Tuesday, April 9, 2013

Why Kids Mess Up Exponents

1. Table of Contents

  1. Table of Contents
  2. Exponent Mistakes + Teacher Explanations
  3. Let's. Try. Evidence.
  4. Results
  5. What's Going On Here?
  6. Sequels

2. Exponent Mistakes + Teacher Explanations

Anyone who teaches exponents is familiar with a series of closely related mistakes:

Students like to treat exponentiation like multiplication. Why?

Thanks to mathmistakes.org, we have a nice collection of attempts to explain this phenomenon.

  • "Most kids just try to cram in the fact that negative exponents do ummm, something to the base. Without some conceptual hangar to place this fat they are left wildly guessing. "
  • "But really, as much as I try not to write these off to “autopilot”, I can’t think of any other explanation."
  • "Often, if you can get a student to slow down and be more present in a problem, they can avoid mistakes they would be prone to make otherwise. We need ways to differentiate true misunderstandings from these sorts of automatic pilot errors."
We've got votes for autopilot, wildly guessing, and the idea that students are operating with an incorrect conceptual model for exponentiation.

How can we sort this out?

3.  Let's. Try. Evidence!

This survey was given to students in a first-year Algebra class. They're studying exponents, but have never seen negative powers before, and they've certainly never seen non-integer powers.

What would you expect these kids to answer? 
  • If you think that kids are wildly guessing, then they ought to report a relatively low level of confidence in their answers. 
  • If you think that kids are mistaken about what exponentiation means, then they shouldn't get the first question right. After all, if they know what exponents mean with one problem, shouldn't they know what they mean just seconds later?
4. Results

This stuff is pretty fascinating. Here's everything, and here are some quick observations:

  • Overall, kids answered 50 to the third question, and had a good deal of confidence behind their answer.
  • Answers to the second question were more varied, but nobody just multiplied the base and the power together, like they did for 3a (or even 4a!).
  • Overall, kids had more confidence with rational exponents than with negative exponents.
(You might be wondering whether these observations are a fluke, which they might be, but they're at least a fluke twice. Here's a repeat of the experiment.)

5. What's Going On Here?

These kids are not guessing. Or they are, and they're lying, because they're telling you that they have confidence in their answers. So you can knock that theory out, it's not what's going on.

These kids are coming into your classroom with ideas about negative and rational exponents. So it's not about rules or memorizing or whatever, these kids have ideas about powers and are pretty confident about them.

These kids do sometimes treat exponentiation as multiplication, even when they're just dealing with plain old positive powers. To me, this supports the idea that when faced with a difficult exponentiation problem, sometimes the mind skips right to multiplication.

These kids are especially confident about rational exponents, where they seemed comfortable answering "50" to the third question. This could be because they've got a fuzzy story they're telling themselves about fractional powers, or it could be because of something more intuitive. Those are your options, and I'm not exactly sure what the difference is between them or how to test for this.

These kids don't just multiply the base and the exponent together when dealing with negative exponents. Instead, they tend to do treat the negative exponent as a positive one and then just tack on the negative to that result. (I suspect that this has something to do with the way we teach kids to do multiplication of negative numbers: do the multiplication, and just tag on the sign at the end.)

Your problem, as an Algebra teacher, is far more serious than autopilot. It's not just thoughtlessness that's responsible for these mistakes. It's a substantive intuition about what the answer to these things should be. It's such a strong intuition that it exists among students who have never even seen these concepts before, in a classroom or (presumably) otherwise.

6. Sequels

The survey file is here, but it's easy enough to make one of your own. I'd love to see what your students respond to math questions that they've never seen before. This all seems like a fruitful way to plumb the images that our kids bring into our classes.

I think the two live explanations for this sort of student work are that (a) kids have explicit, mistaken models about how exponents work and (b) there's a sort of intuition about what exponents should be, and this intuition operates below the level of consciousness.

I don't know how to tease those two possibilities apart, though all my experience in doing and teaching math leads me to think that it's all about intuition, and not really about explicit, mistaken models.

I was surprised that large numbers were sufficient to trigger multiplication of the base and the exponent in a lot of cases. I was also surprised that this didn't happen with negative powers. I'm not exactly sure what to make of that. I wonder what else triggers multiplication.

And does this sort of thing happen with other operations? Is this a general phenomenon or is something special about exponents?

NSF grant proposals in the comments, please. 


  1. I'm thinking maybe because 2^3 means you multiply three 2's together, they think 100^(1/2) means you have 1/2 of a 100.

  2. This would also explain problems with negative exponents - how can you multiply -3 20's together? I think it has to do with the fact that we teach exponents as a way to represent repeated multiplication of the base....

