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, .
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 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 , nor . What does it mean to multiply a single 2? Or no 2′s at all?
Number of doublings makes this more clear. means double once, while 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.
Me: Start with 5 and double it.
Griffin (eight years old): Ten.
Me: Then double it again.
G: 20. Then 40…80…160…2…no…320…640…1280…
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.
Me: What if you doubled one and a half times? What do you think that would be?
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.
Me: Instead of 20?
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:
- Linear interpolation for rational exponents, rather than triggering a multiplication schema, and
- The possibility that (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.