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## Combining like terms…

…is probably looking like an unfortunate name for adding monomials when you see a mistake like this one. Right?

Or wrong? Speak up in the comments. And more thanks to John Weisenfeld for the mistake.

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## Simplifying exponents

How would you help this kid out?

Thanks to Chris Shore for the submission!

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## Negative and Fractional Exponents

I want to share a theory on this mistake:

The student had an association between negative exponents and reciprocals and “half-powers” and square roots. As the student was parsing the problem he “fulfilled his obligation” to use that association on the number. I guess what I’m positing is that the mind works by making a connection, and then remaining in tension until that connection is used in a problem. (I’ve often had the experience of feeling as if there’s an insight that I haven’t used yet in solving a problem, and it’s like having a small weight on my back.) The mind comes to relief at the moment that the insight is used.

The student’s mind made the connection between negative powers and reciprocals and was in tension. He then used this insight at the first opportunity he saw, to relieve himself from the burden of his insight.

Some of you might disagree. For instance, you might think that the student had just memorized some rule poorly, had no conceptual understanding of powers, and gave the answer that he did.

But I think that the answer felt right because he used the fact that he knew. I’d predict that this student would be able to answer $x^{1/2}$ correctly.

If you think that the student just memorized a rule, is there any reason to think that a student would get a question such as $x^{1/2}$ correct?

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## Combining Like Terms + Exponents

Here are 7 mistakes. There represent all of the variety of mistakes from a selection of 36 students. The first two mistakes were repeated by several students, but the last 5 were unique in the sample.

Which of these mistakes would you predict? Which ones surprise you? Can you make sense of them all?

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## Exponents – Positive Integer Powers

Thoughts?

(via Chris Robinson)

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## Negatives

What’s going on in this (reconstructed) student work? Tell a story in the comments.

And then go thank Christopher Danielson for sharing this stuff.

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## Zero and Exponents

This post is brought to you by David Wees. Thanks, David!

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## The Distributive Property of Exponents

Why is this mistake so enticing, and how might you help students avoid it?

[Compare: http://mathmistakes.org/?p=396]

Thanks again to Anna Blinstein for the submission. Follow her! Virtually! Not literally!

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## Cross-Multiplying

What’s the mistake? Diagnose the disease, and find the cure in the comments.

Thanks to Anna Blintsein for the submissions. Go follow her on twitter!