Entries from ParentsTeachingKids

But when you're talking about young kids, at least, all kids hate it, no matter what their skills. I myself find it quite uncomfortable, once I've understood and solved a problem, to go back and start all over again trying to solve the problem a whole different way--or to work through and understand a different solution offered by the book.

My neighbor and I spent some time trying to figure out why this should be so. Why is doing-it-a-different-way so unpleasant? I think this passage from W.W. Sawyer's Prelude to Mathematics explains it:



I think it's unnatural to throw out the pattern you've just discovered & go off to find a whole new one. It goes against the grain.

I think that's the source of the aversion children, 'weak students,' and adult students like me feel to doing it.

it's not the same thing for us

Around the blogosphere I see an awful lot of complaining about weak students and lazy students and recalcitrant students and students who think their learning style matters.

I'm sympathetic, to a point.

That point is here, asking myself why a 'weak student' would not enjoy having his teacher tell him to 'do it another way.'

I strongly suspect that the practice of doing a problem 'another way'--or simply perceiving that two or more different ways of doing a problem exist--is a different thing for the expert than it is for the novice.

I say this because of my own experience. Slowly, I'm turning myself into what is called, I believe, a talented novice. (I'll check the phrase.)

A talented novice is neither fish nor fowl, neither expert nor beginner.

For that very reason, a talented novice can be a terrific teacher.

As I move into talented novice territory, I find that 'doing a problem more than one way' is becoming a whole different thing. It's starting to be fun. It means making connections--extending the pattern--rather than throwing the pattern out and starting again.

back to school with Steven Pinker

Learning math is hard. Children do not spontaneously see a string of beads as elements in a set, or points on a line as numbers. If you give them a bunch of blocks and tell them to do something together, they will exercise their intuitive psychology for all they're worth, but not necessarily their intuitive sense of number. (The better curricula explicitly point out connections across ways of knowing. Children might be told to do every arithmetic problem three different ways: by counting, by drawing diagrams, and by moving segments along a number line.) And without practice that compiles a halting sequence of steps into a mental reflex, a learner will always be building mathematical structures out of the tiniest nuts and bolts, like the watchmaker who never made subassemblies and had to start from scratch every time he put down a watch to answer the phone.

Yes, he's going to have to do a problem he's already solved all over again, solving it a different way.

BUT I'm asking him to use the same methods he used yesterday & the day before. It's not chaos; there's a predictable pattern to the two or three different ways he's going to solve the problem.

I'm trying to give him a stable structure he can hold onto while he does (and I do) the hard work of learning math.

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.

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AnneDwyerOnSingaporeMath 27 Aug 2005 - 00:06 CatherineJohnson


You might want to check out the discussion on teaching things more than one way.

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

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SusanOnMadMinutes 13 Nov 2005 - 18:34 CatherineJohnson


Susan used Mad Minutes with both kids--

I agree about the "mad minute" approach for all levels of math ability. I used it with both kids and I'm glad I did. It just helps with the speed and proficiency.

Gifted kids are notorious for not wanting to memorize anything. It's too boring. I had to make mine do speed drills all through first and second grade. I'm still doing it with my LD 8th grader.

I remember when math kid was learning multiplication in the first grade. He informed me that it was stupid to memorize them because he could always just count the the groups. I said, "Quick, what's 7 X 8?" His eyes went up looking for those groups, and after a couple of seconds of his looking for them and then trying to skip count by 7, I said, "That's why you memorize them."

I think it helps in the confidence department, as well, to get that speed going earlier than if it just came through practice. And if practice is the key, like "exposure" to lots of words was to the whole language crowd, then what exactly happens to the kids who don't get enough of that practice? Where is the place where enough practice crystalizes into math fact proficiency, especially with these "spiraling" curriculums that keep pushing mastery on down the road?

Trailblazers, as well as other NCTM curriculums, never seem to have a plan for the ones left behind. And I guess you can't ever know who is accountable since mastery was never the goal to begin with.

Interestingly, both kids do love the minute drills where they try to beat their last best score, so I never have any trouble giving them to them.

Well, that's two.

I have to say....I simply can't see any reason on the planet why worksheets would be bad (though TRAILBLAZERS explicitly states that worksheets are destructive!)

Given that there's no (apparent) downside, and given that they've worked for other kids, I'm certainly going to be using them with any child whose math education I'm involved in. (I'm starting to pick up a few! It's incredibly fun. More on that later.)

modifying worksheets

That reminds me.

One of the kids who took my Singapore Math class just could not get through a Saxon work sheet--and he didn't improve any over time, either. All the other kids got fast fast. (It really was remarkable.)

This particular boy is BOUNCY; he is one high-energy kid. He just can't stand the thought of a 5-minute worksheet; he probably takes one look at those sheets and sees what I would see if I were contemplating singlehandedly painting a two-story house.

I told his mom: try having him do ONE LINE of the worksheet as fast as he can.

I don't know if she'll get to it, but when they get back from Italy in January, I think I'll experiment with him, and see how he does. Just click the stop watch and tell him to call 'TIME!' when he hits the end of one line.

I bet that will work.

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FromAroundTheEdusphere 06 Sep 2005 - 04:32 CarolynJohnston


Here and there in the "edusphere" I've seen mention of Professor Plum. He's a fellow educational radical (as I've grown to think of people who favor actual instruction in the classroom), and today I checked out his website.

I learned, among other things, that Direct Instruction actually refers to a very specific method of instruction, and to a commercially available set of curricula. It's not just what happens when I Directly Instruct Ben on how to do a math problem, as I had thought. Professor Plum has a lot of material on it here, if you're curious.

But on a quick perusal, I wasn't attracted to Direct Instruction. I couldn't find what I thought was a sufficiently clear description of what Direct Instruction is about. I learned that it is scripted interaction between teachers and children, and that a great deal of teacher training is needed to implement it properly -- all of which statements I've also seen recently in the Connected Math context. I'd like to see more beef, up front and center.

One of Professor Plum's links also took me here, to a site for parents on how to develop contracts for children that help them achieve academic success. I really like this guy's ideas, which are built around a principle I've been using to good effect around here since Ben was a little toddler, namely bribery. It's not really bribery, of course; it's merely setting up a system of targeted incentives intentionally, rather than accidentally setting up the wrong ones haphazardly. There are lots of good suggestions and examples on this website; a lot of detail of the sort that makes you braver about actually implementing his suggestions.

I also did very much like a recent post of Professor Plum's, entitled Basic Features of Effective Instruction. This post is a gem; it summarizes the features of effective teaching very well, I think (I'd love to know whether KTM teachers agree with me on that!). While reading it, it struck me that I hadn't seen teaching methods of any sort described with such clarity since Ben was very young, and I was working with Applied Behavioral Analysts to implement the Lovaas curriculum, which is designed to treat young autism-spectrum children. There is no tougher customer to teach than a very young autistic child; they are extremely disinclined to pay the teacher any attention at all, and they are often not motivated by the things that motivate typical children (like praise and attention). A teacher can't mess around; her message has to be crystal clear, and her incentives have to be right on. Many of the principles he outlines here are typical-kid versions of those one uses in Applied Behavioral Analysis, to decrease confusion and ineffectiveness (and no surprise either, since he has worked with autism spectrum kids in his career). A terrific post.

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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.

That decline has already had an impact on U.S. universities.

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

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InnumeracyPart2 13 Nov 2005 - 19:59 CatherineJohnson


A section of the innumeracy article Carolyn linked to caught my eye:

Wieman says getting students comfortable with math as a way of describing the natural world is a nut he has had trouble cracking. He said methods such as those developed by his Physics Education Technology program can give students without science backgrounds a deep understanding of scientific concepts, "yet when something involves a simple arithmetic calculation, their brains click into this totally different mode."

Steven Pollack, a CU physics professor whose research focus is improving the education of physicists, says the problem also happens in reverse. Physics students often can't conceptualize or explain the results of the equations they so breezily manipulate.

This is something I've been wondering about.

This may sound crazy, but, as a kid, I was reasonably good at math. I got straight A's, I had no trouble learning whatever I was supposed to learn (my one bad moment, in 2nd grade, WHICH I REMEMBER TO THIS DAY, involved--guess what?--fractions).

I took my SATs cold, with no practice, a year after I'd looked at my last math book and got a 620, which put me way up in the top percentile of the nation's 17 year olds at the time. (IIRC, I may have been in the top 95th percentile for girls.)

So....I was reasonably good at math.

That's why when Christopher came home with his 39 on Unit 6 it never crossed my mind I couldn't simply sit down and teach him what he'd missed.

no can do

You all know the end of that story. I discovered I knew practically NOTHING about math.....which is an exaggeration, but is sure the way I felt. I've been obsessively re-teaching myself elementary mathematics ever since, and intend to go on to trig & calculus & and a bit beyond.

So what does it mean to say that I was 'reasonably good' at math?

It means I could set up two-variable algebra problems and solve them in a jif. Thirty years later, I could still do it. Easily.

But I had no idea why setting up two equations (or 3 or 4) worked.

This is why I'm such a fan of the Singapore Math bar models (one of the reasons); it was a bar model that first explained to me what subtraction really meant.

the difference between two numbers

I had a Helen-Keller-at-the-water-pump moment the first time I drew this bar model. I had simply never noticed that the 'number' of boys and girls in Mrs. Johnston's class, up to the number 10, is the same number. The 'extra' five boys are the difference.

For my entire life I had heard the word 'difference' used to name the number you end up with when you subtract one number from another, but I had never, ever realized that 'difference' actually did mean difference. It wasn't just some random word that had gotten attached to the operation somewhere back in the mists of time.

I now point this out to any kids I teach--and they all seem to find it extremely cool, too. I say, and then I repeat frequently, Subtraction is finding the DIFFERENCE between two numbers.

Then I point out that, if you're subtracting 3 from 5, 3 and 5 are the same number until you get past the 3-that-is-inside-the-5.

quick question re: number partition theory

The article on Everyday Math that I linked to yesterday, Weighing the Factors says this is number partition theory.

Is it?

odd man out

Teaching the how-many-boys-and-girls problem to kids, I also point out what would happen in Mrs. Johnston's class if you paired each girl with a boy.

You would have five boys left over.

That seems to make enormous sense to grade school kids, perhaps because they spend their grade school years being assigned partners or buddies to walk in lines with, or go to the bathroom with, etc.

This is obviously the way to teach the concept of even and odd, too. If you have an odd number of kids, there's going to be one child left over when you assign them to teams.

AND this image works great for teaching the idea that you always get an even sum if you add two even numbers or two odd numbers, but you get an odd sum if you add an even & an odd. In my experience so far, kids can instantly see that, when you add two odd numbers, you get two 'odd men out'--and now those two finally have a partner!

why don't numbers & concepts connect more easily?

This brings me back to my original point: somehow, you can learn numerical manipulations, including more advanced numerical manipulations that require you to set up equations and solve them, and not have a clue what it all means.

I don't understand this.

I don't understand how I could have so much fun setting up equations & solving them, and never gain the slightest idea why what I was doing worked.


keywords: conceptual understanding & bar model difference between two numbers comparison of numbers subtraction as comparison subtraction has two meanings


partial product division in Everyday Math
fighting innumeracy at CO
subtraction as the difference between 2 numbers
study sheet: subtracting integers & absolute value
notes on integer, subtraction, & absolute value study sheet

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WilliamKSmithCalculus 16 Sep 2005 - 12:16 CatherineJohnson


Here's another recommendation from Barry Garelick:

Calculus with Analytic Geometry by William K. Smith (also available at Amazon)

I've already ordered my copy.

Have I mentioned I'm planning to take calculus?

Well, I am. I'm planning to take calculus.

But first I have to 're-take' algebra & geometry. Then trig, which I've never studied.

You folks here at ktm are helping me so much. Even though I'm a writer, I can't locate the words to describe what you've given me. The reason I can't 'locate the words,' of course, is that I don't actually know what I'm learning from ktm. I study & absorb what people say, but then forget the source of my new knowledge once it's been assimilated into my store of old knowledge. I'm left with the hazy feeling that 'I'm learning a huge amount from the Commenters at ktm.'

So I'm going to start taking notes. God is in the details.

thank you!

integers! integers!

So Christopher's math class started integers on Monday—a topic he knows virtually nothing about—and he's having a test tomorrow. He is way not prepared, so I'm busy today writing an Integer Lesson. Probably won't be posting much (though I may have a couple of things from Barry.)

I'm taking a moment to make one more plug for Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, though.

I could probably add & subtract integers in my sleep. (Though I did have to do some review last year when I first re-encountered the topic, which I take as a sign that my knowledge was more procedural than conceptual.)

But last night, after working with Christopher for awhile, who was semi-lost (I don't think he could pass a test at this point) the Math Fog rolled in.

This is the good thing about working with people who know less math than you do. Concepts and procedures you thought you understood turn out to be not quite so clear. I assume that's what Bernie meant when he said the other day that he'd realized there were aspects of reciprocals he hadn't thought about (if I've got that wrong, Bernie, I'll change it!) Carolyn has said something similar at times. I'll be asking her about some elementary concept that, for her, is as simple as breathing in and out, and suddenly she'll see why Ben--or anyone else--might get confused.

lost in translation

This is another one of those constructivist insights that's been lost in translation.

For me, and I think for most teachers & writers, teaching or writing about a subject always forces you to understand it far better than you did.

Radical constructivists conclude from this that children should explain all of their answers in words.

I'm pretty sure that's wrong, because math is not language. Math is math. A child who can explain his answer by showing the mathematical steps he took to find it has produced a proper mathematical explanation as far as I'm concerned. (Russian Math & the Chinese teachers in Liping Ma all offer mathematical explanations & demonstrations.)

But what really bothers me about the 'explain your answer in words' business is that it puts the onus on the child to teach himself. The teacher doesn't have to work and fight and struggle to find the right words; the child does. I know that's wrong.

While I'm on the subject, why don't I just go ahead and take umbrage at the suggestion that a child is capable of explaining math in words?

Writing is hard. Writing well is extremely hard. Finding the words to explain any mathematical concept well is a vast and ambitious undertaking in itself, not a toss-off in the middle of a homework assignment or state assessment. (I'm seriously against the extended response (pdf file)requirement that's taken over IL state rubrics. At least, for the time being I am. [update 5-14-06 sorry, link no longer works])

back to Russian Math

I shouldn't be putting words in people's mouths, so if I've misunderstood Bernie or Carolyn I'll issue a CORRECTION.

In the meantime, why don't I just return to quoting myself.

It's true for me that when I work with a child for awhile, I realize I don't understand things as well as I thought (or hoped).

After Christopher went to bed, I got out Mathematics 6 and turned to the section on adding & subtracting integers.

The first thing that struck me was the fact that this topic appears at the very end of the book. Prentice Hall Pre -Algebra^* opens with integers, and I question that. I question it not based on any profound grasp of pre-algebra as a coherent whole. I question it on grounds that Nurk & Telgmaa are geniuses, and they put adding & subtracting integers last, not first.

I'm sure they have their reasons. (I intend to figure out what their reasons were.)

Reading through Nurk & Telgmaa's discussion, I realized why I was confused. I think I realized why Christopher was confused, too. I hope so.

We were both, I believe, stumbling over this type of problem:

5 - (-7) = ?

