- Table of Contents
- Exponent Mistakes + Teacher Explanations
- Let's. Try. Evidence.
- What's Going On Here?
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.
We've got votes for autopilot, wildly guessing, and the idea that students are operating with an incorrect conceptual model for exponentiation.
- "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."
- "I think that this idea is attractive because many students think that raising a number to a power is the same as multiplying the base by the power as in “4^2 = 8.″
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?
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.
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.