the deficiencies of students who lack math proficiency.

update

from Education Week:

The math panel is scheduled to hold its first meeting May 22 at the offices of the National Academy of Sciences, in Washington. That organization is part of the National Academies, a federally chartered, independent organization of scholars that researches a broad range of science, technology, and other issues, including K-12 education, typically by pulling together committees of experts.

Mr. Bush has asked the panel to submit a preliminary report with recommendations to him by Jan. 31, 2007, and a final report by Feb. 28, 2008

Now a list of panelists has been announced that looks good to me:

• Dr. A. Wade Boykin, Professor and Director of the Developmental Psychology Graduate Program in the Department of Psychology, Howard University

• Dr. Francis "Skip" Fennell, Professor of Education, McDaniel College (Md.); President, National Council of Teachers of Mathematics

• Dr. David Geary, Curators' Professor, Department of Psychological Sciences, University of Missouri at Columbia (you can pick up a used copy of his book for \$859 at Amazon)

• Nancy Ichinaga, former Principal, Bennett-Kew Elementary School, Inglewood, Calif.

• Dr. Tom Loveless, Director, Brown Center on Education Policy and Senior Fellow in Governance Studies, The Brookings Institution (one of my favorites)

• Dr. Liping Ma, Senior Scholar for the Advancement of Teaching, Carnegie Foundation

• Dr. Valerie Reyna, Professor of Human Development and Professor of Psychology, Cornell University

• Dr. Robert Siegler, Teresa Heinz Professor of Cognitive Psychology, Department of Psychology, Carnegie Mellon University (interesting study on errors in children's internal number lines - and on SES differences in children just entering school) How Children Develop (textbook) Early Development of Estimation Skills

update 8-4-2006: This statement from the abstract for Siegler's 2006 article on children's learning concerns me: "Overall, contemporary analyses show that learning and development have a great deal in common." The view that development and learning are one and the same is a core tenet of progressive-ed ideology. This may not be Dr. Siegler's meaning, obviously, but I'm guessing that won't matter when Columbia Teachers College cites his work.

• Dr. Sandra Stotsky, Independent researcher and consultant in education; former Senior Associate Commissioner, Massachusetts Department of Education (another favorite)

• Vern Williams, Math Teacher, Longfellow Middle School, Fairfax, Va. (Barry's neighbor and "an excellent middle school math teacher")

• Dr. Hung -Hsi Wu, Professor of Mathematics, University of California at Berkeley

Ex-officio members:

• Dan Berch, National Institute of Child Health and Human Development, National Institutes of Health
• Diane Jones, White House Office of Science and Technology Policy (Barry: "works for the White House on education matters, and was formerly with House Science Committee working the issue of math education")
• Tom Luce, Assistant Secretary, U.S. Department of Education
• Kathie Olsen, Deputy Director, National Science Foundation
• Raymond Simon, Deputy Secretary, U.S. Department of Education
• Grover (Russ) Whitehurst, Director, Institute of Education Sciences, U.S. Department of Education

polite agreement or something we can use?

-- CatherineJohnson - 25 May 2006

AleksForSummer 31 May 2006 - 11:43 CatherineJohnson

This is exciting. I had asked Barry Garelick about ALEKS awhile back, and recently a parent sent him this account:

My child used ALEKS for Algebra and Geometry. IMO, it is a good solid math tutorial. A couple of years ago, I was looking for an inexpensive way for my child's classmates to learn mathematics, as opposed to what they were being taught in school (TERC and CMP). My child was taking an online algebra course (by Academic Systems) through Johns Hopkins' Center for Talented Youth (CTY). When he was about halfway through the course, I suspended the course and had him switch to ALEKS Algebra. The initial assessment indicated that he had mastered about half of the material in ALEKS Algebra, so in this sense it was comparable to the CTY course. Also, after he completed the ALEKS Algebra, I had him complete the CTY algebra course, which he did in about 2 weeks. That's about as fast as you could finish the course since it is linear (presented in a fixed sequence) and with video introductions and problem exercises which must be completed. My child thought the two courses were comparable. He also used ALEKS Geometry last year after he completed his high school "Honors" Geometry course. His initial assessment indicated that he had not mastered about 40% of ALEKS Geometry!

