Entries from EverydayMath

Welcome Independent George!

I started by saying that my principle has become 'teach things more than one way.'

Carolyn, Bernie, Chris & others objected.

I have to say that while 'teaching things more than one way' is a core principle for me at this point, whether rightly or wrongly, I don't really know what I mean by that.

In practice, what I've been doing so far is to teach bar models each and every day, along with, each and every day, the standard American 'symbolic' approach. I had Christopher start with the very first word problem in Primary Mathematics Book 3A, which is the first semester of 3rd grade in Singapore, & do one word problem a day, drawing a bar model to illustrate the problem set-up, and then doing the math using the standard algorithms.

And that's it. Each problem takes him a couple of minutes (a little more when he was starting out).

His 'real' math lesson obviously takes a lot longer.

Another example. A couple of days ago a Saxon 8/7 lesson taught two different ways of prime factoring a number. I threw out one of them, and substituted the RUSSIAN MATH approach, which I insisted he learn, almost entirely because when I learned it I found it incredibly fun to do. Christopher ended up liking it as much as I did.

Then yesterday, after Drew & Marc taught Christopher how to subtract-a-fraction-with-borrowing, I forced him to sit with me and watch while I subtracted the same fraction without borrowing, ending up with a whole number and a negative fraction. Then I subtracted the negative fraction from the positive whole number and voila. Fraction subtracted without borrowing.

I didn't make him do the subtraction-problem-without-borrowing himself, but only because he was in a MOOD. If he hadn't been in a MOOD, I would have insisted he do one or two such problems.

Now, I wouldn't insist he practice this approach to mastery, because it's Clunky, and forcing a child to practice Clunky Subtraction would be Wrong. IMHO. It's wrong because math isn't clunky, or shouldn't be.

The only reason I'd insist he work a couple of Clunky Subtraction problems is to make sure he really saw that the reason we borrow or regroup is that regrouping is an elegant, mathematically powerful way to do things--NOT because we can't subtract a bigger number from a smaller number! I know for a fact that a lot of kids think the reason you borrow-or-regroup is that you can't subtract a larger number from a smaller. Well, I don't want Christopher thinking that.

(I actually vividly remember the day, just this year, when my neighbor showed me that YES YOU CAN subtract 17 from 25 without borrowing. She's a statistician, and yet even she was puzzled for a moment when I asked her, 'Why do you have to borrow?')

The point is that I'm feeling my way, basing a lot of what I do on my own experience of relearning math, and on what I read in Liping Ma or see in the PRIMARY MATH series. I have no idea whether & when what I'm doing is a good idea, and whether or when it's a waste of time.

Here is Anne on PRIMARY MATHEMATICS:

I have been studying the Singapore math textbooks and workbooks. This is what Dr. Ma says the math teachers in China do.

When a new topic is introduced for the first time, there is an illustration which visually explains the topic. It is very simple and straight forward and ties into all the other illustrations that have been used in the book. There is usually a short English explanation and an equation if appropriate. For example, in 1B on the topic of comparing numbers: the illustration is comparing the number of stamps. The first illustration has 3 stamps. There is a cartoon of a child saying the number 3. The second illustration has 4 stamps, but 3 of them are exactly the same as in the first picture. The exact same cartoon child is saying the number 4.

Then, there are more illustrations but with all different types of things. For example, when learning about ten and ones, sometimes the illustration is bundles of sticks, sometimes blocks of ten etc.

Finally, there is a set of problems by themselves with no illustration.

Then, the workbook has all different exercises for the same type of problems. For example, Daniel is working on equivalent fractions in 3B. There are about 5 different exercises on this subject, some with illustrations to help and some without.

Since topics are always introduced in the same way with the same type of illustrations, you can tie back to what was learned before.

Additionally, word problems are very uniform also. For subtraction word problems for one, two and three digit numbers, there will always be one that uses the words more than, one that will use how many left, and one that will be how many of one type of thing.

Also, Singapore math introduces the first multistep problems in 2A, but only in the textbook.

So in Singapore math, the student is introduced to the concept first by visual illustration and then the procedure. And he has learned to do problems in several different ways right from the beginning. No one asks him to do the same problem in a different way but different exercises in the workbook show him how to do different problems in different ways for the same concept.

As for Everyday Math...well, I've been studying that too for comparison. I won't bore you with my rantings here. I have just one example that I think sums things up:

In the Everyday Math journal that students use in class, there are pages of Math Boxes that are review. In the first semester second grade, there are 120 Math Box pages with 6 problems on each page.

In one particular box, there was a problem to count back by 5s starting with 45. And there were spaces to put in the numbers. Then underneath is said, "Can you keep going?" And had this: 0, , .

Well, of course, my daughter had left this blank. Her teacher filled it in for her with -5 and -10.

What possible sense does it make to throw in negative numbers in a problem in second grade?

But that is Everyday Math.

fraction subtraction without borrowing

I received an email today from Joanne Cobasko of Save Our Children from Mediocre Math (SOCMM). She drew my attention to a couple of articles, describing the improvement in California test scores after the new California standards were adopted.

I looked at the attachment and skimmed the second article. It's not a research study (i.e., it would not meet the WWC's standards of evidence for a well-designed study); but it is definitely one situation where Saxon went head-to-head with fuzzy math, and won.

Here's the letter (thanks, Joanne!):

Hi Carolyn:

Both these studies show fantastic classroom results achieved in CA classrooms which are attributed to Saxon Math. I believe Bishop & Hook down play the Saxon Math connection in favor of the "CA Key standards" so as not to promote any particular curriculum over another, they choose to promote the math standards employed.

You will find references to the curriculum in their write ups though.

http://www.nychold.com/talk-hook-040404.pdf
http://www.nychold.com/report-wbwh-040619.pdf

There is also a great district comparison of standardized test results from Manhattan Beach, CA and Palos Verdes, both well to do communities (the comparison was provided to me by Martha Swartz from Mathematically Correct). [Note: Joanne points out that Manhattan Beach uses Saxon Math, and Palos Verdes uses Everyday Math. -- Carolyn]

Palos Verdes has the edge with a 26% Asian population, and one Kumon or other type tutoring facility for every 429 grade 2 - 6 elementary age student (the tutoring info was my informal review of the school population per the state testing info and a print out from the Kumon & other centers indicating their locations within a 5.22 mi radius).

Manhattan Beach, with a 7% Asian population and only 1 KUMON facility in town for 2,1113 grade 2-6 students, outscores Palos Verdes on the 2004 test scores.

Jo Anne Cobasko
Save Our Children from Mediocre Math (SOCMM).

And I've read enough about other countries' curricula to believe this observation:

"The middle school is the crux of the whole problem and really the point where we begin to lose it," says William H. Schmidt, a professor of education at Michigan State University and the U.S. research coordinator for TIMSS. "In math and science, the middle grades are an intellectual wasteland."

Still, I'm not persuaded middle schools are entirely to blame for the middle school slump, necessarily.

Everyday Math in Schaumburg, IL

I'd been meaning to write about this for awhile now.

