How to teach subtraction without linguistic barriers

An ancient post from Out in Left Field, relevant to Students with Autism.

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: 

31

- 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: 

211

- 8

First I apply this renaming 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)  

10111    (eleven in the tens place, useful when subtracting 90)  

11011            (eleven in the hundreds and in the ones place, useful when subtracting 909).  

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