Both Saxon Math 8/7 & Russian Math teach addition & subtraction of integers using the number line. Saxon's lessons were particularly strong, I thought.

But when I tried to untangle myself by resorting to the number line, I got stuck.

Start at zero, move five to the right, then.......then what?

What was my next move? My very next move, without renaming or re-expressing - (- 7) as + 7 ?

I was stuck.

Reading through Mathematics 6 I realized that the problem is something Wayne Wickelgren & his daughter Ingrid have raised: the same letter or sign has been made to stand for two different things.

There are two 'minus signs' in 5 - (-7). One means 'opposite,' and the other means 'subtract.'

One means 'perform an operation' and the other doesn't (I don't think. Is 'taking the opposite of a number' considered an operation? I don't know.)

In any case, for both Christopher and me, 'subtract' and 'take the opposite of' are two different things.

Mathematics 6 has a formal demonstration of the fact that:

5 - 7 = 5 + ( -7 )

This is something I think I figured out on my own many, many years ago. I've been using it ever since to de-confuse myself when dealing with long lines of integers to add & subtract. At some point, if I'm getting confused about whether I can or can't use the commutative or associative properties, I just turn the whole thing into addition.

Reading Mathematics 6 I realized that's what needed to happen with 5 - ( -7):

5 - ( - 7) = 5 + [ - ( - 7) ]

Voila!

Christopher and I both understand that 'the opposite of the opposite' is the number you started with originally; the opposite of the opposite of 7 is 7. (This wasn't an especially hard idea for Christopher, but the number line really nails it down.)

Once you convert '5 minus negative 7' to '5 + the opposite of the opposite of 7' it's in a form Christopher understands, and can do.

AND it's in a form you can perform on the number line, if you like or just want to check.

5 - ( - 7) =

5 + [ - ( - 7) ] =

5 + [ 7 ] =

5 + 7

Once you've converted a 'double negative' subtraction problem into addition, you no longer have an anomaly, The One Subtraction Problem That Cannot Be Done On A Number Line.


We'll see how it goes. This morning I had Christopher quickly rewrite 12 subtraction problems as addition problems. (I haven't explained to him why a subtraction problem can be rewritten as an addition problem, and I don't know whether I'll get to that today. I haven't closely studied Mathematics 6's presentation to see whether I can introduce it 18 hours before the test.

Fortunately, Ed had already introduced the idea that 'subtraction is addition' last night, when he used the addition-of-debt-to-debt (a concept that is not foreign to our household) to show Christopher that:

- 7 - 7 = - 14

I think he had a lesson in Saxon on subtracting a positive from a negative being the same thing as adding a negative to a negative, so he probably had some knowledge to build on before Ed gave him the add-one-debt-to-another example.

It's the minus-minus issue that's throwing him.

I hope.

one last thing

Looking at this, it strikes me I'm also going to have to create some problems that I ask Christopher to 'simplify'—'simplify' defined broadly as 'write it in the simplest possible correct way that will allow you to recognize what the computation is and do it.'

For instance:

-7 + 5

He probably needs some practice rewriting this as 5 - 7.

I'll see.

I'm also going to try to put together an incredibly simple 2 - 1 type problem that he can always solve quickly when he gets jumbled up. Something like this:

1 - ( - 1) = 2

-1 -1 = -2

-1 - ( - 1 ) = 0

He hasn't learned the Polya line about how 'For each complicated problem you can't do, there is a simple problem you also can't do.' I realize it's not clear that you can explicitly teach problem solving, but I'm going to have to try. He's got to learn the strategy of creating a super-simple version of a hard problem in order to see how to deal with the hard problem SOON.






^*new title: Prentice Hall Mathematics: Explorations & Applications

keywords: subtraction negative minus absolute value subtraction is addition integers extended response

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TeachingSubtractionAndIntegers 18 Sep 2005 - 02:42 CatherineJohnson


click on Printable Version to print


What is subtraction?

Subtraction is the ______________ of addition.


When you subtract, you __________ ___________ ___________________ of the number you are subtracting.


An absolute value is always _________________.


1 - 2 = _________

1 - ( - 2 ) = _________

-1 - 2 = _________

-1 + -2 = _________

1 - | 2 | = _________

-1 - | 2 | = _________

-1 - | -2 | = _________


answers
study sheet for class quiz on pages 2 - 16, Prentice Hall Mathematics: Explorations & Applications & Prentice Hall Pre -Algebra

outloud sheets: integers & absolute value
answer key
notes on outloud sheets for integers & absolute values
Carolyn on introducing absolute value
keywords: integers subtraction addition absolute value opposite add study sheet outloud out loud

---

PracticeSheetIntegersSubtractionAbsoluteValue 16 Sep 2005 - 14:38 CatherineJohnson


I wrote up a study sheet for Christopher's test (it's in the next post) & dragged him through it kicking and screaming.

I think it worked, but we'll see.


If you hit 'Printable Version' it prints out great, exactly enough space for answers in big, round middle-school handwriting.

update

Christopher said last night he doesn't like it when I tell people he screams when we do math.

I told him, Stop screaming and I'll be happy to stop telling people.

We are at an impasse.

---

StudySheetIntegersSubtractionAbsoluteValue 16 Sep 2005 - 14:45 CatherineJohnson


(study sheet is here: subtracting integers & absolute value)


Here is how Christopher does this problem:

-1 - ( - 2 )


He pencils in a vertical line across both of the minus signs in the middle, turning them into plus signs:

- 1 + ( + 2 ) =


That works for him every time, no matter what the numbers, and he isn't thrown off by the same problem written with an absolute value:

-1 - | - 2 | =


This reminds me of Carolyn's belief that you need to get math into a child's hand.


For some reason a problem like:

-1 - 2

makes sense to him. He 'sees' that he's adding two negative numbers.

Here, too, however, he does a swoop and swoop thing: he squeezes in a plus sign between the 1 and the second minus sign, like this:

-1+-2 =

Ed's explanation to Christopher that you can think of -1 - 2 as adding two debts -- first you owed 1 dollar, then you borrowed 2 more dollars and you owed 3 -- seems to have been the ticket.

I tried that explanation on a friend of mine who is severely math phobic, and she instantly got it, too. Adding debt to debt is something everyone can grasp! It's EVERYDAY MATH FOR THE MASSES!


From one of Carolyn's first posts:

That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.


swoop and swoop
the craft of math
subtraction as the difference between 2 numbers
outloud study sheet: subtracting integers & absolute value
answer key
notes on integer, subtraction, & absolute value study sheet
Carolyn on introducing absolute value

keywords: integers subtraction addition absolute value opposite add study sheet outloud out loud

---

ILikeMathPart3 17 Sep 2005 - 02:47 CatherineJohnson


I almost forgot!

Monday or Tuesday night, when Christopher was doing one of his first homework assignments from Prentice Hall Mathematics: Explorations & Applications, he saw an illustration on the side of the page with the caption:

The early Egypticans drew pairs of legs walking in different directions to stand for addition and subtraction.

He looked up at me and said happily, "I like math. I just don't like math when you make me do it."


BeingYourChildsFrontalLobes
GreatMomentsInWorldHistory
ProgressReport
ATeachersStory ("I like the idea of math")
BonusPreTeenPost
fun with Saxon Math in the summer
SundaySchool
I like math
I like math, part 2
TheGoodNewsFromHere
GoodNewsBadNews
ImGoingToPlayland
ImportantQuestionFromJoanneCobaskoOfSocmm
ImportantQuestionPart2
OutsmartingTheTests
ConversationsWithKids
Christopher on his 39
I like math, part 3

---

PrenticeHallPreAlgebraQuestion 21 Sep 2005 - 10:24 CatherineJohnson


Well, Christopher managed an 85 on his first math quiz.....but we're gonna need to step up the pace around here.

Ed checked his homework tonight, which prompted vast quantities of screaming and yelling (maybe there's something to that brain periodization business after all), and now reports that Christopher has essentially zero comprehension of how to solve a story problem involving negative numbers. He's just looking at the problem and trying to figure out which operations to do. Ed says he's more or less guessing.

The good news is he got points off for mechanics, failing to put in the degree sign and the like. He would have had an 89 if he'd LABELED EVERYTHING CORRECTLY. So from now on he will label everything correctly.

The bad news is that the teacher is doing what she did last year, which is putting problems on the test they've never done before or even seen in class or on homework. He had two story problems like this one:

The boiling point of oxygen is -297 and the boiling point of nitrogen is -320. How much higher is the boiling point of oxygen?

Here's my question.

Distance, I know, is always expressed as an absolute value.

Is 'distance' on a thermometer the same thing?

Say the question had been written as, How much lower is the boiling point of nitrogen than the boiling point of oxygen?

Would the answer still be 23?

anyone know of a good source of story problems?

To get Christopher through this course, I'm going to need two things:

• LOTS of practice computation problems (integers, fractions, the works)
• LOTS of practice word problems

Does anyone know of good supplies of either?

I am also going to need vast supplies of Iron Will.

teach kids good handwriting in school

Christopher's handwriting was so bad in Kindergarten that his teacher told us he was considered 'at risk' for dyslexia. (Kids with learning disabilities often (usually?) have bad handwriting.)

That was one of those four-star fun-with-childbearing moments. Two autistic kids, and this one's gonna be dyslexic!

Ed pooh-poohed the whole thing (he is the pooh-pooher in the family), and in fact Christopher began reading on his own literally 2 weeks later (THANK YOU, GOD).....and that was the last any of his teachers had to say about handwriting until he had Ms. Duque in 5th grade last year.

So tonight he missed at least one problem on his homework because his handwriting is still so bad he can't read it himself. His test is a mess; I don't know how he managed to do as well as he did given what a visual morass it is.

Two summers ago I researched handwriting programs, and we spent one summer working on handwriting....and then Ms. Duque pushed him on it last year, though she didn't teach it. When my parents went to school, handwriting was taught in formal handwriting-practice programs that worked. (Ever noticed that ALL members of the greatest generation have beautiful handwriting?)

Today there are schools on Long Island that don't even teach cursive anymore, or maybe it's the other way around.

Anyway, handwriting is one of those ROTE NON-CONCEPTUAL NON-CRITICAL-THINKING SKILLS that have been drop-kicked right out of the curriculum. Replaced by character education. Last year the school spent 20 minutes each and every morning for six months doing their No Put Downs program. This year the program's even bigger as far as I can tell. The teachers are all being trained, and one of Christopher's teachers told us on back-to-school night that, thanks to all the character education she would be doing during class time, 'your children will be better people.' [update: This teacher was Mrs. R. 3-25-2006]

The point is, if Christopher is going to speed through these tests, he's going to have to develop fluency not just in math facts & computation, but in handwriting, too.

Well, at least he had those couple of months with me. I have a couple of cursive practice books sitting in his homework file, so maybe I'll pull those out and get started again. One more thing to brawl over.

I feel a rant about character education coming on.

That stuff I just wrote?

That wasn't it.


keywords: character education bullying no putdowns


keywords: good handwriting Write Now

---

MathLessonRepeatingDecimals 22 Sep 2005 - 20:01 CatherineJohnson




My neighbor showed me this yesterday. Naturally no one had ever taught me how to do this, which is par for the course. But she's a statistician & she'd never learned it, either.

I love this. It reminds me of the shenanigans I go through trying to force Microsoft Word to do graphic design.






I've entered this on the Math Lessons page.


other resources

Purple Math

Math Wizz on converting repeating decimal to fraction




update: Saxon meltdown (3-2-06)

Maybe I'm just tired, but I practically had a nervous breakdown tonight trying to convert 0.013333....(repeating decimal) to a fraction.

I just could not get it.

Finally Math Wizz saved me. Of all the websites I looked at, Math Wizz had the simplest, cleanest, & most follow-able explanation.

Math Wizz also has gigantic gifs.

---

LoneRangerOnHandwriting 21 Sep 2005 - 19:49 CatherineJohnson


I'm bringing Lone Ranger's advice on handwriting up front, because it's important.

For one thing, there's research showing teachers give higher grades to papers with good handwriting, whether they're supposed to or not. That makes sense. Any teacher who's spent more than 5 seconds in the classroom will have noticed that children with learning disabilities aren't the ones with the nice, neat handwriting.

Personal anecdote pause: Ed was giving me some grief about my handwriting lessons with Christopher. He thought they were silly. Everyone thinks they're silly.

So then we had a birthday party for Christopher, and one of the kids we invited was a boy who was in the multisensory class, considered 'at risk' for LD, etc.....the handwriting on his birthday card was so erratic and wild it was almost scary. And.....it didn't look all that majorly different from Christopher's handwriting.

Ed took one look at that birthday card next to all the other kids' birthday cards and got with the program.

(He has horrific handwriting himself, btw. He's a leftie.)

Back on topic: the other reason kids need good handwriting is the same reason they need good-everything: they need automaticity. Again, there's RESEARCH SHOWING that kids with good handwriting produce better content in writing essays. (I'm not going to look it up; you'll just have to trust me.)

Kids need to be able to write legibly and quickly.

Last but not least, this is another one of those curriculum decisions that I'm betting favors girls. Every boy I know has had handwriting woes; Christopher's handwriting was so bad he was being pulled out of Kindergarten for free O.T. services.

My understanding is that girls have better fine motor skills.....I could be wrong, but that's my impression.


Here is Lone Ranger:

To remediate handwriting buy some handwriting paper with a dotted center line. Miller Pads and Paper sells some for upper elementary aged kids and it is great stuff. www.millerpadsandpaper.com Now teach him two things. First, all his letters must basically be the same width (there are some exceptions here). Second, all letters must bump the top and bottom lines. Lower case would bump the dotted center line and the base. Upper case and numbers would bump the top and bottom lines. (For some reason kids like the expression "bump the lines") Have him practice on this paper until this new skill is internalized. He will see instant improvement and that is usually quite encouraging. Good luck!

update

I was Googling around, trying to find the terrific penmanship-paper software I had for my PC, and I found this instead: SpellWrite Books 1-4

It's a spelling and handwriting program from Oxford that sounds terrific. I've never seen it, obviously, but I like the idea, and I generally like Oxford Press. Offhand, and knowing next to nothing about the teaching of reading, writing, & spelling (now there's a caveat for you), I like the approach:

SpellWrite is a series of four workbooks for middle and upper primary that assist students to develop their spelling skills. It focuses on the four forms of spelling knowledge: Phonological, Morphological, Visual, and Etymological.

Features:
• Spelling is presented as an integral part of the writing process
• 25 units of three pages designed to provide a full week’s work for students
• Each book in the series includes: Words to Learn, Assessment for the appropriate year, Rhymes to read, Word Bank, Test pages, Revision units, and teacher ideas and suggestions for each activity
• Each unit begins with a rhyme, tongue twister, or word game designed to introduce the relevant sound and excite the children’s interest in words
• Allows teachers to explicitly and systematically teach students the four forms of spelling knowledge
• Covers a variety of text types with a focus on rhyme and poetry.

The books cost $12 apiece. I'm almost tempted to get one, just to see if I can kill two birds with one hand.

I love Megawords, but the student writes everything on the text pages themselves, which frequently don't have enough space.

Which brings me to another Personal Anecdote moment.

