Admit it. For most students, exponent rules can seem really lame. It's easy to confuse them and there's not much real-world application for these things. Textbooks try really hard to make it sound like there is...but they're a bunch of liars. The only applications I could find in my textbook (that weren't related to scientific notation) looked this like:
I don't think this is a bad problem. In fact, I like that it makes you think about area. But this isn't a real-world application because you'd never actually need to find the length of a rectangle like this. (Trust me. I was an engineer for six years. I never had to do anything like this.)
And kids pick up on that lameness. This is one of those lessons, up there with factoring polynomials and simplifying radicals, where students will at some point ask, "When will I ever use this?" This question usually comes up when a particular student is getting frustrated with the lesson and suddenly says to himself, "I don't get this. This is stupid and not worth my time. I'll probably never use this outside of school anyway."
And then we, the teachers, come up with a myriad of excuses: you need it for Algebra 2, you'll need it for the test, you'll need it when you're doing sciencey-kinds of things some day, you'll need it when you're flipping burgers at a rate of 2x^6/x^6 per minute, etc. None of these answers are satisfying to the students.
On top of all of this lameness, we make things worse by explicitly teaching students the rules for exponents. We don't give them the chance to figure them out on their own. I think there is a huge danger in just teaching students rules (or procedures or tricks). If you don't let students develop the meaning behind those rules and procedures, they are subject to error...like when a student tries to multiply two fractions, but for some reason he or she cross-multiplies. Or when they get really good with rise over run, and when you ask them to plot (4, 3), they move 4 up from the origin, and then over 3. Without meaning, rules and procedures become confused and are applied in the wrong situations.
Now, I always thought that I did a good job with teaching exponents. I always explained everything. I showed them how to expand expressions and put them back together.
The logic was there. The kids who were carefully listening "got it". But there were a lot of kids who only got the rule...just add the exponents.
As time went on, and I explained more and more of these rules, students became confused. They didn't really understand what was going on. They didn't know when to add or multiply exponents. They forgot that powers had to have the same base to combine them. They started to think that 2 times 3 was 5!
Here's a nice little graphic that shows how I feel when I teach exponents. You can see what happens as more and more rules get piled on. Students get confused and I'm left wondering, what happened? It was all going so well.
Some teachers are using a different approach. They're allowing students to discover the rules on their own. Just read these sweet posts by Andrew Stadel and Timon Piccini. (It's worth mentioning that both of these posts were inspired by Michael Pershan, who is doing a lot of awesome stuff himself.) I believe that exponents, like many things we teach, are a lot easier to understand if we de-emphasize the rules. And when students understand, they are less likely to ask, "When are we going to use this?", and they might actually enjoy mastering this concept.
Many times, we think we have to show our students every little step before they can do it on their own. Over the last few years, I've discovered that this really isn't the case. Don't believe me? Next time you teach something new, ask yourself if any of your students can figure it out without you explaining it to them. If you believe that at least one of your students can solve it without your help, let them try it. Just shut up, and let them work. Let them come to terms with what you're asking them to do. Let them wrestle with it a bit. Even if they don't get it, at least they're invested in the problem and they have some motivation to learn about its solution.
When you allow students to find their own solutions, they're going to appreciate it much more than you telling them how to do it...if only you'd shut up for once.
Here's the sequel to this post...."I Shall Never Play a Review Game Again!"
