Unit Coordination

Perturbation Instigating Reflection in Students

Hopefully last week you read the blog by Tiffany Friesen in which she discussed perturbation.  In it, she gave a couple of examples from her own experience. Both of the situations that she mentioned were familiar enough to her that she had the capacity to resolve her confusion. She had familiarity with making the cookies in the past, as well as cooking experience, and she knew how to become more familiar with a word in a new context. They were not so ‘out of her league’ that she ignored them and gave up.

For a math student, the situation may relate to an activity that happened in class. We have seen students moved to a state of disequilibrium many times. Perhaps you know a student who can add only as long as they have enough fingers on which to count. That is fine for awhile, but what happens when they have need to add to a sum larger than ten? One move is to slowly progress the student to adding values that sum to eleven. That is a small move that will cause a bit of disequilibrium and require some reflection. The student will feel perturbed (unsettled) at not having quite enough fingers and may reflect on other ways to arrive at their answer. Perhaps they will count something else in their field of vision or perhaps they will imagine an eleventh finger. Some will choose to reuse a finger that they counted previously. The point is that the child is the one who makes the goal of finishing the count and finding a way to achieve their goal. It is also important that the student had some prior knowledge of adding.

With multiple experiences similar to this, the student will construct a stronger concept of number. If the child had been told they were wrong (having only arrived at ten) or had been provided with more material and suggestions from the teacher, they would not make the goal for themselves, experience the disequilibrium, reflect on their options, and build up a stronger concept. It is also the point that a standard algorithm was not taught and that the numbers did not advance too quickly so as to make the problem beyond the capacity of the student’s mathematical abilities.

I once had a colleague that told me if information was in front of her that conflicted with her current understanding, she was immediately perturbed and could not help but reflect upon the situation. The information in front of her related to material for which she had already built an understanding, but conflicted with her understanding in some way. What she needed then was time to reflect and to simmer on the thoughts in her head, check her understanding by reviewing it, read something additional or simply talk it through. How can we provide these opportunities for reflection to our students? What kinds of actions do you take to challenge a student’s understanding in a way that makes reflection inevitable?


When you hear the term partitioning, you might think about partitive division or partitioning a discrete set of objects, like dividing a dozen cookies among four people. Partitioning also applies to continuous intervals. An example would be the task of equally sharing a candy bar among 5 friends. The outcomes for how a child would… Continue Reading

Rich Questions

This month kicked off with a bang! It started with a trip to the beautiful Monterey Peninsula for the California Mathematics Council (CMC), North Section – Asilomar Conference 2016. If you have not been before, switch over to your calendar right now and mark off the first weekend of December for the conference. Then you… Continue Reading

Skip Counting with a New Goal

If you are an elementary school teacher, I am sure that you are already familiar with skip counting. We want students to learn how to count by 2’s, 5’s, 10’s, etc. We think of this as preparing them to understand and efficiently multiply. I have recently learned from the research that if we add a… Continue Reading

Coordinating Units, Part 3: Fractions

When looking at coordinating units, it is important to consider concepts other than just multiplying. One of those is fractions. Fractions would be first among these encountered by a child in school. Mental operations that must develop for a child to understand fractional ideas include partitioning, iterating, and splitting. These developments are not taught, much… Continue Reading

You Can’t Teach Math

A week or so ago I had the pleasure of meeting Dr. Les Steffe, whose research we have been learning from. As he was speaking, he made a statement that I think we notice, but often as an educational system, we tend to ignore. He said, “You can’t teach math.” Now, what did he mean… Continue Reading

Coordinating Units, Part 2

This post continues my September 20, 2016 post, “Coordinating Units: A Brief Introduction”. Last time, I introduced a problem to illustrate the basic differences between additive reasoning and multiplicative reasoning used to solve a problem. I also defined levels of units and what it means to “coordinate units”.  In addition, I said that “in activity”… Continue Reading