I know that ALEKS is used by a number of gifted children to accelerate in math. I know one parent, who coaches his local competition math team and has a child who is exceptionally talented in math (when he was in 8th grade the child had the highest score in his state on the AMC 10). The child uses Stanford's EPGY courses to learn math topics and then uses ALEKS to make sure he hasn't missed anything. That is a very strong endorsement in my book. Although I am most familiar with ALEKS being used by gifted student, it is not intended for gifted students, as is EPGY.

Being neither a mathematician nor a math educator, just a parent trying to provide his child with a decent math education, I heartily recommend ALEKS. A couple of years ago I asked for information about ALEKS on the nyc-hold list and got no response. I even wrote a couple of people offlist and asked them directly what they thought of it, but got little feedback. I hope someone else responds to your post with a more authoritative evaluation. If not, let me suggest that you try their free trial (see the top of the page you linked to). Pick a topic that you are comfortable with and incorrectly answer a couple of questions where you would particularly like to see what material they present and how.

ALEKS used to have a statement on their website to the effect that their program was designed to meet international math standards. I cannot find it now. I think they have intentionally rewritten the history of the development of ALEKS so that it would appear to be based on NCTM standards, "The practical development and implementation of an assessment and teaching system for Arithmetic based on Knowledge Space Theory began in 1992, financed by a 5-year grant from NSF." See Research Behind ALEKS. However, the history they used to have posted indicated that much of the original research began in the mid-80's, as I recall. You can learn more about ALEKS development and content by contacting ALEKS at 714.245.7191. You need to get beyond the customer reps (by asking for technical/content information they can't provide), but once you do the people are very helpful and knowledgeable.

Christopher's going to do ALEKS this summer.

Me, too, maybe. I'm thinking I can use it as a check on my self-teaching. Cost is \$19.95 a month, or \$99.95 for 6 months.

ALEKS as an inexpensive home-assessment tool

I think ALEKS probably works as a low-cost assessment tool for parents, based on this post from ParentPundit, whose daughter was languishing in Everyday Math:

In the summer at the end of 5th grade, I had her try the Aleks computer program in math, www.aleks.com. The Charter School in my town uses it, and I decided to try it for my own daughter. A tutor would have been expensive and less than optimal in this situation because my daughter does get concepts, she just needs more drill (how can most kids hone their number sense if they aren’t ever asked to multiply and divide numbers continuously), and she needs algorithms that have fewer steps so there is less possibility of error (everything that Everyday Math does not provide.)

According to Aleks, my daughter only knew 21% of a traditional 5th grade curriculum – and this was at the end of 5th grade. Talk about having a heart attack! This was soon remedied. My daughter is now in the 6th grade and she has completed the 5th and 6th grade curriculum according to Aleks. I’m looking forward to the tests at the end of the year to see if my intervention worked.

I'm trying to muscle Christian into taking an assessment test on ALEKS, too, but so far he's not enthusiastic. That's fine. The first project is getting him back in Westchester Community College. We got to work on that last night.

Then we'll deal with math.

I'm lying in wait.

I like this:

ALEKS is not a game, a toy, an art project, a virtual reality "space," a chat room, or a set of glitzy graphics with sprinklings of academic material. ALEKS is an interactive tutor which provides real assistance to anyone attempting to learn math.

Glencoe is now distributing ALEKS, which, based in my experience with their Parent-Student Study Guides (available free online), I think speaks well for it. You can still order ALEKS independently, however. No need to show your license.

ALEKS
parent report on ALEKS
ALEKS Graphic
formative assessment on wheels
ParentPundit uses ALEKS to fix Everyday Math
ALEKS question
ALEKS assessment coming right up

Glencoe math materials available free online
Glencoe Pre-Algebra Parent Student Study Guide
Glencoe Algebra 1 Parent Student Study Guide
Glencoe Geometry extra examples

Glencoe writing models available free online*

*good to know about in case your child's teacher is not at liberty to provide models of good student writing

keywords:
Glencoeparentandstudentstudyguide
Glencoewritingmodels
BarryALEKS

-- CatherineJohnson - 31 May 2006

MathTricks 31 May 2006 - 23:43 CatherineJohnson

Came across Math Tricks while looking for Math Teach and Math Learn. This is my favorite so far:

```From: Bob Stanarrow <Straitfromtheheart@aol.com>
To: Teacher2Teacher Public Discussion
Date: 2001030718:38:50
Subject: How to remember how many feet in a mile

I figured out a way to remember how many feet are in a mile.

Just say to yourself 5 tomatoes. Really there are 5,280 feet in a
mile, so you can remember that by saying 5 tomatoes.

5    to   mat  oes
( 5 ) ( 2 )( 8 )( 0 )
```

-- CatherineJohnson - 31 May 2006

AnneDwyerMathBoosterGaps 20 Jul 2006 - 20:12 CatherineJohnson

Here's a gap story from my summer Math Booster Camp.

