One of the most important things I’ve found in teaching so far is the importance of asking meaningful questions to deepen mathematical understanding. I want my students to be mathematical thinkers and to be able to discuss their thinking with me and with their peers. I think that so much of mathematics education has focused on answers and procedures. Instead of that, I want to instill in my students the idea that the journey to get to answers, the thinking, is what doing math is all about. That’s what is important.
At my latest observation, I asked my coordinator, Jon, to focus on the questions I was asking students and how they can influence the thinking of my students. I wanted to analyze how well my questions were actually inspiring thinking within my students and if I could be more effective at this.
One of the main realizations we had was how often I was starting questions with “why?” That was even the whole question a lot of times. Though this is a good start to the way I want students to be thinking, it comes off as a bit aggressive. Jon talked about how there are genuine questions we can ask as teachers and how these are often more effective. For example, we ask a lot of questions like “what is the distance formula?” for repetition purposes, but these aren’t genuine questions. We know the answer. Instead, a genuine question would be “how did you think about that?” We don’t know how our students’ thought processes work and having them explain it is what I would want my ideal student to be able to do– to explain, in detail, how they thought about the math involved.
Another idea we came across was the issue of how to reword questions when students are not responding well. When my students struggled with the answer, it was hard for me to reword it in a different enough way to get them to understand. Then, for time management purposes, I would often give them or lead them into answers that I wanted them to discover or illustrate to me themselves. Having ideas for multiple questions before the lesson will help me to get students to understand what I’m asking. This will take a lot of planning, but will definitely be worth it.
My final “grow” moment from this lesson was about addressing student misconceptions. When students come up with ideas that are incorrect, but common (and understandable), I have a hard time knowing how to respond. I know that saying “no” in any part of my response is something I don’t want to do. This has a lot to do with my opinion that students who hear the word “no” will shut down and not listen to any of the rest of my response. For example, in this particular lesson, I had asked students which side was the hypotenuse and how they knew. One of my students responded and said “I know that side is the hypotenuse because it’s the one that’s diagonal.” My response in the moment was “yeah, but, actually, it doesn’t always have to be. We know it’s the hypotenuse because it’s across from the right angle.” I immediately regretted saying the “yeah, but…” part because I don’t want my students to be confused. Jon and I talked about saying something like “that’s really interesting! Let’s look at this,” showing them an example of something that disproves their idea. In this way, we are not shooting students down, not giving them incorrect ideas, and clearing up misconceptions (that much of the class shared.)
In the future, I want to be conscious of the ideas that we discussed. Firstly, I want to change my question starters and begin asking more genuine questions. I think my students will really respond better to questions about their thinking, rather than “why?” Why questions seem to be looking for the mathematically correct answer of why, which isn’t my goal at all. So, I want to focus more on the goal I have and which questions seem to be getting there. Second, I want to plan, plan, plan. I want to plan multiple questions that address the big ideas of the lesson and how to get students to be able to answer them. This will take practice and experience, but I’m enthusiastic to start trying. Last, I want to spend a minute or two actually addressing misconceptions. I don’t want to just tell them why something is wrong, I want to show them. I want my students to see exactly why the hypotenuse is not always the diagonal line in order to better understand and retain that information. I think all of these things will help me to increase the mathematical thinking of my students. ‘Cause, after all, it’s the journey that’s important, right?