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.


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.
For one hour of my school day, I work with an amazing group of Seniors in a class I have called "Mathematical Thinking" (more about that in a later post, perhaps). Mostly, the class is a mixture of problem-based units and other miscellaneous open-ended puzzles, problems, and mathematical games.

Yesterday, we worked on a puzzle/game called Cartesian Chase. I played a few games against students to demonstrate the rules (we confined ourselves to a 3x7 rectangle) and then just let them play for while. Then, I had them stop and record anything they were noticing in terms of a strategy that seemed to be working. Then, they switched and played with new partners for a while longer. I stopped them again after a few games and had them record updated strategies. We ended with a class a few "undefeated" people playing each other. It quickly became apparent that there was a winning strategy at play.

In the process of all this, here is what I noticed:
- nearly ALL of the students were engaged and playing for the whole time
- students were having fun with each other
- we had a few early conjectures in place about what strategy might be best
- students uncovered structure in the problem, used it to win every time, & were able to clearly explain it
- after the game was "solved," a few students were curious: "what if we added another column?" and "what happens with other board sizes?"

I work hard to bring this same spirit of playfulness to other lessons. I work hard to make every day feel like a puzzle in our class. For some reason, I can never quite bridge that gap in the way I would like. I think I get pretty close most days, but for some reason "how many burger combinations are possible?" still feels more like a math problem and less like a puzzle to students. Maybe it has to do with our intent as teachers? Do we place too much emphasis on students "knowing" something specific by the end of the lesson? Could we set up the task better (slower?) so that it emerges as a puzzle? I have a lot of questions, but I do know that I value what students are learning about themselves as mathematicians and thinkers from a lesson just as much, if not more than, I value students knowing some piece of the thing we call "mathematics."
At the beginning of the year, I started this tradition where each Thursday (after having been to grad school the night before) I would bring in a journal entry or something that I was thinking about and discuss it with the students. I thought it was cool for them to see that their teacher is still learning and constantly thinking about becoming a better teacher (and person). At first, they thought I was a little weird but now it has evolved into this anticipatory event each week that we call "Thoughtful Thursday" (unless the schedule necessitates moving it to the next day, in which case we call it "Filosophical Friday"). Last week my thought was about viewing the world through a mathematical lens. I told them that, often, I have toast in the morning and that the piece of bread I use is so big that I need to cut it in half in order to fit it in the toaster. Naturally, this leaves me with two pieces of bread. I also like to spread Laughing Cow cheese on my toast. The Laughing Cow cheeses come in these awesome little wedges and EVERY time I go to cut the cheese (I hear your snickering….grow up), I ask myself this question:

Where do I need to cut in order to guarantee that I get an equal amount of cheese on each piece of toast?

Of course, you could make a vertical cut and get a close estimation but I'm always more interested in the horizontal cut. Anyways, my point for "Thoughtful Thursday" was simply that looking at the world through a mathematical lens inspires us to 1. ask questions about the world we live in and 2. try to make sense of them in the best way we can. I left it at that.

The next day when the students walked, this happened:
Student: Mr. Meyer…are we gonna solve that cheese problem today?!?
Me: Wait, you mean you actually want to figure it out?
Student: Well….<shrugs shoulders>…yeah
Me: Do you think anyone else is interested?
Student: I don't know.
Me: <Asks the class how many would be interested in attempting the question>
Class: <3/4 of them raise their hand>

How much different would this have been if I just brought in this question and "forced" them to work on it? Not sure. Teaching is fascinating.
I'm reading "Out of the Labyrinth" right now and even though I'm only about 50 pages in, there are some pretty powerful quotes. This one is my favorite:

"To teach it now as if it were A Rule, or (even more intimidating), The Law, is to pretend that what took years of experimenting and ingenuity is as obvious as your nose. And then, because you never really had a chance to understand what was going on, whenever you need this rule again it will come as just that - an arbitrary fiat, enforced by Them. And so the whole integrity of mathematics is compromised. The only reasonable conclusion for a struggling student to draw from such pretense is that he is irremediably stupid, or that Mathematics works in mysterious ways, its wonders to perform."

and further down the page:

"...and so a teacher, who is supposed to develop our powers of inquiry, becomes instead a messenger of Received Truth."

There are many other incredible little tidbits in this book and, so far, it is making a pretty good case for a spot in my preferred reading list. As I read the above passage, I was reminded of something that I have been thinking about a lot lately (and even alluded to at the end of this post). We talk a lot about the importance of context in math education. There are many benefits to situating mathematics into some nice context that students find engaging/relevant (mostly, it seems beneficial to help students see that math can help them describe and understand their world) and there are some sites/people that are doing this very well.

But simply using an exciting context does nothing to remedy the fact that often the "whole integrity of mathematics" can still be compromised. I mean, you can start with a really interesting and perplexing question and still completely miss the boat when it comes to helping students do their own mathematics; it just becomes a better way of teaching "The Law." These types of questions could (and should) drive a whole unit because then we can explore different parts of the problem, honor different approaches and ways of thinking, and ultimately, help students create their own math along the way.
This one has been bugging me and I really wish I had more readers because I think there is a valuable discussion here:

What is the perimeter?

How long is the coastline?

The perplexing question here is essentially the same. This guy talks a lot about the quickness with which we move to abstraction. I'm wondering if maybe that applies here. I like the abstract because, in terms of creating a unit with a nice resolution, it easily comes full circle. I'm not sure you can say the same about the coastline. So…"what's the difference?!?"