I met two retired teachers, a married couple, from Schaumberg, IL at the airport on my first trip to Chicago this summer.

I was working on problems from my Russian Math book, so we got to talking about school & about math, and the wife, who had been a first grade teacher, told me that Schaumberg has been using Everyday Math for 15 years.

They were one of the first districts to try it out, and their students' scores promptly went up by 3 times. So they adopted Everyday Math, and have been using it ever since.

The grade school teachers apparently love E-Math, and the parents don't seem to mind. There was a Schaumberg district mom sitting next to me, who said she couldn't help her daughter with any of her math homework because she didn't understand it. This wasn't a problem; she seemed to think it was natural not to understand anything your 4th grader is doing in math, and not to be able to help with homework. No complaints.

The middle school teachers were another story.

When I asked how the middle school kids were scoring, both grimaced & said, 'Their scores are terrible.'

Then the wife gave me the story on the middle school teachers. 'They don't want to change,' she said. 'They want to keep doing things the same way they've been doing them for 20 years.' Her husband nodded.

They were sure that if the middle school also changed curricula, those students would have high scores, too.

I started to say kids need to know fractions & long division to do algebra, but had to stop when the wife grew visibly alarmed, thrust out both her arms at me hands first, and said emphatically, 'I teach first grade. I don't know anything about that.'


Schaumberg, I learned from my brother-in-law, is the 2nd largest school district in the Chicago area, after Chicago itself.

update

THE STUDENT SHOULD BE THE UNIT OF ANALYSIS!

Tomorrow I'm reading up on Cargo Cults.

update update

parent info night for Carolyn
le rentree
research on middle & elemiddle schools
TIMSS & middle school scores
locker woes & locker instructions
all your children are belong to us
middle school math teacher blogs
Dan K on transition to middle school
Fordham debate on middle school in DC

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ClocksWithoutHands 12 Sep 2005 - 02:02 CarolynJohnston


This just in from Lamprey River, New Hampshire: kids will be learning a new way to tell time this year! This is a news article that parodies itself. From the article:

RAYMOND - Students at Lamprey River Elementary School will learn a new way to tell time this year, thanks to a math program called Everyday Math.

They will be learning to tell time from clocks that have no hands. At least the first time they spiral through telling time.

The research-based and classroom-tested program (which is also recommended by No Child Left Behind) will break with the traditional worksheet-centered approach and embrace a more hands-on strategy.

Except that the hands will actually be off. The clocks. At least to start.

Principal Jane Lacasse says that rather than teaching children basic computation facts, Everyday Math emphasizes concepts and making sure children understand why. The program is centered on a "spiraling curriculum," which means that instead of moving on to a new topic after the old topic has been completed, classroom teachers will keep coming back to topics that had already been studied and expanding on them.

Hence, we'll start in first grade with handless clocks, then add the hour hand, then the minute hand, then the second hand, and in fifth grade we'll take up money, starting with pennies.

For example, Lacasse said, in first grade, children might learn to tell time to the hour, while in second grade, when time-telling is revisited, they will learn to unravel the mystery of the clock to the quarter hour or to the minute.

"The mystery of the clock" used to be taught outright in first grade, didn't it? Never to be spiraled back to again? Why not just ditch the clock completely, send it the way of the slide rule? Telling time is so 20th century.

The school purchased Everyday Math from the McGraw-Hill Publishing Company, investing in new textbooks and workbooks (although assistant principal Dan LeGallo is quick to point out that "this is not a textbook program").

I'll bet they spent real money on those non-textbooks.

The school also had to retrain its personnel.

I'll bet they did! They had to train them to tell time from the handless clocks. Perhaps they told them that there were hands on the clocks that only the most virtuous people could see.

What on earth does it mean to be "recommended by No Child Left Behind"?

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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?

Milgram & Klein links:

• Why Long Division? speech by James Milgram, Stanford University
  
• The Role of Long Division in the K-12 Curriculum
  by David Klein, Department of Mathematics California State University, Northridge and R. James Milgram, Department of Mathematics, Stanford University
  

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

keywords: Columbiajournalismstudent EverdayMatharticle

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WhatIsConstructivism 14 May 2006 - 17:18 CarolynJohnston


AndyJoy asked on this thread: Can someone explain extreme constructivism to me? Is the problem that proponents never want to introduce the standard algorithm for a problem or make children memorize facts?

The short answer is yes, but for the record, here is a fuller explanation. I think the best quick introduction to constructivism and its recent history in U.S. educational practice is Barry Garelick's An A-maze-ing Approach To Math, which appeared in Education Next this year. I'll excerpt a little piece of it to answer Andy's question, entirely without Barry's permission (but hopefully with his blessing).

Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students' not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering (and taking ownership of) the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.

Texts that are based on NCTM's standards focus on concepts and problem solving, but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches "mathematical thinking" rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo without first teaching him to swim.

The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics (PSSM 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.

So how does this fine-sounding idea play out in the classroom? Kids tend to spend too much deriving everything from first principles. What gets sacrificed is time spent learning advanced skills, as Barry shows:

Concept still trumps memorization. Textbooks often make sure students understand what multiplication means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 x 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I've seen 4th and 5th graders, when stumped by a multiplication fact such as 8 x 7, actually sum up 8, 7 times. Constructivists would likely point to a student's going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.

Another case in point was illustrated in an article that appeared last fall in the New York Times. It described a 4th-grade class in Ossining, New York, that used a constructivist approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 x 16 equaled. Using the old-fashioned method (one that is held in disdain because it uses rote memorization and is not discovered by the student) the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.

Now, consider the constructivists' argument for allowing this lack of 'domain knowledge' to persist -- kids develop deeper understanding, 21st century skills, bla bla bla -- after having read KDeRosa's "Terminator essay" on math education.

That essay just puts this nonsense to death, don't you think?

p.s. from Catherine

I found the smart constructivism post.

Here are the 2 best passages.

Smart constructivism says:

A common misconception regarding 'constructivist' theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves. This perspective confuses a theory of pedagogy (teaching) with a theory of knowing. Constructivists assume that all knowledge is constructed from previous knowledge, irrespective of how one is taught (e.g., Cobb, 1940)--even listening to a lecture involves active attempts to construct new knowledge.**

Radical constructivism says:

It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.

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TeacherProtestsEverydayMath 05 Oct 2005 - 20:46 CatherineJohnson

comment left at SOCMM:

I am a third grade teacher and have been trying to tell my administrators that Everyday Math is not an effective math curriculum. I have taught it for three years and the students coming to me have no mastery of basic concepts. It also does not meet our district or state standards, but the administrators will not abandon the curriculum. I struggle with teaching it and then when I started researching the program, I now feel it is my duty to speak out to parents and be an advocate for my students.

Is there a reason why so many states are adopting this mediocre curriculm? Please respond!

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AnimatedLatticeMultiplication 18 Nov 2005 - 15:34 CatherineJohnson

extended problem

What ironclad rule have I just violated?

(I'm betting on Doug for this one.)

time's up

The answer is here.

