This is an interview I conducted with AFEMO, a group of Francophone math teachers in Ontario.
You present your problems in 3 acts. What are these 3 acts and what is the purpose of presenting or solving a problem this way?
The first act is the setup of a problem and attempts to create interest in a particular question. It’s often a video or photo of the world around the student. In the second act students and their teachers “mathematize” the question, adding information, structure, while learning necessary skills for answering it. In the third act, we show the answer as it exists in the world and students see if the math they used was accurate.
What are the advantages of working a problem in three acts?
Many times we give students all kinds of information and structure and skills before they know why they might be interested in it, essentially beginning in the middle of the acts. The three act structure reminds teachers they should give students a reason to care. It also involves students in the challenging act of mathematization whereas many problems come pre-mathematized by the teacher or textbook.
The problems you present make use of technology. They mostly start with an image or video and use few words. Why do you think this is important today in math class?
It’s important because that is how we experience the world as problem solvers ourselves. We typically start with just a question on our minds based on something we’ve seen and only then do we start adding words and information.
What act is essential or is the most important in the 3 acts?
There are only three of them! How can I pick? Each one carries a lot of weight.
There is often a sequel to your 3-act lessons. How would you use this sequel? How important is it? Should they be named 4-act lessons?
Students finish work at different speeds. The sequel then extends the work of the original problem, often times by reversing what was previously known and unknown.
You often ask the students to formulate the questions themselves. How important is that step? What are the criteria of a good question?
Asking students to ask and share their own questions about some context makes it easier for them to see the essential aspects of the context. It activates their own curiosity and prepares them for whatever question I’m about to ask. Personally, I don’t know how to evaluate questions as either good or bad. I welcome any curiosity so long as it’s sincere.
What type of skills or concepts do you think students will develop with this approach? What is different from a more traditional approach in teaching math ?
Students develop the capacity to model the world using mathematics, to take the messy world and decide which parts of it matter to a certain question, to go and get helpful tools. A traditional approach gives students trivial questions, lots of information, and a formula to put the information into. It’s more tedious and less valuable at the same time.
I can see that in presenting this type of problem, a teacher needs to use a lot of creativity. Can a teacher develop his creativity or where can he find good ideas?
I don’t think teachers need to come up with these problems from scratch. It’s important that they, themselves, solve problems mathematically, though. If their only experience of mathematics was being lectured at and completing lengthy sets of exercises, teaching using problems as a vehicle will be very difficult.
How does creativity fit in to mathematics? For the teacher? For the student?
If I had to choose one â€“ creativity or curiosity â€“ I’d take curiosity every time. If teachers have experienced the electric sensation of being puzzled by something and then working to unpuzzle themselves they are more likely to create similar experiences for their students.
In this type of teaching, teacher interventions are critical in guiding studentâ€™s learning. What is or are the main challenge(s) for a secondary teacher? a primary teacher?
One large challenge for both primary and secondary teachers is knowing the best conditions for that intervention. Many teachers simply intervene from the first moment of class. “Today we’re going to learn how to add fractions with unlike denominators.” That kind of thing. Students need experiences before those interventions, experiences where they encounter a need for the intervention or where they approach a question on an informal level first before the teacher intervenes and helps them formalize it.
What first step should a traditionalist teacher that wants to change do?
I think a very productive question we can ask ourselves first is “Why are we teaching this new topic?” And I don’t mean, “What is the real world application of this new topic?” which may not always exist, or even “Why should we care?” I mean more, “Where did this come from?” I mean, “How does this new topic arise from the old topic? How does this new tool fix the limitations of our old tool?” And then the really difficult question, “How can I put my students in a position to experience those limitations before I help them develop the new tool?”
For instance, we solve systems of equations algebraically rather than graphically because the graph doesn’t give us a precise enough answer. So if we put students in a position where precision is really, really important and ask them to solve by graphing, they may then experience a need for algebraic solutions.
For another instance, we develop algorithms because we see the same solution method appear over and over again to the point of boredom and it makes sense to formalize it. It doesn’t make sense to teach students algorithms before they’ve experienced that boredom!
How does your approach in three acts differ from the 3 step approach used in Ontario (mise en situation, exploration, Ã©change mathÃ©matique)?
The Ontario approach seems suited to general problem solving whereas my three act approach aims more narrowly at real-world modeling. This means the first act is somewhat more specific than thinking about previous solutions. It’s intended to generate questions. The third act of my approach, rather than being a general review of solutions, is meant to verify whether or not the world works mathematically, by showing an answer.