I went to Andrew Stadel’s afternoon session at TMC yesterday: Math Mistakes and Error Analysis: Diamonds in the Rough. I got out of it a fun technique to get students thinking, and some new perspective on building deep knowledge. Andrew talked a great deal about the value of mistakes, and…]]>

I like what Andrew has done here, but I wonder if asking students to identify the mistake or the misconception might be useful.

I went to Andrew Stadel’s afternoon session at TMC yesterday: **Math Mistakes and Error Analysis: Diamonds in the Rough.** I got out of it a fun technique to get students thinking, and some new perspective on building deep knowledge.

Andrew talked a great deal about the value of mistakes, and using those mistakes as opportunities for students to learn. His first proposal was simple, and looked like this:

Simple, and an awesome way to get students thinking beyond answer-getting to the mathematical structure. I’m excited to give this a shot in my class this year.

Building off of Andrew’s ideas, I’ve been thinking a great deal about how students come to understand broad, abstract ideas. Dan Willingham has been a source of great thinking here. I’m coming more and more firmly behind the approach that students build abstract understanding through the variety of examples of an idea they encounter, and from…

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

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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