I am curious if there is a way for transference of skills to be acquired over time, especially for a difficult solution technique.
To satisfy my curiosity, I am trying the following scenario. In August I gave my Integrated Math 3 students a task, working with a partner, they had to sequence out the strips of paper hidden in an envelope. I made enough for each to student to work with a partner, one student would move only the words and one student would only move the equations.
The students had an entire period to work, and asked when they finished to take a picture with their device and email it to me. This first round easily took the entire period, as students became familiar with the task and had to make sense of the problem. As students worked, I circulated and asked guiding questions, but offered no direct support. A little over half way through the period, I had two groups that were close an accurate sequencing of the problem, so I prompted that students should walk around and look at what other groups were doing, develop a dialogue, and see if some fresh ideas developed.
By periods end, almost every group had some form of an accurate sequence, and I was quite impressed by their perseverance in this task. Thinking of the SMPs, we had hit three of them hard (SMP 1, SMP 3, and SMP 7), and I really think the students enjoyed it, even if “their brains hurt.”
A month later, I gave them the same envelopes, this time as a warm up problem. After 5 minutes, most groups were done, many of them could recall the flow of the solution, with minor errors; however, what stood out to me was how students were smiling once they felt they were able to do this task again, with much less effort.
A few weeks later, the students come to class they saw the familiar envelopes, they grabbed them and started sequencing the steps without being prompted. Less than three minutes later, the entire class was finished, so I took pictures of each group’s result, and projected them. As a class, I asked the students if they were similar or different than their own, this lead to a great discussion about many of the pieces that were in place, and why certain strips went where.
Following our discussion, I posted two similar problems to the one they had been sequencing to see if their understanding would transfer. The students tended to fall apart in attempting to solve these problems, their understanding of doing and undoing, i.e. the use of inverse operations and a mushy understanding of the problems became apparent, though they were clearly on the path to understanding the sequencing problem.
One thing that shocks me is that there is little connection from the sequencing of the problem to the examples shown, even after guided questioning. Meaning I’m not asking the right questions to uncover their thinking to make these connections, or I didn’t set the stage well enough for them yet, or I jumped ahead of myself and need to revisit at a future time. In any case, my curiosity is not satisfied, I do not know if students are able to transfer their understanding of one way of looking at a problem to another way, or if this task is too cognitively demanding to test this process with. In any case, there is evidence of learning in many other areas, and the collaboration over this task has been a pleasure to observe.
I am curious about your experiences with sequencing, transference, and promoting perseverance in with your students.