OK, I should have read the Comments first

As predicted, Doug takes this one:

"To say that the animation was distracting to these users would be an understatement. It was downright irritating. ...

"... During at least one of our tests, while trying to answer a question about the lowest fares to England, an animated ad appeared with the text of "Lowest Fares To London." Not only did the user not click on the ad, he swore he never saw it. Somehow, the user had "masked" out the animation."

The language here is interesting: 'the user had masked out the animation.

My understanding of frontal lobe function—and I'm not entirely confident of this, so take it with a grain of salt—is that it requires energy to block out distractions.

'Ignoring' something is an action.

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LatticeMultiplicationWithCarrying 22 Oct 2005 - 16:31 CatherineJohnson






This explanation at Math Forum is the clearest I've seen. Perfect.


I'm too tired to think it through right now, but I'm not so sure the final point is right.....

'Doctor Mason,' at Math Forum, says:

Multiplication really takes three steps: multiply, carry, add. The method we typically use does the multiply and carry steps together. The lattice method does all three steps separately, so it's really easier! Centuries ago, the Germans had a method for doing all three steps at once. That method takes a lot of concentration!

But I think the method I left in one of the Comments threads also separates the 3 steps.

Doesn't it?


I'm going to keep an eye out for Dr. Mason.

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EverydayMathFramesAndArrows 23 Oct 2005 - 00:00 CatherineJohnson






from Anne Dwyer:

In a frame and arrow problem, the idea is to either use a rule to fill in the blank boxes or to use the data already in the boxes to get the rule. The trick is that two boxes have to be filled in next to each other so that the students can add or subtract to get the rule.

The problem should have looked like this:

9 ---- 16 ---- ? ------30.

The the students could have subtracted 9 from 16 and get the rule +7. Then they could have added 7 to 16 to fill in the next empty box.

By leaving two boxes in a row empty, the problem took away the students only known way of solving the problem. Adding letters to make it more clear:

9 ---- a ----- b ----- 30.

Since the arrows stand for addding a constant, the problem becomes:

9+c = a
a+c = b
b+c = 30

[T]he actual solution required the construction of and solution to three simultaneous equations. The only other way to do it is, you guessed it, guess and check.

Everyday Math to the rescue

Although TIMSS and other studies (e.g. Fuson, Stigler, & Bartsch, 1988) suggest that U.S. school mathematics lags behind Japanese and German curricula by a year or more, EM lessons are generally a year or more ahead of topics in other U.S. programs. Where traditional mathematics programs are slow and repetitive, Everyday Mathematics picks up the instructional pace, increasing both the depth and the breadth of the mathematics taught.

source:
An Analysis of Everyday Mathematics in Light of the Third International Mathematics and Science Study, by William Carroll UCSMP Elementary Component (pdf file)

OK, I have the answer to this one.

Other countries, when they introduce topics earlier than we do, see to it the students actually learn them.

8th grade algebra in New York state

On the face of it, the new Grade 8 content guidelines for New York state require that algebra be taught in the 8th grade. I know, because I asked Caroline to look them over for me last year. Here's what she had to say:

It's a fairly standard algebra 1 course, I think.

I was excited when I read that, because I thought it meant that even if I didn't manage to get Christopher into Phase 4, he'd be taking algebra in 8th grade anyway, because state standards now required algebra in 8th grade.

I was wrong.

The department chair told us that, yes, they would be teaching 'algebraic concepts' to all 8th graders. (I think she used the word concepts. Something like that.)

But none of the regular-track 8th graders would be prepared to pass Regents A, which tests algebra, after 8th grade. Only the Phase 4 kids would be prepared to take the Regents after 8th grade.

What would the regular track kids do in 9th grade, after being exposed to algebraic concepts in 8th grade?

They would take algebra.

That's the plan.

spiraling is a waste of time

The idea that introducing advanced concepts before students are ready is a good thing is ludicrous on the face of it.

Singapore Math doesn't it.

Saxon Math doesn't do it.

Russian Math doesn't do it.

A good curriculum introduces a new topic at precisely the moment a student is ready to learn it, then makes sure the student does learn it.

That's the secret to high TIMSS scores.

Not dabbling in frames & arrows problems that require a 3-simultaneous equation solution in 3rd grade.

bonus quote

Caroline also said, in the same email:

What the heck does item 8.A.5 mean? I have no clue.

Beats me.

8.A.5 Use physical models to perform operations with polynomials

Guess I'm gonna have to find out before Christopher hits 8th grade.

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LatticeMultiplicationAtIllinoisLoop 08 Dec 2005 - 15:06 CatherineJohnson



Becky C reminded me that Illinois Loop has a scathing review of TRAILBLAZERS posted (I've read it at least twice & am due for another go at it).

When I clicked over to the site I found this:

"Yes, New-Math is multiplying, but I am sorry to report that too many children are not learning to multiply with New-Math. ... Multiplication is not all that difficult if one learns the multiplication tables and the logical, precise algorithm for the process. One day I was teaching traditional multiplication when one of the special education students wanted to show me the process she had been taught. Her problem even shocked me, and luckily I had my camera with me.

source:
New Math Multiples by Linda Schrock Taylor



For some reason I've come to love images of lattice multiplication. I'm forming a collection. Any minute now I'll be bugging J.D., Doug, Dan, and perhaps Carolyn, too, to make me one of my very own!

(Just kidding. I do love looking at them, though.)

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IepsForEveryChild 19 May 2006 - 21:47 CatherineJohnson



Rereading Parent Pundit's post about her daughter's experience with Everyday Math and ALEKS, this passage caught my eye:

...they give a pretest and a posttest for the curriculum. In other words, they give the final at the beginning of the year and at the end of the year to track the learning. My daughter received a 25 at the beginning of her 5th grade year in math, but she only received a 69 at the end of the year....

Clearly, intervention was needed. 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.)

I give Parent Pundit's school—and the authors of Everyday Math—credit for the pre- and post-testing.

My problem is: what comes next?

They give this child a pre-test and she scores 29; they give her a post-test and she scores 69.

And then......nothing.

"Clearly intervention was needed."

I'll say.

Why is intervention the parent's responsbility?

The school has failed to teach this child 5th grade math. When she takes the ALEKS test, the program tells her she knows only 21% of a typical 5th grade curriculum. (I'm wondering whether ALEKS allows people just to take the grade-level tests, and if so, how much they charge. I'll check.)

If this child were classified as having special needs, she would be entitled to be taught the content that is listed on her 'IEP,' which stands for Individualized Education Program.

Of course, in my experience the content on the IEPS doesn't get taught, either, but still.....it's there; the parent has a leg to stand on. (And in my own children's case, in fact it's extremely difficult to know what they are and are not able to learn, though I suspect Engelmann would make short work of some of the IEP meetings we've had.)

But with a typical child with normal intelligence, there's no mystery. She can learn 5th grade math in 5th grade. It's the school's job to teach it to her—and to reteach it if they failed the first time around. If that means providing tutoring or summer classes, so be it. It's the school's failure; the school needs to fix it.

