I find the tone of this article a bit over the top: It Ain’t No Repeated Addition. In it, Keith Devlin (as is SOP for mathematicians) takes a too-strong epistemological stance–multiplication is not repeated addition.

I am much more interested in the nuanced space between provocative stances. For instance, I am much more interested in a question such as, What is gained and lost in defining multiplication in relation to addition versus some other approach?

Exploring this question allows all knowledgeable parties access to the conversation, and it helps us listen to each other. Telling others that they are wrong tends to shut down the conversation, to discourage listening and make people defensive. Through that lens, I can can read Devlin’s piece in a productive way.

In that spirit, I have engaged with our good friend Michael Pershan on the topic of exponents (By the way; go read this piece—it is excellent.). In particular, I have attempted to ask the analogous question about exponentiation as an operation.

In particular, he has been exploring the conditions under which students confound exponentiation with multiplication. As seen in the very common algebra mistake, $100^{0.5}=50$.

I have suggested that perhaps the trouble lies in defining exponentiation as repeated multiplication. After a bit of brainstorming, I came up with an alternate definition: doubling (and tripling, etc.)

What if we think of the powers of 2 not as repeated multiplication, but as number of doublings?

This sounds like a trivial difference, and perhaps it will prove to be. But I think it is more than that.

For instance, repeated multiplication makes me think of $2^{5}$ on its own. But number of doublings suggests to me a starting value (which could be anything) and then we double that value some number of times.

Repeated multiplication doesn’t make clear what to do about $2^{1}$, nor $2^{0}$. What does it mean to multiply a single 2? Or no 2’s at all?

Number of doublings makes this more clear. $2^{1}$ means double once, while $2^{0}$ means do not double your original value.

Rational exponents? Start with mixed numbers and you should be in good shape. One and a half doublings is more than twice what we started with, but less than four times.

What would it be like to start instruction in exponents from the number of doublings perspective instead of from the repeated multiplication perspective?

No better playground for hypothesis testing than a truly blank slate.

Little man knows squat about exponents.

Griffin (eight years old): Ten.

Me: Then double it again.

G: 20. Then 40…80…160…2…no…320…640…1280…

Me: Wow.

G: Then two-thousand…five-hundred-sixty.

Me: Holy cow. I did not know you knew that many doublings!

G: Yeah. That’s all I can do, though. I can’t think of what comes next.

Me: Right. Next would be 5120. But that doesn’t matter. We started with 5. Then you doubled one time to get 10. You doubled two times to get 20. You doubled three times to get 40.

G: Yeah.

Me: What if you doubled one and a half times? What do you think that would be?

G: 15

Me: So if you double 5 one and a half times, you would expect it to be 15?

G: Yeah. Is that right? What would it be?

Me: Wait. I want to know what you think here. I will answer all of your questions after you answer a few more of mine. Why do you say 15?

G: Well, doubling once is 10, then half off the next one would be 5 less, so 15.

G: Yeah.

Me: What if you had zero doublings?

G: Well…it could be 5. Or maybe 0.

Me: What is the thinking behind 5?

G: You don’t double it at all, so it’s just the same.

Me: And what is the thinking behind 0?

G: Adding and timesing with zero…it’s usually zero. So I think it might be that. But it could be 5. What is it?

Me: I promise I’ll answer all your questions in a minute. One more…What if you doubled half a time?

G: Well…I don’t know….Seven and a half, maybe. I don’t know. I like whole numbers better.

The doublings approach led this third grader to:

1. Linear interpolation for rational exponents, rather than triggering a multiplication schema, and
2. The possibility that $2^{0}=1$ (albeit with a low degree of certainty).

These both seem like improvements over the intuitions Michael demonstrates in his piece—intuitions which certainly mesh with the misconceptions with which I am familiar in my middle school and college teaching.

8 responses to “Rational exponents, third-grade style”

1. My brain is working really hard on a thought, but I’m not sure what it is yet. It has something to do with these:
1) Addition and multiplication are operations, maps from R2 to R in fancy terms. You take two numbers and do something and get one number. They have inverses and identities
2) Exponentiation is also an operation, but it has a sided-ness. Maybe that’s just a naive way of saying it’s not commutative, but it seems like we almost more think of exponentiation as a family of functions: 2^x and 3^x and 4^x, rather than thinking of that as an operation on 2 and x, 3 and x, 4 and x, etc. That’s why we don’t really think of an identity element or inverse elements (the thing you raise 2 to to get 2 (identity), or the think you raise 2 to to get 1 (inverse)).
3) You’ve sort of turned the exponentiation operation/function thingy into a very slightly different sort of function. Like, exponentiation is a function on some other (hidden) number (usually 1 but sometimes other things). If 2^5 is “the rule that tells you to double something five times” and you don’t know what that something is, that might be confusing. Or it may not give you an intuition about 2^5 itself (except in relation to other numbers like 2^5(1) = 32, 2^5(5) = 80). Any maybe that’s okay because maybe exponentiation does make more sense in a function universe than an operation universe?

And then a few other thoughts.
4) Naively what happens if you double 5 -2 times could be thought of as un-doubling 5 two times, and you’d get half of half of 5, or 5/4. It’s kind of neat that you could think of un-doubling 5 times being written as 1/2 ^ 5 or 2^-5, if you’re really good at thinking about – as an opposite.
5) Half of a doubling is interesting. If you double half a time, and then double half a time again, should that be the same as plain old doubling?
6) Would you mind asking Griffy about quadrupling at some point (is that a word he would know or pick up on with some examples?) Because half a quadrupling is clearly a doubling, to me. And two doublings is one quadrupling.