Ed's always been a tad skeptical of the Home Spelling project I've got going. Then, on back-to-school night, we were all handed class schedules written out by our kids. Christopher had spelled cafeteria cafitrea.

Ed has got religion

StartWrite software

I found it.

This is a dandy little program, that apparently works on a Mac.....wonder if I can find the original disk down in the basement?

review of StartWrite Handwriting

The great thing about StartWrite is that you can make the lines as dark or light as you like, and you can space them the way you like. You can also print any words at all for tracing. Extremely easy to use.

free online penmanship sheets

handwriting worksheets at Teachnology. These aren't bad. Upper case, lower case, printing, cursive, and numerals.

---

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 keep forgetting to ask.

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--

---

FunBrainNumberLine 22 Sep 2005 - 19:26 CatherineJohnson


Line Jumper, an online number-line competition at FunBrain.

The kids in my Singapore Math class LOVED the FunBrain site. They especially liked the Math Baseball game.

Normally I'm skeptical of online activities (because Christopher seemed to learn nothing from software math facts programs) but the kids I've known really did like these math facts games, and could play them for a lot longer than they'd do a worksheet.

You can also use Math Baseball to teach mental math, because the kids have to do the calculations in their heads, unless they pull out a pad of paper & a pencil.


I had Christopher do the integers worksheet from Saxon Math 8/7 last night. I'm going to have him keep doing it until he can finish it in 5 minutes & get everything right. That will help.

---

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
• downloadable number line 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.

• Computer Lab: Math, Numbers, and Counting HUGE survey of 163 different online math sites for kids. Terrific collection.
  Item number 163: Disaster Math for Kids
  From FEMA! Interactive word problems graded by the FEMA site.

• 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.

• MacTutor History of Mathematics

• 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...)

• Maths Dictionary (British?)

• 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)

• NCTM Illuminations Looks decent. Lots of interactive 'tools': affine recurrence plotter and--Yay! nets!! I've been looking for nets!

• National Library of Virtual Manipulatives for Interactives Mathematics I've visited this site several times before. It's pretty good, as I recall. (If I'm remembering correctly, the pan balance problems are kind of cute & potentially useful.)

• 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 'Diamond Mines' knock-off I spent a couple of years as a Diamond Mine addict.

• 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.

I will carry on saying bad things about the NCTM.

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

---

BestGrammarBook 15 May 2006 - 02:07 CatherineJohnson


I have appointed Susan grammar diva, because....she knows grammar! (And, more to the point, grammar books!)

Susan, what book should I order RIGHT THIS MINUTE?

Christopher got a 63 on his grammar test, because he 'mixed up subject and predicate.'

I can't take it.

He's ELEVEN.

And he doesn't know subject & predicate.

So.....which one of the books you told me about should I get NOW. I need something with MAXIMUM direct instruction, MAXIMUM coherence (if possible), and PRACTICE EXERCISES.

Sigh.


Another commenter once recommended the Shurley grammar series--how involved is this series?

(Does anyone know?)

Can I fit it in with everything else?

---

OwlPurdueUniversityOnlineWritingLab 23 Sep 2005 - 20:27 CatherineJohnson


For future reference -

The OWL - Online Writing Lab - at Purdue University is a fantastic resource.

I'm posting the link on the Math Supplements page.

---

GrammarSchool 14 May 2006 - 15:09 CatherineJohnson


So, yes, I am now in the grammar instruction business, too.

Ed asked Christopher last night what the subject and predicate were in the sentence, I ate too much food, and Christopher didn't have a clue.

He flat out couldn't say what the subject was, and he thought the predicate was 'too much food.' Then, when Ed corrected him, he sobbed for 15 minutes.

Middle school stinks.

We're only....3 weeks in? Already I've got at least 4 crying children stories, 4 that I can remember, anyway; there may have been more. Today Christopher's close friend M. started crying when the math teacher docked him a point on his math test for telling his twin brother, 'It's easy, you can do it.'

M. protested that he had only been telling his brother he could do the test, and the teacher said that didn't matter, he could have been cheating.

So back to grammar, Christopher has no clue what a subject and a predicate are. He rejected outright Ed's claim that 'I' was the subject: How can 'I' be a subject??????' Then collapsed into sobs brought on by the sudden realization that the reason he 'put the line in the wrong place' was that he didn't know where the subject ended and the predicate began. A classic example of a child not knowing what he doesn't know, which Willingham has written about. (Why Students Think They Understand—When They Don’t and How To Help Students See When Their Knowledge is Superficial or Incomplete)

I'm guessing Christopher probably thinks 'subject' means 'topic,' as in the topic of an article or book; and, by extension, 'predicate' means the topic of the second half of the sentence. Which would pretty much rule out pronouns & verbs as subjects & predicates, respectively.

Christopher is 11.

His school has two hours of 'English language arts' a day, TWO. And in two hours a day this teacher--this tenured, health insuranced, pensioned individual--did not manage to teach Christopher what a subject and a predicate are.

Teaching math is hard. I'm not going to be wildly critical of a math teacher who is trying. (A math teacher who docks a twin a point because he might have been cheating is another story.)

But teaching subject and predicate to a bright child with a good attention faculty whose strength is English language arts.......

Rolling off a log.

And I'm the one who's going to be doing the rolling.

I'm not happy.

update

I just thank God I started teaching Christopher spelling when I did.

---

FunbrainMadLib 23 Sep 2005 - 23:12 CatherineJohnson


Carolyn suggested I try Christopher on some Mad Libs, to teach him the parts of speech.

I'd never heard of Mad Libs!

OK, that's not right.

I'd heard of Mad Libs. Didn't know what they were; especially didn't know they were fun things that could teach parts of speech. fyi, I am aware of the fact that I have just written two incomplete sentences joined by a semi-colon. It's a secret technique of mine.

Turns out Funbrain has extremely fun Mad Libs.








Here's an Eleanor Rigby Mad Lib.

The Grammar Bible

I've mentioned this book before, but I'll do it again: The Grammar Bible: Everything You Always Wanted to Know About Grammar but Didn't Know Whom to Ask by Michael Strumpf is supposed to be the best book out there.

I heard about it from my editor at Holt, who said Strumpf ran a web site on grammar for years; grammar was his life. Copy editors all over New York would call him up on the phone to ask him how to edit sentences. Finally he wrote a book, and this is it.

I read the chapter on subjects & predicates and then taught it to Christopher--it was great.

He has a rhyme: the subject & the predicate are the name and the claim.

I always like rhymes.


I've got grammar stuff posted on the Math Supplements page for safekeeping.

Steps to Good Grammar

OK, I did it. I ordered Steps to Good Grammar. Has raves on Amazon & B&N.

I'm gonna get Rod & Staff, too.

Given that we've managed to miss THE FIRST TWO SUNDAYS OF SUNDAY SCHOOL, I figure we need it.

---

DougsNumberLines 25 Sep 2005 - 14:03 CatherineJohnson


I can't thank Doug enough for his number lines--and ALL OF YOU SHOULD USE THEM, TOO!

Adding & subtracting positive & negative numbers is one of those areas that can be severely procedural if you have a good memory, which most children do as far as I can tell.

Christopher was already becoming entirely 'procedural' with his integer problems: if he saw two minus signs in a row he automatically penciled in little vertical lines and turned them into two plus signs. Then he added.

ooops--must take Christopher to his playdate

will finish this when I get back

I love these number lines.

back again

Well, I'm back from my Excellent Adventure at Barnes & Noble with Andrew, who is obsessed with pulling Arthur books off of their shelves and lining them up on the floor, while I apologize to clerks & customers.

Now he's shrieking at the top of his lungs & jumping as hard as can on his bedroom floor upstairs, which has shaken loose all the lightbulbs in the kitchen light fixture, which means someone will have to climb up on a ladder, take down the fixture, and screw the bulbs back in.

I'm in a nuts-to-autism mood.

back on topic

as I was saying.....Christopher's memory is good enough that he's reaching the point of procedural fluency with integer computations, and that's got its bad side as well as its good side, because I'm sure he still has no idea how to do the problem I posted a couple of days ago, the one asking what the difference was between the boiling point of oxygen and the boiling point of nitrogen.


[pause]


This is grueling. I've just spent 15 minutes trying to deal with Andrew, who has slapped himself on both sides of the head so hard that he'll be bruised again. We're both trembling. I loathe this disorder.


I'm going to make one more stab at writing about Doug's number lines before I go have my own nervous breakdown.

What I'm trying to say is that it's clear Christopher is reaching the point where he's going to have procedural fluency with virtually no conceptual understanding, and Doug's number lines are the answer.

Number lines are to integer problems what bar models are to story problems. Perfect.

I'm going to have Christopher do a page of Doug's number lines every day for a few weeks, and I'm going to do them myself.

I was telling Ed about all of this, and he said number lines were essential in the GED math course he taught to high school drop-outs in Newark years ago. He was pretty successful with those kids, and number lines were a big part of it. These kids--young adults, actually--were years and years behind in math; they'd never, ever gotten it. Ed had to have visual ways of teaching math to them, he said, or he couldn't have done it.

I haven't got all of the number lines posted yet, but will get to it soon. Right now one page is here, at the top of the Comments thread.

number lines in your head

I'll have to track this down, but one of the neuroscientists who studies math argues that we have number lines--a kind of number line, though not a formal visual image of a number line--in our heads; number lines are essentially there already, along with basic counting & very simple fractions such as 1/2. (I'll have to see whether this particular researcher thinks animals have number lines, too.)

This is all the more reason to use number lines frequently, I think. Any time you can hook a new concept to something a child already has inside his head you've got an advantage.

Vision in Elementary Mathematics by W. W. Sawyer

While we're on the subject of visual models, I've been reading Sawyer's book, which my neighbor gave me for my birthday. It's challenging, but incredibly useful & rich.

Andrew is better

Lately Andrew seems to be having low blood sugar crises. Either that, or dehydration, or both. He has an autistic eating disorder on top of everything else, and won't drink water or milk, etc....and eats only a couple of foods. So he's chronically low on calories, nutrients, and fluids.

I forced grape juice down him 10 minutes ago, and now he's making cheerful noises. His face is bruised, and he's urinated all over his TV stand, videos, and whichever of my books he'd lined up as part of the still life.

BUT, he's OK.

I feel like Annie Sullivan after breakfast.

He folded his napkin.


(So is Kumon Math sounding like just the ticket for Andrew?? I say yes!)

---

GrammarQuestion 14 May 2006 - 15:10 CatherineJohnson


What is the complete subject of this sentence?

While taking the dog for a walk, she stepped in poop.

Thank you in advance.

---

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.

---

AnyNumberCanBeAFraction 01 Oct 2005 - 05:52 CatherineJohnson









Steve, on the thread need for speed thread, pointed out that any number can be a fraction, and when I said I ought to put together a worksheet on this subject for Christopher, Dan directed me to this frame, DimWksheet010.ppt. of his dimensional dominoes!

It's wonderful.

I'm going to have Christopher do it.

Which reminds me, yet again, I have got to get Doug Sundseth's number lines attached. I've used them two nights in a row, and today I sent a bunch in to Christopher's teacher, in case she wants to use them with the kids.

math ed is a riveting subject

Obviously, I've become obsessed with math education. I'm constantly trying to figure out what it is about math that makes it confusing, and what one can do to make it less confusing.

Liping Ma talks a lot about fragmented knowledge, and cognitive scientists all wrestle with the problem of expertise, which means the ability to generalize what you know to novel problems and solve them.

I've noticed (I may be quoting others without realizing it) that one of the problems with the 'novice' stage of learning is a kind of over-solidity of numbers, a thingness.

Doesn't Freud talk about children first playing with words as if they were things?

Does he say the same of numbers?

I don't remember.

In any case, what I've seen in myself, and in Christopher, is that numbers are too-solid. Both Saxon & Singapore spend a great deal of time conveying the idea that numbers are fluid, in a away, blinking constellations that can be one thing one moment (-10, say) and another in the next (-5 + -5, or -20/2, or any of an infinite number of combinations & expressions).

The Everyday Math article called this 'number partition theory,' and I haven't been able to figure out whether it is or is not number partition theory, but for my purposes, at the moment, it doesn't matter. Just knowing that the number 10 doesn't have the stability of a chair or a tree or a car is a big help.

So I've been trying to convey this to Christopher.

Dolciani's classic algebra text, btw, opens with this idea. '6 + 4' is another expression for '10.' Ten is not the answer to '6 + 4,' but another expression of '6 + 4'. The difference is huge.

Saxon 8/7 constantly uses the word 'Simplify' to mean 'Find the answer,' which I think is excellent. One day Christopher actually said, 'When he says simplify, he means find the answer.' And I thought that was fine. He's getting the idea that simplify and answer are synonyms.

generalizing knowledge

I'm wondering whether making numbers less thing-y for a child might help him or her to generalize a bit more easily, or a bit sooner -- or at least help him to generalize when he's practiced enough that he/she ought to be generalizing.

---

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

integers problems from RUSSIAN MATH

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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.

---

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

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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)

---

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.

---

INeedAPlan 18 Sep 2006 - 17:21 CarolynJohnston




(blank Saxon answer sheets are attached in the Comments thread.)



Ben is not liking his new math plan. He's been doing Saxon math with an aide for around 3 weeks now, and is doing okay, but he complains a lot about wanting to go back to Ms. Math Teacher's class. She's a nice teacher, that's one thing, but also the other kids are there and he doesn't like feeling singled out. Especially not when the singling-out means he's working a lot harder than he used to in the regular class.

The other problem we are having is one we could be having with any math curriculum; Ben is trying to whip through his math homework, as fast as he can, in order to finish as quickly as possible. As a result, he makes a lot of careless mistakes. I really want to get him working more meticulously, because any work habit Ben acquires will be very hard to get rid of. Here is the plan I've implemented in order to get him checking his work more carefully. It's behavioral, a reward system; I've used behavior plans with Ben since he was a young spacey toddler because if I design them well, they work.

With Saxon 8/7, most of the homework on any given evening is review; it's mostly stuff that he could do correctly the first time if he were careful. So I've taken the risk of telling him that I expect him to get it 85% correct the first time through. If he does that, he gets rewarded (the current favorite reward is getting to watch The Simpsons on DVD when he's done with homework). If not, he does without. But I'm not crazy about this plan, and I'm looking for a replacement.

One thing I don't like is that the reward is tied directly to performance. If he ever gets a homework that is on a topic he finds more difficult than usual, the 85% plan isn't going to work very well.

The other thing I don't like about it is that it isn't working very well. He still makes careless errors the first time through his work. When he is done with that first pass through, and I say "you'd better double check your work now before you hand it in," he usually passes up the opportunity to check his work again. Checking Your Work Again is the brussels sprouts of schoolwork; I remember loathing it myself. Apparently it is SO loathsome that he'd rather risk losing his Simpson's reward than Check His Work Again..

I call this an almost-failed intervention. It's not a total failure; his error rate has dropped considerably since we started, but his first-time-through success rate is topping off at just under 85%. He still has a baseline careless error rate of around 10%, I'd say; he usually has one or two problems he's skipped over completely. You really want a behavioral plan, too, that the kid accomplishes successfully most of the time, not one where he's always just barely missing the target.

Can anyone think of a better metric for success?

