Monday, July 14, 2008

How to teach subtraction without linguistic barriers

Too often, Reform Math lets language get in the way, whether in its convoluted, poorly written directions, its convoluted, poorly written word problems, or in its relentless demands that children explain their answers.

But, for students whose math skills far exceed their language skills, even traditional math poses problems.  Consider the term "borrow," as in "borrow 1 from the 10's digit."  And consider the autistic spectrum child who understands neither the word "borrow," nor the underlying (socially-grounded) concept.

In the course of helping my autistic son realize his mathematical potential, I've thought long and hard about how to simplify and mathematize the accompanying language.

Now, in teaching regrouping to my daughter, I'm revisiting what I came up with for my son.

We start by exploiting a common counting error:

"Twenty-one, twenty-two, twenty-three, ....,twenty-eight, twenty-nine, twenty-ten, twenty-eleven, twenty-twelve, twenty-thirteen,...."

Then we look at a particular problem:

-  8

I let my child notice how you can't subtract 8 from 1.

Then I remind him or her of the counting error, and discuss how thirty-one is the same as twenty-eleven.  Then I have him or her rewrite the problem accordingly:

2 11
-  8

First I apply this to the most straight forward problems (where the top number is between 30 and 99). 

Then I introduce the teens:  "onety-one, onety-two, onety-three,... onety-eight, onety-nine, onety-ten, onety-eleven, onety-twelve..."  (My daughter now regularly--tongue in check--refers to 11 as "onety-one", and 21 as "onety onety").

Then I introduce the ones:  "zeroty-one, zeroty-two, zeroty-three..., zeroty ten, "zeroty eleven." (And my daughter renames 11 as "zeroty onety").

Then I introduce, via 90, numbers over 100:  "ninety, tenty, eleventy, twelvety..."

Next I translate specific numbers in the hundreds:  705 is "six hundred and ninety fifteen;" 821 is "seven hundred and twelvety-one" or "seven hundred and eleventy eleven."

Lastly I introduce numbers over 1000, which don't sound so odd to our ears in translation: "ten hundred, eleven hundred, ..."

Finally, I have my child translate specific numbers in the thousands:  1111 is "eleven hundred and eleven" (for carrying from the thousands place to the hundreds place), "ten hundred and eleventy one" (for carrying from the hundreds place to the tens place), or "eleven hundred and zeroty eleven" (for carrying both from the thousands place to the hundreds place and from the tens place to the ones place."

Or, translating directly into numbers, one can write 1111 as:
1111    (eleven in the hundreds place, useful when subtracting 900)
1011(eleven in the tens place, useful when subtracting 90)
11011  (eleven in the hundreds and ones place, useful when subtracting 909)

For my quirky kids, all of this has been surprisingly straightforward.


Anonymous said...

It is me, Liz, from I Speak of Dreams. Just now, I'm not signed into Google, so I show up as anonymous.

Lefty, I've bookmarked this post to get back to soon, but a short comment now.

I'm taking a M.Ed class in k-8 math this summer. I was dreading it, as the prof. has a reputation as a die-hard constructivist.

To my surprise, she is a dedicated mathematician, and is chockful of how to effectively teach kids.

Thursday's class was on the effective use of manipulatives. One phrase: "manipulatives on their own are useless." She went on to say, model, and critique: "You have to help (model and give accurate feedback) the children move from the concrete (the manipulatives) to the semi-abstract (pictorial representations) to the abstract (using numerals)" Another phrase: "Real mathematicians don't say "borrow". They say "regroup".

Another epiphany was the idea that in a subtraction problem, there's only one number: the starting number. The other two numbers are contained within the starting number. Sheesh! Why wasn't that obvious?

OK, I've spent about 20 minutes trying to explain this with words. I'm going to have to shoot a video.

In the mean time, I recommend thoughtful use of manipulatives (a 1-20 number line for single-digit addition and subtraction) and "base 10 manipulatives" for 2+ digit addition and subtraction.

lefty said...

Interesting. I'd love to learn more, esp about recommended uses of base 10 manipulatives. In fact, I have no issues with manipulatives for basic arithmetic. I think they're a great way to teach place value; in fact, that's how my rather traditional teachers taught me way back when (Popsicle sticks). And it's also how my sister learned in kindergarten when we spent a year in France. (They actually combined manipulatives with number renaming, such that "quatre-vingts" became "huitante").

The odd thing about my (odd) kids, though, is that they seem learn place value without manipulatives--without going through the concrete stage.

But I'm sure the number line will come in handy when we get to negative numbers.

Lsquared said...

Montessori teachers use "exchange" instead of either regroup or rename or carry/borrow, as in: you can exchange a dollar for 10 dimes, or you can exchange 1 hundred for 10 tens. (Montessori also has really cool manipulatives for base 10--they start with usual base 10 blocks, but after that there are 2-3 more layers of slightly more abstract manipulatives before kids do stuff with just numbers.)

I really like your trick with the twenty-eleven stuff, though. I'll have to think about where I can use that.

lefty said...

Lsquared, I agree-- "exchange" seems a much more transparent term that borrow or regroup. And using currency dominations seems like a really good way to teach place value.

Speaking of this, I've recently seen how Singapore Math uses dollars and cents to introduce decimals, as early as 2nd grade.

Tthe abstract Montessori manipulatives sound interesting. What are they like?