This mother was in the same position I was in at the end of 4th grade. My child was failing; the problem was the school's, not his or mine. (In his case the problem was almost certainly the teacher, who I liked very much, but who apparently just could not teach math at that early stage of her career. The school didn't give her tenure, which was the right move. But children who lost a year of math in 4th grade weren't given any help or remediation. No one came to parents of these children and said: Your child failed to learn math this year, because his teacher was inexperienced and didn't manage to teach the subject to mastery. Here's what we're going to do to re-teach the material he missed.

American schools, by and large, teach for coverage.

Not for mastery.


free assessment at ALEKS?

It looks like ALEKS offers a free assessment. (I haven't tried to use it, because I'm not sure I can run the test twice on one computer, and I'm most interested to see where Christopher scores.)

If this assessment really is free, and is easy to use, it could be a useful tool in talking to teachers and administrators.

What we really need is our own simple-to-administer, at-home assessment, 'rolling' assessment tools.

I'd like to be able to send my school a report each month on where Christopher is in the curriculum.

Of course, that's another project.

report cards for the school

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SheddingTearsOverEverydayMath 06 Dec 2005 - 21:02 CatherineJohnson



from Joanne Cobasko of SOCCM:

The email below came from a CVUSD parent. Names and sexual identities have been changed to protect the innocent and guilty.





Another Everyday Math Crying Story - Discovery method strikes out again

Working with my 2nd grader on her math homework - she becomes frustrated because she isn't being taught the algorithms that are needed to solve the problem. She sometimes gets so frustrated she cries. (name of curriculum director) should be fired for installing this crappy math curriculum. The one problem she cried on was:


91 — unknown = 45.


The teacher didn't instruct her to put the numbers in a column and subtract:

  91
 -45
  __
  46

How are the kids supposed to know how to do this without being taught? How is a kid supposed to solve the problem? My daughter's classmate wanted to construct a number grid writing all the numbers between 45 and 91 to try to solve it, her mother said. These poor kids whose parents are not helping them with math at home are going to be lost.

Heresay at (School using Everyday Math) is that the teachers aren't making the kids do the ridiculous algorithms EM teaches.

Supposedly a kid in Mrs B's 5th grade class who was a straight A student in math is now getting D's because Mrs. B isn't explaining the EM method well enough. What a shame these kids have to suffer through EM.

You know the kids will say they like the program because they just play games and don't memorize the math facts.

If you follow the logic of EM, why then should the kids have to memorize spelling words if they can just use spell check on the computer?

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ChangingDeckChairsOnTheTitanic 10 Jan 2006 - 16:21 CatherineJohnson


from Illinois Loop comes word that &mdash:

At one point, Oak Park District 97 used the merely mediocre Scott Foresman Addison Wesley "Math" series.

Then the district jumped into that land of fuzzy math with both feet by adopting Math Trailblazers.

In May 2005, a parent reported to us that D97 "is leaving Trailblazers behind in Fall 2005 to go to Everyday Math for grades 1-6."

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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

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SpirallingStories 07 Feb 2006 - 16:40 CatherineJohnson



I'm pulling parents' experiences together into one post.


Math Trailblazers

A parent here told Ed that in 2nd grade TRAILBLAZERS teaches kids how to construct graphs.

Then, in 3rd grade, TRAILBLAZERS teaches kids how to construct graphs again — the exact same lesson — except that, this time around, they teach the kids TO LABEL THE AXES. (fyi: She wasn't sure what the grade span was; it could have been 3rd to 4th.)



Everyday Math

My cousin describes her experience with Everyday Math:

Chicago Math gives you advanced math problems sprinkled in with the elementary math your child is learning. They slip it in.

They would have you guess at the answers for the advanced problems, but then they never gave you the answers so you didn’t know if you guessed right or not. You’re always a work in progress with Chicago Math. So you never get a definite answer. And you never had a sense of completion or success on a day-to-day basis.

But my pet peeve was that it sped you along at a rapid pace and you never mastered the material that you left the page before. When my daughter was in the 2nd grade one work page would be coins; the next day you’d be dealing with weather; the next day you’d be dealing with problem solving. My daughter had no sense of what a quarter or a dime was.

When I was taught math, each day you built on what you knew. When you did the coins you learned a penny, a nickel, a quarter. You kept going. Telling time, same thing. You work on time until you get it. You don’t just have a flash of it one day.

In Chicago Math you had one page on one topic, then you went on to something completely different on the next page. There was no repetition. It was irresponsible, very ungrounded.

Mike Feinberg of KIPP on spiral curricula
Steve and Susan J on spiral curricula
acceleration versus remediation
parents' stories about spiralling curricula

-- CatherineJohnson - 06 Feb 2006

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RoteKnowledgeInEverydayMath 23 Feb 2006 - 12:06 CatherineJohnson



Great comments on the advice from a top high school student thread. (This was the student whose father countered his son's rote learning of math by having him derive every formula he used.)


from Steve:

Perhaps you don't have to derive everything, but you do have to be able to understand and explain why you can do something using basic definitions and rules. And don't forget mastery. There is linkage.

The biggest fallacy of the latest fad math is that it teaches conceptual understanding. It does no such thing. My son is taking 4th grade Everyday Math, which is supposed to be one of the better fad math curricula. One of its rapid spiral "Home Link" assignments lately was a "Fraction of" sheet with problems like:

4/5 of 25 is _

This is before they know anything about multiplying fractions. How does the teacher explain how to do this problem? You take the whole number (25), divide it by the number on the bottom of the fraction (5) (notice that it is evenly divisible), and then multiply it by the number on top of the fraction. All rote understanding. Perhaps the students have to try and get a Zen-like understanding of four-fifths of 25. But then what do they do with a problem like:

4/5 of 7/8 ?

or

4/5 of 2.3 ?

Another Home Link spiral sheet talked about something like "Part of One" ?!? which is supposed to be the opposite of the example above:

20 is 4/5 of _

Once again, my son was taught a rote procedure for solving this problem.

I am getting really sick of this modern math conceptual understanding rubbish. Can anyone give an example of teaching real mathematical understanding in MathLand, TERC, Everyday Math, CMP, or their cronies? The so-called problem with traditional math was that it was all about drill and kill and no understanding. Well, nowadays, modern fad math does neither.

One of our school committee members told me that her younger daughter (in 6th grade, I think) doesn't have the math skills that her older brothers had at her age, but she has better conceptual understanding because of MathLand and CMP. I honestly don't have a clue what that means.



from Susan:

Exactly. Problems like these show up in Saxon in the form of word problems with the aim being to "see" the numerator and denominator in the form of a bar model, or to practice and clarify unerstanding the role of the numerator (3/8's of the girls had blonde hair, what fraction of the girls did not have blonde hair.) There are several problems like that, but I there is no rote procedure to solve them except in drawing out the vertical bar model to see the segments.

The Saxon version seems to be shooting for conceptual understanding without anyone locking in the procedure of dividing by the denominator, then multiplying by the numerator as the most efficient way of doing it.

