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13 Sep 2005 - 14:03

## conceptual understanding vs numbers

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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From the article you quoted:

"Everyday Mathematics has a saying, 'trust the spiral,'" said a teacher in a suburb of Philadelphia who has been using Everyday Mathematics for four years. "But I think that's just a hoax." Because she helps teachers throughout the school with the program, she sees how it works in all the grades. "The child that isn't getting it in second grade doesn't get it in third grade," she said.

This makes me furious. They come up with some little 'saying', a bit of jargon, that implies that they have some secret wisdom about the effectiveness of spiraling that you don't have. And then it comes to naught because, of course, it was a lie and a cheat to begin with. No problem for them -- just for the kids they failed to teach.

-- CarolynJohnston - 13 Sep 2005

Spiraling doesn't mean that you allow students to go on to the next grade without understanding the required material. It's not circling, but that's how many schools implement spiraling. You are supposed to cover the same (plus new) material, but at a greater depth. When I was in school, I had a course called Algebra I and then a course called Algebra II. One could call that spiraling.

However, many know that nowadays spiraling is another way to implement social promotion and developmentally appropriate ideas. Our public schools have these detailed documents describing how this works. It's called ITRE. 'I' for the year that the material is introduced, 'T' is for the year in which the material is taught. 'R' is for the year in which the material is refreshed, and 'E' is the year before which the material has to be mastered. This is not spiraling on to new, advanced material. It is circling. The document even specified that 20 percent of the students are allowed to continue past the 'T' year without understanding the material. In a full-inclusion, child-centered, mixed-ability classroom, what are the better students doing when the 20 percenters are brought up to speed? What happens, however, is that the material is just "refreshed". (rehashed?)

-- SteveH - 13 Sep 2005

I felt the same way when I read that. It's horrifying; it really is.

Plus it totally ignores the 'ageing' & 'maturing' factor in learning math. I've got to get a post up about this. The longer you KNOW something in math, the deeper that piece of knowledge becomes, as you encounter new knowledge, and new extensions of it. This is why, I believe, you can't 'cram' math. The element of time is critical.

I saw that a couple of weeks ago with Christopher & his friends Drew & Marc.

Drew & Marc have been in Phase 4 math from the get-go, so they learned material a full year before Christopher did.

When I tried to teach Christopher the Saxon 8/7 lesson on subtraction-of-fractions with borrowing, he couldn't do it--and he got upset.

Drew & Marc happened to be coming over that day, and I asked them if they could do it.

Sure enough, they could. They'd been taught back in 4th grade, when Christopher was languishing in Phase 3. Not only did they remember how to do it, they did a good job of explaining to Christopher (or at least demonstrating) why they were doing what they were doing.

I'm pretty sure I had already taught Christopher a lesson on this, and his Phase 4 teacher this year had done so, as well, but he hadn't understood.

Drew & Marc had been TAUGHT (taught 'til they mastered it) how to subtract fractions with borrowing in 4th grade, and they'd had a full year to let that knowledge sink in and take hold.

-- CatherineJohnson - 13 Sep 2005

Steve's right. I was shocked when I saw one of my favorite curricula--either Saxon or Singapore (I think Singapore)--described as 'spiraling.'

I was used to 'spiraling' meant 'circling.'

What Singapore (I think!) meant was SPIRALING.

A concept is taught to mastery in 1st grade, then revisited and taught to mastery again in 2nd grade, with new concepts built on the earlier concepts.

-- CatherineJohnson - 13 Sep 2005

In that sense though, every curriculum is spiraling. For example, if a kid is taught how the distrib property applies to mental math in 3rd grade, then how it applies to multidigit multiplication in 4th, then how it applies to algebra in middle school, isn't that essentially the same concept being revisited? The difference is that the idea is taught to mastery in that new context, every time.

What E-math means by spiraling, conversely, is literally circling!

-- CarolynJohnston - 13 Sep 2005

Wow, I'm playing pretty fast and loose with the boldface today.

-- CarolynJohnston - 13 Sep 2005

one possible reason for not having understood
the "difference" of two numbers as the amount
by which they differ: many students
learn to take it for granted that math terminology
is intrinsically obscure.

and this is perfectly natural ...
it really is easier to say,
and to a certain extent, to understand, e.g.,
"multiply this by this; then add this"
(with appropriate waves of the pencil) than
"quotient times divisor plus remainder".
and students will tend to prefer the former
for this (very good) reason.

careful teachers will habitually use
lots of technical terms anyway
(though probably not exclusively) ...
we want them to know (not only "how do i check
the answer in a long division", but also)
what the words "quotient" and "remainder" mean
(so that we can use them in other contexts).
if there's world enough and time, one should
ensure that the student(s) can (correctly!)
use the vocabulary actively.

now, in the same example, "remainder" (obviously?)
refers, just as in natural english, to
"something still there" (something "left over";
what have you), but "quotient" has no such
natural language association (unless one knows latin):
it's just an arbitrary sound to be memorized.
if you pile up a lot of those in a short time,
a beginner should be forgiven for feeling that
understanding natural language isn't much help
in understanding whatever mathematics may be at hand.

but it gets worse. i suppose the reason i chose
the example i did is precisely that "divisor"
and "dividend" sit somewhere between
"plain english" and "arbitrary sound" --
it's pretty easy to guess that they've got
something to do with "division" ...
but which is which? i resented this when
i first encountered it: why should anyone
learn this teacher talk?

