This post is my contribution for Week 3 of the ExploreMTBoS Blogging Initiative. Join in on the fun!

Last week, my honors course began graphing compound inequalities on number lines. During the first day of our work with the concept, I used a series of everyday experiences to drive the lesson (speed limits and minimums on expressways, percentage ranges for letter grades, etc.). From the success most students had with these tasks, I was comfortable turning them loose on a mess of problems I posted around the room.

As I circulated about the classroom to talk with students, I kept noticing errors with the directions of comparison symbols, choice of symbols, shading of points, and directions of arrows. I was surprised at the mistakes students were making since they accurately solved context-based tasks earlier in class. When I asked students questions to guide problem solving, I began to notice a common trend among students that was causing the errors. I first asked students what questions they were asking themselves about the graphs or problems. I got puzzled looks or responses like, “None,” to my inquiry.

Now I knew the root of the mistakes being made: self-questioning.

The next day of class, I told my class about my realization. I shared how I kept asking students questions like, “What are we comparing in this problem? What quantities are being used in the comparison? Is the inequality true when x is more/less than this number? Can x be between these numbers? Is the inequality true when x is one of the points?” I asked in a blind survey how many people asked themselves these questions. Without surprise, few students raised hands.

I was humbled by this experience for many reasons. First, I’ve been trying to instill this need to self-question during problem solving from the start of the year. Second, students just got into the habit of asking themselves questions when writing equations for word problems. Third, I felt like it made everything from the previous day worthless. If students did not question themselves during the introductory activities, I questioned if my lesson (for how intuitive I thought it was with its grounding in familiar contexts) developed student thinking abilities alongside content knowledge.

After seeing the sparse amount of hands raised, I proceeded to write the questions I posed on the board for students to reference. I directed students to ask themselves these questions for the practice activities that followed, but I stressed that even simpler questions like, “What are we finding? What information do I need? How is this related to that?” should become a normal part of their thinking for all problems. Inequalities became pretty simple for students as the day progressed, since they were taking the time to ask themselves questions about the problems they faced.

My experience with questioning this week stirred within me a deeper thought I have about teaching. If teachers are aiming to develop thinking abilities and reasoning strategies in students, how do we develop these qualities besides repetitive modeling? In my experience this week, I ended up modeling self-questioning and directing students to self-question later in the day. How can we create situations and experiences with mathematical content that naturally push students to pose questions about their thinking about given problems? Bryan Anderson mentioned noticing and wondering as a means of developing student questioning abilities, but most of those questions are directed outward (formulating problems) rather than inward (the metacognitive task of asking yourself a question about a predetermined question or task). I would appreciate any thoughts or comments on this subject.

[cross posted to trigotometry]

I suggest giving them problems which they have never encountered before. These do not have to be difficult.

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See the previous post on this site – popcorn.

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This post reminds me of a persistent concern I have as a teacher and as a dad. I have an intellectual faith in the idea of modeling appropriate behavior. Unfortunately, both with my children and my students, too often I see behaviors modeled regularly are abandoned when I am not by their side or in front of them reminding them of these behaviors. I often tell my students that their number one job as problem solvers is to train themselves to ask guiding questions. I point out that when we are together I rarely tell them much, I ask them guiding questions so that they don’t need much information from me. This, to me, is probably THE great challenge of teaching, how to help students develop their own strategies to support their own learning.

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I couldn’t have said it better myself. Now to investigate ways to get students to actively recognize and build these strategies… I think the metacognitive aspect is a whole another ballgame in itself (especially when you’re working with students who are just starting to understand what thinking about thinking is).

Thanks for the comment!

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