^{2}a squared number is what I call a 'dangerous concept', as in a little, which I read as less than complete, knowledge is a dangerous thing. 3 * 3 = 9. 9 is a point on the number line. For smart people this makes no difference but I think this concept contributes to student's difficulties with units - as explained in the original power-weight ratio post.

Another way this can be seen is if someone says how silly negative - or in some case imaginary - numbers are. "How can you have -3 of somethimg?" is a variation of the popular refrain. This hints to a larger misunderstanding of numbers. When I heard this in school, I would ask my classmates, "Can you give me 3?" The key here being you can only give me three of something: 3 pencils, 3 rubbers, 3 rulers. The unit reifies the number.

And so it is with multiplication that 3units * 3units = 9units

^{2}. It is the unit that becomes a square, not the numbers. If kids were thought that numbers were raised to the power of two but units were squared, wouldn't that help them to grasp the concept.

Along a similar theme, my next PWR post will be on denominonsense.

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