AIMS Center

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?

Meet the Children Where They Are: Episode 2

Narrator: “Last time on Meet the Children Where They Are: Mr. Unnamed nerdy-looking stereotypical math Research Associate (RA) is down in the mud getting ready to do some math with an unsuspecting 5-year-old child named Bob. Can he find Bob’s ZPC without damaging the child for life? Let’s head back to the construction site where… Continue Reading

Professional Learning with an Online PLC

Teachers must spend time collaborating, sharing experiences, and reflecting about what they are learning to assure deep, rich professional growth. Those who participate in long-term professional learning projects participate in and establish ways to collaborate, share, and reflect when meeting face to face. Equally important, are effective ways to do the same when some of… Continue Reading


When Dr. Thiessen first discussed his ideas about launching the AIMS Center for Math and Science Education with me, he suggested that our motto should be:      “Know the Math; Know the Science; Know the Research.” And, he said, even more importantly, we can never forget that:      “We believe in children’s knowledge!” I have been working… Continue Reading

Mind the Black Box

Here at the AIMS Center, a central focus of our attention is the mathematical thinking of children. It should not be surprising that children do not think like adults. While as adults we agree in theory, our actions consistently seem to contradict this truth. The habitual act of laying our own mathematical thinking onto children… Continue Reading

A Welcome Shift

The Fall semester (I have been in education so long I don’t see seasons as much as school terms), is one that is full of conferences and opportunities to reach out into the broader educational community. In my dual roles between AIMS and FPU, I end up at a significant number of conferences. This Fall… Continue Reading

Finding My Home

I am inspired to search out new ways to improve the Professional Learning Division at The AIMS Center. I continually think about teachers in North America and how dedicated, unselfish, and committed they are to the students they teach. At AIMS, we want to provide opportunities to assist classroom teachers to be their very best.… Continue Reading

Fingers as Math Tools

In my last blog, I reflected on my experience at the California Mathematics Council’s Southern Conference in Palm Springs and our presentation, “Don’t be Quick to Count On!”.  Referenced throughout the conference were the Mathematics Teaching Practices from NCTM.  One of these practices that resonates with the work of my team of research associates is,… Continue Reading


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