Stumbling blocks in math - the way it is written and explained
By Murray Bourne, 17 Dec 2006
I came across this interesting math bashing [no longer available] by some guy who appears to have graduated from high school and was a math award winner. [Warning: Some naughty words in his post - if you are easily offended, don't go there.] His name is impossible to find on his blog.
Anyway, he had some good points:
Let's start with exponents, the way teachers say it.
"The exponent is the amount of times you multiply the number by itself."
You CANNOT multiply 2 by itself with -3 twos (2 to the power of -3). There would be -3 twos in the equation, which is not the same as -2×-2×-2. It's just not possible.
Yep, I'm with him there. But this next one is not as big a problem as he makes out, I feel:
2 the the power of 2 is not two times it self twice. That's 2×2×2, which is correctly known as 2 to the power of 3
2 to the power of 1 is not two times itself once. That's 2×2, which is correctly known as 2 to the power of 2.
To get over this seeming difficulty, he suggests:
"The exponent is the amount of times the number comes up in the equation where it is being multiplied by itself."
Hmm - I can see the problem. "Multiply 3 by itself once" would mean "3 × 3". Fair enough.
"Multiply 3 by itself twice" could mean 3 × 3 = 9 and 3 × 3 = 9 again, so we would have 18. Or maybe just two 9's, with no indication of what to do with them.
But I think he is splitting hairs, since most students don't stumble on this issue, at least for positive integral exponents. They would certainly stumble more if we started to write 21 = 2 × 2. His earlier point about negative exponents is quite right, though.
YOU CANNOT HAVE A [...] SQUARE ROOT OF TWO IF THE NUMBER GOES ON FOREVER AND NEVER ENDS BECAUSE WHEN YOU MULTIPLY THE NUMBER BY ITSELF IT WILL ONLY BE CLOSE TO TWO SO TWO HAS NO [...] SQUARE ROOT.
Neither does any other number that is not a perfect square, because squares are [...] perfect.
There is a similar conceptual difficulty for the case of 1/3 = 0.3333 (repeater). If we add up 3 of those, we get:
1/3 + 1/3 + 1/3 = 0.9999... (repeater). It's not 1, but is "taken to be 1".
Anyway, math is a game. There are a set of rules that we need to play that game by so we have some consistency. So things like following patterns to get definitions for zero and negative exponents is fine, as long as we are all playing by consistent rules.
But misunderstandings that arise because the game is not explained well is a big problem. How many math teachers still say things like "Let a be for apples, b is for bananas, c is for carrots, etc..."? No, a should always stand for the number of items, not the item itself.
Thanks, whoever you are at quietness-awaits, for sparking me to think more about this topic. I have made some appropriate revisions on the Integral Exponents page of my site.
See the 8 Comments below.