# Blog Archives

## Teaching Through Concrete Examples: The Intermediate Value Theorem

I’ve been getting pretty into cognitive science lately. I realize some of it is useless, and a lot of the rest of it is made up of kind of common sense things once you really think about it, but regardless, I have found it so helpful to put scientific names and research to intuitions I have in the classroom. One of the ideas that I have really liked (from Daniel T Willingham’s Why Don’t Students Like School?) is that we learn everything by connecting it to things we already know, and much of what we already know is concrete. **Thus, the more you can teach through concrete examples, the more likely students are to learn the material. **

### EXAMPLE: Speed, iPhone prices and the Intermediate Value Theorem

This year, while teaching the Intermediate Value Theorem in AP Calculus, I did not start with the theorem itself, as I always find that language so intimidating for what is actually a simple idea. Instead I started with this:

I showed them a video of a speedometer that cuts out for about 10 seconds in the middle (ah, you’re dizzy and you pass out for a second at the wheel!). Before the cut out spot, the car was going 60 mph, and after it was going 100 mph. I then asked the to tell me:

**What was a speed that you are 100% sure that you must have gone in the time in between? Why?****What was a speed that you could have gone in the time between, but you aren’t 100% sure? Why?**

We talked about this for a few minutes, letting the students argue a bit about their thoughts and came to an agreement as a class. Then I put up a new picture that showed the original iPhone prices at some intervals. It started at $599, a few months later was $399, and then two years later was $99. Then I asked very similar questions:

**What was a price that you are 100% sure that the iPhone must have had in the time in between? Why?****What was a price that the iPhone might have had in the time between, but you aren’t 100% sure? Why?**

Again, I let them argue for a bit and discuss. After we had settled on answers, I asked what was different about the situation, keeping in mind that we had already discussed continuity in the class, but I had never mentioned this in this situation. Students said wonderful things like *“To get from one price to another, the iPhone doesn’t have to pass through the other prices”* and *“Prices can can jump whereas speeds can’t”* and I let them continue to do that until one student finally realized **“Speed is continuous, whereas price is not!”**

Prepped with the ideas of theorem, we took the speed situation and translated it into a mathematical theorem before looking at the actual Intermediate Value Theorem. It took about 10-15 minutes of class, which was well worth having a strong conceptual understanding of the theorem. Students still struggled mightily with proving anything with the theorem (as they have in proving anything mathematically both this year and in previous years – any advice there?) but the conceptual development of the idea was not only quicker, but I think stickier.

*Isn’t that better than starting with this?*