I use "Problems of the Week" (or POWs) with my class every week. As far as I know, the Interactive Mathematics Program authors coined the term, but the problems themselves are usually well-known (?) puzzles from the history of mathematics. Students have always responded well to these types of problems...even students that don't always respond well to other types of tasks in the classroom. This has always been interesting to me, but it has really come onto my radar since I started my action research on habits of mind and agency. It has led me to believe that the tasks we use, teacher expectations about student outcomes, and what we value as teachers all have pretty powerful effects on the preservation of student agency.

Tasks

In 5 Practices for Orchestrating Productive Mathematics Discussions, Margaret Smith and Mary Kay Stein provide a useful table for sorting tasks by cognitive demand (see below). In particular, they outline two types of "higher-level demand" tasks: "procedures with connections" and "doing mathematics."
I have become really interested by the difference between the two. I have included two tasks below that I used in class recently. The first is modified from an IMP textbook and is, in my opinion, "procedures with connections." The second is a "Problem of the Week" borrowed from 'Thinking Mathematically' which, in my opinion, is "doing mathematics."

Task 1
(Procedures with Connections)

Context: Students have been working on a unit problem about whether or not a penny dropped from the Empire State Building would kill someone if it hit them on the head. We dropped a ball from the roof of our school, modeled it, and found that this equation was useful.

Task 2
(Doing Mathematics)

Students are pretty interested in the unit question about the penny, but the second task was by far more popular and definitely engaged more students in a variety of ways. Today I asked my Seniors, "How many of you would be happy if the class was only POWs?" Out of about 25, all but two said that would like that. The two that said they wouldn't like it cited "preparation for college" as their reason for not wanting that.

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.

Teacher Expectations

In 'Task 1' above, I think it is clear to the student that this is a typical math problem designed to get them to understand a particular topic or concept. In other words, it is clear from the outset to them that they are expected to get to a certain place by the end of the task. It has given them something external to value themselves against. Now, all of a sudden, if they understand it they are smart and if they don't they are not smart (I'm hypothesizing here about what the task is implicitly saying to students).

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

Standards and teacher expectations put pressure on students to 'know' certain things on a specific schedule. When we (teachers/curriculum/whomever) set this finish line in advance, it changes a learning environment. It becomes about measurement and judgement. I would just much rather put the priority on student thinking, student confidence, and student agency.

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.
 


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