## Theorems Without Thought

Conceptual understanding is overrated. It’s clearly better to teach rote algorithms without comprehension, so long as they can be repeated with minimal effort. Right?

I learned this poem a long time ago…

Ours is not to reason why;
Just invert and multiply!

But this one was new to me…

Minus times minus is equal to plus.
The reasons for this we need not discuss.

Here’s a limerick that provides the formula for a sphere without explanation…

A great circle inside a sphere
Has an area, it would appear,
Exactly one quarter —
Not longer or shorter —
Of that of the sphere. Is that clear?

And I really like this poem of a woman who doesn’t believe everything she’s told…

“Since day follows night,” they told Fawn,
“It is darkness that causes the dawn.”
But Fawn said her cat
Followed her, too, and that
Didn’t prove that the cat’s caused by Fawn.

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• 1. xander  |  November 11, 2010 at 4:42 am

This actually sparks what I think is an interesting area for discussion. Are there times when it is appropriate to provide an algorithm without explanation?

The “negative times negative is positive” rule is, I think, a reasonable case study. In the US, we tend to introduce operations with negative numbers before we start teaching algebra. When they first encounter multiplication with negative numbers, most students don’t really have the background to understand why two negatives should produce a positive. Is the problem that we are introducing negative numbers too soon? or is it okay to give them a hard-and-fast rule for the short term, with the understanding that, if they continue in mathematics, they will get an explanation? or is there a good way to explain the reasoning behind the rule without resorting to knowledge that they can’t be expected to have?

xander

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