I was thinking of ways to get struggling students to classify alternate interior, alternate exterior, etc. angles. I thought about having the kids make up the names themselves! What a great way to start a dialogue! I imagined the discussion, arguments, suggestions, etc. and couldn't wait to start. I would give them the vocabulary for corresponding angles and show them those pairs, then they were on their own.

It was less than ideal. In fact, if someone had walked into the classroom while my kids were thinking of names, they would have thought I was a teacher with no plan and no control. Lots of "What are we supposed to do?" "Name them what?" "What do you mean?" I visited each group and coaxed them along. "Left middle" said one group for angles 4 and 5. Someone else just decided to name them "Fifi". 


So, I gathered the suggestions and read them to the class. Then I said, "Are you ready for the real names? Let me know if they make sense." I began naming the angle pairs slowly. For angles 4 and 5 I said "Same side...(ooohhhh, the class murmered)...interior." (Some said "huh?", then said "oh!".) I went through naming each angle pair. I thought the activity was pretty much a bust...until the kids started having to classify the angles. The confusion, perplexity, and struggle that looked so messy to the outside eye was working wonders on the inside. They paused before naming any angle pair, I could see their minds working.


I'm beginning to realize that deep thought on the part of the student does not necessarily look like what I thought it would look like. I had imagined quiet contemplation, maybe some head nodding and a finger on the chin (imagine the thinker pose here). Instead, I am coming to realize that for some kids, thinking is noisy and active. I need to be comfortable with that. I also need to be confident enough with it so that if someone does walk into my classroom, I don't immediately start trying to shush the kids. Noise = thinking in my class and for my current students. It's working so far for me!

I never anticipated the number of "aha" moments I would have by encouraging my students to talk or write about what they are thinking. The most recent was a simple warm-up on combining like terms. A student thought that 2x + 3x = 5x^2. Lately I've been asking students to give me "convincing arguments" about concepts. High school students like to argue, so I thought I'd go with it. So, my question to students was "Give me a convincing argument as to why 2x + 3x not equal to 5x^2?" That's a hard question to answer. Most students responded about the rule being that you shouldn't add the variables. They are "like terms" so x + x won't equal x^2. From there I was inspired by the "always, sometimes, never" questioning from the Mathematics Assessment Project. I asked "Is 2x + 3x = 5x^2 always, sometimes, or never true? Justify your reasoning and give counterexamples." Most of the students said it was never true, some said it was always true. I had to explain how to find counterexamples and we started substituting values for x. One student noted that x = 1 made the equation true. And then we puzzled about what else would make it true.

I grabbed Desmos, and showed students that 2x and 3x could be graphed using y = 2x and y = 3x. Two lines...so, what happens when you add two lines? You just get a different line...but it is still a line. The students predicted a shift, but instead the line just got steeper. 

Then we graphed y = 5x^2 and saw the parabola. Quite clearly students saw that 2x + 3x cannot be 5x^2 because you can't add two lines and get a parabola. However, they also saw that 2x + 3x = 5x^2 was sometimes true...when the graphs intersect. We substituted x = 0 and x = 1 to show that was true.

Students rarely think about an equation being false, let alone sometimes true. What a great way to show them visually what is going on. It wouldn't have happened unless I had been asking my students to give me convincing arguments, and using formative assessment to learn what they knew. I know that if students can make me curious, I will have more success making them curious.