Thanks for sharing this (and thanks to your guinea pig as well)

2. Christopher

Max, I am probably gonna need another whole post to dig into the lovely ideas you raise. Thanks for the substantive reply.

Quickly, I have done some work with my College Algebra students in the past around exponentiation as an operation. I should report my activities and findings on that. Thanks for the reminder to do so.

I thought about the half-doubling-twice idea while Griffin and I were talking, but it just didn’t seem like a natural question to ask. I haven’t worked out whether that is because of the nature of the starting value of 5, or whether it was because of the nature of the task and where he is at.

3. Jo in OKC

Slightly confused (but that may be normal).

What exponential is doubling 5 supposed to represent?
It sounds like what you have him calculating is
5 x 2 = 10 = 5 x 2^1
5 x 2 x 2 = 20 = 5 x 2^2
5 x 2 x 2 x 2 = 40 = 5 x 2^3
5 x 2 x 2 x 2 x 2 = 80 = 5 x 2^4
5 x 2 x 2 x 2 x 2 x 2 = 160 = 5 x 2^5
5 x 2 x 2 x 2 x 2 x 2 x 2 = 320 = 5 x 2^6
5 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 640 = 5 x 2^7
5 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1280 = 5 x 2^8
5 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2560 = 5 x 2^9
etc.

Is that what you wanted? I guess it’s the 5 that’s confusing for me because the 5 isn’t really related to the exponential at all.

So, the exponent of the 2 indicates the number of doublings of the other factor. That only works for a description of powers of 2. Powers of 3 would be the number of triplings? Powers of 4 would be the number of quadruplings?

What about situations where you just have the exponential with no other factor. What would you say you’re doubling/tripling/quadrupling/etc.?

I can see how you can reason from this to 1.5 doublings. But, I’m not sure that I can get to half a doubling is the same as multiplying by square root of 2.

And, negative exponents don’t make sense in my brain with this method. But that may be my grown-up used-to-the-standard-explanation brain.

4. Christopher

Right. I chose the 5 in order for it not to be related to the exponential part of things. Your question, Jo, hits on something that I think is important:

What about situations where you just have the exponential with no other factor. What would you say you’re doubling/tripling/quadrupling/etc.?

I guess the question at hand here is, What sorts of intuitions do children have about exponentiation that we can build upon? Subquestion, I suppose, is What are some mathematically honest stories we can tell about exponentiation in relation to these intuitions?

On that first question, it is well documented that children learn to double numbers long before they master other multiplication facts, and Griffin’s long list of the doublings of 5 attests to this. So if doubling is an intuition on which we can build exponentiation, then I wanted to see what the affordances of that intuition might be.
In order to explore doubling, you need something to double. Five leads to 10, 20, 40 and 80; all of which I expected Griffin to be able to do, so it seemed a useful starting place.

I hypothesize that exponentiation on its own (i.e. $2^{4}$ instead of $5\cdot2^{4}$) is a later topic, rather than an entry point. That hypothesis then causes Max to wonder about the nature of the mathematical story I am constructing for Griffin—What IS exponentiation, exactly?

On your last questions, I am curious about what sorts of generalizations Griffin will be able to make about doubling. What will he say when I ask him, What would half a doubling of 100 be? And what will he say when I ask him What if I had -1 doublings of 5? I don’t know. But I’m going to find out.

For the moment, let’s imagine that he says 150 is half of a doubling of 100. That feels like a better starting place than 50 does. Both are wrong, of course. But the former is about precision, while the latter is about something more profound.

5. I like it — the exponential with no factor becomes a special case, rather than a defining feature.

It was hard for me to orient myself to it (I went through the process Jo describes, and had to use paper!). I interpret my difficulty switching representations as evidence that even my own understanding is not as strong as I’d like to believe. And if even a couple of years of calculus leaves me that rigid in my thinking, then it supports the idea that alternative approaches could be helpful!

6. l hodge

Interesting post! I did not get Devlin’s point at first, but have become more sympathetic to it the more I think about it. I think the repeated addition approach maybe leads towards thinking of multiplication as discrete whereas scaling does not. I have often found students that would naturally think to multiply to get a result when the numbers are integers, but not when the numbers are decimals (even with calculators available).

Grant Wiggins had a post making the analogy between teaching grammar and teaching Algebra. I tend to agree that what passes for algebra is often taken in by the student as an obscure language with grammar rules to remember. The thinking is more verbally oriented than mathematically.

When you have a mathematical conversation with your children, you are speaking in a familiar language (English), so nothing is lost in translation between an obscure symbolic language. Your child is speculating on doubling 1.5 times using incorrect, but solid mathematical reasoning.

I think students faced with an unfamiliar expression are often speculating on a possible grammatical rule rather than speculating mathematically (as your child did). What seems most reasonable from a grammatical point of view (100^.5 is almost the same as 100*.5) may not be reasonable at all from a mathematical point of view. Most of us make a big deal about how exponentiation is different than multiplication – look how fast it grows – etc. But, I think students very easily fall back into that verbal mode of thinking when they see symbolic notation.