Normal math isn't enough....

The Problem:  Caleb initially had a hard time grasping borrowing because he already understood negative numbers, and I had learned borrowing when I was 7 or 8 using the phrase, "you can't do that--the bottom number is too big" and I used that on him, which perplexed him. Too big for what? Why can't you do it? It just produces a negative answer--that doesn't make it impossible!

Two solutions:
Eventually he got it when I changed it to, "It's against the rules to have a negative number in the ones column in this kind of problem."

In the process, we managed to develop a front-end method of two-digit subtraction that doesn't use borrowing ever, and that allows negative numbers in the ones column. (Consequently, this is actually totally useless for teaching math to most 8 year olds because you have to understand negative numbers in order for it to work! Alas....nothing game-changing here.)

Here's how it works:

Sample problem:

 34
-27
-----

First you do the ten's column:  3 tens minus 2 tens is 1 ten, or 10.
Now do the ones column: 4 - 7 is -3.
Now add the two answers: 10 + -3= 7

For me, it's faster than "cross out the three, write a 2. Put a one in front of the 4. 14 - 7 = 7."

Caleb hasn't tried it with 3-digit numbers yet, but for 2-digit numbers, it makes it so he (and I) can do problems in his head that normally you'd have to do on paper (with borrowing, etc).

I never thought I'd have fun playing with numbers, but doing it again as a grown up, with the pressure off (no grades!), I'm really enjoying it. Part of that might be because I'm using Saxon math with Caleb, and Saxon makes sense, teaches tricks right up front, and makes it really easy (not that I ever struggled in math; I was really quite good at it. But I never understood the math behind the numbers. I approached it the same way I did reading music and cooking: the figures on the page give you specific instructions; if you follow them, you get the right answer and who cares WHY it works. It's kinda fun to understand why now.). Part of it might just be maturity.

I wonder if it works with 3-digit subtraction. Let's try it.

 721
-386
------

7 hundreds minus 3 hundreds is 4 hundreds.
2 tens minus 8 tens is - 6 tens
so....340
now 1-6=-5.
335.

I did it in my head.

Whoa.

Funny thing: when I tried to check my answer in my head but using traditional borrowing math, I got the answer wrong and had to get out a calculator. And then work through it again to figure out where I messed up.