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# “Two cubed is eight, but seven squared is fourteen.”

1. They do know the definition of exponents. It’s written a line above. They did it a line above.
2. They’re doing this with confidence. There aren’t erased numbers. This isn’t slow thinking. This is just what kids think seven squared ought to be.

By defining exponents in terms of multiplication while offering no other images or models for what exponentiation does, we create a default model for exponents that sticks with people forever. When mentally taxed — either with a tough multiplication, or with an unusual power — kids revert back to this default model. They’ll do this especially in high school, and they’ll get questions wrong on tests and all sorts of other things not because they’re being sloppy, but because this default model is constantly lurking in their minds.

Incidentally, I asked these kids why they think about multiplication when they see powers, and this is what they said:

I’ve written about a lot of this stuff before. See here, especially, where I shared the high school versions of this mistake.

Now that all of this has been established, the next step needs to be finding a curricular approach that doesn’t rely as heavily on the “repeated multiplication” model for exponents. We need to build a distinctive set of images and intuitions that are native to exponents so that our kids aren’t always defaulting into multiplication when they have to think hard about math.

This is work that I’ve started, in a post titled “Exponents Without Repeated Multiplication.”  I’ll send you there for the details, but I stake out two major claims about exponents education:

1. Much in the way that arrays support early multiplication work, geometric notions of area and volume can serve as the bedrock of an exponents education
2. We tend to think of four, not five, major operations of arithmetic, but we need to start thinking about exponents as on par with all the others and taking care to build them thoughtfully throughout the entire curriculum.

Beyond all of this, these exponents mistakes serve as a big reminder about the nature of learning, teaching and knowledge. The big, big lesson of all of this is that knowing/not-knowing is not clean and it’s not binary. There are degrees of knowing something. Would you say that these students don’t yet understand what exponents mean? What does that even mean, given the contradictory evidence we have in front of us.

But, then, what does it mean to understand something at all?

## 4 replies on ““Two cubed is eight, but seven squared is fourteen.””

Michael Paul Goldenbergsays:

One thing that MIGHT account for the frequent error on 7^2 by kids who mostly seem okay with the other problems is that they don’t know what 7×7 is immediately, but they do know 7×2, so that default isn’t just a conceptual misunderstanding but one grounded in how accessible a given fact is. If they are generally confident about positive integer exponents with small or “easy” (e.g., 10) bases, and then hit one that doesn’t immediately come to mind (e.g., 7 x 7 = 49), then they pull up the first “related” fact that they do know involving 7 and 2.

I could be utterly off-track (and there’s little guarantee that my theory explains the error in EVERY case), but it’s one shot that might hold up under further inquiry.

I love what you’re doing here and I am going to give you a shout out on my blog. I am surprised at exponents in grade 3. We did not touch them until grade 5; now we’ve eliminated everything there except powers of 10. Curious to know what curriculum you use and just how exponents fit into its sequence.

I think that in my ideal world, kids would be coming into 5th grade already being comfortable with the language of squaring and cubing, and ready to extend that language to all integers.

Zenosays:

Some thoughts:

All of the students got 2^3 = 8. I suspect they’ve seen that example frequently in class and didn’t have to do any calculation. They just knew, as a recallable fact, that 2^3 = 8.

All but one student got 10^4 = 10,000. The one who got it wrong had an extra zero. It seems possible that the students knew the special fact that the exponent in a power of ten indicates the number of zeros. Again, no actual calculation involved, just adding zeros.

All the student mistakenly got 7^2 = 14. This might look like they’re just multiplying the exponent and the base, but maybe they’re not. Perhaps they know that 7^2 involves combining two 7s but they’ve mistakenly used addition (7+7=14) rather than multiplication (7*7=49) to combine them. Their thinking might be something like: “The exponent means I have two sevens. Two sevens is fourteen, so the answer is fourteen.” Note the student comment that “10^3 I think 10+10+10”.

Three of the students got 3^4=81. The two that got it wrong got 36 and 108. They didn’t simply multiply the base and exponent. It looks as if what they might have done is 3*3*4=36 and 3*3*3*4=108. They seem to have started off correctly with successive multiplications by the base, but then for some reason used the exponent as the last multiplier. Is it possible they just lost track of what they were doing? To do the iteration they need to keep the multiplier (base), the running total, the loop count (multiplications done), and the loop limit (exponent) all in mind. Perhaps at the end they confused the loop limit with the multiplier for the last multiplication.