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

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I asked Ralph Raimi this same question. He theorizes that addition facts are drilled more than subtraction facts. If subtraction facts were drilled more, kids would know the answers more readily. That's his theory anyway.

-- BarryGarelick - 20 Jul 2005

Some of it, though, is definitely the left-to-right business.

Ms. Duque, Christopher's teacher (who I hope will come write for us after she's settled in in LA) walked me thru a division problem to show me how complicated the left-to-right business is for kids.

She said when she asks, 'What is 10 divided by 2?' they will right 10 first, as the divisor, and 2 second, inside the division box, as the dividend.

They do this because they are used to working left to right.

It would be interesting to see if subtraction is harder for Israeli kids whose native language is Hebrew (which reads right to left, yes?)

-- CatherineJohnson - 20 Jul 2005

OK, memory is returning......the reason left-to-right versus right-to-left is a problem is that it increases the demands on working memory.

With addition you go smoothly right to left.

With subtraction w/borrowing you have to veer over to the left to 'borrow' the ten and then veer back over to the right.

At the micro level you're probably doubling the amount of mental work....(or somewhere around there).

-- CatherineJohnson - 21 Jul 2005

With subtraction with borrowing, you may actually have to veer several places over to the left, if all the intervening numbers are zeros. (We've had some trouble with that).

-- CarolynJohnston - 21 Jul 2005

I would like to add an OBJECTIVE comment, knowing well that it would not be viewed as such by the KTM readership--which, in and of itself, makes it much more objective:

Christopher sounds to me like a burgeoning mathematician.

Why do I say this? Because, for one thing, numerate cognition--the idea of thinking primariy in quantitative terms on any one topic is rare (and, thus, rarely applied) in our everyday lives. What is most rare, however, and could be said to belong exclusively to the field of mathematics, is the practice of numerate metacognition--thinking about your (thinking in numbers).

To ask the question Christopher did, it seems to me, one must consider oneself above mathematics. That is, one must see mathematics as it is--as a human enterprise, born of a human mind and extended by each human mind that follows it.

More often than not, our students are slaves to mathematics, incessantly asking of it whether we are worthy--Is this correct? Did I perform this calculation correctly?

Certainly "cinemathematics," as I have posted on before (http://mathandtext.blogspot.com/2005/05/cinemathematics.html) comes into play when we assess the relative difficulties of procedural addition and subtraction.

But, leaving that aside, I think it appropriate that we stop and take note of what a childlike imagination (and childlike questions) can accomplish in the realm of mathematics.

-- JdFisher - 21 Jul 2005

A simple suggestion as to why it's harder:

Think about counting forward and backward. Which do we do most? (Count forward)

Now think of addition as 'short cut' to counting forward.

Subtraction is a 'short cut' to counting backward.

Once you learn how to add (forward) through the next ten, you can add quickly.

So once you learn how to subtract through the ten, that should help.

Just my two cents.

-- CarolynMorgan - 21 Jul 2005

Christopher sounds to me like a burgeoning mathematician.

Wow, J.D.

You just made my day.

As 'fearless' as I am generally speaking when it comes to confronting any subject you have to struggle to learn.....I feel constantly at sea.

I don't understand math at all myself, although I can see that I understand it more than I did. (Bernie once told me that learning math would be good for my humility.)

But with Christopher it's even worse, because I don't understand children's learning processes well; I don't understand how to perceive what they're good or or what they're going to be good at, and he's so much better at social-studies-type subjects that sometimes I wonder if I'm just on a crazy, neurotic, value-what-you're-not-good-at-instead-of-valuing-what-you-are-good-at quest.

I guess what I'm saying is that I frequently wonder why I do the things i do.

HEY!

THAT'S METACOGNITION, ISN'T IT???

THAT'S A GOOD THING!

Boy, I am off today.

You know--I'm definitely going to have to FINALLY get some of the 'core posts' up that Carolyn & I both have talked about forever.....in fact, why don't I do that now.

It relates directly to what you're saying.

-- CatherineJohnson - 21 Jul 2005

OH!

Goodness.

I didn't see your remark about metacognition--I didn't mean to sound like I was poking fun at you.

I've been tramping through the NRC book on learning & the brain, and I'm in the middle of the metacognition section.

