In fact I once had this sort of mathematical experience.

Somehow, while memorizing tables of arithmetic in the first grade, I learned that 11 - 6 = 7. This equation didn't come up very often in elementary school arithmetic, and even more seldom in high school algebra, and so I seldom got any math questions marked wrong. Then one day at university, I received back a Math 300 homework assignment on which I'd casually asserted that 11 - 6 - 7. My TA had drawn a red circle around the statement, punctuating it with three question marks and the loss of a single point.

I was confused. There was nothing wrong with 11 - 6 = 7. Why would my TA have deducted a point? Everyone knew that 11 - 6 = 7, because it was just the reverse of 7 + 6 = wait-a-minute-here.

Pen. Paper. I grabbed eleven coins and carefully counted six of them away. There were not seven of them left. I started writing down remembered subtraction problems. 11 - 4 = 7. 11 - 5 = 6. 11 - 6 = 7. 11 - 7 = 4. One of these sums was clearly not like the others. I tried addition, and found that both 7 + 6 = 13 and 6 + 7 = 13.

The evidence was overwhelming. I was convinced. Confused, yes—fascinated by where my error could have come from, and how I could have held onto it so long—but convinced. I set to work memorizing 11 - 6 = 5 instead.

It didn't entirely take. Twenty years later, the equation 11 - 6 = 7 still feels so right and familiar and uncontroversial that I've had to memorize 11 - 6 = stop. I know the answer is probably either 5 or 7, but I work it out manually every time.

Comment author:byrnema
03 August 2010 03:40:47PM
5 points
[-]

I don't know if the American elementary curriculum is better than it was (I hope so) but this mistake is less likely to happen now. My niece in 2nd grade is learning different methods of 'knowing' arithmetic. They still memorize tables, but they also spend a lot of time practicing what they call 'strategies for learning the addition facts'.

For example..

11-6 = (10-6)+1 = 5 is the compensation approach.

and 11-6 = 10-5 is the equal additions approach.

They also spend a lot of time doing mental math. I'm impressed with how different things are, and hope that students are doing better with this more empirical, constructivist approach. (My niece is good at math anyway, so I don't know if she's getting more out of it than average.)

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 math—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)

OldIn fact I once had this sort of mathematical experience.

Somehow, while memorizing tables of arithmetic in the first grade, I learned that 11 - 6 = 7. This equation didn't come up very often in elementary school arithmetic, and even more seldom in high school algebra, and so I seldom got any math questions marked wrong. Then one day at university, I received back a Math 300 homework assignment on which I'd casually asserted that 11 - 6 - 7. My TA had drawn a red circle around the statement, punctuating it with three question marks and the loss of a single point.

I was confused. There was nothing wrong with 11 - 6 = 7. Why would my TA have deducted a point? Everyone knew that 11 - 6 = 7, because it was just the reverse of 7 + 6 = wait-a-minute-here.

Pen. Paper. I grabbed eleven coins and carefully counted six of them away. There were not seven of them left. I started writing down remembered subtraction problems. 11 - 4 = 7. 11 - 5 = 6. 11 - 6 = 7. 11 - 7 = 4. One of these sums was clearly not like the others. I tried addition, and found that both 7 + 6 = 13 and 6 + 7 = 13.

The evidence was overwhelming. I was convinced. Confused, yes—fascinated by where my error could have come from, and how I could have held onto it so long—but convinced. I set to work memorizing 11 - 6 = 5 instead.

It didn't entirely take. Twenty years later, the equation 11 - 6 = 7 still feels so

rightand familiar and uncontroversial that I've had to memorize 11 - 6 = stop. I know the answer is probably either 5 or 7, but I work it out manually every time.I don't know if the American elementary curriculum is better than it was (I hope so) but this mistake is less likely to happen now. My niece in 2nd grade is learning different methods of 'knowing' arithmetic. They still memorize tables, but they also spend a lot of time practicing what they call 'strategies for learning the addition facts'.

For example..

11-6 = (10-6)+1 = 5 is the compensation approach.

and 11-6 = 10-5 is the equal additions approach.

They also spend a lot of time doing mental math. I'm impressed with how different things are, and hope that students are doing better with this more empirical, constructivist approach. (My niece is good at math anyway, so I don't know if she's getting more out of it than average.)

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

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 math—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!