Multiplying fractions as a procedure comes quite a bit later. While using the word "of" in previous chapters as another word for "times," this chapter is where they first bring it to learn and practice in a straightforward, rote way. The timing, I think, makes a big difference and is probably less confusing. My son has learned all of this with no bumps in understanding. One piece just fits into the next.

Cancelling is not mentioned at this point. Reducing, at this point, only happens in the answer. I'm dying to just teach this to him, but I'm sure I'd be piling on too much, too soon, and I've learned my lessons the hard way about doing that. A couple of chapters later is the GCF chapter, so I know that's why that next step is postponed a bit.

It's just hard to be an adult and go backwards. I want to show him the easy way when he needs to soak in the new stuff a little at a time.



aside from Catherine:

I am ONE with Susan on this point.

I've mentioned that I worked every single problem in Primary Mathematics Challenging Word Problems Book 3, and that I'm doing every problem in Saxon 8/7 as well.

That's a lot of bar models.

As a direct result, my brain has changed. When I read a fraction problem, bar models pop into my mind's eye unbidden.

I imagine some of you will feel skeptical about this, but for me that's a good thing.

Also, Steve's question — But then what do they do with a problem like: 4/5 of 7/8 ? or 4/5 of 2.3? — has an answer. In Singapore & Saxon the sequence of fraction problems is such that you 'see' that you need a common denominator — that is, you need a bar model divided into 40 segments.

I can't remember whether either Saxon or Singapore asks students to draw bar models of a problem like 4/5 x 7/8 — judging by the fraction problems I just did in KUMON Level F, for god's sake, Singapore may.

Saxon would, I think, use bar models to have kids do a problem like 2/3 x 3/5.

You see from what you've drawn that 2/3 'of' 3/5 is 6/15 which equals 2/5.

The funny thing is that, because I'd learned the multiplication operation with zero conceptual understanding attached (zero conceptual understanding of how the algorithm worked, I mean) I spent quite a long time being befuddled by the computation itself.

I just couldn't 'get' why you multiply the numerators and the denominators. You guys all tried to teach me & I still didn't get it! (I should rustle up those posts. Rudbeckia sent me a wonderful explanation; Dan created a graphic that everyone loved & I was the only person who didn't understand - - - )

Finally, the idea that 'clicked' for me came to me in the car.

This will sound incredibly unschooled & dumb....but tant pis. (French for t**** s***.)

I'd always sort of 'gotten' the idea that you multiply the numerators for the same reason you always multiply; you're finding '2 of 3' or '2 sets of 3' which is six.

But I couldn't put that together with why you multiplied the denominators.

Finally I realized that the denominators are divisors. 2/3 of 3/5 means you are successively dividing 3/5 by 3; you're doing two divisions in a row.

Two divisions in a row means you're dividing by 2 x the factor. (If you divide 12 by 2 and then divide the quotient by 2 again, you're dividing 12 by 2 x 2.)

I don't think this works as a verbal explanation for somebody still trying to figure this out, but it probably makes sense to all of you.

I have NO idea why I was so stumped by this.

I'm guessing I spent too many years overlearning the algorithm without the faintest idea why the algorithm worked. But I don't know.



conceptual understanding without skills

I think I do know what conceptual understanding without skills may be.

I think it would be quite possible to gain conceptual understanding of fraction multiplication — including conceptual understanding of problems like 4/5 x 7/8 — without acquiring procedural fluency in the multiplication algorithm.

It might even be possible to gain conceptual understanding of fraction multiplication with very limited understanding of factors.

I think it's the same observation Ken made a little while ago, after giving his son a Rubik's cube for Christmas.

It's possible to understand the directions for how to solve a Rubik's cube.

Actually solving the Rubik's cube is a different story.



from Kathy:

The biggest fallacy of the latest fad math is that it teaches conceptual understanding. It does no such thing.

Good-I was looking for an excuse to post my latest rant. My daughter is also in 4th grade Everyday Math. Tomorrow is her Unit 6 test, which covers long division, coordinate grids, something called "turns", map coordinates, angle measuring and drawing with a protractor, and word problems where you have to interpret a remainder. And all these topics relate to each other in what way??? With Meg's issues, all this jumping around is very confusing. She has figured out long division, thanks to much practice and tons of supplementation, but all this other stuff is causing much confusion.

Just for "fun" I was able to find the 4th and 5th grade Math texts I used back in the mid-70's and started to do a quick comparison to Everyday Math.

The thing that jumps out immediately is the sheer number of practice problems from my old books. For example, there are over 300 long division problems in the 4th grade 1970's text, just in the division chapter. They return to previously taught concepts in "Keeping Fit" sections which appear at least twice in every chapter.

How many practice division problems in EM's division unit? 20.

Measuring angles with a protractor wasn't introduced until 5th grade, and there's just 1 lesson on it, logically in the Geometry chapter. And decimals didn't appear until the very end of the 5th grade book; in EM they appear in 3rd grade, often through the introduction of problems which the kids are never taught to do.



from Carolyn:

Oh, Kathy, you're bringing it all back to me. 4th grade Everyday Math was the absolute worst, perhaps mostly because it was new and horrible to me, but also because 4th grade is a year when you have to learn and master so many critical things -- fractions, long division, multidigit multiplication. And there is all the jumping around in topics, and never never enough practice, and topics introduced in advance of their being taught.

I have to go lie down now.



a fraction problem from Intensive Practice 3B

Ms. Martinez ordered a pizza. The boys ate 2/5 of the pizza while the girls ate 1/2 of it. One of the boys, Mike, said that all of them ate 3/7 of the whole pizza. Was Mike correct?

3B is second semester, third grade; this problem comes from the 'Take the Challenge' section, so it's considered difficult.

Unfortunately, I sold my copy of 3B, so I can't look to see how kids are taught to solve such problems.

I'd put money on it they draw bar models along with using the addition algorithm, however.


advice from a top high school student
rote knowledge in Everyday Math



-- CatherineJohnson - 20 Feb 2006

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MeapScores 16 Mar 2006 - 14:05 CatherineJohnson



from Anne Dwyer:

We just got back the results for our MEAP tests.

The kids usually take them at the end of the year. This year they took them in Oct.

For math, here are my elementary school results: Grade 3: 89.2 exceed standards Grade 4: 62.9% exceed standards Grade 5: 43.4% exceed standards

This tells me that the kids are good in math until they hit anything above addition and subtraction. Then they hit the wall. The higher the number of skills required, the worse they do.

The school has been using Everyday Math for 8 years. Curriculum director says everything's fine.



point of comparison

At the fourth-grade level, the U.S. did reasonably well on the TIMSS exam. Our students scored above the international average in both math and science. In science, in fact, we came very close to being number one in the world; our fourth-graders were second only to the South Koreans. In mathematics, on the other hand, our performance was only decent; it was above average, though not in the top tier of countries. (Detailed findings, including tables and graphs, can be found on our Web site, http://ustimss.msu.edu, or at the U.S. Department of Education’s TIMSS Web site, http://nces.ed.gov/timss).

