Exactly where [a lesson] moves depends on such complex factors as the structures of those present, the context, and what has been anticipated. It may move toward more formulated understandings, if such formulation is relevant to the play space or if it becomes part of a further exploration. It may simply move to other sorts of activities. This, of course, is not to say that we should just allow whatever might happen to happen, thus abandoning our responsibilities as teachers. Rather, it is to say that we cannot make others think the way we think or know what we know, but we can create those openings where we can interactively and jointly move toward deeper understandings of a shared situation.
(Davis, 1996, p. 238-39)
My Grade 9 students are currently working on recognizing, analyzing, graphing, and solving problems involving linear relations. Linear relations lend themselves so naturally to describing patterns à la www.visualpatterns.org, and this is precisely how we got our toes wet in the topic: For several days, my students had been analyzing, extending, and (productively) arguing about a variety of linear and non-linear patterns. The intention of these first few lessons was to have students develop (or, in some cases, refine) an understanding of constant and non-constant change and to connect patterns in pictures to patterns in tables of values.
As the students began to connect ideas, I looked to develop an activity that gave students an opportunity to apply the generalizations emerging from the phenomena that we were playing with. Melanie Kong must have read my mind, because just as I was thinking about this next lesson, she shared a link to Nat Banting‘s Relation Stations task. In a nutshell, based on their recent experiences with constantly-growing patterns, groups of students are to create their own linear patterns. The three rules Nat provides for this activity are:
- You must use your tiles to model the first three stages of your pattern. (I lifted the restriction of needing to use tiles and allowed students to create patterns on whiteboards.)
- You cannot copy or repeat a pattern we have already seen.
- Your pattern must have a starting point and a constant change.
After the groups create their patterns, the students circulate around the room, drawing the next two stages of the patterns and filling out a table of values for each.
I had been planning a similar activity, but mine didn’t involve students circulating to study each other’s work. I’m glad to have made this change, because I found this to be a great opportunity to simultaneously validate students’ mathematical work, encourage appreciation for their peers’ thinking, and to challenge their own. (I now recognize that I need to provide similar opportunities more often in my classroom.) In their work on this task, my students created some lovely patterns of various degrees of complexity, including the following:
Even more so than I had hoped, the task provided an incredible range of opportunities for discussion that spilled over well into the next day. (What a humbling, rewarding, and fun experience it is to be surprised by students’ creativity.) Notice, for instance, the second pattern, where the number of squares is decreasing rather than increasing. What does Stage 6 look like? Notice also the third pattern, where the relation is either linear or quadratic, depending on what you count:
The main reason for this post, though, was a similarly ambiguous pattern (excuse my finger):
(Yes, I do notice the student doodling in the middle, and yes, I am a very recent convert to one marker per group.)
I brought this pattern to the next class to use as an opener. The students had had a long weekend, so we needed a refresher on the ideas we had been discussing last week. To be honest, I chose this pattern among the others that the students had created mainly because it was unique in the shapes it used – a combination of triangles and hexagons rather than squares, as every other group had drawn. I did not anticipate what would come to fall out of this pattern.
I set the students to work, with the instructions being to describe the pattern in words, in a table, and in an equation. I had expected that they would count the number of shapes, as in every other pattern we had studied. (I would be curious as to what your first instinct was!) Some of the groups did count the total number of shapes in each step, which is what the creators of this pattern had done. Others counted the number of triangles. One group decided they would count the number of sides/lines, including those on the inside. At first, I worried about the diverging paths the groups were taking, especially because some of those paths led them to non-linear results (this was supposed to be a review of linear relations, after all), and briefly contemplated instructing them to all count the number of shapes. But there was something magical about telling students who had likely waited for instructions for most of their school careers that it was up to them to decide where they would take the problem. And so, in Nat’s words, I let the problem drift.
As a group, we first brainstormed the different things we could count in this pattern; my students came up with at least 5. We described the more obvious patterns, after which I set the students to work on describing how the number of lines changed from step to step. (I happily scrapped the next pattern I was going to have the students work on.) What better way to solidify understanding of constant growth (linearity) than by contrasting it with an example of non-constant growth (non-linearity)? A student then mentioned that we could count the number of hexagons, and we argued over whether the pattern 1 (Step 1), 1 (Step 2), 1 (Step 3)… was linear. But then, another student piped up to say that the number of hexagons was increasing – indeed, take a look again at the third shape!
Although I hadn’t planned on it, we ended up studying this single pattern, whose depth slowly revealed itself before our eyes, for the greater part of the lesson, and I think we learned as much about Mathematics (its playful side, its emergence from human imagination) as we did about the mathematics (linear relations, constant growth). Looking back, this was one of those rare hours during which it was truly my students who were leading me through the curricular landscape (again, Nat’s words, not mine), rather than the other way around. The experience re-minded me of the need to remain flexible in my teaching – to move away from the image of teaching mathematics as a one-way path to an unpredictable and ultimately unknowable landscape, and to always (always) leave space in planning for students, lest we “freeze the body of knowledge that is otherwise dynamic, vibrant, and alive” (Max van Manen, as cited in Davis, 1996, p. 101). As you may have noticed through my references, this lesson brought to life for me several key ideas in Davis’ Teaching Mathematics: Toward a Sound Alternative (a work of art or a piece of poetry as much as a book about teaching mathematics). In particular, more than any other lesson so far, the experience has re-minded me that
There is no predicting what ideas will come up, what interests will emerge, what insights will arise. [A] “plan” is thus best thought of as a series of prompts or nudges to encourage movement through a mathematical space. It is not a scheme to be implemented, but a series of possible entry points for teaching action. It is, then, merely a starting place for a continuous process of anticipating; it is more along the lines of a strategy for an as yet unplayed game than an algorithm for reaching a particular destination. It is a way of stepping into the current of a curriculum.
Davis, 1996, p. 127
Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York, NY: Routledge.