KSA #05 Differentiation · KSA #06 Planning · KSA #07 Student Needs · KSA #11 Assessment · KSA #15 Professional Learning

Open Problems: Learning to Manage Variation & Working Memory

Mason and Johnston-Wilder (2006) began their article with the following: “the purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitized to notice and competent to carry out” (p. 25). I read this quote and I was hooked. I think this sentence phrases what I imagine to be all of our goals as teachers. Of course there is a place for rote learning, which we have discussed, but when designing a task, that is intended to be more than rote practice, I would think we all wish for them to be a “fruitful activity that leads to a transformation”.

The article discusses the perception of the task. In my experience the most important learner perception is whether or not the task has any relevance to real life. When I give my students a task, if they see how the skill they are developing will one day help them in life, there is more buy in. I would argue this is the first step to making a task meaningful.

Next on my list would be the classroom ethos and ways of working. We have talked lots about the need for a safe classroom community. If students do not feel safe, supported or like they can ask questions / take risks, how can we expect them to learn? I really liked how Mason and Johnston-Wilder (2006) discussed the conjecturing atmosphere, which is necessary for students to think mathematically. Students should feel safe enough to think out loud even when they are uncertain and supported by their peers when their idea is mulled over by the rest of the class. This comes back to seeing the students in our classroom as collective learners and not a collection of learners (Davis & Simmt, 2006).

In our PLC today we talked about the idea of scaffolded tasks, and how this hinders the learning opportunity for students. We talked about how the scaffold, structure, opportunities for students to notice differences and similarities, or to discern, needs to take place before hand through various lessons and activities. However, the general consensus from the staff was that a task should not be so scaffolded that students are able to follow steps, or able to complete pieces without having to put in much thought. We need to ensure that if we have given our students a task, that it’s structure doesn’t make it so that the learners are not getting anything out of it. The tasks that we provide our students need to challenge our students to think and not reinforce trained behaviours. Through this work we can develop a relational understanding in our students versus the instrumental understanding developed when students are “taught how to solve problems” (Skemp, 1976).

If anything it’s probably become abundantly clear that there is no correct way to structure mathematical inquiry in meaningful ways. As teachers we need to ensure that:

  • we are creating a classroom community that fosters collaboration and risk taking,
  • our tasks are authentic, rigorous and connected to real life experiences so that students see value in their education, and
  • we are effectively applying the variation theory of learning within our daily lessons, so that students are able to discern and learn the intended object of learning

These factors will help to support students develop a strong relational understanding so that they are able engage meaningfully in mathematical inquiry.

Until next time,
Mr. B

Davis, B. & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics teachers (need to) know. Educational Studies in Mathematics 61, 293-319.

Mason, J. & Johnston-Wilder, Sue. (2006). Mathematical tasks. In Designing and using mathematical tasks (pp. 25-68). St. Albans, England: Tarquin.

Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.


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