There was an era when I spent a great deal of time documenting the activities of my practice on this blog. My students sponsored many an inspiration and innovation as I learned (and re-learned, and re–re-learned) how to teach mathematics. Times have changed globally (the math teacher blogging community has largely disbanded) and locally (my young family keeps me busier than ever), but my students continue to notice interesting things and make meaning in innovative ways–I swear. I’ve been chewing on one such occasion for a month now and decided to use this space to get it off my chest.
It all began in the same way that most interesting observations begin–during a lesson on simplifying rational expressions.
I launched into, what I felt to be, a nuanced demonstration of how expressions all have numerical values once you decide the values of the variables. I built two expressions (designed so that one was actually a simplified version of the other). Students chose values, did calculations and then noticed that there were moments where they were multiplying and dividing by the same numbers. This feels like too much unnecessary work, right? This was all to get to the payoff: We can cancel these terms to make the expression simpler. Genius, right?
A few moments later, one delightful student made the following comment:
So these are just fractions? You could have led with that.
I immediately knew she was right. I got so lost in the mechanics while trying to connect the expressions to their (potential) values, that I forgot to connect the topic to the students. I then got to thinking: There are lots of times in our continuum of secondary school mathematics where a “new” topic is an elaboration or extension of an elementary topic. I should use this to my advantage to help students make meaning. 1
Fast forward 2 weeks.
I’m planning a lesson on trigonometric equations. I’m exhausted. The students are exhausted. It’s two weeks until Winter break. I just want to tell them what to do with a few examples, toss them our snazzy unit circle manipulatives (to check the “innovation” box), and move along. Until the words of my other student reminded me:
These trig equations are just equations. You should lead with that.
So I did.
I took a few trig equations, removed the trig functions and replaced them with “x” and asked students to solve them and share their process with their partners. (Here’s the simple handout I made). I was intentional to build in time for students to share their ideas with one another, but the introductory task was as direct as could be. As I moved around, I heard exactly what I wanted to hear:
- One student commented on how she remembered being intimidated by these in Grade 9, but now they are automatic.
- Two students debated which “move” they should do first to isolate the variable, and whether move order mattered.
- One reminded their partner to check their answer in case it was extraneous, and then we talked about if linear equations can have extraneous roots.
All great stuff. Then I simply asked them to re-write their equation to the right of the previous one except, this time, to replace the “x” with “cosx” or “sinx”. They were instructed to solve this “new” equation, and I waited to see if they would re-do all the steps or simply jump directly to the solution they just found. As the murmurs picked up around the room, one student exclaimed:
I’m not doing this all again. That variable doesn’t phase me!
Bingo. We now had the soundbite I needed to enter into conversation about the nature of trigonometric equations, but I also had so much more information. I had a built-in diagnostic on how developed our concept of “equation” is, I had an idea of the “go to” order that the students like to use in order to solve an equation, and students had a connection to their own mathematical past. Win-win-win (in my books).
It was a simple reminder from my most important source–my students. The best lessons aren’t always intricate; if a topic is familiar, lead with something that will tease out this familiarity.
NatBanting