With this second post, I’d like to start writing about the sorts of things I hope to write about most – what I’m actually doing in my classroom, the recipes I’m designing and using to try to make the smartest cookies I can. Everything I do is very much a work in progress. I welcome all suggestions, tips, resources, and nudges in new directions!
That said, here’s what I’ve been doing for the last two days in my Algebra 2 class.
Two and a half years ago, when I began teaching, a colleague gave me this article to read. It’s called Habits of Mind: An Organizing Principle for Mathematics Curriculum. It opened my mind to new ways of thinking about math education and helped to solidify and put into words (and legitimate – someone else actually does this??) the ways of thinking about math education that had already been floating around in my head. I love the way it’s set up and the language it uses.
Many ideas from this article stuck with me, but I think the habit of mind that did most was, “Students should be tinkerers.” Here’s what the authors have to say about mathematical tinkering:
Tinkering really is at the heart of mathematical research. Students should develop the habit of taking ideas apart and putting them back together. When they do this, they should want to see what happens if something is left out or if the pieces are put back in a different way.
I love incorporating tinkering into the work we do in class because it’s easy to begin, but the sky’s the limit. It’s also a low-pressure activity – students aren’t expected to produce anything other than something interesting. The word “tinker” also, to me, implies lots of room for student choice and creativity. Tinkering shouldn’t be prescribed or predetermined – it’s playful and individual. Finally, tinkering is constructive. By tinkering, students are building their own little mathematical monsters with which they can surprise, entertain, and wow each other. And, because they chose how to tinker, those monsters are theirs, to take ownership and be proud of.
I said I’d write about what I did in class, so I’ll stop there and start talking about the tinkering we did in my Algebra 2 class yesterday and today.
The assignment I gave my Algebra 2 students at the beginning of class was to start with the function y=x^2 and choose a single change to make to it. Some of the examples of small changes I gave were: add or subtract a constant; multiply or divide by a constant; change the exponent on x or y; and, add or subtract terms with some power of x or y. Then, I told them to make small changes in the change (say, add 1, then 2, then 3, then 4 to the equation) and graph each resulting function. How does the particular change chosen affect the graph of the function?
Some background: We just had a week off from school. Before the break, we’d been doing a lot of work with graphing parabolas in “vertex form.” When planning classes for this week at the end of the break, I wanted to refresh their thinking about the relationship between a relation and its graph before beginning to work on completing the square. I really wanted to reinforce the idea that small changes in a relation will make changes in the relation’s graph that we can predict.
A few students immediately chose a single, simple change to make to the relation and started graphing, as per the assignment. At least half of the students made a bunch of really bizarre changes and got a crazy graph. I expected this. Only the most conscientious student could be presented with an extremely powerful graphing program and not type in the weirdest thing he or she could think of. While graphing insane equations wasn’t the point, I didn’t want to squelch creativity. They really were asking, “What happens if I do this?” as they graphed their bizarre equations, which was the goal of the assignment. And the things they were creating were quite interesting.
So, instead of redirecting them by saying, “Now, now, little Johnny, don’t do that, do the assignment,” I stopped at each student’s computer and said, “Wow! What a crazy cool thing you made! How did you do that? And, what do you think will happen if you make this tiny change?”
While this strategy may have made the assignment take longer (which I have the luxury of allowing), I think it led to much more creativity and student ownership over their work. By the end of our second class on this assignment, everyone had completed the work they were asked to do – but in their own way. Several students wrote paragraphs describing the graphs’ transformations. Others plotted many graphs on the same screen and labeled them – one even started using Photoshop to color-code them. Two students challenged each other to an artistic graph-making duel. One of those students nested almost 100 sixth-degree curves to create a wave-like figure; another graphed various versions of y=x^(2x-c), for different values of c, which made a picture that looked like fireworks. Each student had a unique piece of work that he or she was proud of. I’m really looking forward to the sharing session we’re going to have in class today.