Yesterday was the first day for my intermediate algebra class, which meets for a 2-hour block. Students spent the first hour solving linear equations (a review topic) with no instruction. We followed up with a class discussion regarding which problems they struggled with, and got students to offer their advice for those problems. It was a great start – students being responsible, communication, positive classroom atmosphere. I also had opportunities to sneak in some growth mindset ideas: speed is not important, mistakes make our brains grow, it is important to think and actively participate rather than be passive and watch.

My favorite part of the class came at the end of the first hour. I walked to the board and wrote the equation |x + 4| – 6 = 3. Then I told the class “This is an absolute value equation and I can tell you how to solve it, but it will be more beneficial if you try to find a solution without me telling you how first.” So I asked them to find a number that worked as a solution. After a couple moments I could see that several students had found a value, and when I asked for it one student gave me x = 5. I asked her how she found it and she told me that she knew that 9 – 6 = 3, so she knew that |x + 4| had to be 9. She followed up by saying that she knew that 5 + 4 = 9, so 5 was a solution.

Perfect! The reasoning was outstanding, and she explained her thought process so well. Then I told my class that there was a second solution, and I wanted them to find it. After a couple more moments I saw that several students thought they had it, and were explaining their reasoning to the students around them. I asked for a solution, as well as an explanation. One students told me that his solution was x = -13, and he knew that -9 also has an absolute value of 9 so he just had to figure out what made x + 4 = -9.

So, my students were able to come up with the reasoning for the procedure to solve an absolute value equation (isolate the absolute value, rewrite as two equations, …) BEFORE I showed them to procedure. That level of understanding will help them when it comes time to using the procedure.

I’m afraid that many of us are so worried about covering all the material that we are not allowing our students the time and space to think. And that robs them of chances to develop deeper conceptual (or visual) understanding. Part of my plan to improve my teaching is making a conscious effort to allow my students to discover more by letting them think and experiment.