And, clearly, Checking One's Work Again from the beginning, at the end of an assignment, is too disgusting to contemplate. How do you get a kid to Check His Work Again on a problem-by-problem basis?

---

OnlineLessonInUnitMultipliers 25 Oct 2005 - 19:36 CatherineJohnson



unit multipliers

---

AnotherDistributivePropertyQuestion 31 Oct 2005 - 04:21 CatherineJohnson



Quick question.

I mentioned Christopher has a wicked time trying to simplify this type of expression:

5 ( 6 - x )

Even worse is an expression like:

-5 ( 6 - x )

or:

-5 [ 6 - ( -x) ]

I've been trying to figure out how to teach and explain this. (I didn't get to show him either Doug's or J.D.'s graphics last night, because he had to study for his English test, and do math homework....which reminds me, I have a question about last night's homework, too.)

Back to the distributive property.

It's correct to say that we are distributing multiplication over addition, right?

We're distributing an operation?

That's what I had thought we were doing, but when I went Googling around about it, I found some dissenters.

                                  

Unfortunately, when I say we are distributing multiplication over addition, Christopher gets even more confused by the minus-minus aspect of 'addition' when what he's looking at is a subtraction.

I've written some 'Out loud' sheets on the concept of addition being subtraction of the opposite....but if anyone has other ideas, please let me know.

I'm having trouble breaking this problem down into its component parts.

Should I have him, at this point, rewrite the question as addition of the opposite?

That's what I'm thinking at the moment, but I'm not at all confident this won't introduce even more confusion and angst.

finding   x - 15 = 30

Here's my other question.

Christopher came home last night with a bunch of simple equations to solve.

He knows how to solve all of them using inverse operations, because he practiced that a lot in Saxon Math.

The teacher told them they couldn't do it that way. She wanted them to do it this way:


   x - 15 = 30
      +15  +15
____________
   x + 0  = 45

       x = 45

So naturally we had a whole battle royale about that. Christopher didn't understand the teacher's explanation, and forgot to bring his notebook home, so I had no idea what he was supposed to do. When I said he needed to call a friend and find out he exploded; when I called one of his friends to find out what he was supposed to do he triple-exploded.

His plan was to just find the answers the way he always does, write them in the blank, and take the half-credit.

Here's my question.

To me it seems like a good idea for Christopher to do a bunch of problems the way his teacher showed him.

But why is that?

How do I explain that using the inverse operation is different, sometimes, from isolating the variable?

And is that what we still call it?

Isolating the variable?

I just remembered

Last question:

a = b

b = a

Is this an official property?

Or is it just obvious?


I ask, because I have two Out loud sheets based on this principle.

The first one is full of problems like this:

5 + (-2 ) = _____

The answer is:

5 - 2


Then I have a second sheet filled with the opposite problem:

5 - 2 = _____

The answer is:

5 + (-2 )


I plan to ask Christopher why, if 5 + ( -2 ) can be rewritten as 5 - 2, then 5 - 2 can be rewritten as 5 + ( -2 ).

And I intend for him to answer that if a = b, then b = a.

All a & b have done is switch sides.

But is that right?

Or is this an Official Property?

update from J.D.

You know, it strikes me that this is another Out loud sheet.

I probably better write up a lot of simple multiplication problems, and have Christopher tell me how & where the distributive property is used in the algorithm.

keeping track of graphics

I'm trying to get all these incredible graphics stored where people can find them, which I have taken to mean stored in multiple locations:

Book-style index
Math lessons
Our favorite supplements

I also have a page devoted to Carolyn's math explanations (seriously behind, unfortunately, but I'm working on it).

I do need a Kitchen Table Math intern.

I bet I could rustle one up.

I'm not quite as behind on this project as I am on others (I realized today, I should just go ahead and post Dan's fraction-multiplication graphic now, before I've had time to sit down & study it...).

In any case, these contributions should all be findable.

---

IsSaxonPlusSingaporeTooMuch 07 Nov 2005 - 23:47 CarolynJohnston


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

I found this site several weeks ago and I LOVE IT! I started homeschooling my two sons last year after taking them out of public school. I have been using Saxon math. Last year they were in second and third grade and I had them in Saxon 2 and 3. This year, I have them both in Saxon 5/4. I like the Saxon program because it seems to be very thorough and they have plenty of practice. Neither I nor they are very strong in mental math and I have wondered about supplementing Saxon with Singapore Math. I'd like some advice on this. Would it be overkill? To let you know about where they are now: It takes them about an hour a day to do their math lessons. They are at lesson 28 in Saxon. (It's all pretty much review—nothing they haven't had before.) They have had four tests and have done well on all of them. (They both scored 100 percent on the first three.) My older son knows his multiplication facts through 12s pretty well. My younger son is shakier on these and hasn't learned sixes, eights and twelves. I tried giving them the Singapore 3a placement test and they just couldn't do it. I started giving them the Singapore 2a placement test and they are handling that fine (though with a lot of complaints because they have to THINK about what to do in the word problems.) They both like to have me walk them through problems instead of making a stab at it on their own. Thanks in advance for any help anyone can give me. Diane

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).

---

HowToWriteAlgebraEquations 05 Nov 2005 - 03:08 CatherineJohnson



Christopher's friend Marc just asked me for help writing equations for word problems.

Here's the question.

Is it the case that you can't write something like:

3 + 5 = x


Russian Math says the convention is to have the variable on the lefthand side of the equation.

However, Prentice-Hall seems to want not only the variable on the left-hand side, but also one of the numbers.

To pass muster you'd have to write:

x - 3 = 5


That seems wrong to me, and in fact, Pre-Algebra: An Accelerated Course by Mary Dolciani has one equation in the answer key in which the variable is isolated on the left side of the equation.

What is the convention?

Meanwhile Marc's dad told him, 'Just write the equation in the most complicated way you can think how.'

Marc is good at math, but even he was a little befuddled by that.

I came up with: Write it whatever way makes sense, then flip it.

That worked for Marc, so he is now writing the equation the way it makes sense, then inversing it.

"I can inverse it," were his exact words.

I have a sneaking suspicion this is the kind of homework scene that led people to think Reform Math might be a good idea.....

---

EmailToMathTeacher 08 Oct 2006 - 22:14 CatherineJohnson

Hi—

I think Christopher probably did poorly on yesterday’s test, which is distressing. When the test comes home I’ll have him re-do all the problems he missed, and I’ll write worksheets containing similar problems for him to do as well.

We are very committed to Christopher learning to mastery every topic you teach.

Christopher says the test included a number of very long equations to simplify.

That’s great; the kids should be able to simplify long equations. But he hasn’t had any long equations to simplify in his homework, and unfortunately it didn’t occur to me to write such problems myself until Sunday night, when it was already too late. (I’ve written several sheets of practice problems for this chapter.)

I’m really hoping you can send homework at the difficulty level of the items that will be on the test. Kids only learn through practice, and a test isn’t practice!

Thanks—

Catherine

P.S. This is funny. I just pulled up my Chapter Two worksheets, and on the very first page I have written:

Distributive property to do list:

Write some long, complicated equations incorporating all the properties

Also—
I’m attaching my Chapter Two worksheets. Feel free to use them if you like, but be sure to check the answer sheets yourself—

Christopher was having a lot of trouble distributing a factor over subtraction, so I focused on the various permutations of distribution over subtraction.

I also used the technique used in MATHEMATICS 6 & in KUMON, which is to create problem sets in which a student does the same thing over and over again before doing any mixed practice:

The first column of problems distributes a positive factor over subtraction.
The second column distributes a negative factor over addition.
The third column distributes a negative factor over subtraction.
The fourth column distributes a negative factor over an expression with two opposites.

Last but not least, I'm sending my ‘Out loud’ subtraction sheets. Those were very helpful, so you might want to give them to the other kids. I’ve started doing ‘Out loud’ sheets, because it’s a technique used by Mathematics 6, the award-winning Russian textbook.

Enjoy!

to send or not to send, that is the question

Ed read this and said, 'Don't you want to wait 'til you see the test, and find out if Christopher is right about the long problems?'

I think normally that would be good advice.

But in this case I'm going to email first & ask questions later.

I've mentioned that there was a lot of parent furor over this course last year. A major part of the problem—perhaps the problem—was that the tests contained material far more difficult than anything the kids had seen or done in or out of class.

That may be fine in college. (I don't see why it's good there, either, but ktm readers will have informed opinions on this, and I don't.)

It's not good teaching in 6th grade.

Christopher is taking a class in pre-algebra, and the school's job is clear.

The school's job is to teach pre-algebra and make sure the kids learn it.

So my thinking is:

• Christopher is most likely to be right, which means the sooner Ms. Kahl hears from me the better.
• If he's wrong, that's important in and of itself, and is information Ms. Kahl should have. Why is a committed student who's clearly working hard perceiving the test incorrectly?
• Christopher's situation aside, the words 'teach to mastery' probably cannot be spoken often enough. Spoken, written, emailed, tattooed to one's forehead: Teach to mastery.
  This is The Message.

I'm hitting SEND.

question

Does it make sense to have the kids simplifying very long equations at this stage?

To me, it seems as if maybe we're getting ahead of the game, but I don't know. (I'm thinking the kids need more practice on the component parts of equations....but, as I say, I'm just not sure.)

I'm serious about having Christopher learn to mastery every topic the teacher covers. I don't question her authority to decide content—especially since the course content has been excellent so far, apart from the Extended Problems, that is, and even those are probably coming under control. They did their last extended problem in class, and the kids were able to manage it on their own. That's as it should be.

I'm curious what math-savvy readers & teachers think.

---

HomeschooledPreTeens 10 Nov 2005 - 15:41 CatherineJohnson



What are homeschooled pre-teens like?

Are they as hostile to their parents as public school pre-teens?

Does anyone know?

---

HomeschoolersTalkAboutPreTeens 10 Nov 2005 - 21:44 CatherineJohnson



My neighbor told me, back when we first met nearly 7 years ago, that every homeschooled child she'd ever met was a nice kid.

I never forgot that.

Now, having read Steinberg's Beyond the Classroom: Why School Reform Has Failed and What Parents Need to Do at the very moment Christopher is changing from child to adolescent, I'm remembering what she said.

Yesterday I asked what homeschooled pre-teens are like.

Here are some answers.


♥  ♥  ♥  ♥  ♥


My sons are 9 and 8 now; they were 8 and 7 when I started homeschooling them last year. My oldest son was acting a lot like what I remembered from the middle school years: his friends and teachers had more control and authority over him (in his mind) than his parents. What did we know about his life? He spent the best part of his day with other people. Homework was an (almost) daily battle from kindergarten through third grade with him. If I suggested a way of doing something that was different from what he'd learned at school, or what he thought he'd been taught, he would object that my way was not what his teacher told him. The implication was that SHE was in charge, not me.

I figured that if I homeschooled him, at least I'd be fresh and alert for our daily schoolwork battles instead of waiting until the end of the day for them when I was tired. I also figured he couldn't point an authority who wasn't there (the teacher or the textbook that he couldn't bring home) when he disagreed with me.

In fact, this has worked for our family. We attend a small co-op of about 10 families who meet once a week and do projects for science, history and study various fun electives. The other kids are not as sarcastic to their parents as the general public schooled population seems to be. (Of course, I'm looking at a very small sample.)

I think we've been releasing our kids too soon into the wide world. I know I did. Both of my sons were in daycare or with sitters 40+ hours a week from the time they were each about 3 months old until they started kindergarten. I was "outsourcing parenting" but I was working and didn't know what else to do at the time. My kids were spending their days with people who had no real stake in their lives.

I may not always be able to homeschool my kids, and it is absolutely NOT EASY, but, for now, I am grateful that I can do it and I think they will benefit academically, emotionally, and, yes, socially.

Maybe this goes without saying, but, you never know, so I'll say it anyway: I do not think myself or my children superior to anyone who does not homeschool. I am only able to do this by the grace of God. I'm just doing the best that I can do right now.

-- DianeAustin - 10 Nov 2005


I watched a homeschooled 10 year old come up to his mom and give her a big, snuggly, bury your head in mom's neck kinda hug—in public (McD's), AND in front of a dozen other homeschooled kids.

I've seen even older homeschooled kids enjoy playing cards with 6 and 7 year olds.

Being exposed to these types of kids really cememted the idea of homeschooling for me.

BTW, if you bring your child home from public school to homeschool, you need to read up on DE-Schooling.

-- NicksMama - 10 Nov 2005


Homeschooled kids are nowhere near as hostile. Not only are they less hostile in their pre-teens, but also less hostile in their teens. Not that it would be all smooth sailing for homeschooling parents, but having the kind of close relationship that you can really only get when you are together for a lot of the time really eases some of the tension.

Try reading John Taylor Gatto for information on the way school was designed to create disharmony between family members and contributed to the more extreme problems for teens today. The Odysseus Group

-- SamanthaRawson - 10 Nov 2005


Well, from what I've heard from the other moms at the Well-Trained Mind message boards, the kids are still spacey at that age, no matter where they learn. They turn 12 and act like their brains have rolled under the bed with the dust bunnies, not to be found again for 2-3 years. It's not uncommon for them to forget how to do stuff they mastered three years before. They discover the opposite sex. Some of them get moody, etc.

I have a friend IRL whose oldest is Mr. Know-It-All now that he's a teenager. He's very bright, and enjoys letting everyone else know it. :-P

I hope my kids won't be as insufferable as I was at that age. Of course, my folks hope for just the opposite to occur. ;-D

-- BrendaM - 10 Nov 2005



Steinberg on effective parenting
Steinberg is actually an expert on adolescent development, not education (which I think accounts for some of the ways in which he goes wrong, IMO).

I'm hoping to get around to writing about his study of effective parents, but for now the good news is that probably most of your ideas about what makes an effective parent have been born out by lots of research.

Steinberg talks about the 3 basic kinds of parents who turn up in study after study after study:

• permissive
• authoritarian
• authoritative

Judging by the Comments left here, ktm readers are in the 3rd category, and that's a Good Thing, as Martha would say.

Kids with authoritative parents are in lots better shape going into the Teen Peer Disaster Zone that is an American public school.

The horrifying thing, however, is that effective parenting is more effective for some kids than others.

Specifically, authoritative parenting for white kids packs a wallop. Not enough of a wallop, IMO, but a wallop nonetheless. You can actually define the difference in terms of a child's grades. The exact same child, in terms of SES, parent education, etc. will be getting Cs if his parents are either authoritarian or permissive, Bs if his parents are authoritative.

For black parents, the situation is completely different. The same black child with authoritarian or permissive parents earns Cs; the same child with authoritative parents earns a C+.

Asian parents, good, bad, and in between, have an even smaller difference on their children's grades, but in their case it doesn't matter. The Asian child of effective parents earns just one-quarter letter grade higher, overall, than the Asian child of ineffective parents. Asian kids have one peer culture open to them, and that's Brainy Asian Kid Culture.


the kids I know who've turned out well
I know a lot of them. These kids are glaringly from authoritative homes—and, I would add, from strongly authoritative homes.

They've been sat on.

Interestingly, all of Christopher's friends but one comes from an authoritative home (and the one who doesn't is the one we complain about!)

They've got seriously intense moms, that's for sure.