As a way of introduction, I have to say that this has been my favorite set of classes for Math Boosters. I am teaching a 6-11 year old group and a middle school group.

My middle school group is great. They are all math brains and they are all there because they want to be. They range from 4th grade to 7th grade.

Here's what happened today and it illustrates just what is wrong with Everyday Math:

Today I reviewed multiplication and division of fractions. I gave them a problem set that included Exercise 305 from Russian Math. Note that these are meant to be done out loud with no paper but they look like this: 5(1+1/5). The students were having trouble with them, so I put them on the board and said that they could add the fractions inside the parenthesis and multiply by 5 or they could use the distributive property.

They gave me a totally blank look. I asked them if they had ever heard of the distributive property. They said they hadn't.

So I demonstrated the distributive property. I expected them to say something like, oh, I've seen that but I didn't know what it was called.

But not one of them did. Here is a class of seven math brains, all of whom have been trained with Everyday Math, who have never been taught the distributive property.

My new math equation: Spiralling curriculum + no content = big gaps

-- CatherineJohnson - 20 Jul 2006

HowToTeachYourselfArithmetic 21 Sep 2006 - 21:51 CatherineJohnson

This may seem like a strange question coming from me.....can you teach yourself arithmetic?

UPDATE 10-19-2006: The answer is yes. You can. Christian is doing it now. Starting in Saxon Math 5/4.

I ask because Christian just got his placement test results — he passed reading!

I don't think we can credit the Yonkers school system for that, but the Mamaroneck schools may have had a hand in it. I say "may" because Christian's mom is college educated and has always subscribed to the New York Times, which meant that as a child Christian, like the rest of us it seems, was getting most of his vocabulary and exposure to print at home. He went to Mamaroneck schools through middle school, then moved to Yonkers where his 12th grade English teacher used the same book Mamaroneck used in 7th.

So I'm not giving Yonkers a lot of credit.

The bad news is math. We're looking at a pre-algebra placement.

(Can we sue the schools for not teaching yet?)

I'm in no mood to pay for two zero-credit remedial courses at Westchester Community College, and I don't know whether financial aid exists for pre-algebra. Even if it does, Christian needs two semesters' worth of remedial math (pre-algebra and high school algebra) before he can take a math or science course for credit. If that's what he has to do, then that's what he has to do, but he also has to support himself and stay motivated. The college completion stats don't show a lot of people who have to take two semester's worth of remedial math making it through.

I'd like to find another way if possible.

Naturally I'm thinking Saxon. Christopher's been moseying through Saxon Algebra 1/2 this summer. He's up to Lesson 15 and he's been getting all the answers right. Christian could probably teach himself fractions, decimals, and percents using Algegbra 1/2.

On the other hand, the Saxon books are huge. "Huge" meaning long and time-consuming. Long and time-consuming may be the only way to go here, seeing as how there's no royal road to geometry. But if anyone has thoughts, I'd like to hear.

update: I've just realized I'm going to have to get Christian to take the Saxon placement test.

If Saxon puts him into 8/7 or 7/6....I'm going to have to find another way.

computers & test anxiety

Christian says his mother was shocked that he passed the reading test.

I didn't get that at all until he told me he's always had a hard time taking tests. It sounds like he has some test anxiety; plus he's got some kind of fine motor "issue" (Carolyn's favorite word!) that tripped him up for years. He was classified special needs, along with all the other black kids, and his mom was constantly trying to get the school to provide him with a keyboard. Plus he's lefthanded.

So basically, he's never been able to take tests.

Apparently the reason he did well on the WCC test was that it's done on a computer terminal. He took the Accuplacer test, which I gather is being used in colleges all over the country. I had no idea the College Board is also in the remedial placement testing business. Apparently there's a whole Accuplacer test prep world out there, too. (It's aways worse than you think.)

