Consider the math mistakes below. Not real samples of student work (for that, go here), but real mistakes. I’ve seen each one. I think you’ll recognize them.

**Homework**

Answer questions 1 and 2.

1. What math mistake did each student make?

2. What are some implications for our work?

Good. Now answer questions 3 and 4.

3. What role did memorization of the times table play?

4. What are some implications for the conversations we *could* be having?

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I shall take this moment to get on my exponents soapbox.

Two of these mistakes essentially have to do with a kid’s comfort with exponents. Exponents are important enough in high school that we need to be a lot more thoughtful about how we introduce them. The best elementary and middle school curricula have kids playing with a variety of non-algorithmic models for the four basic operations. We need to start thinking about the five basic operations instead. And we need to start developing a hierarchy of models that will come before and after a kid’s formalization of exponents as repeated multiplication.

I assume your soapbox is a perfect cube.

I like this. What are you thinking here? Geometric representations (squares/cubes)? In MS, we may start here but by the end of that lesson we’re multipying numbers by themselves. Thinking of “number of factors” instead? (This idea yours? Maybe Danielson’s?) Tell me more (or drop a link) about these hierarchies.

Each of the Big Four has an inverse. So if we add exponentiation…

Question 3 was a light bulb moment for me. Memorizing (rather than reasoning with) times tables as a possible explanation for the prevalence of mistakes like 3^2 = 6 had never occurred to me. Thanks!

A deep association between 3, 2, and 6 getting in the way? Could be. Why 3^2 = 6 and not 5^2 = 10? This is puzzling. The inconsistency of multiplying the base by the exponent makes it easy for students/teachers to brush it off as a silly mistake (“brain fart” in my classroom) but there’s more to it than that. Even students who do not make this mistake with whole numbers will think 100^1/2 = 50 when introduced to rational exponents. We

wantto multiply.Perhaps as much as the quick memorization of times tables, the quick recall (or glance at the formula chart) of the Pythagorean Theorem, without a solid conceptual understanding of what it means would lead to some easily fixable errors you present here. Great, thought-provoking questions. I like this activity.

Thanks, Geoff. I hadn’t connected the automatic recall of the times table to that of the Pythagorean theorem.

I think the issue here is each student is following an algorithm rather than really displaying an understanding of how Pythagorean Theorem works. I no longer even mention a^2+b^2=c^2 when teaching Pythagorean, as can be observed in this video: https://www.youtube.com/watch?v=9755B-aLI2E&feature=c4-overview&list=UUKZYfQiLaYwyM-zVsTIbuZg

thanks for sharing, I might have students respond via google forms and then (hopefully remember to) send it your way.