Thursday, May 1, 2014

They Need To Talk About Powers Before They See Exponents

A power of 5
Is multiplication really repeated addition?

In a recent post, Christopher Danielson argues that this is the wrong question to ask.
It is a strong and presumptuous claim to say what an idea is. In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.
Similarly, I argue that while there's nothing wrong with thinking of exponents as repeated multiplication, it's not the best way to start thinking about them. Instead, we need a different conception of exponents to build on.

What is that conception? Writing about multiplication, Danielson says "Multiplicative structure is captured better by this idea: A times B means A groups of B."

What's the analogy for exponents? "A to the power of B means, ummm, B nestings of A? B recursions of A? B doublings/triplings of A?"

With multiplication, we can rely on the informal language of groupings to ground multiplication. But every day experience provides less clay to be shaped into the precise language of exponents. Presumably, this is what makes exponents so hard for kids and teachers of kids.

All of this leads me to two claims about teaching exponents meaningfully:

  1. "A to the power of B means B _______s of A." Effective teaching of exponents will ensure that kids have something to fill in that blank. 
  2. The best noun to fill that blank is simply "powers." As in "A to the power of B means B powers of A." 
(I'd say that a "power of A" is an element in the Geometric sequence of A.)

What follows from this is that we have to teach kids what powers are before we can teach them about exponentiation. 

There are no special obstacles standing in the way of this task, though. "Power" is essentially a new piece of mathematical language, and that means our job is going to be giving our kids lots and lots of opportunities to talk about powers before showing them exponents.

How do you create situations where kids end up talking about powers? That's the most important teaching problem here, and it's the one that deserves careful and creative thought. I'm sure you have ideas -- I'll share mine soon.

(In case you're keeping score: I'm now recanting this post and its enthusiasm about grounding all of this in Geometry.)


  1. I don't like using the word "power" twice in your definition. It makes the sentence feel like circular reasoning. How about, "A to the power of B means the Bth fractal of A." That's what your drawing of five fives of five makes me think of: fractals. The next stage would be to replace each of those dots with sets of five, which would make 5^4, or five fives of five fives.

    Thanks for this post! For the first time since Keith Devlin's original "Multiplication Is Not Repeated Addition" post way back when, I can imagine an intuitive way of introducing powers as something more meaningful than "repeated multiplication."

    1. Great thoughts, Denise! I especially appreciate your careful thinking about definitions.

      I'd like to think more about whether my definition suffers from a problematic circularity.

      "A to the power of B means the Bth power of A."

      This isn't exactly circular, since we're defining "to the power" in terms of "power of A." The question is, what do we ground "power of A" in?

      I'd say that something is a "power of A" if it appears in a certain kind of sequence. A power sequence? That has nice connections to higher math, but it's understandably unsatisfying.

      I'm tempted to say that, as a pedagogical matter, we can define that sequence imprecisely and informally. If pushed to be more formal, though, I'd give you instructions on how to construct the sequence, i.e. start with a number of A dots, and then replace each of the A dots with A dots, etc.

      Does that work any better, to your eye?

    2. This visual also gives an intuitive reason for 5^0 = 1 not zero. The zeroth stage of the fractal would be everything merged into a single dot.

      I don't know how we could extend the image to fractional exponents, however. My mind balks at cutting the dot into fifths--that doesn't seem like a natural extension of the fractal idea. Time to introduce infinite series?

    3. I actually think that fractional exponents aren't so bad. Considering every 1/2 power in the sequence means introducing an extra step between each element in the power sequence. So 2^(3/2) power is what fills in the blank in 2, ___, 4.

      It's true: dots will be hard to work with here. But defining this all in terms of a sequence is a significant advantage over the repeated multiplication definition, as far as fractional powers goes, I think.

    4. Perhaps "circular" isn't the right term. What I meant is, using the same word to define itself tells me nothing. "A to the gargle of B means the Bth gargle of A." If I don't remember what a gargle is, how does this help me? And if I do remember, why bother to say it?

  2. I look forward to hearing how you explain "Considering every 1/2 power in the sequence means introducing an extra step between each element in the power sequence" to 4th graders (or high school students) in a way that doesn't lead to linear interpolations!

    1. Can I cheat and pick an easy case?

      Here are the powers of 4:

      1, 4, 16, 64, 256

      But does anything go in between the powers? If anything did, it would have to be the 1/2 powers.

      1, __, 4, __, 16, __, 64, __, 256

      What's the 5/2 power of 4?

    2. Pattern-sniffing with easy numbers? And then moving from that to the definition of a fractional power, I suppose? As a visual thinker, I don't find that anywhere near as satisfying as your five fives of five illustration above.

    3. How about this: I'll ask my 4th graders what the 2 1/2 power of 4 should be, and then we'll take a look at their work?

    4. Sounds interesting. I just asked my 9th grade algebra 1 student, who has studied exponent rules but hasn't fully internalized them. She didn't recognize this as "that type" of question.

      Her first reaction was linear: "It should be somewhere in the middle."

      But when I asked her to pick a specific value, she thought a little more and said, "Since multiplication grows faster the farther you go, it should be a little less than the middle number between 16 and 64."

      We found the middle number as 40, so I asked her how much less, and she resisted answering. Perhaps she felt I was quizzing her, which wasn't fair outside math class? So I said, "Well, what number just feels right?" and she decided on 32: "Because I play a lot of Minecraft, and 16, 32, 64 feels right."

      So apparently, our kids do have a better feel for powers of 2 and 4 than I ever had at their age :)