Doing the test on the computer made Christian feel as if he wasn't doing a test. He was the second person finished; he just whipped through it.

ALEKS?

This is making me wonder whether ALEKS might be a good idea for Christian.

I'm certain Christian has math baggage (scroll down for Rudbeckia, Steve H, Carolyn, & Susan) and it seems pretty clear that looking at math on a computer will help him "break set."

On the other hand, I've been using ALEKS for a few weeks and while I find it highly motivating - addictive, almost - I don't find it highly illuminating. It's pretty much the ultimate in fragmented content, and the program offers no "metacognitive pointers" as Saxon does. You're on your own.

By "metacognitive pointer" I mean the kind of pointers people give when they're telling a delivery person how to get to their house. ALEKS doesn't give pointers. ALEKS just gives you the procedure, along with a lot of hyperlinks to other pages filled with other procedures & definitions, and that's the end of it. It's like learning algebra from Hal.

Years ago, when I interviewed nearly 100 couples for a book on marriage, I ended up dividing people into two categories:

• people who give good directions

• people who don't *

People who give good directions always tell you where you're going to be tempted to go wrong, how to tell if you have gone wrong, and what to do about it when you realize you did go wrong. A really good direction giver will say "You can't really see the driveway from the road, so if you get to the traffic light across from the church and the Sunoco station you've missed it."

That kind of thing. That's what the Saxon books do. Saxon lessons routinely tell students what mistakes they're likely to make and how not to make them. Often these pointers give you greater insight into the topics you've been studying.

Saxon Algebra isn't going to be addictive for most people.

But it is illuminating.

Any thoughts?

* When I first met Temple, she made exactly the same observation & for the same reasons.

Christianlearnsmath

-- CatherineJohnson - 31 Aug 2006

TheBadGetsNormal 04 Sep 2006 - 20:07 CatherineJohnson

I just noticed this passage in the middle of an article on parents helping with middle school homework:

Brainfuse.com also offers online tutor help, primarily to schools through the federal No Child Left Behind legislation. If a school has failed to make adequate progress under the law for two or more years, the school can choose from a state-approved tutoring company, with Brainfuse among them, said Francesco Lecciso, director for the company.

Brainfuse now has contracts with school districts in Los Angeles, Chicago, New York and other smaller districts, as well the Queens public library system.

“There’s clearly times students need a face-to-face tutor, but sometimes he needs anonymity,” Mr. Lecciso said. “Online, a student might be more willing to ask the same question eight times in a row, or to admit he doesn’t know how to do long division even though he’s in 7th grade.”

So we buy fuzzy math curricula that don't teach long division, then we pay professional tutoring companies NCLB funds to teach long division to embarrassed 7th graders.* By the time this phenomenon finds its way into the Times, it seems perfectly natural.

This reminds me of the 1st grade teacher I met at O'Hare a year ago who told me that the first graders in her district had been doing great ever since they brought in Everyday Math, but the junior high kids were a mess. She thought the answer was for the junior high kids to have a constructivist curriculum, too.

Of course, that probably is the answer.

If every kid in the district had a fuzzy book, the subject of long division would never come up.

Danwei Chu

* If they're lucky. We have friends from L.A. whose 20-year old college son still "can't really do long division."

-- CatherineJohnson - 03 Sep 2006

DougSundsethOnRightIsoscelesTriangles 15 Sep 2006 - 16:00 CatherineJohnson

"Why do these triangles come up everywhere?"

An isoceles right triangle is a 45-45-90 triangle, and is a pretty obvious sort of triangle to draw. It's also what you get if you cut a square in half on the diagonal. You can also cut such a triangle in half through its 90 degree interior angle and get a similar triangle (and you can repeat this procedure infinitely). Here the sides have lengths of one, one, and square-root-of-two.

The sine of 30 degrees is 0.5, which is one of those easy-to-remember numbers. And the result is a triangle with side-lengths of one, two, and square-root-of-three.

Both of these are used pretty often in carpentry, in part because they're both easy to construct with a straight-edge and divider.

"In what context?"

Math books. (Oh, as noted, they're used pretty extensively in real life, too, but that's the important answer for a math student.)

"Are there lots of shortest-distance-between-two-points problems?"

They're more used in the context of trigonometry, I'd say, since figuring the distance between two points on a plane is pretty trivial once you know the Pythagorean Theorem.

