Friday, May 29, 2015

Connecting Quadratic Representations

I always introduce linear functions with the idea of a growing pattern. Students are asked to describe growth in patterns of coloured squares, predict the values of future stages, and design their own patterns that grow linearly. Fawn's VisualPatterns is a perfect tool for this.

While stumbling around Visual Patterns with my Grade 9s, we happened upon a pattern that was quadratic. The students asked to give it a try, but we couldn't quite find a rule that worked at every stage. While I knew this would happen, the students showed a large amount of staying power with the task. The pattern growth was an engaging hook. After a conversation about what made this pattern ugly (the non-constant growth), we looked at the growing square.

It was then that I decided to try and insert patterns into the introductory unit on quadratics--a notoriously dry and abstract topic (at least in my class). I developed a series of quadratic patterns and animated their growth on Vine. They can be found on the Vine page on this blog. (Look for Quadratic Patterns). Each one is represented as a yellow set of blocks, but then as coloured blocks to illustrate possible ways to visualize the growth.

The plan was to play with patterns, develop quadratic functions, and then use the blocks to find the shape of the new, non-linear graph.

How I Structured Class:

I randomly grouped the students into groups of three, and started them off with a linear pattern from my Vine page. Once we discussed how to generate a relation, I asked them to copy down the first three stages of this quadratic pattern:

I then gave them each a handful of tiles, and asked them to model stages one through six. This took a little bit of trading as some groups became obsessed with having the colours match in all the squares. This (while slightly OCD) came in handy in visualizing the graph later.

Once all their stages were built, I collected residual blocks, and then ask them to do the unthinkable--destroy their patterns. But keep the stages separate. I drew a Cartesian Plane on the board and told them to use the tiles to build a sort of bar chart on their tables.

The results were the shapes of the graphs of parabolas.
Here, the +2 is represented by one red block on the bottom and one on the top of each stack. 
After stage seven, the pattern had outgrown the table
Some decided that, to save space, they would stack their blocks.
This stack preserves the +2 with the use of two blue foundation blocks on each stack.
This quickly became the method of choice throughout the class.

OCD colour matching comes in handy for pattern visualization
The cool thing was the conversations that emerged from the blending of the representations. Students began to use the metaphors of scaffolding that was holding up the graph, and roller coaster supports while the graph was the track. Some began to comment that it is good they didn't go past six, because it would have got out of hand. It provided an excellent image for them about the increasing slope of a parabola.

We talked about the cause of the rapid growth, and settled on the square. If we wanted it to grow slower, we would need to minimize the effect of the square. This brought the idea of width into play. We asked what would happen if negative stages could be represented. We used the function we built to help us conceptualize a "stage -1". It didn't take long until they had the concepts of vertex and axis of symmetry. (Although formal verbiage only became established when it became necessary for efficient communication).

I found myself referring to the activity when we were talking about the concept of range. We recalled how the patterns quickly out grew the tables, and would have continued on if we would have added stages. This helped them anchor the idea of an infinite range.

Quadratics occupy a large chunk of our grade 11 math curriculum. This was a nice way to connect a representation commonly used for linear relations and functions in the previous grades (that of the growing pattern) to the graphical and algebraic representations of quadratics.

Dan Allen captures the activity in 5 seconds. (The lesson now loops full circle)

Alex Overwijk posts about his experience with a similar exploration


Thursday, May 7, 2015

On Collective Consciousness and Individual Epiphanies

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. 

Why is that so big?
Make it smaller!
It won't fit, so change the scale.
The square is making everything too large.

The large output values perturbed the groups' thinking. Some handled it by changing the scale while others chose inputs that made the square as small as possible. I wanted groups to do the latter, but some resolved the problem as a matter of scale. They changed the problem, and I gave them the licence to do so

I wanted them to get at the idea of a vertex, a lowest point, or a turning point. I asked them why the graph was turning around, and because they had been given the opportunity to experiment with choice, they knew that the exponent was creating positive outputs from negative values.

This was perfect. It was exactly what I wanted them to get out of it, and I had harnessed collective action to get there. 

I would have been more than happy to distill these experiences into grade-level competencies, but then I took one more stroll around to discuss students' work with them. One group had a table of values on their board:

I commented on the growth of the y-values. One student said the following:

"We noticed that the growth wasn't constant, but it did grow constantly"

It was this moment that I pulled out my phone and wrote this quote down, because it clicked. She had described differential calculus. 

I took the time to act collectively with my students and it couldn't have paid off more... 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. 

Where these personal knowings landed, I could only assume based on our interactions.

For me, the personal knowing was centred around the connections between tables of values and calculus. The pattern they showed me occurred because the rate of change of a quadratic function is not constant, but does grow constantly--linearly. I had pulled the idea of a first derivative from a lesson on introductory quadratics. 

Circling back to the point of the post: I saw great learning from the groups. It wasn't all identical, and that enriched the fabric of the lesson (and the intended aims). From the space opened for collectivity, I pulled out a personal meaning--one deeper than I ever would have anticipated. 

This is the very essence of education. In a system obsessed with individual scores on specific competencies, we lose sight of the fact that deep meaning is pulled out of collectivity. It isn't one or the other. Curating a collective consciousness in the classroom allows students to build understanding in context as they change their problems with their actions on them, but that doesn't preclude them from creating powerful, personal meanings. 

The episode provides an illustration that even though collective student action softens control on what students digest mathematically, it doesn't mean that the classroom events provide only group knowing and lack personal meaning.