I would like to begin with a conjecture:

*The amount of collective action in a learning system is inversely related to the possible degree of curricular specificity.*

The mathematical action of a group of learners centred on a particular task gives rise to a unique way of being with the problem, but also reinvents the problem.

In short, what emerges from collectivity is not tidy.

How can I justify curating a collective of learners, when school is so interested in individuals?

Learners commerce on a central path of mathematical learning while acting on a problem, but each take away personal, enacted knowings from the process as well. Collective consciousness grows as agents interact, but we live in a system that values individual learning–often in a very narrow sense. Although I cannot be sure where the problem will go, students will become more mathematical by acting on it.

This wondering has been pushed to the forefront of my thought by two events today. First, a Skype call with a graduate supervisor regarding the nature of collective consciousness and its relation to the outcomes-based school system we teach in. Second, a moment of personal significance from a Grade 11 class on quadratics.

Here is what happened and why I think it illustrates the essence of education:

I have a group of grade 11 students who have never seen quadratic functions. In a effort to tether the idea to linear functions, I organized them into random groups of three, gave each group a large whiteboard, and asked them to graph the function using ordered pairs:

I anticipated students beginning their table with “x= -3” because the values -3 through 3 were commonly used in our study of linear functions. I purposely gave them a quadratic that returned large numbers for the first few inputs.

I watched as the groups began organizing themselves around the task. It wasn’t long until each group developed a personality. Some groups divided inputs among themselves, and built a joint table of values. Others worked through the arithmetic together. Conversations around input choice and error correction began as their pattern-finding skills took over.

*I gave them the licence to do so*.

*“We noticed that the growth wasn’t constant, but it did grow constantly”*

*for me*! They didn’t know it, but I pulled an extremely valuable

*individual*knowing from our

*collective*knowing. They centred their group work on the idea of change. From there, they looked at the symmetry of change and how it created a parabola. This is valuable work. Each had encountered the math and created personal coherence from the task as well.

*what*students digest mathematically, it doesn’t mean that the classroom events provide only group knowing and lack personal meaning.