I don't know very much about the American curriculum, having grown up with the Canadian one. But I also didn't pay very much attention in math class. I preferred to read the textbook myself, early in the year, and then play around with as many derivations and theorems as I could figure out, occasionally popping my head above water long enough for a test.

I wrote and memorized my own subtraction tables, and invented a baroque and complicated system for writing negative numbers -- for example, 1 - 2 = 9-with-a-circle-around-it, and 5 - 17 = 8-with-two-circles-around-it. Really this is the sort of mistake which could only have happened to me. :)

I'm glad that they're teaching these sort of strategies in US schools. My experience tutoring elementary school math (my son attended an alternative school in which parents all volunteered their own skills & experience) is that every kid has a slightly different conception of how numbers interact. The most important thing I could teach them was that every consistent way of approaching math is correct; if you don't understand the textbook's prescription for subtracting, there are dozens of other right ways to think about the problem; it doesn't matter how you get to the answer as long as you follow the axioms.

I never bothered to memorize trig equivalences. Instead, I just reduced sine, cosine, and tangent (and their inverses) to ratios of the sides of a triangle, and then used the Pythagorean theorem.

Well, it's so much easier and more robust that way! Instead of a long list of confoundingly similar equations, you're left with a single clear understanding of why trigonometry works. After that you can memorize a few formulas as shortcuts if it helps.

Of course this principle completely breaks down when you start working with a child who's already convinced that they can't do mathâ€”or with a group of 30 kids at once, a third of whose mathematical intuitions will be far enough from the textbook norm that no one teacher has enough time to guide them through to that first epiphany.

Comment author:orthonormal
03 August 2010 04:52:11PM
1 point
[-]

it doesn't matter how you get to the answer as long as you follow the axioms.

Well, it does also matter in practice that you can communicate effectively (a lesson I had to learn myself at that age). But learning how to translate from an idiosyncratic system into a standard one can be a source of even better learning, so I agree that kids should not be discouraged from inventing nonstandard but valid systems.

## Comments (390)

OldI don't know very much about the American curriculum, having grown up with the Canadian one. But I also didn't pay very much attention in math class. I preferred to read the textbook myself, early in the year, and then play around with as many derivations and theorems as I could figure out, occasionally popping my head above water long enough for a test.

I wrote and memorized my own subtraction tables, and invented a baroque and complicated system for writing negative numbers -- for example, 1 - 2 = 9-with-a-circle-around-it, and 5 - 17 = 8-with-two-circles-around-it. Really this is the sort of mistake which could only have happened to me. :)

I'm glad that they're teaching these sort of strategies in US schools. My experience tutoring elementary school math (my son attended an alternative school in which parents all volunteered their own skills & experience) is that every kid has a slightly different conception of how numbers interact. The most important thing I could teach them was that

everyconsistent way of approaching math is correct; if you don't understand the textbook's prescription for subtracting, there are dozens of other right ways to think about the problem; it doesn't matter how you get to the answer as long as you follow the axioms.I never bothered to memorize trig equivalences. Instead, I just reduced sine, cosine, and tangent (and their inverses) to ratios of the sides of a triangle, and then used the Pythagorean theorem.

Well, it's so much easier and more robust that way! Instead of a long list of confoundingly similar equations, you're left with a single clear understanding of why trigonometry works. After that you can memorize a few formulas as shortcuts if it helps.

Of course this principle completely breaks down when you start working with a child who's already convinced that they can't do mathâ€”or with a group of 30 kids at once, a third of whose mathematical intuitions will be far enough from the textbook norm that no one teacher has enough time to guide them through to that first epiphany.

Well, it does also matter in practice that you can communicate effectively (a lesson I had to learn myself at that age). But learning how to translate from an idiosyncratic system into a standard one can be a source of even better learning, so I agree that kids should not be discouraged from inventing nonstandard but valid systems.

Your method of subtraction is similar to being the p-adic numbers, you might want to look them up!