(Procedures with Connections)
I think there is a lot here about the set up and structure of tasks in relation to student agency, but I haven't unraveled that yet. I would love to hear your thoughts.
In 'Task 2," the situation is much different. The problem is pitched like a puzzle. There is a clear question, but the solution to that question is not the end of the problem. The problem ends (potentially) when students create a piece of mathematics to describe what they are noticing. I think this makes things a lot different. This task isn't designed to get students to understand something specific. It is designed to get the DOING mathematics and thinking mathematically. There is no pressure here. I don't feel like I need to hold everyone accountable to something. We can just think together and wherever we end up....that is where we will be.
What We Value
I think we should ask ourselves..."Why teach math?" I can't help but think that a lot of the things students 'learn' in school will soon be forgotten. What they might remember, though, is what our classes and teaching taught them about themselves. I want students emerge from high school trusting their own thinking and having confidence in their ability to figure things out. I'm beginning to think that sometimes the tasks we choose and our expectations for students might get in the way of that.