-- CatherineJohnson - 21 Jul 2005

To ask the question Christopher did, it seems to me, one must consider oneself above mathematics. That is, one must see mathematics as it is--as a human enterprise, born of a human mind and extended by each human mind that follows it.

Yes, I think that is an example of metacognition....

-- CatherineJohnson - 21 Jul 2005

Carolyn

I hate to say this, but I'm not quite following....

Can you re-state?

I know that's a lot of work!

btw, I'm thinking this thread should be our second submission to EDUCATION WONKS.

-- CatherineJohnson - 21 Jul 2005

Carolyn M -- you might really be right about that.

She's saying that addition is easier than subtraction because, at base, we have more practice counting forward than backward.

Addition is "souped-up counting forward", and subtraction is souped-up counting backward.

Catherine

It's hard to get a clear picture of a 10-year-old's talents -- Christopher could be a late bloomer (and trust me, I know this).

I'm pretty sure that math will be part of Ben's future, but some nights he struggles so much I have my doubts, and the next night he'll ace everything. You need to do a lot of averaging to get a good picture.

I would guess you'll really need a few more years working with Christopher before you'll have a clear picture of where his talents lie. I would actually guess that Christopher is going to be one of the lucky ones who can do anything he wants. You may be perceiving him as less mathematically talented than he is, simply because his verbal skills are exceptional (I would really expect them to be over the top, knowing you and Ed).

-- CarolynJohnston - 21 Jul 2005

I've rethought this and I really do think that counting backwards is the key.

Try these:

Start at 34 (or 53, etc.) and count backwards by 2. Start anywhere. (But twos are too easy, so move on.)

Then count backwards by 3's, 4's, etc.

Two minutes a day. It could be oral drill, it could be written, a 30 second mental math warm-up type activity. But it's got to get to where students pass 10's with larger and larger numbers.

I do those written "30 second" activities a lot, but I don't think I've ever done counting backwards. I've never thought of it as making subtraction as easy as addition.

I'm going to try it this coming year. Thank you KTM.

-- CarolynMorgan - 22 Jul 2005

Saxon does this quite a bit (we're in 6/5). I used to just ignore that part, but I've been doing it faithfully with my one son and I think it makes a difference. Saxon (so far, anyway) has you count up and then back. My son always slows down quite a bit, but it does seem to get easier as we do it. So far the counting has been pretty easy (2's and 5's alot), but the other day he had to count by 5's but starting at 1.

Just asking him to do something different kind of shook him up, but he did grab on to the pattern quicker than I thought he would have. He also isn't freaking out at the mere mention of these exercises. He's got a sense now that these problems are going to be within his grasp, so his confidence seems to be rising. I wish I'd been doing this all along instead of here and there.

I do think the "backwardness" of it is what throws him off, or the "taking away."

Someone once said that divison is like a betrayal to the LD child. They think they're done with subtraction and there it is again.

-- SusanS - 22 Jul 2005

Addition is "souped-up counting forward", and subtraction is souped-up counting backward.

I love it!

That's going on the Wit & Wisdom page!

-- CatherineJohnson - 23 Jul 2005

Carolyn M may be right.

After she brought up the difficulty in couting backwards, I remembered that 'mental status' tests often require you to count backwards.

Here's the only one I could find online quickly:

The Short Portable Mental Status Questionnaire (SPMSQ)
1. What are the date, month, and year?
2. What is the day of the week?
3. What is the name of this place?
4. What is your phone number?
5. How old are you?
6. When were you born?
7. Who is the current president?
8. Who was the president before him?
9. What was your mother's maiden name?
10. Can you count backward from 20 by 3's?

SCORING:*

0-2 errors: normal mental functioning

3-4 errors: mild cognitive impairment

5-7 errors: moderate cognitive impairment

8 or more errors: severe cognitive impairment

*One more error is allowed in the scoring if a patient has had a grade school education or less.
*One less error is allowed if the patient has had education beyond the high school level.

Source: Pfeiffer, E. (1975). A short portable mental status questionnaire for the assessment of organic brain deficit in elderly patients. Journal of American Geriatrics Society. 23, 433-41.
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-- CatherineJohnson - 23 Jul 2005

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