Welcome to the Blogarithm! We’re exploring ideas raised by the Standards for Mathematical Practice in the Common Core State Standards for the middle grades.

I noticed something while watching an eighth-grade class. The teacher was getting the students to notice the rule about adding exponents, asking them to figure out 2^{4 }x 2^{3} and other similar cases. When they saw it was 2^{7}, and that you could, in general, just add the exponents with the same base, she asked them why it is always true. One student said, “Because you are just multiplying 2 times itself 3 times and then 5 more times, so it’s 8 times total.” They then did it for division (e.g., 2^{7 }÷ 2^{3} = 2^{4}), which was just a rearrangement of the same thing.

Later, the teacher was talking about 2^{0} and negative exponents like 2^{-}^{3}. A student asked why 2^{0} is 1 because shouldn’t 2 times itself 0 times be 0? Hmm. The teacher fielded this pretty well by asking the class what 2^{3 }÷ 2^{3} was, based on the subtraction rule. They said that it was 2^{0} and noticed that it also had to be 1 because it was 8 ÷ 8. So it makes sense to say 2^{0} is 1. Crisis averted.

This weird bootstrapping thing happened. Students—

- started with a definition (exponentiation as repeated multiplication)
- discovered a rule and justified that it
*had to *always work, based on this definition (addition/subtraction rule of exponents)
- generalized the rule to a new domain on which the definition, and hence the justification, does not even make sense (0 and negative integer exponents)
*created* new mathematical objects (0 and negative integer exponents), defining the old thing on this new domain. The new definition didn’t logically have to* *be that way; it’s just *nice* for it to be that way because it continues the rule we’ve gotten fond of.

This bootstrapping thing shows a big difference between two kinds of arguments that can show up in the math classroom as students engage in Practice 3: “This claim *must be* true based on this other thing we know” vs. “We *ought to* define this new thing in this way because then it allows us to generalize a rule we like.”

I also think the bootstrapping reveals some underlying structure to the math curriculum that emerges as we alternately generalize and justify. I think it might happen all over the place. For instance, in elementary school, teachers justify the distributive property (2 + 5) x 6 = 2 x 6 + 5 x 6 based on the definition of multiplication as grouping or repeated addition (2 groups of 6 and 5 groups of 6 is just 2 + 5 groups of 6). To generalize the distributive property to (2 + 5) x
√ 3 , we need a new definition of multiplication as, say, scaling (because what the heck is
√ 3 groups?). Can anyone think of other examples?

*Rob Ely is an associate professor in the Department of Mathematics at the University of Idaho. He likes researching student reasoning and how it relates to historical reasoning, particularly with the infinite, infinitesimal, and algebra. He likes playing hammered dulcimer, bridge, and fetch.*