Teaching For Understanding

I want my students to believe that math makes sense, and to understand why different rules, strategies and techniques work. I work hard to come up with explanations that are clear, concise and accessible. I comb blogs for clever ways of framing and illustrating difficult concepts. Yet students still don’t get it — they take useful tools and apply them in the wrong places, make silly mistakes that illustrate a lack of understanding of the structure of a tool, and, endlessly, they forget.

I’ve spent plenty of time trying to figure out the best way to introduce a topic so that students understand the why. Should I start with a proof that builds off of what they already know? Should I have them look at a bunch of examples and extrapolate? There’s lots of different paths here, and I’m skeptical there’s a “right answer”, only different approaches that are best paired with different topics.

I’ve found what I think is something better. I’ve spent much of my energy this year exploring what it looks like to spiral learning — spacing and interleaving practice to improve student retention. It struck me that, while my approach to problem solving and application was making what seemed like effective use of spaced practice, I wasn’t doing the same with my approach to student understanding.

I teach a concept as I normally do, trying my best to suit my explanation to the topic and my students’ prior knowledge of it. Then, a few days later, my warm-up is for students to explain why that mathematical rule is true. I was a bit nervous about it falling flat, but students were engaged. Many wrote pretty hollow explanations at first, but discussing them as a class moved understanding forward, and students seemed to enjoy being able to improve upon one another’s explanations — maybe not the best reason for engagement, but I’ll take it.

Two Examples

I introduced the Binomial Theorem with Bob Lochel’s jigsaw activity. The activity as Bob wrote it has two parts. First, students break into four groups, and expand (x+1)^4, find the number of ways to flip a coin 4 times and get a certain number of heads, write out Pascal’s Triangle and record the 4th row, and find 4 choose 0, 4 choose 1, etc. Then they come back together, and whoa! They’re all the same!

Here, Bob has a handout asking students to draw different connections between the activities from the jigsaw. Instead, I saved this, and offered a rudimentary proof of the theorem, and we discussed as a class and worked through a few examples. Then, two days later, we returned to the activity, and students answered the follow-up questions — why are combinations and coin flips the same? What do they have to do with Pascal’s Triangle? Instead of cramming all of the connections and explanation into one day, they needed to recall and reinforce those ideas later. It was rusty — which underscores why it was worth doing — but it felt like a useful amount of time to wait, where it was still doable, but forcing them to really think.

Second, I’m teaching polynomials in a different class. One of the big ideas I want students to leave with is the idea that a term (x-a) in a function in factored form corresponds to an x-intercept at (a,0) because, no matter what the rest of the function looks like, when we plug a in for x, the output will be 0. I first introduced polynomials in this way as a simple extension of quadratics, and students seemed to get the idea. But after a few days of getting lost in the weeds of polynomial division, factoring, and end behavior, that was an easy idea to forget. Sure enough, when I started class a few days later with the warm-up, “Why does the function y=x(x-a)(x-b) have to have x-intercepts at (0,0), (a,0), and (b,0), most explanations were along the lines of “because those are the x-intercepts”. But there were plenty of useful pieces out there, and we put them together.

Again, the fact that this was really hard for students reinforces for me that it’s worth doing. It’s easy to pretend that engaging, clever, mathematically sound justifications will magically make students remember them forever, but that seems unlikely to work for most students. And it’s also easy to blame students for forgetting. I’m pretty excited about this approach to student understanding, and I’m considering adding this to my usual routine of warm-ups. In the meantime, I want to try it out a few more times, especially with my Algebra-II class, and see if the growth in understanding sticks around.