Counting Cogs: Power Equalization Part 2

In the complex instruction teaching model, and in many other articles, TED talks, etc. that I’ve seen lately, the idea of shifting power from the teacher to the students is central to creating a classroom where students are fully engaging with challenging and interesting math topics. When I wrote about my colleague helping her student to recover and “save face” in front of the math class, this is one step towards a classroom where the balance of power either shifts permanently or shifts frequently. When we don’t give students the answer (as if we are the only ones with access to the “answers”), then we fundamentally shift the thinking to one of finding the “answers”. This is the first step.

This first step can be a difficult one. Many in the math world think that there IS only one answer and that there should be only one expert in the room. I disagree. I believe, fundamentally, that all of us have math expertise in us. I don’t think I’d be a Math Fairy otherwise. I believe in the magic of math being something that we all can find. So, if you’re not there yet, if you prefer to be the expert, I would just challenge you to try ONE time to give the kids a problem or challenge to which you do not already know the answer. Don’t figure it out ahead of time. Don’t make an answer key. Figure it out with the students as if you are all a team working to solve together. See what happens. Maybe you’ll hate it. Maybe you’ll love it and be willing to try a little more.

Here’s an example of what I mean, Part 2 of exploring Power Equalization in the classroom…Counting Cogs.

If you colored ONE tooth on each cog, which pairs of cogs let the colored tooth go into every “gap” on the other cog? Why? What patterns do you notice?

I was looking for an interesting challenge for my students to explore Greatest Common Factors and Least Common Multiples. Naturally, because I love Jo Boaler’s work, my first place to look was the Stanford University youcubed website. There I found an investigation called “Counting Cogs”. I rewrote the directions to provide some scaffolding for the process of cutting, organizing and getting started. Then I printed the cogs and we were off! I did not figure the problem out ahead of time, although I did have an idea of what I thought might happen. On the first day of what has turned out to be a three-day activity the kids worked in their table groups of 3 or 4 students to cut out the cogs (with teeth of 4 through 12), color them and start trying to figure out the challenge:

Immediately, we started having questions, confusions and wonderings. I realized that I didn’t even understand fully how to turn the cogs! My complex instruction mentor was visiting that day so she and I both started talking with groups and trying things out. We figured it out with help from the kids and then I was able to help subsequent groups figure out how to roll the cogs. In the past I would have freaked out that we were investigating something that I wasn’t even clear on. Not anymore. I know that making mistakes is great for brain growth. (Thank you Youcubed!) I also know that when kids need to work hard to figure out exactly what the question is asking, they are building resiliency and problem-solving attack skills.

So that step was resolved and then we started turning cogs, recording results and looking for patterns. As I was reminding the students at the beginning of day 2 how to turn the cogs to see if the two cogs worked together as per the essential question, I got really excited myself about exploring these relationships. I had made an incorrect assumption that the cogs would work together if they had common factors. Turns out it’s just the opposite but it took exploring and trying lots of different pairs to start seeing that. On our Day 2 the kids spent the class period turning cog pairs and recording results. Almost all the kids quickly began to see that consecutive number pairs would work together. My response to them was, “why?”, which is a question that I continued to ask myself.

On Day 3 I asked the kids to stop investigating and try to gather together what they had observed so far. We talked a lot about conjectures and evidence for those conjectures. Our focus for this last day was Math Practice standard 3 about creating arguments and critiquing reasoning. We spent the second half of class with each group coming up to share their ideas and then taking questions from the class. Such great ideas! Such great questions! I was super proud of my sixth graders….creating and defending arguments can be challenging even for adults but they did so well. Here are some examples of what they wrote on their final product:

• 5 times 2 is 10 so 5 and 2 cannot work
• Our second conjecture is that executive (they meant consecutive!) numbers, like 4 and 5, work.
• I think that if one of the numbers goes into the other one, then it won’t work. We proved that if one of the numbers goes into the other one, then the pair doesn’t work together because we tested all the combinations.
• Halfs of numbers don’t work because they go to the same spot every half time around.
• One thing we notice is more “yes” than “no”!
• Numbers that are double to each other don’t work but 11 and 7 work with everything.
• When you work with numbers that are half of each other they don’t work and they keep coming back to the same place.
• At least one of them has to be an odd number or it won’t work.
• Numbers that 3 can go into don’t work like 6 and 9 or 9 and 12.
• 7 and 11 are able to go into all the cogs. This may be because they have no common factors under 12.

I believe that our work on tasks such as this Counting Cogs activity is shifting the power paradigm in our classroom because:

• I was not the “keeper of the answer”. Students had to reach their own conclusions and understanding.
• We were all working together, helping each other and sharing ideas.
• Each team shared their thinking and accepted questions from the class, as if they  were the teacher/expert(s).