Nice piece, Jason. Often our students will make enough mistakes for us to “capitalize” on the ‘negative space’. Really it’s a good reminder of the value of mistakes.

I’ve been troubled ever since David Cox asked why students needed to learn the names of the algebraic properties (associative, identity, and so forth). Certainly I see misunderstandings: students confronted with

$latex (3 + x) + 2$

not aware they can combine the 3 and 2 with associative and commutative properties, or the classic

$latex frac{x+2}{x}$

leading the student to cancel the *x* terms rather than consider the distributive property. (*)

Clearly these things are being taught, but what’s going awry when they are used in practice? And why do students learn the names of these things?

It struck me that students only get taught the definition in a positive sense, memorizing (for example) that

$latex a + b = b + a$

and identifying 2 + 4 = 4 + 2 or (2 + x) + 3 = 3 + (2 + x) as the commutative property, and leaving…

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