The set-up was that students were separated into groups of 4. Two students in each group were designated as the mathematicians, and the other two students were the recorders. They were given four minutes, during which the two mathematicians were supposed to talk through a problem, and the recorders were supposed to write down everything that the mathematicians said. I made strong suggestions that the mathematicians should say everything that comes to mind, not worrying about whether they said something wrong. And the recorders should not speak during this time. They only write down what the other two are saying. After these four minutes, I would call time, and the mathematicians and recorders switch roles. Now the people who were previously recording become the mathematicians, and they start working on the problem, and the mathematicians have to turn their mouths off and become listeners and recorders for another four-minute round.
A bunch of math educators got together one Saturday afternoon and did this activity, led by Avery, with the following prompt (credited to @mjfenton and @klreneau):
9+16=25 and 9*16=144When Avery started the first timer, the room immediately burst with frantic, excited discussion among the first sets of mathematicians. After three minutes, the discussion immediately switched into a new set of frantic, excited voices. When the second three minutes were up, people had a hard time stopping their work on the problem. We had a lively, fun discussion about the virtues and challenges in the activity. I was so pumped to try this in my class.
10+10+5=25 and 10*10*5=500
1+1+1+...+1=25 and 1*1*1*…*1=1
What is the largest product you can create using numbers that sum to 25?
The following week, I did this activity in one section of Geometry and again in two sections of Elementary Algebra.
In Geometry, my burst experiment was fairly specific: find the angle measurements of the spiraling pentagons. I had them play with the pentagons for a good 10 minutes with no guidance, and they made wonderful patterns, and then I put them into groups for the burst and set them on the task of figuring out the measurements for making the pentagons. I had only only seven students in class that day, so they were in a group of 4 and a group of 3 (with group of 3 always with one recorder and two mathematicians). The discussion was not very lively, and it didn't get very far. We had discussed the polygon angle sum theorem at the end of the previous lecture, and I figured this was a perfect follow-up activity, yet nobody really brought the angle sums into the discussion. I felt like the whole activity was a bust, and I was prepared to go in to class the next day and have a discussion about taking risks and learning from failures. However, I asked my geometry students about their experience with the burst, and apparently it wasn't a failure after all. I think what it was was very different than the experience of trying this out in a room full of already-engaged, already-math-minded math teachers. Some of my disappointment, honestly, was just the difference of feeling. When we did this at the tweet-up, I recall being immediately energized and engaged, and somewhat fed by the energy of the whole room frantically thinking through the problem. And my students were much more timid, quiet, and slow in the process. That made me think that they weren't hooked. After discussing it with them, I think it's just that they were slower to ramp up. They all (7 of them) felt like it was a valuable and energizing experience. Still, this experience offered me the opportunity to talk to them about failures and how important they are, and how vital it is that they learn how to accept and use their failures, yet how difficult it is for me, as a teacher, to try new things at the risk of failing in front of a room full of expectant students.
The following week I tried a burst in both sections of Elementary Algebra, using exactly the same "break up 25 to make the largest product" problem that we did in the tweet-up. In the first class they seem to spend the entire first four minutes just trying to figure out what the problem was, and they were only able to actually engage in the question for the second four minutes. Some expressed frustration at the time limit, saying they felt rushed.
In the second class, I changed the way that I introduced the problem, mostly by writing real-time and talking my way through the "here's one way to break up 25, and here's the product of the breakdown; and here's another way we can break up 25..." rather than just showing them a pre-made slide and talking them through the problem. I also extended the time to 5 minutes per round. There was still the phenomenon where the first group was mostly, quietly, timidly just trying to figure out what the question was asking. But when the timer went off and they switched roles, there was a noticeable sound increase in the room. As if the recorders were biding their time and just seething, waiting for their chance to talk as mathematicians in the second round.
Interesting after-discussion in the second class, and I decided that I had to write down what they said. For some reason I was inspired to ask them, did they feel like they were doing math. Almost unanimously, the class shook their head, "nooooo." After asking why not and getting responses, I decided I needed to write down what I was hearing:
- it was fun
- there weren’t any formulas
- we weren’t being graded
- there wasn’t a right answer
- it was simple arithmetic
- it was competitive and collaborative (math is something you do by yourself)
- there wasn’t pressure
and at some point I asked if anyone thought that it did feel like math, and about 6 students emphatically nodded and raised hands. And I got this:
- numbers
- it got confusing
- I got frustrated, and math usually does that to me
- we used calculators
- we were doing calculations
- we used strategies
- we had to collaborate
These lists made me sad nearly to the point of tears. It was like the piñata of all of the challenges of teaching math to adults, broken wide open. I guess the happy bit of it was that we were able to have a discussion about how sad and destructive and incorrect many of these myths of mathematics are.
I'm still thinking about this activity, and wondering how good it is. I haven't tried it again since then. One big caution I have is around the speed issue. I'm extremely careful to emphasize to students that speed is not a necessary ingredient for being a good mathematician, and this activity does have a rush aspect to it. In some ways, I think the speed problem is balanced by a suggestion that "maybe you won't reach an answer at all," and that the point is to have unfiltered discussions. Having the time limit, I think, gives more freedom for the unfiltered discussions. Yet some students did express some anxiety over the time limit. So I'm wondering if there's another way of presenting it to help relieve that anxiety. It might be helped just by doing the activity on a semi-regular basis, so they can get in the habit of having the discussions, knowing that there's no Real Goal, other than to have the discussion. I also think that I need to modify the geometry question. There's something much more accessible and open about the sum/product question than there is in the geometry question.
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