    1. My bet is that it has nothing to do with that sort of explicit thinking about powers. My bet is that 100^0.5 just feels right to kids.

      But I can't figure out how to test this idea. I could ask kids to explain themselves, but even if they have an explicit story to tell, that could just be their attempts to justify what feels right.

  3. Wow, that first survey response on the scribd page is exciting! I'd love to be able to see inside that kid's head. Clearly on 2a they've picked up some messages about multiplying with negative numbers and 0. They added a 0 and a minus sign to the answer on the question before. I wonder if the answer would change if the first question was 5^3 and the second was 20^-3, so they wouldn't use the answer to question 1 to do question 2.

    All the responses are interesting. Thanks for posting them. I wonder if the way we write exponents has anything to do with the students' confusion, and if we could find a way that left them less room to switch to multiplication in their brains. Maybe if we used the ^ symbol. Then it would be a different symbol rather than just placement. Students sometimes mix up multiplication and addition, especially at the beginning, but it's much less common than multiplication/exponent mistakes, in my experience.

    (Note: I've never taught middle school or high school, so my experience is just random tutoring and college teaching, but since when has a lack of expertise ever stopped anyone on the internet?)

  4. I love the opportunity you're giving all of us to join the physics teachers in focusing our teaching on confronting misconceptions rather than teaching as though students are blank slates. I think exponent rules are a great place for me to start practicing that approach. And the first step in confronting those misconceptions is to get students to articulate exactly what their conceptions are.

    With your permission, I think I may make a version of your questions with multiple-choice answers including most of the common ones from your results here, and then let kids pick answers and have them argue with each other about who is right. I wonder what will come of that?

    1. Permission is unnecessary, but still granted. You'll share your results with us, right?

  5. This is great! I agree that there is something very interesting going on in exponent mistakes, and your evidence reinforces my feeling that it is not just carelessness or autopilot. There seems to be something operating like "interpret it analogously to things you already know" - which is usually a good rule, but which goes awry with exponents. But why it goes wrong in these specific ways is puzzling.

    So many kids will go to 2^3=2*3=8 right away (and I teach college!), which puzzles me: if x^y=x*y then why go to the bother of making a new symbol?

    I have a pet theory that the odd notation for exponents is partly to blame; that an in-line notation consistent with addition and multiplication (for example, "calculator notation", e.g. 2^3) would reduce these mistakes. But I haven't yet put my guess to a systematic test.

    1. Regarding the odd notation: Notice that the second survey ditches the notation and asks kids to evaluate "64 to the 0.5 power," and things of that sort. I found the same results when I wrote the sentences out, as opposed to using the notation.

      This doesn't kill your theory, of course. It could be that the pull of the notation is so strong that it reinforces a confusion between multiplication and exponentiation. At the same time, it knocks out of contention a crude version of what you're saying, where kids are just literally confusing the exponent notation for multiplication.

  6. Great stuff. The misconception literature - especially in science - says to me that in the face of uncertainty we resort to 'common sense' intuitions, as you suggest e.g. it is colder in winter because the earth is farther away form the sun (the famous science one). I have always felt it was vital to consider something that in our egocentrism we forget: most modern big ideas are counter-intuitive; they are not at all common sense. I think this is why your theory is onto the truth of the matter. It's counter-intuitive, for example, that 1.03567394 is smaller than 1.04; it's counter-intuitive that a negative times a negative is positive.

    1. Grant or Michael or anyone,

      Any particularly good readings in the "misconception literature?"

    2. Hi Michael, I haven't been reading long so I'm not sure if these are already familiar to you. In case they aren't, you might start with Halloun and Hestenes' paper "Common sense concepts about motion" (1985), which includes lots of excerpts from interview with students. Also consider "The initial knowledge state of college physics students" (same year and authors) where they explore the ways people respond when their initial concepts are in conflict with new information.

      Derek Muller (of Veritasium video fame) tackles the same subject, and Frank Noschese blogs about it.

      Both Hestenes and Muller are heavy on the "elicit-confront-resolve" model of responding to misconceptions. For something hopeful, different, and sometimes painfully challenging, read what Brian Frank has to say about it. Especially these:

      I Said I Didn't Want to Talk About Misconception

      Misconceptions Listening

      Misconceptions Misconceived

  7. This is great stuff Michael. Thank you for getting us started on this.

  8. This year, I've tried to use the structure of the multiplication operation in a chart... illustrating the recursive pattern of ×2 for each row going down the chart (next=now×2), therefore, ÷2 going up up a row. For most of these 8th gr kids, it establishes the pattern & structure of the exponent, the math, & the value of the number. The 2^0 & neg exponents are not as scary. We don't spend much time w/ fractional exp. But some are seeing 2^2.5 is more than 4, but less than 8. It's a start.

