In my last blog I talked about tasks I chose for my students based off of their mathematical thinking. I was able to do that because the research on student adaptive pedagogy developed a progression that allowed me as a teacher to look for some classic behavior. Last time I wrote about students who use fixed finger patterns or reused fingers. There are so many ways students use their fingers, but they don’t all mean the same thing. In fact, how they use their fingers can allow me as the teacher to understand their thinking.
The next significant classic behavior I noticed in my students is when they only said the number words for the first addend. If they were solving 47 + 8, they would say typically under their breath, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11….47.” Lifting fingers sequentially, they would continue to count, “48, 49, 50, 51, 52, 53, 54, 55.” Now they did not need to reuse fingers for the first addend. They simply had to say the number words. This process may seem like a small thing for us as adults, but it is an exciting step towards the abstract for students. I remember when I was first teaching I would tell my students that mathematicians loved to look for efficiency. Students are on this journey of stripping away everything they don’t need in order to solve these addition tasks.
We usually call this type of method “counting on.” When a student can solve a problem like 47+8 in the manner described above, I know that this student is ready to work with more abstract material. Check out this video of Arya counting on.
At the point a student has begun to count on, they are ready for “change unknown” tasks. Too often we are presenting tasks for students that don’t make sense for them. Below is a video of a child solving for an unknown change. Look at how she is using her fingers.
It appears to be very similar to how a student might have used them in the “result unknown” addition task, but there is so much more going on. Knowing that it is not a good fit to present a “change unknown” task to my students who reuse fingers is so critical for both my students and me. I don’t want to give students the message that math doesn’t make sense, and they should only do what I do. That is what I think happens to students all the time because we don’t realize they aren’t ready for that type of problem. The progression allowed me to be in the “sweet spot” of all my students. This empowered us all to learn and grow.