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CongratulationsBen 17 Nov 2005 - 18:44 CatherineJohnson









(I'm sorry. I know this is blinking. But I had to do it.)

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SusanOnBeingYourChildsSecretary 08 Oct 2006 - 22:14 CatherineJohnson



great comment from Susan


I have no idea what work you "show" for greater than/less that questions.

You might want to write down managable questions like what you put up above for your meeting with her, which I know you'll be having soon. That's perfectly legitimate and it will get you clear and perhaps make her realize that she's not so clear. She has to tell them what she means or the book must have had them doing it that way unless it is some standard way of doing it that everyone knows about.

We were having similar issues with not having enough homework for the work being asked to be done. We've had to use the other books I have for extra practice.


children don't know what they don't know

Again, children don't know what they don't know. They don't know about flexible/inflexible knowledge. They don't know how much is enough. An experienced teacher whose had children bomb on sections would probably anticipate problems with certain chapters. My son's algebra teacher is a veteran. He has stretched and redone some chapters with extra practice. After 25+ years of teaching math he knows exactly what's going to happen and when he can trust the text and when he can't. Even with that, some kids aren't going to make it and I still have the feeling it has more to do with not having enough practice.


the parent as personal assistant

I have had to become his personal secretary because of the school's expectations of him regarding homework and projects and deadlines. He is given all kinds of things to do with all kinds of deadlines and no real guidance on how to manage his time. Many of these things are lacking in specificity. I have to make him pull out his assignments and go through them one by one. If he can't explain something I ask why he didn't write down more so that he would understand it when he got home. We've had much whining and crying over this, but he's starting, finally, to realize that I am going to look at it when he gets home and it must make sense. Just my hammering away at the assignment book and his responsiblity to accurately get his work written down thoroughly has started to make him realize what he has to do to succeed, but that is a gargantuan assignment in and of itself.

I seriously don't remember this kind of juggling of assignments myself much before high school, so it irritates me that I have to take so much time to teach him how to even write it down properly.

I think as a parent you can point out these kinds of murky expectations by the teacher (like the show your work problem) and that they need to be clarified better.

Test-taking has been more difficult for my son, too. There's a stamina and a maturity needed that's a little different than is required for the quizzes. We were doing great on the quizzes, but tanking on the tests. We've talked it through with him and he's improving, but he still isn't as strong on them as he is on the quizzes.

It sounds like you are trying to turn it into a Life Lesson about perseverence and I think you are so smart to do so. Like you said, quitting soccer is no big deal, but he needs to see that some things he can't quit and that it will be alright. They really think it's the end of the world.

With all that blasted "character" stuff they're teaching, you'd think they'd include some of what he's going through.


the veteran

My son's algebra teacher is a veteran. He has stretched and redone some chapters with extra practice. After 25+ years of teaching math he knows exactly what's going to happen and when he can trust the text and when he can't.


This is exactly my concern with Ms. Kahl.

She is, I think, a 2-year veteran, and last year was a trial by fire.

Plus she's up for tenure this year, and while I don't know whether she should have tenure or not, I don't feel that she shouldn't. I know what a tenure year is like; we went through two years of he**. I'd have to feel strongly that she's in the wrong business to want to make Ms. Kahl's tenure year more stressful than it already is.

Christopher has said to me, several times, 'Ms. Kahl is a good teacher,' or 'Ms. Kahl is a pretty good teacher.'

Ms. Kahl isn't a crowd-pleaser; I'd be stunned to learn that she plays to the kids in any way, or grooms fans.

So if Christopher is telling me she's a good teacher, one thing he's not saying is that she's a narcisstic teacher winning love from kids. Plus he doesn't love her. He sees her as a good teacher who wants him to do well.

She's someone who might be a terrific teacher in 5 years' time.


chipperness restored

OK, Christopher just walked in chipper as usual; so far so good.

He's in particularly good spirits because they had another bomb threat today, so they had to walk down the hill to the Main Street School and mill around with their friends until The Danger Had Passed.

That's two bomb threats this fall, both at the middle school, and both, oddly enough, starting in the girl's restroom. "They always come from the girls' restroom," Christopher says.

I know my school didn't have bomb threats in the girls' restroom when I was a kid.

So we finished up with the bomb threat and segued to the subject of, "Do you have my Feature Story?"

"Yes, why?"

"Mr. Fried wants to see it."

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HelicopterParentsPart3 10 Jan 2006 - 13:43 CatherineJohnson








source:
Staying Within the Lines on Homework help


I spent years reading about how women (or blacks) internalized the culture's view of them.

Ed reminded me yesterday that this is called false consciousness.

Parents have false consciousness.

Here's an article, written by a parent, all about the Bad Things Parents Do when their children go to school. The author lives here in Westchester; she's in one of the river towns. Hastings, Dobbs, or Ardsley, can't remember which. That makes her a neighbor.

LISA JACOBSON runs a tutoring business, Inspirica, in Manhattan, and she has seen parents at their worst, their most enmeshed, their pushiest. Parents who do their children's art projects for them, so the third-grade classroom looks, she said, "like a gallery at MoMA." Parents who tinker with science labs and correct math homework and edit English essays until the child does not recognize more than a comma in an opening sentence.

Gee. It's those Pushy Parents again. The ones I keep hearing about here in Irvington.

I wonder why all those Pushy Parents are spending hours of their lives doing their children's art projects.

Might it be because if they don't do their child's projects the child will be given a large, prominently displayed 'D' for all the world to see, called up to the teacher's desk, asked loudly, 'Are you even trying to do the work?' and sent off to the cafeteria to be taunted by the entire 6th grade class?

I wonder.

Back when my sons were younger, the rule was that they did the "content" and I would help out with the cutting and the coloring. It just didn't seem worth the extra hours they would spend wrestling with scissors and crayons. So after my older son drew his poster for social studies intricately mapping the route from the school to his house, I colored the roads black and the treetops green. And once he had completed his essay for French about the Arc de Triomphe, I took a razor and cobbled a three-dimensional model of that landmark from foam-backed board. (For the record, he lost points for neatness on the map poster I colored, and while his French essay earned an A, my foam representation got only a B.)

Does this passage offer a clue?

A parent-created art project earns a B.

Question.

What grade does a child-created art project earn?

As it happens, I have the answer, since I've just run that experiment.

Here's how it comes out. Other parents stay up all night doing their child's feature story/persuasive essay/major research product. (Seriously. One mother told me she had to pull an all-nighter to get it done. Good for her. She's as furious at Mrs. Roth as I am, btw, and has been hovering on the brink of Going To The Principal for some weeks now.)

Your child writes his own feature story.

Your child receives a bright red D, is berated in front of his classmates, is taunted at recess, spends a week crying at home every night.

Meanwhile you drop work on your Actual Job, the one you need to pay your monster property taxes to support the school, in order to steal time to launch a major offensive against the school you're working so hard to support.

Question.

What was the smart play here?

Stay up all night writing your child's paper and be done with it, or let your child write his own paper, after which all he** breaks loose and you get to spend the next 6 weeks dealing with it. And that's 6 weeks if you're lucky.

On the one hand, I am well positioned to help with their writing. Not to do it for them, but to read what they write and send them back to revise. On the other hand, is that helping or hurting? Can a teacher, however well intentioned, possibly give scores of children the same attention that I can give my own? Am I cheating my boys more by stepping in or standing back? Should the roles of parent and professional ever be mixed?

False consciousness!

The Core Question is not Should the roles of parent and professional ever be mixed?

The Core Question is What is my child learning at school, if anything?

My fifth grader's teacher has specifically asked us not to help," said Jacqueline Ghosen, who also has a fourth grader, and who is more than able to help with math because she teaches business classes at the University at Buffalo School of Management. "Her thought is that if the children are not getting the concept, she is not teaching it well," she said. "But if our child gets it wrong, regardless of whose fault it is, he still gets a lower homework grade. Also, if he is the only one who didn't get the concept, she is not going to reteach it."

That's a problem, alright.

Two words: formative assessment

So every night Ms. Ghosen and her husband spend at least three hours reviewing their sons' math, one equation at a time, telling them how many problems are wrong and sending the children back to find the mistakes themselves.

A big, fat, red 'A' to Ms. Ghosen and her husband for logical reasoning.

If the teacher isn't teaching to mastery, somebody has to.

Who's it going to be?

Other teachers have the opposite request: they want parents to take the reins. Ms. Jacobson recalls a recent parent-teacher conference where she was told "that the only way to keep kids achieving at the high level expected by the school district is to teach at school and then have the kids go home and be drilled and helped and tutored by the parents."

Another big, fat, red 'A' to Ms. Jacobson's teacher for logical reasoning.

This teacher would no doubt thrive in a DI system.

She is not teaching in a DI system.

So she's leveled with the parents. If the school isn't teaching to mastery somebody has to do it.

Unless you have a live-in tutor (that's another story) it's going to be you. Us. The parents.

The real story here, the story that should have been written, is the story of why the schools aren't teaching to mastery.

She's looking at the symptom of school failure.

Not the source.


p.s.

I just spent a couple of seconds looking at that picture.

It's great, isn't it?

Totally undermines the article, something I've seen more than once.

Here we have an anxious child, bewildered by the indecipherable schoolwork he's supposed to complete at home, on his own, with neither competent instruction nor help. The teacher has written some stuff on the board, or the child and a couple of classmates have discovered some stuff in a small group, and now he's supposed to know it.

And here we have a mother glaring at the books her school has sent home—glaring from clear across the room. She's also looking semi-bewildered, but bewildered in a mad way, not a say way.

Wait! she's saying. Is it a 'feature story'? Is it a 'persuasive essay'? Is it a 'major research PRODUCT'?

Plus, she's so ticked off she has apparently acquired the ability to project herself across the room telepathically, double in size, and change colors; she's so ticked off she's turning into THE HULK.

I could send this out as a Christmas picture.


Of course the good news is that parents who possess supernatural powers terrify school administrators.


a personality change, too

Plus the mom was a happy, nice, non-hovering, non-helicopter parent before she got a look at the incomprehensible junk they sent home for her child to do.

I think the TIMES should forget about writing articles, and just have the artists draw the stories.


helicopter parents, part 1
helicopter parents, part 2
helicopter parents, part 3
helicopter parents at the AFT
news from nowhere, part 6 (AP students)
helicopter parents of the word, unite
helicopter parents of the world, unite part 2a (t-shirts)
MiddleWeb says hovering is good

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NewPlanForPractice 19 May 2006 - 16:31 CatherineJohnson



I've got some great Comments to pull up front, and will get to those ASAP.

But first, I wanted to put this out there in case any of you have ideas.

Christopher's math course is now officially a disaster.

For Christopher, it's pure memorization of fragmented procedures that have nothing to do with each other. That's how he's experiencing it.

We'll talk to the teacher and the principal, and they'll do what they can. But it won't be enough. We need a do-over.

So we've moved into the minimize-the-damage phase. We have to get Christopher through the course in one piece, so I can reteach the material next summer. We have to make sure he doesn't get a C, D, or F, and we have to prevent him from deciding he hates math, if possible. (Actually, he could take a C and stay in the track, I think. I'll find out.)

This weekend, trying to think what we need to do just to get through 'til spring, I decided to start giving Christopher timed practice. The tests are 'plug and chug'; they're about speed and accuracy and the ability to memorize huge quantities of (seemingly) unrelated material.

Here's what I came up with; I'd appreciate any feedback you might have.

I went through the sections of the chapter that would be on the quiz and pulled out each component task that was either assumed or taught by the book.

There were 21 separate skills in 4 Lessons.

Then I wrote out 5 or 6 problems in each of the 21 categories, and had Christopher do them while I timed him on my running watch.

Side note: on top of everything else, he's now developing test anxiety. Just what we need.

We pointed out he doesn't have test anxiety for KUMON, and he didn't have test anxiety with Saxon......so he won't have test anxiety for pre-algebra, either, if he knows the material cold.

So.

He did his 6-problem timed sets, and I checked them.

If he could do them top-speed and get everything right, we moved on. I'll have to figure out some way to fit in distributed practice, since this stuff is going to be out of his head the minute he finishes the test. But he looked like he was OK for the quiz.

If he made a lot of mistakes, we did another set.

We also talked strategy.

We told him he's very fast, much faster than he needs to be. He did one set of 6 calculations—reducing fractions to lowest terms—in 21 seconds. In fact he has close to two minutes per problem, and if he's doing 6 calculations in 21 seconds he's going to make mistakes he can't afford.

I did that with KUMON at first. I pounded through the worksheets as fast as I possibly could, and I made lots more mistakes than I needed to. Then I read somewhere, possibly in one of the TIMSS studies, that this is yet another difference between U.S. & Japanese students. U.S. students sprint through their tests and finish every problem, getting lots of things wrong; Japanese students apparently follow a systematic, and more deliberate strategy of doing problems they know how to do, and getting those right. I guess it's a quality over quantity thing...In any case, it's definitely a useful skill to learn how to pace yourself. I had no idea about this until I started doing KUMON worksheets. Sometimes, now, I deliberately slow myself down.

So we started coaching him on slowing down.

We also told him to look at the quiz when he gets it, find the questions he can do, and do them. Skip anything that stumps you; just go past it. Come back if you have time. Etc.

He's due home from school any minute, so I'll ask him how the quiz went. Then we'll see how he did when his teacher grades it.

But if any of you have thoughts on teaching a child to take timed tests on material he doesn't understand and learned only a few days before, I'm all ears.

Thanks.


existential question

Ed is champing at the bit to ask the principal exactly how he sees kids going from TRAILBLAZERS to this course.

I'm so horrified by the whole thing, I don't even want to watch a school administrator try to handle that question.


good news

Christopher just came in saying the quiz was "extremely easy."

'Extremely easy' means he knew the material and did well. At least, so far 'extremely easy' has always meant that he did well. His ability to judge his performance breaks down in the middling realm, but at the extremes, he seems to know.

We'll see.

The teacher gave him scratch paper.

um.....I mean scrap paper.

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CommentsToCome 15 Dec 2005 - 20:33 CatherineJohnson



I have a boatload of Comments to get pulled up front.....which means it's going to take awhile.

I thought I'd mention that the reason I pull Comments up front is that a) I don't want casual visitors to miss the super-meaty ones and b) once a Comment is on the front page it's part of the Category thread, so anyone reading that thread will be sure to see it. (All Comments stay connected to the original blooki posts, but a person reading through the KUMON category, say, isn't necessarily going to have the patience to click on each post individually so he/she can read each Comments thread individually.

So these things need to come up front.....

I've finally begun disciplining myself to KEEP A LIST, and here's what I've got at the moment:

• boys hate journaling

• JD (& others) on Houghton Mifflin Math & long division

• Susan on grading handwriting (Christopher is going to be reading this one out loud)

• Steve on epiphany

• Suzuki

• teaching to mastery

• Rudbeckia Hirta on finding stats on colleges "Random factoid (before I disappear into a cloud of office hours, reviews, calming of panic, and then grading): if you want a statistical profile of a college/university (like graduation rates, etc.) search their web page for the Office of Institutional Research and look for the Common Data Set."