By eighth grade, however, the U.S. dropped to the international average, slightly above average in science and slightly below average in mathematics. In other words, just four years along in our educational system, our scores fell to average or even below average. The decline continues so that by the end of secondary school our performance is near the bottom of the international distribution. In both math and science, our typical graduating senior outperformed students in only two other countries: Cyprus and South Africa.

Some people might ask, “What difference does it make if we can’t do fancy math problems?” It does make a difference. A typical item on the TIMSS 12th-grade math test shows a rectangular wrapped present, provides its height, width, and length, as well as the amount of ribbon needed to tie a bow, and asks how much total ribbon would be needed to wrap the present and include a bow. Students simply need to trace logically around the package, adding the separate lengths so as to go around in two directions and then add the length needed for the bow. Only one-third of U.S. graduating seniors can do this problem, however. This is serious.

source:
William Schmidt
A Coherent Curriculum: The Case of Mathematics (pdf file)
AMERICAN EDUCATOR

update: 4th graders aren't great, either

Jo Anne Cobasko left a link to the recent AIR study (pdf file) finding that in fact our 4th graders aren't doing especially well, either.

Despite a widely held belief that U.S. students do well in mathematics in grade school but decline precipitously in high school, a new study comparing the math skills of students in industrialized nations finds that U.S. students in 4th and 8th grade perform consistently below most of their peers around the world and continue that trend into high school.

Thanks, Jo Anne —



-- CatherineJohnson - 15 Mar 2006

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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

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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 something different.

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
1st grade division word problems

2nd grade worked bar model, Primary Mathematics
2nd grade word problems





update from Mark Roulo

I'll add a personal anecdote.

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

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EverydayMathInNewYork 01 May 2006 - 23:08 CatherineJohnson

To see the defects of Everyday Mathematics, one need only examine its treatment of paper-and-pencil subtraction of two multi-digit numbers.

Most adults will have learned to write the smaller number below the larger one, lined up at the right, and write down the result of the subtraction right to left, doing whatever "borrowing" is needed mentally. This is not taught in Everyday Mathematics. Instead, the Everyday Mathematics pupil is exposed to five different subtraction methods, each of them viewed as suitable for the same task. The gymnastics employed to avoid simple methods is truly breathtaking.

There is in Everyday Mathematics a "trade first" variant of the traditional method: Borrow first in all columns where it is needed, recording the intermediate results, and then do the subtractions. There is a "counting up method": count up from the smaller to the larger number, first by ones, then tens, and so on, and then the odd remainder, and then in a second pass, add up the addends. (Example: If we do 425 - 48 then the second stage involves adding up 2 + 50 + 300 + 25 to obtain 377.) The third standard method is left to right subtraction, the way one might well do the problem mentally, but carried out with paper and pencil. The fourth approach is a "partial differences method": subtract in each column separately, keeping track of the sign if a borrow would be needed, and then combine the results by mental arithmetic. Finally there is the "same change rule": change both numbers by the same amount so that the smaller number ends in one or more zeroes and the problem is easier. Addition, multiplication, and division likewise have multiple standard methods in Everyday Mathematics, and in all of this, true fluency in the basic operations appears not to be an aim of the program.

One can well imagine how a pupil who already has excellent mastery of arithmetic can enjoy seeing and understanding how the multiple methods of Everyday Mathematics all lead to the same correct result. The danger of this profusion of methods for pupils who are not so comfortable with the basics is also easily imagined. These pupils, and some teachers and parents as well, will be hopelessly confused. Combine that with the easy tolerance of calculators in Everyday Mathematics and one can foresee that entire classrooms will throw up their hands and rely on the calculator for arithmetic, never to achieve the facility with number and operation that they'll need to advance beyond the grade school level.

Everyday Mathematics combines the very defective treatment of basic arithmetic with some quite sophisticated content elsewhere, resulting in a strange mixed bag that ought never have been selected for city-wide use in the elementary schools. Mr. Klein would do well to reverse himself and listen to the advice about successful curricula that mathematicians and others have provided to him and his staff. As one example, I would suggest that the chancellor look hard at the documented results of the Saxon Math curriculum. He can start by perusing the input provided by New York City HOLD to the Chair of the Children First Numeracy Working Group and available through the NYC HOLD Web pages.

source:
NY Sun editorial
NYC HOLD

Braams' observation that seeing other methods of subtraction would be illuminating for people who've mastered the standard algorithms is certainly true of me. I found all that stuff fun a year or so ago, when I was seeing it for the first time.

Though I can't say I was amused by lattice multiplication.





false dichotomy? for some kids, but not others?

In theory, MATH TRAILBLAZERS is designed to tilt back towards fluency in the math facts, and, I think, in the standard algorithms.

The curriculum also assumes that all children will gain fluency incidentally, rather than through formal practice or drill and kill:

• Early emphasis on problem solving. We believe that children must indeed learn their math facts, but we de-emphasize rote memorization and the frequent administration of timed tests. Both of these can produce undesirable results. Instead, our primary goal is that students learn that they can find answers using strategies they understand.

• Ongoing practice. Work on the math facts is distributed throughout the curriculum, especially in the Daily Practice and Problems and in the games. This practice for facility, however, takes place only after students have a conceptual understanding of the operations and have achieved proficiency with strategies for solving basic fact problems. Delaying practice in this way means that less practice is required for facility with the number facts.

• Gradual and systematic introduction of facts. Students study the facts in small groups that can be solved by a single strategy. Early on, for example, they study facts that can be solved by counting on 1, 2, or 3. Students first work on simple strategies for easy facts, and then progress to more sophisticated strategies and harder facts.

Conceptual understanding first, procedural knowledge second. This is Founding Law in MATH TRAILBLAZERS. It doesn't make a lot of sense to me, speaking as a Naive Relearner of math. Neither does the idea that one would always teach procedures first, and concepts second, although I think this would be different if I were teaching a high-end autistic child....and I wouldn't be surprised to learn that other children do best learning and practicing procedures first....

I don't know.

Now that I'm well into my third Saxon book, I think it's fair to say that Saxon joins the conceptual with the procedural wherever possible. In Saxon 6/5 (5th grade), Saxon even tries to teach the division of fractions conceptually as well as procedurally. Speaking for myself & Christopher, I'd have to admit that the book fails miserably, but they give it a good go.




[note: this is an animated gif.
To see the motion you have to
release the scroll bar.]


-- CatherineJohnson - 01 May 2006

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MissingTopicsAndSkillsInEverydayMath 26 May 2006 - 13:17 CatherineJohnson



golden oldies


Review of the Everyday Mathematics Curriculum and Its Missing Topics and Skills (pdf file)


-- CatherineJohnson - 14 May 2006

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HowToGetParentBuyInPart2 27 May 2006 - 02:30 CatherineJohnson










source:
Getting Your Math Message Out to Parents








how to get parent buy in, part 1

newsletter excerpt
Getting Your Math Message Out to Parents (pdf file)



-- CatherineJohnson - 26 May 2006

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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

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NctmReformsAgain 14 Sep 2006 - 16:52 CatherineJohnson




In today's Wall Street Journal ($):

Arithmetic Problem
New Report Urges Return to Basics In Teaching Math
Critics of 'Fuzzy' Methods Cheer Educators' Findings;
Drills Without Calculators Taking Cues From Singapore
By JOHN HECHINGER
September 12, 2006; Page A1

The nation's math teachers, on the front lines of a 17-year curriculum war, are getting some new marching orders: Make sure students learn the basics.

