There should be no such thing as boring mathematics. – Edsger Dijkstra
In looking at the results of the math mindset surveys my students completed last quarter, I was pleasantly surprised that, compared to a mix of responses to the statement “Math is creative.” on the pre-survey, my students (a mix of pre- and in-service teachers in a graduate level math methods course) took at the start of the course, 100% of them agreed (with 60% of them feeling strongly about this) that math is creative. What excites me most is that this shift was not in regard to their dispositions toward mathematics, but in regard to the very nature of mathematics itself. It got me thinking about the choices I made this past quarter that may have contributed to this shift.
When thinking about how I would approach the math methods course when I was first assigned to teach it three years ago, I knew right away that teaching through problem-solving would be its focus. I knew that, just like me, many of my students have learned to become math teachers by apprenticeship – watching their own teachers, who often used the gradual release of responsibility (GRR) approach for teaching math. That is, when introducing the day’s math concept, they worked out one example for the class, then worked out a few more examples with the help of the class, and finally assigned an independent practice worksheet. I am not knocking this method at all, I have and still do use it myself, but I felt that students were exposed to it so often that they needed to learn something different from me, specifically that the structure of “I do”, “we do”, and “you do” is a flexible one and can follow an order of “you do”, “we do”, then “I do”, to support inquiry rather than modeling.
The process of acclimating my students with teaching through problem-solving in my math methods course begins when with me assigning the Orchestrating discussions article by Smith, Hughes, Engle & Stein, which introduces them to the 5 practices (anticipating, monitoring, selecting, sequencing, and connecting) and the Bag of Marbles problem. Then, using GRR to model a lesson that utilizes teaching through problem-solving, we work through the border tiling problem. I have done both the paper and Desmos versions of this lesson in the past. I like using this problem as the first representation of teaching through problem-solving because it’s a straightforward problem that generates many different solution paths and students are usually impressed with the unique thinking of their classmates, so different than their first approach.
The other problem I use to model teaching through problem-solving is the mathematical tug-of-war problem. I love this problem because it does not use any numbers – it consists of pictures and short descriptions that can be read out loud, acted out, and modified to fit the needs of the students in the class. This is a math problem that has characters; it is funny; it is inviting to students who do not see mathematics as engaging and accessible. I encourage the students in my math methods course to ask silly questions, and they eagerly discuss what may have happened that resulted in acrobats and grandmas being involved in a tug-of-war. Are these the grandmas of the acrobats themselves? Are the grandas all retired acrobats? How come there are no grandpas? Whose dog is Ivan anyways?
Using these two problems to model mathematics instruction positions mathematics as creative, accessible, and appealing to different types of students. After I model, I ask students to plan a lesson, utilizing teaching through problem-solving, that they teach to the class in the following weeks. Last quarter my students engaged the class using problems such as Domino Effect, Stained Glass, Ms. Pac-man, Bed Bath & Beyond, and this problem that stumped many on the internet. It is no wonder that after participating in these lessons as teachers, learners, and critical friends, all of my students walked away believing that math is creative.
What have you done to position mathematics as creative for your students? Share in the comments below!
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