What does mathematics look like to outsiders? It’s often presented as a polished, existing entity to be memorised, rather than as a messy and creative exploration of one’s own ideas. Reuben Hersh used a metaphor of mathematics having a ‘front’ and a ‘back’. The ‘front’ represents the formal, precise, and abstract mathematics encountered in textbooks and lectures, akin to the formal dining room in a fancy restaurant. In contrast, the ‘back’ is informal, intuitive, and tentative, similar to the chaotic scenes in a restaurant’s kitchen where chefs experiment, taste and tweak. Hans Freudenthal used similarly evocative language, describing the mathematics experienced by most students as the ‘fossilised remains’ of reasoning processes, bearing only partial resemblance to the real thing.
These metaphors provoke important questions about the mathematical experiences we offer students. Are they merely encountering the ‘fossilised remains’ of mathematical reasoning, as suggested by Hans Freudenthal, or are they embarking on a journey to unearth the full and vibrant landscape of mathematics for themselves?
The image below is how I picture mathematical thinking. This is the Design Squiggle, created by Damien Newman to illustrate the process of designing things like personal computers, desktop software, and graphical user interfaces. It’s meant to convey the feeling of the creative process, of starting with mess and uncertainty and ending with focus and clarity. When applied to the process of mathematical thinking, the left-hand-side—the ‘mess’—is us initially exploring why something is as it is. The right-hand-side—the ‘clarity’—is us eventually explaining why it is as it is.
On the squiggle below I’ve overlaid a number of verbs that I associate with mathematical reasoning. Don’t get too hung up on their precise placement. We can see the features of explaining (generalising, proving, justifying). We can also see features of exploring (experimenting, playing and specialising). How many of these are students encountering in our mathematics classes? I’ve also highlighted the verbs mentioned in the definition of the reasoning proficiency in the Australian Curriculum: Mathematics (AC:M). Take a look at these verbs. What are your thoughts about the verbs that aren’t highlighted?
I want to start by considering ‘play’. Playing with mathematical ideas is a cornerstone of mathematical reasoning. Francis Su captures this beautifully in his book. He says “Doing math properly is engaging in a kind of play: having fun with ideas that emerge when you explore patterns, and cultivating wonder about how things work.” One could argue that ‘play’ is encapsulated in the word ‘experimenting’, but I think play is a more joyful word that invokes a sense of following one’s own curiosity — and I’d like to hear us use it more often in relation to mathematics.
Another word that I think deserves more spotlight in mathematics education is ‘intuition’. Leone Burton interviewed a number of professional mathematicians about their practices, and reported that the overwhelming majority “recognised something important which might be called intuition, insight, or less frequently, instinct at play when they were coming to know mathematics”. Intuition is often the initial spark that gets a mathematician started on a problem. This intuition is then subjected to verification and examination through reasoning and proof. And it’s not a one-way street. The process is more of a dance; a zigzag between intuition and formal verification. This interplay, this back-and-forth, is the essence of mathematical discovery and reasoning.
‘Convincing’ is a particularly layered mathematical action. In their book ‘Thinking Mathematically’, John Mason, Leone Burton and Kaye Stacey talk about three steps of convincing:
Convince yourself – play with an idea until it starts to make sense to you
Convince a friend – put your thoughts in order to see if it is plausible to a friendly ear
Convince an enemy – present your argument to someone who is skeptical and eager to expose the cracks in your argument.
Cultivating these skills in our students is challenging, especially as we want them to eventually internalise the process and be able to be sceptical and look for holes in their own reasoning.
Infused throughout the whole process is sense making. Alan Schoenfeld emphasises the importance of approaching mathematics as a sense-making discipline. Students should expect mathematics to make sense, and as sense-making becomes the norm, something beautiful happens. They start building their own ideas, forming connections, developing insights that are uniquely theirs, and becoming empowered to be mathematical explorers.
What can we do as teachers? One thing we can do is use good tasks. If we only feed students a diet of routine exercises, then there is nothing to chew on and nothing to reason about. The tasks we choose should encourage all forms of reasoning, as depicted on the squiggle, although there may be times when we focus more closely on developing a specific aspect of mathematical reasoning. However, it goes well beyond task selection.
We should aim to foster an environment where curiosity thrives, where mistakes are part of the learning process, and where listening becomes an art form, attentive to the thoughts, uncertainties, and insights of students. Our aim is to encourage students to dig in and discover the ‘how’ and ‘why’ of mathematics for themselves.
Each of the verbs on the squiggle deserves careful thought on how it can be incorporated into our teaching. Tracy Zager’s book is one of the few I know that considers how the practices of mathematicians — taking risks, making mistakes, asking questions, using intuition, and more — might make their way into our classrooms.
Mathematics is often encountered as a curriculum of dry nouns. What would it mean to focus instead on the rich verbs? I think it could transform the very landscape of learning, shifting the emphasis beyond teaching mathematics to nurturing mathematicians. How might we do this? I’d love to hear your thoughts.
References
Burton, L. (1999). Why Is Intuition So Important to Mathematicians but Missing from Mathematics Education? For the Learning of Mathematics, 19(3), 27–32.
Hersh, R. (1997). What is mathematics, really? Oxford University Press. [Pages 35-37 which contain the ‘front and back’ metaphor can be read for free on Google Books, and it’s well worth going directly to the source!]
Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed). Pearson.
Schoenfeld, A. H. (2022). Why Are Learning and Teaching Mathematics So Difficult? In M. Danesi (Ed.), Handbook of Cognitive Mathematics (pp. 1–35). Springer International Publishing. https://doi.org/10.1007/978-3-030-44982-7_10-1
Su, F. (2020). Mathematics for Human Flourishing. Yale University Press (p 50).
Zager, T. (2017). Becoming the math teacher you wish you’d had: ideas and strategies from vibrant classrooms. Stenhouse Publishers.