In a report to be released today, the National Council of Teachers of Mathematics, which represents 100,000 educators from prekindergarten through college, will give ammunition to traditionalists who believe schools should focus heavily and early on teaching such fundamentals as multiplication tables and long division.

The council's advice is striking because in 1989 it touched off the so-called math wars by promoting open-ended problem solving over drilling. Back then, it recommended that students as young as those in kindergarten use calculators in class.

Those recommendations horrified many educators, especially college math professors alarmed by a rising tide of freshmen needing remediation. The council's 1989 report influenced textbooks and led to what are commonly called "reform math" programs, which are used in school systems across the country.

The new approach puzzled many parents. For example, to solve a basic division problem, 120 divided by 40, students might cross off groups of circles to "discover" that the answer was three.

Infuriated parents dubbed it "fuzzy math" and launched a countermovement. The council says its earlier views had been widely misunderstood and were never intended to excuse students from learning multiplication tables and other fundamentals.

Nevertheless, the council's new guidelines constitute "a remarkable reversal, and it's about time," says Ralph Raimi, a University of Rochester math professor.

Francis Fennell, the council's president, says the latest guidelines move closer to the curriculum of Asian countries such as Singapore, whose students tend to perform better on international tests.

So maybe it wasn't such a great idea after all for IUFSD to ban my Singapore Math course.



new timeline

According to their report, "Curriculum Focal Points," which is subtitled "A Quest for Coherence," students, by second grade, should "develop quick recall of basic addition facts and related subtraction facts." By fourth grade, the report says, students should be fluent with "multiplication and division facts" and should start working with decimals and fractions. By fifth, they should know the "standard algorithm" for division -- in other words, long division -- and should start adding and subtracting decimals and fractions. By sixth grade, students should be moving on to multiplication and division of fractions and decimals. By seventh and eighth grades, they should use algebra to solve linear equations.

Here's the Singapore sequence.




Lutherans turning into Catholics

A recent study by the Thomas B. Fordham Foundation, a Washington nonprofit group, found that only two dozen states specified that students needed to know the multiplication tables. Many allowed calculators in early grades.

Chester E. Finn Jr., the foundation's president and a former top official at the U.S. Department of Education, blamed the earlier math-council guidelines for state standards that neglect the basics. He described the new advice as a "sea change," saying that "it's a little bit like Lutherans deciding to become Catholics after the Reformation."

Understanding math, rather than parroting answers to poorly understood equations, was the goal of the council's controversial 1989 standards. Those guidelines called on teachers to promote estimation, rather than precise answers. For example, an elementary-school student tackling the problem 4,783 divided by 13 should instead divide 4,800 by 12 to arrive at "about 400," the 1989 report said. The council said this approach would enable children using calculators to "decide whether the correct keys were pressed and whether the calculator result is reasonable."

"The calculator renders obsolete much of the complex pencil-and-paper proficiency traditionally emphasized in mathematics courses," the council said then. In 2000, in another report, the council backed away somewhat from that position.

Still, in response to the earlier recommendations, many school systems required children to describe in writing the reasoning behind their answers. Some parents complained that students ended up writing about math, rather than doing it.

As the debate heated up, concern grew about U.S. students' math competence. In 2003, Trends in International Mathematics and Science Study, a test that compares student achievement in many countries, ranked U.S. students just 15th in eighth-grade math skills, behind both Australia and the Slovak Republic. Singapore ranked No. 1, followed by South Korea and Hong Kong. Fueling concern about the quality of elementary and high-school instruction: one in five U.S. college freshmen now need a remedial math course, according to the National Science Board.

low-income students

This is very exciting. The AIR report (pdf file) led me to believe that Singapore Math had been a flop in low-income schools because the student mobility is so high (and see Hirsch on this subject, too):

If school systems adopt the math council's new approach, their classes might resemble those at Garfield Elementary School in Revere, Mass., just north of Boston. Three-quarters of Garfield's students receive free and reduced lunches, and many are the children of recent immigrants from such countries as Brazil, Cambodia and El Salvador.

Three years ago, Garfield started using Singapore Math, a curriculum modeled on that country's official program and now used in about 300 school systems in the U.S. Many school systems and parents regard Singapore Math as an antidote for "reform math" programs that arose from the math council's earlier recommendations.

According to preliminary results, the percentage of Garfield students failing the math portion of the fourth-grade state achievement test last year fell to 7% from 23% in 2005. Those rated advanced or proficient rose to 43% from 40%.

Last week, a fourth-grade class at Garfield opened its lesson with Singapore's "mental math," a 10-minute warm-up requiring students to recall facts and solve computation questions without pencil and paper. "In your heads, take the denominator of the fraction three-quarters, take the next odd number that follows that number. Add to that number, the number of ounces in a cup. What is nine less than that number?" asked teacher Janis Halloran. A sea of hands shot up. (The answer: four.)

Ms. Halloran then moved on to simple pencil-and-paper algebra problems. "The sum of two numbers is 63," one problem reads. "The smaller number is half the bigger number. What is the smaller number? What is the bigger number?" (The answers: 21 and 42.)

In this class, the students didn't use the lettered variables that are so prevalent in standard algebraic equations. Instead, they arrived at answers using Cuisenaire rods, sticks of varying colors and lengths that they manipulate into patterns on the tops of their desks. The children use the rods to learn about the relationship between multiplication and geometry. The goal: a visceral and deep understanding of math concepts.

"It just makes everything easier for you," says fifth-grader Jailene Paz, 10 years old.

Cuisinaire rods for bar models!

That's so cool!




TERC time

The Singapore Math curriculum differs sharply from reform math programs, which often ask students to "discover" on their own the way to perform multiplication and division and other operations, and have come to be known as "constructivist" math.

One reform math program, "Investigations in Number, Data and Space," is used in 800 school systems and has become a lightning rod for critics. TERC, a Cambridge, Mass., nonprofit organization, developed that program, and Pearson Scott Foresman, a unit of Pearson PLC, London, distributes it to schools.

parents don't get it part 1

Ken Mayer, a spokesman for TERC, says many parents have a "misconception" that Investigations doesn't value computation. He says many school systems, such as Boston's, have seen gains in test scores using the program. "Fluency with number facts is critical," he says.

parents don't get it part 2

Polle Zellweger and her husband, Jock Mackinlay, both computer scientists, moved to Bellevue, Wash., from Palo Alto, Calif., two years ago so their two children could attend its highly regarded public schools. She and her husband grew suspicious of the school's Investigations program. This summer, they had both children take a California grade-level achievement test, and both answered only about 70% of the questions correctly. Ms. Zellweger and her husband started tutoring their children an hour a day to catch up.