    Thanks for the ideas & observations.

  9. I'm very intrigued by the 7th student who wrote that 20^-3 is 20/20/20. It seems that he or she might have actually thought that a negative exponent meant the opposite of multiplying the base repeatedly...dividing repeatedly. Had they written 1/20/20/20, they would have been right.
    I try to show students this pattern as you transition from positive to negative integer exponents. Is it possible that this student has also been exposed to this idea in the past? Or did he or she develop this idea independently? If so, that's pretty cool.
    I think the last problem confused many students because they're used to exponents of 2, 3, and 4. I think the number 43 seems so foreign as an exponent that it confused them.

  10. I think Jennifer is on the right track here.

    The kids seem to know that 2^3 means you multiply three 2s together, which gives you 8, so 2^3 is 8. And I'd guess they also know that 2^2 means you multiply two 2s together, which gives you 4, so 2^2 is 4.

    What about 2^1? Well, that would mean you multiply one 2 together. Of course, you only have one 2, so you don't actually multiply because you need at least two numbers to do a multiply. But you have the one 2, so that's the result. 2^1 is 2.

    So far, so good.

    Now, what about 2^0? Well, that would mean you multiply zero 2s together. But if you have zero 2s, you have nothing to multiply. If fact, you have nothing at all. So that's the result. 2^0 is nothing, or in other words, 2^0 is 0.

    And what about 2^(1/2)? Well, that would mean you multiply one-half 2 together. As before, you don't actually multiply because you need at least two numbers to do a multiply. Here you have only one-half 2, so that's the result. Of course, one-half 2 is 1, so 2^(1/2) is 1.

    This all make perfectly good sense, doesn't it? Why is it wrong? Or is it?

    I'd be interested to know how the kids would deal with 10^(2-1/2). By the reasoning used above, that would mean you multiply two and one-half 10s together. Multiplying the two 10s together gives you 100 and then multiplying in the one-half 10 (which is 5, of course) would give you 500. So 2^(2-1/2) would be 500. Note that this is a different result than you'd get by just multiplying the base and exponent.

    Anyone willing to investigate this question?

  11. Oops. There are some typos in my previous post. In particular, near the end it reads "2^(2-1/2) would be 500" where it should have read "10^(2-1/2) would be 500". Sorry for any confusion.

    I'd also like to add a comment on how this reasoning might be extended to negative exponents.

    What would 2^(-3) be? Well, that would mean you multiply negative three 2s together. But that doesn't really make any sense because negative numbers aren't counting numbers. What could it possibly mean to have "negative three" 2s? Well, perhaps it could mean you have three 2s which are negative, or in other words, you have three -2s. Then you could multiply them together to get -8. So 2^(-3) would be -8. It's kind of a stretch, but what else could it mean to have "negative three" 2s? Note that this is a different result than you'd get by just multiplying the base and exponent.

    What about the case where kids think 9^2 is 18? Are they just multiplying the base and exponent? Possibly, but another explanation would be that they know 9^2 involves combining two 9s, and they're mistakenly using addition to combine them rather than multiplication.

  12. I think you are right on intuition... one thing I find myself doing that has been a reliable guide to me is to go away from the symbolic definition and remember what the graph of any exponential function looks like, which asymptotically approach 0 from above as you go left, always cross at (0,1), and climb like crazy as you move right. That description, along with the picture, is how I gauge what the values should approximately be. They do not answer the questions of "why does x^0 = 1 for any base x?" but after learning the base reasons and accepting the reasons, I just reference this picture in my head. So with this I know that anytime the exponent is less than 0, the answer should be between 0 and 1. And when the exponent is 0.5, the answer should be between 1 and whatever the base is, but probably a bit closer to the base because the curve shoots up pretty fast when the base is anything greater than 5.

    My input here doesn't do anything to further answer how to teach this concept to avoid bad models for students, but it is a tool to use either afterwards, or possibly in conjunction to being learned. If you can justify this picture to them, and get them to use it as a tool, then at least they can get ballpark answers, just not the symbolic manipulation simplifications...

  13. First, I have to say that I really appreciate this discussion and that it led to the exponent mistakes worksheet, which I used last year and loved!!! As I plan for next year I was thinking about your confidence surveys so came back to read through this post.

    This is incredibly picky I realize, but I would caution you to be careful about using 50 to the third when you mean 50 to the one-third. I realize that if we were talking about 50 to the third you would most likely say 50 cubed, but I think this can be really confusing to students. We wouldn't say 50 to the fourth and mean 50^1/4.