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The 2° th post

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MTMSPicCaptionWelcome 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 24 x 23 and other similar cases. When they saw it was 27, 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., 27 ÷ 23 = 24), which was just a rearrangement of the same thing.

Later, the teacher was talking about 20 and negative exponents like 2-3. A student asked why 20 is 1 because shouldn’t 2 times itself 0 times be 0? Hmm. The teacher fielded this pretty well by asking the class what 23 ÷ 23 was, based on the subtraction rule. They said that it was 20 and noticed that it also had to be 1 because it was 8 ÷ 8. So it makes sense to say 20 is 1. Crisis averted.

This weird bootstrapping thing happened. Students—

  1. started with a definition (exponentiation as repeated multiplication)
  2. discovered a rule and justified that it had to always work, based on this definition (addition/subtraction rule of exponents)
  3. 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)
  4. 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?

RobElyRob 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. 


I find it interesting that the teacher was working with operations with exponents before dealing with negative exponents and the exponent of 0. Seems a little cart-before-the-horse.

I generally introduce the exponent of 0 (and negative exponents) by showing a progression of 2^3, 2^2, 2^1, (all of which they are very familiar with by then), and going on to 2^0 and 2^(-1), etc. Seeing that reducing an exponent involves division tends to make the concept of negative exponents more concrete. They can then see that 2^(-3) means you essentially "get rid of multiplication of 2 three times", i.e. divide by 2, three times (or divide by 8). An exponent of 0 means you are not multiplying any 2s, so you are still at the starting point (which, in multiplication, is 1)

On a housekeeping note, I love the addition of the blog. But where's the link to the RSS feed?
Posted by: PeterG_07374 at 2/18/2014 7:21 PM


Hi Peter - Thanks for the suggestion! We have added an RSS feed to all blogs. You can find the link to subscribe in the left-hand side bar.
Posted by: LaikenJ_62015 at 2/20/2014 3:02 PM


Peter, good call -- I think that's how I would do it too. But notice that the same issue happens anyway: The reason 2^2 = 4 (or 2^1 = 2) is based on the definition of exponents: "2 times itself twice". But the reason 2^0 = 1 is different: "We want the pattern (exponent goes down by one every time we divide by 2) to continue, so we are going to DEFINE 2^0 to be 1." I think students might not understand that these are two different kinds of reasons.
Posted by: RobertE_93912 at 2/20/2014 3:32 PM


Hello All,
What a fantastic idea for the name of this blog! I love it! Concerning negative exponents, I think the 0th power could be easily explained by using powers of 10 and place value. Since each place in the place value system is 10 times bigger or smaller than the one to the left or right of the previous, and we can write 10 as 10^1, and 100 as 10^2, students usually can then understand why 10^0=1 and 10^-1 = 1/10-- following the pattern.
Posted by: SarahS_04014 at 2/23/2014 4:47 PM


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