"It was a really weird feeling," says their daughter, Molly Mackinlay, 15. "I do really well in school. I am getting A-pluses in math classes. Then, I take a math test from a different state, and I'm not able to finish half the questions."

Eric McDowell, who oversees Bellevue's math curriculum, says parents misunderstand Investigations.

If it weren't for the parents, teaching would be a great job.




math wars and war wars

In the Alpine School District in Utah, parent Oak Norton, an accountant, has gathered petitions from 1,000 families to protest the use of Investigations. His complaints began more than two years ago, when he discovered at a parent conference that his oldest child, then in third grade, wasn't being taught the multiplication tables.

Barry Graff, a top Alpine school administrator, says the system has added more traditional computation exercises. Over the next year, Alpine plans to give each school a choice between Investigations or a more conventional approach. Mr. Graff, who says Alpine test scores tend to be at or above state averages, expects critics to keep up the attacks and welcomes the national math council's efforts to provide grade-by-grade guidance on what children should learn.

"Other than the war in Iraq, I don't think there's anything more controversial to bring up than math," he says. "The debate will drive us eventually to be in the right place."

wow

I bet things are hopping over at math-teach & math-learn.

[pause]

hmm

No action thus far.

Once Wayne Bishop posts this baby, we'll be in a shooting war.

update: Bishop's got it!

let the fun begin



what Singapore students can do at the end of 7th grade

-- CatherineJohnson - 12 Sep 2006

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LatticeMultiplicationInChicagoTribune 23 Sep 2006 - 21:46 CatherineJohnson









This graphic accompanies an article Susan S linked to: Latest `new math' idea gets back to the basics.

I'm not crazy about the opening, though it could be worse:

On one side sit fundamentalists, who prefer old-fashioned drilling and a focus on the basics. On the other side are the so-called "new math" proponents, who care more about understanding the concepts than performing the calculations.

The idea that "basics" and "comprehension" are two separate things, as opposed to two stages of the same thing, never goes well for those of us who've read some cognitive science.

This passage is very helpful, however:

One Downstate high school math teacher stood up Wednesday after [NCTM Executive Director] Fennell's presentation and complained that the widespread use of "new math" and a reliance on calculators has resulted in his students not knowing how to perform advanced math skills.

The man declined to give his name, saying he feared reprisal.

"I've seen abandonment of quick recall and that means kids arrive in my class and I have to backtrack and teach them the basics," he told his fellow teachers. "I hope [this document] addresses that and convinces more people to go back to the old way of doing things."

I love it that you've got the NCTM telling everyone there isn't any math war at the same time you've got high school math teachers standing up at NCTM conventions & saying they fear reprisals.

Nope, no math war here. Just a friendly reprisal or two.

-- CatherineJohnson - 21 Sep 2006

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NationalMathAdvisoryPanelLinks 21 Nov 2006 - 18:07 CatherineJohnson




meetings

• First Meeting (pdf file)

• Second Meeting (pdf file)

• Third Meeting (pdf file)
  

email updates

about the panel

homepage




where you can find links

I'm posting links to the Math Panel homepage, transcripts, & ktm posts here:

• recommended reading page
  

• book-style index
  

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.


panel members w/links
Polite agreement or something we can use?
National Math Panel announcement
National Math Panel update
short story by Vern Williams

nationalmathematicsadvisorypanel


-- CatherineJohnson - 07 Nov 2006

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ReTeachers 04 Dec 2006 - 15:08 CatherineJohnson




from NYC HOLD:

Letter #1

I am a physician who was initially [a] mathematics major in college. I just found your Web site today and wish I had known about it 6 years ago when my oldest daughter began kindergarten in District 2. It was not until third grade that I realized just how little math she was learning, and how behind she was in basic skills. According to her teachers, everything was fine, but then no testing or assessment was done, other than the state wide tests - and I recently discovered that our teachers do not even get access to their student's individual results!

Needless to say we've been struggling ever since - we tried Kumon, had a private tutor, and have used the McGraw Hill Math series workbooks as well -. Unfortunately, attempts other than tutoring, such as workbooks don't teach -they just provide for practice of taught skills.

My daughter is a bright child, but math is not "natural" for her. She has never done well "inventing" her own ways to do problems, and I have been stymied as to why the school was not teaching her how to add, subtract, multiply and divide, etc. Once taught, and after sufficient practice, she gets it. I was never asked to invent math - I was taught the ways that math scholars developed over the centuries - why are we asking our children to reinvent the wheel? Our best year of math at PS --- was 5th grade, when all the teachers were new to NYC, all three having come from teaching in the Midwest. Suddenly there were worksheets coming home and quizzes in class, and my daughter had a sense of what she was expected to know, and I did too. All too late as far as I am concerned. Now in sixth grade we are still catching up I have now ordered the Saxon series to aid me in teaching both her and her 2nd grade sister (who unlike her older sister gets math almost intuitively, so it will be much easier going)

At any rate your mission statement summarizes everything I have been saying to friends and other parents at school for the past 4 years. I too, am convinced that our school has for far too long been taking credit for the extra work that the parents are doing in math - this is why our children are doing well, not because of the curriculum!

Christopher tells me he is to bring in $11 for a state test prep booklet.

ELA? I asked. The ELA test is coming up in January.

No. Not the ELA.

Math.

Christopher is to take $11 to school to purchase a state test prep booklet for math.

uh-oh

Houston we have a problem.

I don't feel like spending $11 on a state test prep booklet.

Why don't I feel like spending $11 on a state test prep booklet, you ask.

I don't feel like spending $11 on a state test prep booklet because:

• in preparation for the state test last year class spent 3 days writing about failure and disappointment in their math journals

• resulting in: class using only a couple of pages in the state test prep booklet

• resulting in: parent forced to assign bulk of pages in the state test prep booklet

• resulting in: screaming, yelling, crying

• also resulting in: parent having to work bulk of problems in the state test prep booklet herself, because answers to state test prep booklet were supplied only to teachers, not to parents
  

In conclusion, I don't feel like spending $11 on a Top Secret state test prep booklet because if I'm going to do the school's job I want the school's materials — all of the school's materials, not just the materials the school sees fit to share with me.

Actually, even if the school did see fit to make my reteaching life a tad easier, I'm not sure I'd want to shell out $11 for a state test prep booklet.

Last year's actual grade 7 math test is here.

The sample tests teachers use for training on the scoring rubric are here. (I believe that's what these tests are.)

So we've got two sets of actual New York state math assessments with answers available free online.

Why spend $11 for a booklet?

Wait!

Don't tell me!

I have the answer to that!

If the school doesn't have to pay for it out of the school budget, it's not really money.

Table of Contents NY math standards
New York State standards: grade C
sample problems, NY state test
"vacation"
impending doom
math journal, day 2
math journal, day 3
thank you, Saxon Math
don't study for the test
state test

math standards- intermediate (pdf file)

State Assessment
State Assessment mathemathics
sample state tests 2005
State Tests 2006



-- CatherineJohnson - 30 Nov 2006

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LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson




I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006

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