• Doug on 'the margins'

• J.D. email

• Verghis on KUMON honor roll

If there are things I've forgotten, let me know.


other

Since I'm posting a public to-do list, I also need to:

• locate Ken's reading test & post links everywhere

• post links to FERPA (thank you, Rudbeckia)

• post links to the Rewards Reading Series, which both Dan and Smartest Tractor have mentioned (Smartest Tractor has purchased SOPRIS' writing program, IIRC)

• post ALL links to reading/writing materials on the how-to-teach-writing page

• collect the science-teaching links from.....was it today? (it's all a blur!)

I should probably go ahead and buy DON'T MAKE ME THINK....

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ThankYouKtmContributors 17 Dec 2005 - 22:23 CatherineJohnson



I was telling Carolyn the other night how rhetorically powerful it had been to open our meeting with the principal by contrasting IMS's Grade Contract to the contracts posted by Ken and Smartest Tractor.

Those documents anchored & defined the encounter.

Carolyn said, 'Well, you've been putting a huge amount of energy into Kitchen Table Math [true], and it's coming back to you."

It sure is.

If I remember to do it, I'll start taking notes on my daily experience reading Contributors' notes. Offhand, I'd say I learn something new every day. Frequently, I find a new way of seeing something I already know, which is exactly what I need, and what I'm looking for.

By the way, I've given everyone a promotion from Commenter to Contributor. Congratulations! (ummm....Carolyn, OK with you?)

I'm guessing the answer is yes, but since I have a Rule against making Unilateral Decisions, I should say that in my own mind I've given everyone a promotion from Commenter to Contributor. I'm an expert on this kind of promtion, btw. This is the kind of promotion where you get to do even more work for the same amount of zero-money you were already pulling down to do the work you were already doing. (Have I mentioned I spent 7 years as a trustee of the National Alliance for Autism Research?)

In other words, this is the kind of promotion your basic PTSA-Mom spends a lot of time getting, the difference being that you guys get way more appreciation than PTSA moms ever get. Trust me. The world of Volunteer Moms is the world of No Good Deed Goes Unpunished. Like, um, agreeing to teach Singapore Math because the chair of the After-school program wants her son to take the course and then having the Superintendent tell the president of the PTSA that you're undermining TRAILBLAZERS.

oops!

off-topic!

A tiny riplet of suppurating rage escaped me there.

Sorry.

I'm joking. (I really am. Today is a very good day.)



Rudbeckia and Susan J on the greater-than and less-than signs

It just happened again.

I asked for thoughts about assessming flexible/inflexible knowledge at home, and Rudbeckia and Susan J left these comments:

I'm wondering if Christopher is having trouble with the comparisons (deciding which is bigger / smaller) or with the notation (which symbol to use).

[and then:]

I'm asking if he has trouble deciding which symbol to use while simultaneously remembering which number he had decided was larger.

[here's Susan J:]

I'm with Rudbeckia. My sister is an experienced OT and a generally sane and competent person who is required to use > and < in her reports of patient progress. She cannot remember which one means which and she starts screaming if I try to tell her how to remember. She's solved this problem with a sign over her desk with the these two symbols matched to what they mean in words.

By the way, if I call them "left angle bracket" and "right angle bracket" when I'm talking about some computer terminology, she has no problem at all telling them apart.

I'm blown away by both these observations.

Seriously.

I've known what the greater & less than symbols mean for so long, they're devoid of meaning, mystery, or possibility. (Though I expect that to change as I move into algebra.)

The idea that anyone could see them as anything other than mundane and obvious is new to me.

My favorite definition of art, btw, came from the Russian constructivist (any relation to our current constructivists? I have no idea).

They said art was the familiar made strange.

I often have that experience reading Contributors' posts.

I love it.



greater-than, less-than, the mnemonic

In case there's anyone who doesn't know this (I didn't), these days they teach kids that the 'big' part of the greater-than/less-than signs always points toward the big number, while the point of the sign points to the small number.

2 < 3
3 > 2

That has worked well for Christopher, though now Rudbeckia's got me wondering....



learned something new, too

Susan J also said:

BTW, being a programmer, I'm not at all sure whether 0.9066 and 0.906600 are equal or not. They may look equal on a print-out but not be equal inside the computer.

[here's Doug:]

Whether 0.9066 and 0.906600 are equal or not, they aren't equivalent. The second implies two orders of magnitude better precision. I'd probably use ≅.

Of course, in a math class, numbers are all presumed to be precise unless explicitly noted otherwise, so = is the correct symbol here.

Cool.

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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

---

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

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HomeschoolCurriculumForAProdigy 19 May 2006 - 21:18 CarolynJohnston


Here's a letter that LoneRanger found on another list and posted on our requests page. It's an interesting story about a homeschool mom with a brainy kid who sort of rolled her own math curriculum. My advice would be "don't try this at home". I don't really like the bit about 'real math not being sequential little increments' and 'the usual order is arbitrary pedagogy', either; it's a mistake to try to make broad sweeping generalizations from your experience with a single child.

But with those caveats, it's kind of interesting to see what she did.

My son is now 19 and a freshman at MIT majoring in math and computer science. He is extremely talented at math. He has never been to school. We've never worried about doing anything sequentially. I've always used the Tetris model of homeschooling with the random pieces falling from the sky, never the traditional beads-on-a-string approach. Here's what we did for math. It might not work for less-mathy kids, but it sure worked for us. It is sort of like learning a foreign language by moving to the country and having to survive, rather than learning a foreign language by studying 10 new vocabulary words a day and introducing 2 verb tenses per semester. I think S would have done much less if we had insisted on a sequential approach. "Real math" is not sequential little increments. "The usual order" is something totally arbitrary that pedagogues came up with.

Elementary PK-5
I used to say we had a "game-based curriculum". We played a lot of dice, card and board games that involved math. Get "Games for Math" by Peggy Kaye for ideas. We owned every math computer game on the market back then. The best ones are the ones where you need to use math to play the game. I avoided the ones where you are drilled for a few problems, and then "rewarded" by shooting a few spaceships or something. We talked a lot about mathy stuff in the car or waiting for food in restaurants. There was no agenda to it. We would just talk about anything off the top of my head to keep him entertained. We did not use a traditional math textbook with formal homework assignments until 9th grade, when he did AP calculus. But we used a lot of children's math books from the library ("Number Devil", "Al Gebra", etc). I also had all of those "What Your Nth Grader Should Know" books, and I went thru the math sections like a check list to make sure he had all of it. We bought all the Doug Downing "Easy Way" books (algebra, trig, calculus) and I read them to him like story books. Whenever he got bogged down in the explanations, I just skipped ahead to where the story picked up. A year or 2 later I would go through it again, with the explanations, a year or two after that we would make the final pass and include all the footnotes and end-of chapter problems. These all overlapped. We did the first pass on the algebra one in 3th grade, and the first pass on the trig and calc ones in 4th grade. We did all the teaching company videos. But I never asked him to do any problems with them. He would just watch them. Also the old Square One on PBS. He never memorized the multiplication tables (neither did I). I taught him the little tricks I figured out as a kid to compute them on the spot, and he invented some new ones. The tricks involve a much higher level of mathematical understanding than memorizing the table, plus they also work for bigger numbers than 12X12.

Middle School 6-8
At the beginning of 6th grade we discovered Mathcounts, and our lives changed. The math was an exact match for him. The problems are extremely challenging, varied, interesting, and combine multiple areas of math in one problem. They sometimes require stuff like combinatorics and number theory that I didn't see into college. Competition math is totally non-sequential. When newcomers start on Mathcounts, they just jump in the stream wherever everyone else is working. At first they can't do it. It takes about 100 problems of a solution book or person walking them through how to do it before they start to get the hang of it. Then, the first years can maybe get 20% of the problems. The second year, the problems don't get any harder, but the kids are better at it. Maybe they can get 50%. Third year maybe 70%. There is a tremendous amount of preparation material out there for contests. We have a huge supply of problem books with solutions, for contests at various levels. S learned by doing problems, with all the math mixed together.

High School Camp
- the summer after 8th grade S started attending USA-Canada Mathcamp. 5 weeks of math-nerd paradise! A huge smorgasbord of offerings at all levels. Kids pick and choose what they feel like doing every day. Some classes are one-lecture long, others might be 2 weeks, or the entire 5 weeks long. Topology, Field Theory, Optics, Problem Solving, Cryptography. A total mish-mash of subjects that the staff is interested in (often has Ph.D.s in) and makes available to the kids. It is totally non-linear. No pre-reqs, homework sets, exams, grades. The kids just jump in it and play around. Everyone is thrilled to be there and looks forward to it all year. 9th Grade - S had already had 3 passes thru Calculus the Easy Way spread over 5 years, plus watched some calculus videos. I got a bunch of AP review books and old AP problems. He worked through those, then took the AP Calculus BC exam at the end of 9th grade. I bought a calculus textbook, but he never used it. 10th grade - he did distance learning courses for Multivariable Calculus and Linear Algebra. This was his first experience actually working through a math textbook and dealing with homework sets and exams. He HATED that aspect of it (too much drill-and-kill even at this level). I had to force him to sit down and do it while he bitched bitterly, particularly on the multivariable which didn't involve as much new material as we had assumed. But he got through, and made As. He also read an AP Statistics review book and took the AP exam. 11th grade - started auditing grad level math courses at UT (Abstract Algebra, Algebraic Topology). 12th grade - Audited 4 grad courses. Because of conflicts with college visits, he was able to do varying amounts of the work, including attending. But he got something significant out of each of them. Also self studied differential equations using lectures and materials from MIT's OpenCourseWare site (free).

College Applications
We ended up with 10 AP scores (all self-study), 4 scores on SAT II subject tests (in addition to the regular SAT I scores), 3 grades from UT distance learning courses, 2 letters from UT profs stating what his grade would have been in their grad course if he had been allowed to register. I put everything together on one big master transcript. I included the things he studied at home, sort of arbitrarily bundled into "courses". I did not include any parent assigned grades. The transcript is 2 pages. There is an additional 5 page document with course descriptions (textbooks used, etc. Max few lines per course). There is also a one-page "school profile" describing our general educational philosophy. It makes it clear that we were homeschooling in order to attain the highest possible level of academic success, not for any religious reason. It also makes clear that the student had plenty of opportunity for social interactions. We got a rec letter from one of his math profs at UT, from one of the coaches of the USA Computing Olympiad, and from a homeschool parent who taught several classes that included my son. He also had an extensive list of national and some international awards in math, physics, computer science.

At MIT
We had a transcript and copy of the syllabus for the two UT distance learning courses he took, but he did not get automatic credit for any of the college level math he had done other than AP calculus. He was able to get credit for multivariable calculus by taking an MIT exam during freshman orientation. He expects to get credit for differential equations the same way, but they require kids to submit a semester's worth of homework assignments before they allowed to take the exam, so he hasn't gotten around to it yet. There are other courses that he has covered and could almost certainly get credit for by taking MIT's exam (linear algebra, topology). But he has decided not to bother since they are not required for the particular math major he is going for, and he expects to have plenty of credits. MIT is fairly loose about prereqs, so he won't have to repeat anything he has already covered.

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SteveOnTeachingWritingPart2 10 Jan 2006 - 14:00 CatherineJohnson



I'm trying to pull together the Writing thread for my neighbor, and just re-discovered this Comment from Steve:

I just helped my son (4th grade) complete his report/map/craft project on Chirstmas in Greece. (All of the kids had a different country.) As with his other projects, the problem is that the school doesn't prepare them to do the job. They may talk a little bit about what to do, but they don't see what goes on at home. The kids just can't do the project by themselves. If I let him do the project all by himself, it would be horrible, take FOREVER, he would learn very little, and he would get a poor grade. I end up doing the teacher's job. I don't do it for him, but he needed major help in organization, reducing the information down to a reasonable size, and putting it all into his own words. No parent I know likes school projects like dioramas, research reports, and other thematic displays of educational pedagogy and feel-good-ness. Perhaps they expect and want parental involvement?!? I'm more than willing to do my part, but, I really don't want to do their job. Please don't ask me again to practice basic math with my son at home.

There are so many fantastic Comments on this site. I've got a list to pull 'up front,' and am going to carve out some time today to get started, at least. The archived entries on how to teach writing are here.

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MathWarsAtHome 17 Jan 2006 - 20:36 CatherineJohnson



Christopher is sitting witih his dad practicing math & screaming & shrieking.

I just heard him shout, 'I thought you were nicer than Mommy!!!!'

Yeah. That's what everyone thought.

-- CatherineJohnson - 15 Jan 2006

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ConstructivismAtHome 16 Jan 2006 - 21:24 CatherineJohnson



You will all want to order this book right away.

Teaching Your Child to Love Learning: A Guide to Doing Projects at Home






-- CatherineJohnson - 16 Jan 2006

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SentenceCombining 18 Jan 2006 - 16:43 CatherineJohnson



....speaking of books coming in the mail, my copy of Don Killgallon's Sentence Composing for Middle School arrived today. (Killgallon's website)

I don't exactly know what sentence combining is, but I have a Bayesian conviction it's going to be the answer to my Writing-Instruction problems at the sentence level, thanks to this fellow:

Grammar teaching and writing skills: the research evidence

Richard Hudson (dick@ling.ucl.ac.uk)

Dept of Phonetics and Linguistics, UCL, Gower Street, London WC1E 6BT

Does a training in 'formal grammar' improve a child's ability to write? At one time it was taken for granted that the answer was yes, so children were taught grammatical analysis as part of the effort to improve their writing. However when educational researchers sought evidence for the expected effects, the results were negative; for example, one of the classic experiments concluded: "It seems safe to infer that the study of English grammar had a negligible or even harmful effect upon the correctness of children's writing in the early part of the five secondary schools." (Harris 1962) A number of studies in the 60s and 70s have since been accepted as 'classic' support for the view that grammar teaching does nothing for children's writing. By the late 60s the dominant view in both the UK and the USA, and possibly throughout the English-speaking world, was that "most children cannot learn grammar and ... even to those who can it is of little value." (Thompson 1969) No doubt this view fitted the spirit of the times both in English teaching (where grammar was seen as a shackle on children's imagination) and in linguistics (where Chomsky was arguing that grammatical competence develops 'naturally' according to an innate programme, so teaching is simply irrelevant).

Since then much has changed in both the UK and the USA, and the pendulum seems to be on the return swing. It would be naive to think that the pendulum is driven by academic research - indeed, there has been very little research on grammar and writing since the flurry in the 60s and 70s; rather it reflects very general attitude changes in education and more generally throughout society. However the result is that there is now much more enthusiasm in some educational circles for the idea that conscious grammar (resulting from formal teaching) could have the useful benefit of improving writing.....

What, then, does the published research really say about the effects of grammar teaching?

[snip]

Grammar teaching could be surreptitious, as it were, with a clear underlying theory of grammar but minimal use of grammatical terminology. This is in fact how a lot of grammar teaching has been done; and in particular there is a well-recognised activity called 'sentence combining' which seems to be widely used in the USA. There is some evidence, apparently good, that this kind of activity benefits children's writing (Abrahamson 1977; Barton 1997; Hillocks 1986; Mellon 1969; O'Hare 1973), and in some studies it turned out that this kind of grammar teaching produced better results than more traditional teaching of grammatical analysis. For example, " Hillocks surveys the many studies of the effects of sentence combining, and finds them overwhelmingly POSITIVE at all levels (grade 2 to adult). 60% show significant gains in syntactic maturity; 30% non-significant gains; 10% no gains." (Weaver 1996, reporting Hillocks (1986)).