"Also....do surveyors use right triangles a lot?"

It's the core of classical surveying. The right angle is normally that between the vertical and the horizontal. With that, a known distance (from where you are to where you are measuring), and a measured angle, you can find the height of whatever you are looking at. (Triangles have three sides and three angles. One of the basic theorems in trig states that with any three of those six pieces of information including at least one length, you can find the other three.)

golly

I feel proud of that final question ("do surveyors use right triangles a lot").

I have no idea how surveyors do their work, and I haven't had a full course in geometry this time around. (Geometry is integrated into algebra in Saxon Math.) Plus I've never in my life studied trig.

The fact that it struck me as likely right triangles would be part of surveying is good.

I suspect this is an example of how comprehension and conceptual understanding come about. Often it's a process of things starting to "seem" true or likely-to-be-true; comprehension comes in glimmers and glimpses.

On the other hand, I do wish, often, that math textbooks would include the kind of explanation Doug just wrote. I'm constantly wanting to know where my studies are leading: what do real live mathematicians and engineers do with this stuf?

I don't need - or want at this point - lots of "real world" applications. What I want is a road map.

Saxon does some of this, and what he does do is terrific. But it's not enough. I suspect that Saxon assumes the teacher will provide the road map exactly as Doug has done here, through answering student questions as they arise. If so, he's correct that a textbook writer isn't particularly good at anticipating this kind of question. (Bob Koegel explained this principal to me once. He told me that research had shown that when teachers selected which words autistic children should learn they always selected the wrong ones. The children themselves know better which words they must acquire now — and are ready and capable of acquiring. That insight has never left me.)

All of this reminds me that I should check Mary Dolciani's & Harold Jacobs' books. Both authors are (said to be) strong on cluing students in to where their work is leading.

-- CatherineJohnson - 14 Sep 2006

FardellsNookesAndHides 16 Sep 2006 - 02:32 CatherineJohnson

from the Yahoo Group called Homework Help:

My son and I both came up with different answers to this problem...I am rusty at math but still think that I am right:)

How many fardells are in 8 hides?...
2 fardells = 1 nooke
4 nookes =1 yard
4 yards = 1 hide

This is a medieval Britain story problem. I got 256 fardells as my answer...sure seems like a lot but it's hard to visualize :)

Help!

Start teaching those unit multipliers now!

I'm going to see if Christopher can do this.

[pause]

Nope, he couldn't. He got 32. "I did it in my head."

After he got his wrong answer I started him out with unit multipliers by writing:

8 hides X

on the left side of the paper.

Then I asked him what needed to be in the denominator of the next ratio. (Ratio? Rate? What's the proper term?)

He knew it had to be hides, but he couldn't figure out what would be in the numerator.

sigh

About 5 seconds after that he got with the program, wrote out the series of ratios, and said, "Now what do I do, multiply?"

me: "Haven't you written out a string of multiplications?"

Christopher: "Yes."

me: "Then multiply."

He got 256.

Every year for the past 4 or 5 years we've gone to the U.S. Open.

And every year Ed has to explain the entire scoring system in tennis all over again.

One year between exposures to content is too long.

Next year I plan to remember "All," as in "15 All," and "Deuce," as in .... as in I don't know when people say deuce. I've forgotten. I just know that once you say "deuce" the players have to win by two points. At least, I think that's what I remember.

I'm going to remember "all" because "all" means everyone and "all" is a tie. All have the same score.

OK, so basically what I'm going to remember is "All."

Also, I think I'm going to remember that men play 6 sets.

Or maybe 5.

See what I mean?

One year is too long.

-- CatherineJohnson - 14 Sep 2006

meetings

homepage

I'm posting links to the Math Panel homepage, transcripts, & ktm posts here:

You can find both pages on the menu to the left.

If all else fails you can search posts using the keyword nationalmathematicsadvisorypanel with no spaces between words. (Works pretty well with spaces, too.)

I'm thinking this is about as findable and redundant as I can make the links now...unfortunately, you will have to remember some constellation of the words "national mathematics advisory panel" to find these links (that could be iffy for me these days....)

But I think I've just raised the odds of re-finding the transcript links considerably.

Polite agreement or something we can use?
National Math Panel announcement
National Math Panel update
short story by Vern Williams