Deschooling math

This morning Anya and I sprawled on the floor with the Magnadoodle and I wrote out a math problem: 12 + 14, written vertically. We haven't done math in a month, but there was a brief time when she could do two-digit addition. In fact, she did a few pages worth. She'd draw a vertical line to separate the tens and ones, and add each column separately.

But this morning she couldn't remember how to add 12 and 14, and I knew why. I had known all along that she wasn't understanding the tens and ones places, and she didn't see that 12 + 14 is just 10 + 10 + 2 + 4, or 20 + 6, or 26. She certainly didn't see that our vertical addition algorithm is nothing more than a short-cut for this regrouping process. So of course she didn't remember it. It takes more than three or four worksheets to memorize an algorithm that makes no sense to you.

Interestingly, this morning she did attempt to break the numbers down into parts that were easier to add. She divided both 12 and 14 in half, and tried to add 6 + 7 + 6 + 7. Unfortunately, she was then stuck with 13 + 13, which got her nowhere-- but it was the right idea.

Then she told me she was going to write down the "close answer" (close as in near). She said "It won't be exactly right but it's the close answer." And she wrote 24 underneath, because she knows two dozen is 24 and since 14 is close to 12, the answer must be close to 24.

We did another problem, 35 + 43, and Anya tried to figure it out by determining how many fives there were in 43 and then trying to count up from 35 by that many fives. She did count that it took eight 5s to get to 40, but got frustrated by not knowing what to do with the 3, and somehow that idea fizzled out.

All of these attempts (and other variations she tried) required more thought and more advanced math, such as dividing 43 by 5, than if she had remembered the cookbook rules. If kids memorize those rules too early, and spend their remaining time on mere quality control (speed and accuracy), this prevents them from understanding math. But most classrooms aren't designed for allowing the natural logic of arithmetic to be discovered. Math manipulatives are often used in early grades, but when "the rules" must be memorized before many kids are developmentally ready to deduce those rules or truly understand them, the manipulatives thing becomes lip service. Kids will learn to get the right answer the same way that I make my car run without knowing a thing about the engine. Follow the rules and forget what's happening inside the black box-- just like working the assembly line, not coincidentally.

Going back to this morning-- at another point we were playing with marbles. I made a grid of 12 marbles, in 3 rows of 4, and showed it to Anya. If you look at it that way, it's obvious why 3 x 4 is the same as 4 x 3. One way it's 3 rows of 4 and the other way it's 4 columns of 3, but it's 12 marbles either way. (This also comes up with legos: is it a 2 x 4 piece, or a 4 x 2 piece? Either way it's got 8 dots.) I also showed Anya why 3 x 3 is called "3 squared," since you literally make a 3 by 3 square. Anya got some marbles and quickly figured out that she couldn't take ten marbles and make a multiplication problem using 4, because you can't make even rows of 4 with exactly 10 marbles. (I resisted using the term "divisible," since it won't sink in until she's run into this issue a number of times.) And a while back, she figured out (by playing with plastic circles cut into different fractions) that you can't make a fraction equal to one half unless you have an even number of pieces, which means the number on the bottom has to be even.

In contrast, I don't think I understood multiple-digit arithmetic until at least three years after I had, as far as the school was concerned, mastered these concepts. One day I was thinking about dimes and pennies and I suddenly realized what borrowing and carrying were all about, that it was simply a matter of changing in too many pennies or changing a dime for needed pennies. It would have been a lot easier to have had that revelation at the start. Another example: it was only in the last year that I realized why the area of a right triangle is 1/2 the base times the height, and that is the simplest thing imaginable. A right triangle is always half a rectangle, and what's the formula for the area of a rectangle? Duh. It's embarrassing, but see, I had no trouble memorizing the formula, and then I just never thought about it again. I'd see a right triangle and "1/2bh" would leap to mind, and that was the end of that thought process. So, at best the rules are unrelated to real understanding; at worst, they prevent real comprehension.

One of the more common questions voiced by those who don't homeschool is the whole "What about math?" thing. I suspect they believe memorization and repetition is the only way to learn math because, again, the rules don't make any sense and can't be arrived at organically. I suppose this is why most people hate story problems: because story problems make it harder to determine which set of memorized rules to apply. They try to make you think about the concepts involved, but many people never had a chance to grasp those concepts.