Why should these exercises be so much more successful than traditional analysis? It seems reasonable to assume that it is at least in part because they are exercises in the production of language, and specifically in the production of written language, so they feed much more directly into the child's growing repertoire of productive skills than exercises in grammatical analysis do. In short, they are more closely integrated into the teaching of writing, so the skills acquired in isolation are more likely to transfer directly into a usable skill. However this conclusion does not necessarily rule out the possibility of transfer from grammatical analysis under the right conditions.

This makes sense to me, so I'm going with it.

5 reasons:

• it makes sense

• I'm a writer, so my intuition about what works in writing instruction is probably worth listening to

• I used to teach writing, so my intuition that sentence combining makes sense is, again, probably worth listening to

• KUMON Reading uses sentence combining

• sentence combining seems somewhat analogous to the way Ben Franklin taught himself to write

We need a Bayesian Rating Scale

That way, we could assign numerical values to the question of, Just how strongly do I think I guessed right?

Here's a possibility:

On a scale of 1 to 7, 1 being 'no clue' and 7 being 'death and taxes, how certain do I feel that sentence-combining will make Christopher a better writer?

6

or

6.5

I'm not feeling a lot of doubt here.



I love this

back to Hudson:

In conclusion, the idea that grammar teaching improves children's writing skills is much better supported by the available research than is commonly supposed. However there is no denying the need for more research in this area, so we finish with quotations (from Walmsley 1984) by two of the twentieth century's most distinguished psychologists who have taken an interest in this question.

Robert Thouless (1969:211):
"If a small part of the research effort that has been put into demonstrating the uselessness of grammar ... had been distributed over a wider field, more might be known about how skill in the use of English can best be developed."

John Carroll (1958:324):
"I am reasonably sure that unless the student gets a feeling for sentence patterning ... his own sentence patterns will show many obvious defects. Research on the effectiveness of teaching English grammar in improving English composition has been mainly negative, but until this research has been repeated with improved methods of teaching English grammar, I will remain unconvinced that grammar is useless in this respect."

I went on a Sentence-Combining treasure hunt on Amazon, and came up with Don Kilgallon as the likeliest prospect. Just glancing through the middle school book, it seems like exactly what I want.

From the back of the book:

With the first edition of his book, Don Killgallon changed the way thousands of high school English teachers and their students look at language, literature, and writing by focusing on the sentence. In this revised edition, Killgallon presents the same proven methodology but offers all-new writing exercises designed specifically for the middle school student.

Unlike traditional grammar books that emphasize the parsing of sentences, this worktext asks students to imitate the sentence styles of professional writers, making the sentence composition process an enjoyable and challenging one. Killgallon teaches subliminally, nontechnically--the ways real writers compose their sentences, the ways students subsequently intuit within their own writing.

Designed to produce sentence maturity and variety, the worktext offers extensive practice in four sentence-manipulating techniques: sentence unscrambling, sentence imitating, sentence combining, and sentence expanding. All of the activities are based on model sentences written by widely respected authors. They are designed to teach students structures they should but seldom use. The rationale is that imitation and practice are as valuable in gaining competence and confidence in written language production as they are in oral language production.

Since the practices have proven successful for the great majority of students who have used them in all kinds of schools, it's demonstrably true that Sentence Composing can work anywhere--in any school, with any student.

I believe it.

Kilgallon has written books for all grade levels.

















Bayesian statistics & false positives
Bayes & the human mind
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale

sentence combining
Smartest Tractor on Killgallon & 5 ways to combine sentences

Bayesianprobability


-- CatherineJohnson - 17 Jan 2006

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AProblemForCarolynAndBen 21 Jan 2006 - 15:17 CatherineJohnson



I've been thinking about Carolyn's at peace with cross multiplication post:

...lately we've been dealing with problems like d/36 = 7/4, and whatever he learned about cross-multiplication last summer is gone.

I've wanted to teach Ben to do them using isolating-the-variable methods. It's difficult to get Ben to take this step-by-step approach. One problem is that he always wants to do the problems in Saxon 'the easy way' -- and I can't tell you what the easy way is, for Ben, because from where I sit, it looks as though the answers just come to him. It's a bit uncanny. I think he visualizes the proportions, somehow.

doing things the easy way

First of all, there's one thing to be said for sending your child to a school that will stomp him with D's and D+'s.

If you don't reach the point of actual School Refusal (we were getting close there), the kid is motivated not to get more D's and D+'s. Brute force has its uses.

Once your child has had his ego mulched, there's a chance he'll listen to reason.

For all of you parents-of-little-ones, let me add that it's not much of a chance.


here's the sequence:

1. big test coming up on material not taught to mastery, not learned, and not comprehended in any way, shape, or form

2. parent points out, logically, that child has not mastered, learned, or comprehended material; therefore he will need to study a lot

3. child rolls eyes, screams, yells, sobs, claims stomachache, headache, exhaustion, etc.

4. parent persists in patient, logical explanation

5. child rolls eyes, screams, yells, etc.

6. parent persists

7. child screams, yells

8. parent loses control of patient, logical tone

9. child screams, yells

10. parent levels threat to take away allowance/PlayStation/TV

11. child says, 'I don't care'

12. parent asks 'do I need to tell your father?'

13. child says, 'Daddy never yells'

14. parent, etc.

15. child, etc.

16. parent, etc.

17. child, etc.


Talk about scripted instruction.

Notice that throughout this exchange, no actual math is getting done. In our house we call this running out the clock.

My point is: the day the 'D+' comes back on the test we have crying.

By the time the next test is upon us, the D+ is history.

Ed and I are still traumatized; Christopher has moved on.

Nevertheless, since Christopher got his D (make that D's plural as of yesterday) he is willing to listen to us when we tell him to write out each step of the solution.



I love this problem:

-x = -5



I don't know where I found it; not Prentice-Hall, I'm sure. I think it was in a workbook — maybe Instructional Fair or Kelley Wingate.

I'm wondering whether this might be a good problem to do with Ben.

Here's what I love.

• First of all, it's incredibly simple visually and mathematically. The load on working memory has to be small.

• Second, you can show, in just one problem, that adding, subtracting, multiplying, and dividing are still related even though you're solving equations, not learning inverse operations.

• Third, you can almost use it the way people use simple proofs and demonstrations; it's so simple and pristine you can ask your child to write out the steps on more than one occasion - and to justify each step, and to solve it using addition/subtraction AND multiplication/division.

• Fourth - somebody else will have to fill this in. There's something about this problem that starkly shows the 'equalness' of equations, as well as the do-the-same-thing-to-both sides principle. I can't express it, but I can see it.

• Fifth, it demonstrates the fact that a 'minus sign' is the same thing as a coefficient of -1.

Christopher can't write out the steps to solve this problem — although he does now 'see' that the answer is x = 5 — but I'm going to see to it that he can.



solving via addition & subtraction:

-x = -5

-x + x = -5 + x

0 = -5 + x

0 + 5 = -5 + x + 5

5 = x



solving via multiplication and division:

-x = -5

(-1)(-x) = (-1)(-5)

x = 5



Wal Mart homeschooling

wow

Wal-Mart has a huge selection of books for homeschoolers.

They have a bunch of Instructional Fair books. I like the pre-algebra workbook.






Illinois Loop likes Kelley Wingate's workbooks, and I see that at least one Core Knowledge course uses her series.

Wal-Mart is apparently selling a cheaper version of the IF workbook I have, which costs $12.99. Wal-Mart's is $2.81.

You can find Kelley Wingate in various places. Homeschooling Supply has the pre-algebra book for $12.31.

These things aren't cheap.

I should talley up all the money I've spent on materials for re-teaching math.

Not to mention the cost in lost work time.

-- CatherineJohnson - 21 Jan 2006

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StickingPointsInMath 30 Jan 2006 - 20:09 CarolynJohnston


Rick Garlikov, the subject of the other day's post on TeachingBinaryLikeSocrates, has a mentoring service for students and parents in Alabama. You can pay to have your child mentored and tutored, or you can pay to be mentored in teaching your own kid. I love the latter idea, actually. Teach a parent to fish, and he'll eat for a lifetime.

Anyway, on his web page about parent mentoring, Rick has a section in which he discusses the areas in math where kids tend to flounder (you have to scroll way down to see it). He also has a section about verbal subjects where kids tend to flounder, but we'll cover that in a different post. Here's Rick's list of sticking points in math education:

• Understanding counting by groups, such as groups of two, five, and ten
• Seeing numerical relationships in general and knowing to look for them
• Place-value and adding/subtracting that requires regrouping or what used to be called borrowing and carrying
• Understanding multiplication and division
• Fractions
• Decimals rate/time/distance problems
• What algebra is about; how it works in general
• Geometry proofs and theorems and their point

Although certainly this is a largely correct list, is it the most useful possible list for parents and teachers to be on the lookout for difficult spots in their kids' math educations?

Can we narrow it down more precisely than saying (in effect) "everything about fractions"? For example, I've noticed that kids tend to have no problem at all multiplying fractions; they do the obvious thing, and it's also the correct thing. It's adding and subtracting fractions that's difficult for kids, because the obvious thing is not the correct thing.

Can we narrow down more precisely what's hard for kids, and what isn't, about algebra?

What does he mean by 'seeing numerical relationships in general', and do we agree that it's a sticking point? My experience was that counting by groups was not especially difficult for Ben, and I never got the impression it was hard for his compatriots, either.

Are there topics that didn't make it onto Rick's list?

Weigh in, and I'll collate all the input and try to put together a comprehensive list.

-- CarolynJohnston - 25 Jan 2006

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ParentsMentoringChildrenInSingapore 16 Sep 2006 - 20:27 CatherineJohnson



OK, I know I'm supposed to be typing Steve's algebra lessons into Equation Editor so mere mortals like me can read them, and also doing the same for my own Comment for Carolyn's Sticking Points post.

Plus I've probably got a civil servant or two who need bullying this afternoon (where does David Allen say to put 'bully civil servants' on the Master List?)

But all of that can wait!

BECAUSE FIRST I HAVE TO WRITE MY SLAVE PARENTS IN SINGAPORE POST!



don't encourage me

brief pause for Character Analysis

My mom used to always say something funny.

She is stubborn as a mule, and she used to say, 'I can be led, but I can't be pushed.'

That's me to the nth.

I can be led. In fact, I like to be led; I make a good second in command.

But don't push me.

Until last night, I hadn't thought of myself, not consciously, as a person who gets her way by bullying civil servants. Sure, I was willing to raise holy he** if I thought my kid was being hurt. But I hadn't thought of this as bullying civil servants. Given the relative unequalness of the match — me against a small school district — it seemed more like.....um........

hmm. The only image coming to mind is Butch Cassidy and the Sundance Kid deciding to shoot it out with the entire Mexican Army.

Sure, you do it. They're out there shooting at you, you're inside your cave bleeding to death, so what the hell? You go down fighting.

I'm joking.

Anyways, now that ktm guest has put it this way I'm thinking......bullying civil servants.

Bullying civil servants to get my way.

yes

I like it!

I want to get my way!

Until last night, I didn't know I was the bully.

I thought I was the helpless parent-victim getting clobbered by fuzzy math & ZERO teaching to mastery.

If I can be the bully and get my way — THAT IS FANTASTIC NEWS!

YAY, ME!

Now all I need is somebody to tell me who I need to bully to get Direct Instruction, Formative Assessment, and Teaching to Mastery established as the formal educational policy in the Irvington Union Free School District.



speaking of getting my way

This is funny.

We're back at tennis lessons, and I ran into the mom I always see there who has a daughter in Mrs. Roth's class, and who's had the same view of her we had (and knew other parents in our boat as well.)

She said, 'You guys must have really done someting. Mrs. Roth is completely changed. All the girls who didn't like her love her now.'

That's what Christopher's been hearing, too.

His chums are constantly telling him, 'Mrs. Roth's so much nicer now that you're gone. She's so much happier.'

He's been hearing this every day, poor thing.

I finally told him, 'Mrs. Roth's not nicer because you're gone. Mrs. Roth is nicer because the principal told her to be nicer.' Then I told him not to be saying this to all the other kids. It's time for them to go back to beating each other up about who has a crush on who. Either that, or just resume milling around randomly calling each other names like they used to.

This is my cue to say again how much we like our principal, Scott Fried. I think I mentioned that Ed said, after our meeting about Mrs. Roth, 'He handled the situation perfectly.' I quote that because Ed's been an administrator for so long himself.

After the meeting, Scott made the call the next day.

Christopher has been moved to a teacher — Mrs. Kozak — who is lovely, and who seems to be making a special effort to get him perked up and de-traumatized. It's working. Christopher comes home every day and reads his notes from her class out loud to me! Plus Mrs. Kozak is teaching spelling, grammar, and writing-as-a-process; they write drafts under supervision, get comments on their drafts, then revise. She even told Christopher he has to come in for extra help with his handwriting. Glory Hallelujah.

Another thing: from a distance, it appears that Mr. Fried worked well with Mrs. Roth, too. The fact that children who were unhappy in her class now have no complaints at all is very impressive. How often do you see a bad situation turn around on a dime?

Seeing as how Irvington teachers & perhaps administrators may be reading this site, I'll say one thing more.

Other parents have been told, by their children, that Mrs. Roth typically chooses one child to pick on.

A child in Mrs. Roth's class has told Christopher that he has become Mrs. Roth's target now that Christopher is gone.

I don't know whether that's true.

But I imagine Mr. Fried is keeping an eye on the situation.



next up

So that was the story at tennis.

Next thing, I heard from my friend J., who called to chat. We hadn't talked since summer. One thing led to another, and she told me Christopher had been called a 'BOCY' at school. She was aghast.

I already knew Christopher had been called a BOCY, and, frankly, I don't care. The kid who called him a BOCY is one of his best friends, and the two of them are going to have to figure it out for themselves.

I did mention to Scott that he may want to take a look at how much BOCY-calling is going on, if only because Andrew will be there next year, which means IMS is going to have some INTENSE BOCY BEHAVIOR HAPPENING ON A DAILY BASIS.

Forewarned is forearmed.

Anyways, I already knew about the BOCY business, but it turned out J. didn't know that her own son had been called the n-word.

Her son told Christopher about it, which is how I knew, but hadn't told his mom because he figured she'd go ballistic.

He was right.

J. called up the principal, and the conversation went something like this:

J.: My friend Catherine told me so-and-so called my son the n-word.

PRINCIPAL: 'Catherine? Catherine Johnson?'

J.: 'Catherine's a good friend of mine.'

PRINCIPAL: 'Did she call you?'

J.: 'No, I called her because we hadn't talked in a long time. If I hadn't called Catherine, I wouldn't have even known it happened!' [J. is very cool.]

PRINCIPAL: etc.

So the next day the principal called J.'s son, M., into his office to interview him about what had happened:

PRINCIPAL: What happened, exactly?

M.: tells story

PRINCIPAL: Who did you tell?

M.: Chris Berenson

PRINCIPAL: Why did you tell Chris Berenson?

M.: Because he's my friend.

PRINCIPAL: etc.


