Point of Inflection

Do you try to make your lessons relevant to your students and their lives?  Do you struggle to find ways your students will actually use your math in the future?  I suggest a reframe: make lessons relevant to your students by immersing them in mathematical situations during class.  Don’t worry too much if you can’t find something in their life outside of class.

My text book refers to bank accounts to make exponential equations relevant, and so does yours, probably.  Some problems with this example are obvious: my account is compounded monthly instead of continuously, I put money in and take money out at weird intervals, etc, so this math actually does not apply without modification.  Some problems are more subtle: you don’t need to know about exponential equations to pick a bank account, you just pick the one with the highest interest rate.  My students have very little money and differences in interest are on the order of pennies per year.  My students don’t have bank accounts.  My students don’t care about bank accounts.

The most subtle problem (that I’ve found, anyway) is that even students who do care about bank accounts aren’t opening bank accounts at the moment I am trying to teach them about exponential functions.  Even if the example is relevant to their outside life, when they are in my class they are not in their outside life.  The best I can do is “imagine you’re choosing a bank account” or maybe “find three bank accounts online.”  What I want is a student faced with a problem that he wants to solve during our class.

Give up on outside relevance in favor of inside relevance.  Pass out super balls.

This idea is from CPM’s Algebra 2: Connections.  Ask students, “how high will your ball bounce?”  They will have a mental crisis as they think of all the reasons this question is unanswerable in its current form.  Already they are thinking about a problem during my class.  Better yet, they are pretty fluent in bouncing already – they have bounced a lot of things in their day, you know?  They have some vocabulary to describe what’s going to happen, and a framework in which to place guesses.

You get the idea: it turns out that the height of bounce x follows a path almost exactly exponential, and you can ask students how high the second bounce will be, and how high the tenth bounce will be.  Please note that this has nothing to do with anything that “matters” in “real life.”  This activity makes exponential functions relevant by showing kids how exponential functions can improve their fluency about bouncing.  Exponential functions give them the power to, with absurd accuracy, guess how high the fifth bounce will be – and everyone likes that.  Increased fluency about reality is relevant to everyone.

So I’ve traded outside relevance for inside relevance, and the biggest difference is that I am teaching during the problem-solving process instead of before the problem-solving process.  The kids who learn about bank accounts won’t use that math until they are opening a bank account, and at that point they can’t get help from me or from classmates.  The kids who learn about bouncy balls use the math at the same time they are learning the math, and so their ideas are immediately challenged and reinforced by results.  It’s better!