There is a lot to be said about the problem solving model that sits at the front of the Ontario Math Curriculum. It’s common place to hear and see teachers discussing and using a model, or a similar model, in their classrooms. It seems for some, it works, or, at least, it gives a framework for teachers to assist students to work through a math problem.
However, it never fully sat with me quite comfortably. First of all, I will make many non-friends writing this, but, it created zombie-like problem solvers. I witnessed many students who would be messy, take risks, and think during their math problem solving. The four step problem solving model was introduced, and suddenly, I found these creative math thinkers felt like they had to follow a path, or a formula, to solve a problem. They weren’t thinking as richly and deeply, and, in fact, were rushing to get through the steps rather than taking their time, and really questioning their math choices.
I have re-looked, re-examined, and re-thought of this model many a time. In fact, it has been the focus of math workshops and district meetings I have attended, and I not once could say that following this model in a linear manner made any of my students any better at math.
Don’t get me wrong: I see it’s purpose. I have found it most helpful with students who struggle to organize their mathematical thinking, and I present it more of a way of thinking, or a framework, they might want to consider if they find they can’t focus on the math, and are getting jumbled in the process of their mathematics. I never ‘forced’ a student to use it, as I always felt that solving a math problem — or any real problem in life, really — isn’t linear. Many times we might jump to carrying out what we think is right, find it didn’t work, then go back to realize we probably didn’t quite understand.
I recently revisited the front matter of the the Ontario Curriculum to really investigate the process expectations, as I was thinking to myself:
Don’t we go through processes of thinking while we are solving mathematics tasks, and also taking steps to do so? Could the two not be intertwined?
But, then I saw this:
A problem solving model hasn’t been re-investigated since 1945?
So now my head is really spinning. I wanted to know a few things:
- What do students really do and think while they are solving a complex mathematical problem? Do they naturally go through ‘steps’?
- What processes do they consider, or act through, when solving a math problem?
- How do we bring a problem solving model back to the actual mathematics?
In my mind, I started to visualize what I had noticed my students were doing during math. Over the months, I have been able to listen to many math conversations, and sat in on many math problem solving sessions with my students.
Right away, I wanted to depict how mathematics is a flow; people move in and out of their thinking process depending on what they are working on.
I feel that this is what my students do in the classroom while they are working on math. More than that, this is what I would like my students to consider as they solve a math problem.
I decided to take it one step further: I asked them to consider their thinking while they were working on some math.
I created a Google Form, and asked students to reflect what they were thinking before, during, and after their math task. Here are some results.
You can see the top two for before were noticing and wondering the math information, and choosing a tool. This was great — this means they were actively processing the problem, and selecting a tool (which is also one of the aforementioned process expectations).
During the problem, students were fixing mistakes and changing their ideas. This means students were actively seeking math connections, and finding and fixing their misunderstandings. Using a tool is still high on their choice, which means that tool must help them represent their thinking.
Students did reflect at the end, as the Polya Problem Solving Model states, which in the re-vamped model is embedded in the questioning themselves area — students, even at the end, are ensuring their mathematical ideas are complete.
I was really excited to see my student responses, and I am going to continue to check in with them as they work through math.
What do you think of my re-vamped Mathematical Thinking Flow Chart (because I am not sure what else to call it)?
Would you use this in your classroom? Do you see your students doing this?