J. told me, 'You guys must have really shaken things up around there.'

I love it!

We're everywhere!

There's no escaping us!

THERE'S NO ESCAPING US BECAUSE WE LIKE TO GET OUR WAY BY BULLYING CIVIL SERVANTS!



is it still Wednesday? not Friday?

That's hard to believe.

This is the kind of post I write on Fridays, after a week of Maintaining Composure & gobbling up so many frontal lobe resources there's nothing left.

OK, back to business.

That being:

SLAVE PARENTS IN SINGAPORE

Carolyn mentioned the subject of parent mentoring, which reminded me of an LA Times article I read about Singapore way back when, before there was Kitchen Table Math.

So if we were to crib from the valedictorian of nations, what would we find? A school system based on two credos: one very American—competition—and one unimaginable in the U.S.—total government control. For students, this means high-pressure exams at the end of grades four, six, 10 and 12 that help determine not only what classes they take but, ultimately, whether they will wind up as doctors or cabdrivers. For schools, the pressure is to attract the best students—who have their pick of campuses.

Then there is:

A national curriculum. In Singapore, there are road maps for instruction at every level, molding tests, tutoring and teacher training. The documents are amazingly concise—eighth-grade math is covered in 10 pages, listing 19 topics within algebra, geometry, etc. (Students, for example, must be able to calculate the "volume and surface area of sphere, pyramid and cone.") By contrast, American eighth-graders race through 30 or more topics, learning them so superficially that they have to be repeated over and over.

Involved parents. Here, that doesn’t mean just showing up for Back to School Night. Parents get on waiting lists for the best tutors, who charge $300 a month. They buy two sets of books to ensure that one is always available for homework. Hundreds pay $300 to attend 30 hours of weekend training so they can understand changes in math instruction. "As parents, we think of always buying the best computers, giving them the best tutors, to play it safe, you know, so they can score high on their examinations," says Siew Yok as she purchased software so her 12-year-old daughter could cram to qualify for prestigious Raffles Girls School.

So here we have it, the Secret of their Success:

• parents get on waiting lists for the best tutors

• parents pay $300 to attend 30 hours of weekend training so they can understand changes in math instruction (now there's a potential revenue stream the folks at EVERYDAY MATH haven't thought of)

• parents buy 2 sets of books

That pretty much describes me to a 't.'

• hiring the best tutors = KUMON

• spending $300 on a weekend seminar = writing Kitchen Table Math so I can learn math & how to teach it from Carolyn & the resident KTM Math Brains

• buying 2 sets of books = buying 2 sets of books, one set via taxes, one set via American Express payments to Amazon.com

Assuming this article is true, in Singapore the job of seeing to it children actually learn what the teachers are teaching belongs to the parents.

Good thing I live in America.

If I lived in Singapore I'd be getting caned on a regular basis.


Asians in Great Neck



-- CatherineJohnson - 25 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

---

FormativeAssessmentAndRichardNixon 16 Sep 2006 - 21:05 CatherineJohnson



I was trying to pull together the various posts & comments on gaps and gapology when I discovered that one of the many benefits of formative assessment is that FA allows you to:

a) discover gaps

and

b) get rid of gaps

Feedback given as part of formative assessment helps learners become aware of any gaps that exist between their desired goal and their current knowledge, understanding, or skill and guides them through actions necessary to obtain the goal (Ramaprasad, 1983; Sadler, 1989)

The importance of this idea should have been obvious, yet I wasn't thinking about gaps when I first began looking into formative assessment.

So I was sitting here thinking about formative assessment and gaps when I flashed on the famous Howard Baker line about Richard Nixon:



What did the President know, and when did he know it?


That's formative assessment for gaps.








key words: gapology
James Milgram on long division & time
can you cram math: learning a year of math in 2 months
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 in a nutshell
formative assessment and Richard Nixon
Terminator



-- CatherineJohnson - 31 Jan 2006

---

GoodAdvice 01 Feb 2006 - 20:53 CatherineJohnson



I was just straightening up my computer files, and I came across this piece of fantastic advice on how to work word problems in algebra.

Unfortunately, I didn't record the author of this advice.

I think this is the text of an email Carolyn's husband, Bernie Johnston sent me back before we started writing ktm. But it may have been posted by Steve....(I'm inclined to think it's Bernie, not Steve, because Steve prefers 'isolate the variable' to 'undo what's been done to X,' assuming I read his response to Carolyn's post on the subject correctly.)

I'm sure one of them will recognize this. [update: Bernie wrote it]

This reminds me of a thought I had last week. For many many algebraic or calculus computations I have a phrase that runs through my head and tells me how to proceed. If I forget what to do I simply conjure up that phrase in my head. When I've tried to tutor people I've noticed that they frequently get stuck at exactly the point where the phrase would be useful but they have no phrase in their heads. When I think back upon where the phrase came from I realize that in many cases it came from my high school algebra teacher.

For example, when you start an algebra word problem, it's very difficult to know where to begin. There are a lot of words and the potential complexity of the problem is enormous. If you spend a lot of time using the front part of your brain to search for an appropriate path you may never find it. The phrase "when you don't know something, give it a name" is essentially the secret of algebra. This allows you to mentally grasp a particular thread of the problem which you can then follow through to the proper conclusion: the problem space has been cut way down in size.

Another one he gave me:

"When you have an equation with variables on the bottom, clear the denominators".

"Put the x's on one side and the numbers on the other."

"Undo whatever has been done to x."

These phrases are not singing rhymes but they are quite useful. The idea that certain procedures must be memorized or learned by rote is highly unfashionable these days but I think absolutely necessary.

words to remember

When you don't know something, give it a name.

When you have an equation with variables on the bottom, clear the denominators.

Put the x's on one side and the numbers on the other.

Undo whatever has been done to x.



words to remember from Vlorbik

Include the units.

Word problems have word answers.




V is right; including the units & writing word answers to word problems this is a VERY good habit to get into, right up front. I'm forcing myself to remember to do this.

Interestingly, Christopher isn't hugely resistant to including the units. I thought he would be, because he's resistant to everything.

Just goes to show how distant the middle-school brain is from the grown-up brain. Christopher seems to view 'including the units' as Obviously Something A Person Should Do.

I wonder if it's the relative hyperspecificity of the child's brain. He may feel like an answer of '5,' when what is meant is '5 cents,' really truly isn't 5 cents.

Don't know.






key words: good advice on algebra word problems good advice on how to solve algebra word problems
understanding basic algebra moves (& Comments)
good advice on solving algebra word problems

-- CatherineJohnson - 01 Feb 2006

---

TimeTimer 08 Feb 2006 - 19:59 CatherineJohnson



My Time Timer came today.

So far, I love it.

I ordered it because I've always been drawn to Time Timers in special needs catalogues, and because The Organized Student, by Donna Goldberg, says students need to be directly taught what time is.

That struck a chord. Like your basic middle schooler, I don't really know how much time any given task takes to do, either.




my problems and welcome to them

One of the Stupid Things I've been doing for — oh, the last year or so — is going to bed too late because I start doing math too late, and can't stop.

Many people would look at this problem and say, "Why don't you start doing your math earlier?"

Good question.

For some reason, I always plan to start doing my math earlier.....and sometimes I do do my math earlier.

But then, somehow, even when I do start my math earlier, I still end up going to bed too late.

This is why I need a Time Timer.



how long is a nap?

OK, so there I am, sleep-deprived & sitting at my desk facing massive piles of Stuff that needs managing.

And I'm tired.

I'm so tired I obviously need to take a nap, BUT I don't take a nap because I DON'T HAVE TIME TO TAKE A NAP....

And so, tragically, I remained rooted at my desk, getting nothing done, because I'm too tired to get things done but I don't have enough time to take a nap because I have too much stuff to get done, etc.

So today, contemplating my groovy new Time Timer, I decided: let's just SEE how long it takes to take a nap.

I took it upstairs with me, set the dial for 30 minutes, got in bed & went to sleep.

Fifteen minutes later I woke up.

Fifteen minutes.

That's not very long.

I can easily spare 15 minutes to take a nap when I'M NOT GETTING ANYTHING DONE ANYWAY.



why do I need a Time Timer for this?

In theory, I could use an ordinary analog kitchen timer, as Google Master pointed out.

Sure, a Time Timer has obvious advantages.

A Time Timer is almost silent; a kitchen timer ticks loudly.

A Time Timer has no alarm when time runs out; a kitchen timer is designed to blast you out of your skin.

Still, these are details. There's no logical reason I have to have a Time Timer instead of a kitchen timer.

Except: the very fact that I have not been able to bring myself to use a kitchen timer as a Time Timer is undoubtedly evidence that I belong to the class of persons who would benefit from the purchase of a Time Timer.

(key words: environmental dependency; frontal lobe function; ADHD ... in case you were wondering)



snippets from the Time Timer materials:

"Since using the Time Timer, my meetings have never been more efficient or effective. People actually want to attend because I don't waste their time."
- Corporate Department Head

"...perfect idea for people with alzheimers disease who constantly ask their caregivers 'how much longer' type questions..."
- Family Member/Caregiver

"I have seven students (six with autism and one who has Downs Syndrome.) I like using the Time Timer with my students because it does not disturb the others, some of whom are extremely noise sensitive."
- Special Education Classroom Teacher

"The Time Timer really helped me keep track of time at my AP tests."
- High School Student

The booklet says Time Timers are good for timing time-outs, transitions, and 'how much play-time is left.'




Time Timer and motivation

I think the Time Timer is going to help.

Christopher definitely has no concept how much time things take. He'll tell us he has 'nothing' for homework, when it's really 'something'; he just has no clue. I have no clue, either, but I'm at a higher level of cluelessness, to paraphrase Charles. THE ORGANIZED STUDENT suggests using the Time Timer to teach kids how long homework takes, and that's what I'll do, for Christopher as well as for me.

So far today I've seen that the Time Timer has a useful anti-procrastination effect.

I'm in 'waiting' mode at the moment — waiting for phone calls, waiting for feedback, waiting to see what comes next — it stinks.

While I'm always inclined to do what I want to do instead of what I'm supposed to do, Waiting Mode makes everything much, much worse. Waiting Mode makes even structured procrastination hard to do. Since half my productivity is based on Structured Procrastination, this is bad.

So this morning, I set the Time Timer for half an hour.....and then I did productive stuff for half an hour.

The reason I could do productive stuff for half an hour, instead of spinning my wheels NOT thinking about the telephone, was that I could see how fast that half hour was passing me by.

I didn't want to lose it.





My Time Timer isn't nearly this beautiful.



some books that have changed my life
the answer to all of Doug's problems
productivity question
what is an hour? Time Timers
my Time Timer came - how long is a nap?
Time Timer says no!



-- CatherineJohnson - 08 Feb 2006

---

AlphaSmartReducedPrice 23 Jul 2006 - 11:18 CatherineJohnson



I talked to the folks at AlphaSmart today & learned that the price has been reduced 30%. $199 (which is what I paid for mine years ago) down to $139.

The price is reduced because the company may discontinue AlphaSmarts; the woman on the phone wasn't sure.

This news prompted me to buy one for Andrew on the spot. (Impulse purchase alert.)

I hope they don't discontinue the AlphaSmart, but the possibility that they might is reason to buy one before they do, not reason to move on to the new, improved Neo or Dana (though the Neo probably is an improvement).

Ed bought a Dana as soon as it came out and has had problems. I can't remember now whether his problems — losing his research notes from a trip to France — were the machine's fault, or his, but I have a memory the problem was in the machine...

UPDATE 7-23-2006: The original AlphaSmart is no longer shown on the site. Now they're just selling the Neo for $249 and the Dana for $429. I should have bought another AlphaSmart while I had the chance.

When I turned on my AlphaSmart for the first time in at least 2 years last weekend, everything was still there where I left it.

I'd guess that whatever bugs the Dana had at first have been worked out, but I know the AlphaSmarts can live in a backpack.

I also got a new keypad for 25 bucks, AND — once you get going with a completely un-thought-through semi-major purchase, you may as well go for broke — I also purchased two 10-dollar cloth slip covers so as to avoid a repeat of the gummy keyboard mishap.

I'm thrilled Andrew will have his own machine. He can be liberated from doing his addition problems at school with stamps and inkpad. Talk about inefficient.



Neo

from the website:

Features:

Neo is a rugged and lightweight tool that can be used anywhere, with 700 hours or more of operation on 3 AA alkaline batteries or 200 hours on a charge. Instant on/off and autosave eliminate startup delays and accidental data loss.

Affordable and expandable.

Neo offers the lowest cost of ownership compared to other computing technology. Plus, with the extensible SmartApplet architecture, new functionality can easily be added so you can get more from your Neo investment.

Connectable.

Easily synchronize data with a home, classroom or office PC.

Built-in word processor.

Easily synchronize data with a home, classroom or office.

AlphaWord Plus, a full-featured word processor that provides:

• Eight active file spaces for one-key file access
• Named files for convenient file management
• Spell-Checking and Thesaurus
• User dictionary for adding additional words and terms
• Linked files for rubrics, homework instructions, or reference materials
• Find/replace and word count
• Spanish-English word lookup
• Built-in help system for quick access to command reference

The large screen and new font technology display up to twice as much text as the AlphaSmart 3000. Students can save hundreds of pages of text with room for SmartApplets “ software programs extending classroom versatility.

prices

• AlphaSmart $139
  
• Neo by AlphaSmart $249
  
• Neo Rechargeable $269
• Dana by AlphaSmart $379 new price: $429
  
• Dana Wireless $429
  

AlphaSmart
AlphaSmart reviews
AlphaSmart (& letter to LA Times)
AlphaSmart & Andrew & KUMON
AlphaSmarts reduced 30%
AlphaSmart to the rescue
the joys of primitive computing
the joys of primitive computing



-- CatherineJohnson - 10 Feb 2006

---

PrimitiveComputing 15 Feb 2006 - 01:44 CatherineJohnson



At least 2 ktm Contributors have ordered AlphaSmarts since I started obsessing about the things last week.

No question I missed my calling in life. I was supposed to be a Travelling Salesman.

This reminds me of the time my mom and I interviewed her Uncle George, who was the patriarch of the family, to the extent that we had a patriarch, which we didn't.

Uncle George was an engineer. He worked all over the world, in Saudi Arabia, South America — everywhere. He has incredible stories of his wife giving birth in the middle of South American revolutions.

Anyways, we were talking about his father, my Grandad McCammon, a Methodist minister who was president of the first Methodist college in Illinois.

Uncle George said (paraphrasing), 'Dad wasn't really a religious man. He was a salesman.'

I just about fell out of my chair.

I'd been wondering about that.

I like religion myself, and try to 'be religious,' but it doesn't come naturally. It's something I have to work at (and it tends to be something I put off working at.)

Tearing around the internet grabbing folks' arms and urging them to BUY THIS REDUCED-PRICE ALPHASMART NOW! is what comes naturally.

Blood will out.

In case you're wondering, the reason my Granddad McCammon became president of the first Methodist college in IL was that he'd raised enough money to build a Methodist Fellowship Center (I think that's what it's called) at the U. of Ill. Folks had been trying to raise the funds for awhile without much luck. When my Granddad took over, he got the money.

That's selling.

His reward was to be named president of the Methodist College.