Today was a hard day. The theme was learning all of the things my students don’t understand that I wish they did. My school loves data — looking at numbers quantifying what students know and don’t know. But so much more important than any number is the experience of standing next to a student and watching them experience mathematics.
Big picture: We spent the day exploring linear equations in Desmos. It was the first day that students were using the Desmos graphing calculator. Goal 1 was for students to gain some familiarity with how to use it — typing fractions and exponents, working with multiple equations, changing the window, using sliders. Goal 2 was for students to use those tools to deepen their understanding of slopes, intercepts, and linear equations.
Many of my students did a great job with the assignment, and learned a ton. Many of the questions were easy for almost everyone. But the places that students got stuck worry me, especially as I near the end of our unit on linear equations. Three examples:
One: I asked students to graph the equation below. The directions said: “Graph y=1/3x+15. Drag the screen so you can see the graph. What is the y-intercept?
I was pretty proud of this question. It seemed like a great way to get students to manipulate the window in Desmos while reinforcing the idea of y-intercept and where that graph actually exists. I think about a third of my students were completely frozen on this question, and asked me, in varying tones of anger and confusion, why the graph wasn’t showing up.
Two: I asked students to graph the equation below and find the slope. The directions said: “Graph 3x-6y=12. What is the slope of this line?”
I thought this was a great way to remind students that lines come in many different forms, and some forms are easier to work with than others. I even thought they’d laugh a bit, as they’ve converted lots of equations into y = mx + b form, and this was such an easier way to do things. Instead, responses from many students, again somewhere between anger and confusion: “It doesn’t tell me the slope.” This was the one that crushed me the most. It seemed like a glowing neon sign saying “YOUR STUDENTS MEMORIZED HOW TO FIND SLOPE FROM WORD PROBLEMS, GRAPHS, TABLES, EQUATIONS, AND A FEW DATA POINTS, BUT THEY HAVE NO IDEA WHAT IT ACTUALLY MEANS. I actually think many more would’ve gotten this question right if I’d asked it without Desmos.
Three: Finally, I asked students to type in y=mx+b and turn on the sliders. I asked them what happened when they changed the value of b.
I’m less mad about this one, but I was still surprised by how many students said it moved the graph from left to right. They’re not wrong, and that relationship is a fascinating one, but I think this is another sign that, while my students can regurgitate y-intercept and initial value on command, they don’t really understand what it means.
Thoughts: I don’t know how seriously to take these. I think some students were rattled by answering questions in a different way. More were confused because they expected Desmos to be a black box that spat out any answer they needed, without requiring them to do any thinking. Some struggled moving between the questions and the calculator without losing their thought. And the fact that the last time we used computers was to do Central Park, which they all loved, made this seem more frustrating than it needed to be.
I’m not sure how best to respond to what I learned today, but it reminded me that one of the most important things I can do as a teacher is confront myself, as often and as faithfully as I can, with my students mistakes and gaps in understanding, to better understand who they are and how to teach them.