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.

Square roots.

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.

8 Comments on “Stumbling blocks in math - the way it is written and explained”

  1. kontan says:

    LOL, thanks for pointing out the freaky code on my site! Odd.

    Now for your post...I didn't go to the guys site. No time to split hairs with him you know...Reading what you have posted, I have to wonder if the guys is a teacher. Sometimes teachers have to say things in a way that will seem absurd to a nonteacher. I agree that sometimes what is said seems inaccurate, but if you are like me and have difficulty with mathematical concepts, someone breaking it down into simplistic wording helps to make it sink in. JMO. Once you figure out the basic then you can understand the actual concept and get the wording right.

    Thanks for stopping by

  2. Salem SAID says:

    Everything he wrote is right, in a way. In mathematics the same notion takes on many forms and exists at many different levels. Also, like Bohr says, the opposite of a false proposition is a valid proposition but the opposite of an important idea is an important idea...this is all funny. Here is what I think concretely,

    1)mathematics is about experience more than knowledge, like chess.
    2)mathematics needs commitment, if someone doesn't want to give math a chance to explain its apparent contradictions, they will not understand.

  3. arup dasgupta says:

    i agree with what has been said about a teacher's practical difficulties in lower classes.
    Why look on Maths alone? Consider Geography:

    *motion of earth : would you tell a student that motion of earth is verrrrrrrry complex (because it is)
    *rotates on its own axis
    *revolves around sun
    *entire solar system on a spiral arm of galaxy
    *galaxy itself rotating
    *galaxy itself moving away (at least by big bang theory)

    so what do YOU teach???

  4. Murray says:

    Arup - Thanks for the extra dimension concerning geography.

    Seems like this is a common dilemma - how much to reveal and when...

  5. Richard Miller says:

    so,,, how does a 50 something year old who NEVER took algebra in high school 30 some odd years ago do algebraic equations? how do you read it?

  6. Murray says:

    Hi Richard. There are many people in your situation here in the IntMath community. And most of them are surviving just fine!

    A key thing is - never give up, and as this article says, don't be too concerned (or confused) by the notation.

  7. Alex says:

    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?.

    just wanna suggest dat 1/3+1/3+1/3 does equal to 1
    Proof:
    Let x=0.9999...
    so 10x=9.9999...
    10x-x=9
    x=1
    In calculus there's another way to prove this by sequence
    it is not taken to be 1
    if you have to forcefully say that, then every number would not be a whole number
    eg: 2=4/3+2/3=1.3333...+0.6666...=1.9999...

  8. Murray says:

    Hi Alex. Your response reminds me of the joke:

    An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a quarter of a beer.

    The bartender says, "you're all idiots", and pours two beers.

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