probably i also resented terms like "sum" and "product"
for what might very well have appeared to me
to be the same reasons (i've already learned
perfectly good words ("add" and "multiply")
that allow me to speak intelligibly about these issues!
get off my case with the mumbo-jumbo and tell me
something i don't know already!) ... well,
i've come around where "sum" and "product" are concerned
but still feel that "divisor" and "dividend"
may very well be best avoided (sometimes).
(i usually say "denominator" and "numerator"
at least in part because i can explain
what these names mean [what kind? how many?]
... if only in a different context from "long" division.)

all that said. it's very helpful if there's always
a dictionary close by and the teacher shows by example
that this doggone thing is the holy scripture
as far as scholarly pursuits (like mathematics!) are concerned.
yours "fastly" and loosely with "quotes" and italics,
http://vlorbik.com

-- VlorbikDotCom - 13 Sep 2005

Back to your point about the meaning of concepts, it's a great part of the difficulty of mathematics that it functions on two levels, the formal and the meaningful. Nobody gets it all; everyone has lacunae in their knowledge. Many great mathematicians function on purely a formal level and it never bothers them. It's rather an autistic trait. It's also a trait common to children.

It seems, more generally, to be a standard aspect of mental development as one ages that one begins to draw connections between things and to ascribe greater meaning to things, rather than simply to take them at face value, within their own reference system.

Personally, it has become almost impossible for me to read a pure math book any more, because I have come to realize that almost all of them are purely formal and have no connection whatsoever to the real world. They start with some assumptions and work them out very formally and very elegantly for several hundred pages, but in the end they have the heft of a wasps' nest. They crumble in the fingers.

Regarding the Singapore math bar models, they are good. But there are literally thousands of things like this which must be grasped, idea by idea, in order to "understand" mathematics. Each such grasping requires some effort, some questioning, some unconventional thought, and no one has infinite energy. "The journey to the mountain begins with the first step." There really isn't anything to be done about it except to pick up the backpack and begin the journey. It has always seemed to me that most attempts at math "reform" are simply trying to square the proverbial circle, to make the journey shorter and easier when it cannot possibly be made shorter and easier. Ergo, the only thing they can really do is to remove things from the curriculum, giving the appearance of shorter and easier. That works fine as long as there are no standardized tests and no comparison to foreign countries which are not trying to perform acts of legerdemain, but simply want to get the job done right. Like Americans, in a more naive age, prided themselves on.

-- BernieJohnston - 13 Sep 2005

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

Is it?"

Yes. A partition of a positive integer is a representation of the integer as a sum of positive integers where the order of the summands is not considered. So 1 + 1 + 3 and 1 + 3 + 1 are both the same partition of 5. There are 7 different partitions of 5. This concept turns out to be quite useful in some areas of mathematics.

-- KtmGuest - 13 Sep 2005

-- VlorbikDotCom - 13 Sep 2005

hmmm ... empty comment.
hopefully the least interesting comment
i've posted on this thread so far.

it doesn't look much like "partition theory"
to these jaded eyes (as i wanted to've said);
that theory would begin at the level of
"exercise 1: find the 7 partitions of 5"
(already implicit in KtmGuest?'s post ... right?).

-- VlorbikDotCom - 13 Sep 2005

"There really isn't anything to be done about it except to pick up the backpack and begin the journey."

My first impression of modern progressive education was the overwhelming feeling that they were doing everything to avoid buckling down and getting to work. Everything had to be play learning, indirect learning, or learning by osmosis. All learning has to be fun and natural. Any other type of learning is rote and/or destructive to one's educational psyche. Therefore, they can't tell the difference between developmentally inappropriate and needing a swift kick in the rear end. I guess the second is developmentally inappropriate by definition.

-- SteveH - 14 Sep 2005

in the same example, "remainder" (obviously?) refers, just as in natural english, to "something still there" (something "left over"; what have you), but "quotient" has no such natural language association (unless one knows latin): it's just an arbitrary sound to be memorized

Interesting.

Yes, I'm constantly bumping into this issue, both with myself & with Christopher.

I've come to appreciate having formal mathematical terms to use for things; it helps me realize that, yes, 'difference' is something unique.....it's not just 'take away from.'

I spent my whole life thinking of the 'difference' as, in fact, a remainder.

-- CatherineJohnson - 14 Sep 2005

Back to your point about the meaning of concepts, it's a great part of the difficulty of mathematics that it functions on two levels, the formal and the meaningful.

This is always interesting.....because I find that there are occasions when the 'formal' feels 'meaningful,' and is sufficient.

It's quite strange, and I can't explain it.

At other times a 'formal' demonstration (I hope I'm using the term the way you are) doesn't work at all. The concept or procedure continues to seem like 'magic.' (That continues to be my benchmark for whether I do or do not understand something. If the concept or procedure seems like magic, then I don't.)

I really have no idea what conceptual understanding of math is in the phenomenological sense of the term 'understanding' (and I hope I'm using that word right, too).

What I mean is: I have no idea what it is that makes one FEEL that one has achieved conceptual understanding.

btw, this all reminds me a bit of various discussion of OCD I've read calling it the 'doubting disease.'

One researcher said that in OCD something has gone awry in the 'checking' system in the brain, so that the sufferer checks & checks & checks again but never FEELS he has checked. The off-switch is damaged, or the 'meta-cognition' switch, or some such.

I have no idea what my I-know-this-piece-of-math switch is, or what trips it.

-- CatherineJohnson - 14 Sep 2005