Thinking like a mathematician – Wonder in Mathematics

What does it mean to think like a mathematician? We often emphasise the verbs of mathematical thinking, such as reasoning and problem solving, but today I want to shift the focus to the practicalities of mathematical thought. Where does this thinking happen? What mental images do mathematicians form as they engage with mathematical ideas? And how are these captured and explored?

In my last blog post, I introduced Leone Burton’s research on how mathematicians come to know mathematics. In this post, we’ll dig deeper into her findings, focusing on where and how mathematical thinking happens, and how these insights might apply to educators.

Where does mathematical thinking happen?

In his book The Mathematics of Human Flourishing, Francis Su wrote that “mathematics makes the mind its playground.” While he was emphasising the importance of exploration and play in doing mathematics, his words also remind us that mathematical thinking isn’t confined to specific places. Mathematics can happen anywhere.

So, where does mathematical thinking happen when mathematicians don’t have access to ways to record their work? Burton posed this question to the mathematicians in her study, and their responses were so delightful that I’ve quoted this section of her book verbatim.¹

They gave me variants on the following although the elaboration of “in bed”, below, was unique to one mathematician:

I do think walking so when I am stuck I do go for a walk;

I typically solve problems in the bath;

When I am thinking about a problem, I have an infinite board in front of my eyes and I draw pictures on it. It is about 2 feet before my head, and I write on it with my finger. I read it, do calculations. I find it very difficult to do it without using my finger. That is the way I generally do work … typically I do it at night in bed. I think, principally I imagine, algebra and write down the equations. I do some sort of calculation. I could quite easily go on for five hours doing this in my head.

And perhaps most worrying was: I do all my thinking in the car.

Aeroplanes featured in a few of the stories: My Australian colleague proves theorems on aeroplanes in his head.

My best places for mathematical thinking are usually when I’m engaged in menial tasks — doing the dishes, gardening, showering, and yes, driving. In fact, given the well-known ‘shower effect‘ in creative thinking, I was surprised that showers weren’t mentioned. Australian mathematician Geordie Williamson recounts being in one in Beijing when he had an idea: “It’s the only time in my mathematical career that I’ve been hit by something like a lightning bolt. I immediately got out of the shower and I forgot to dry myself and I started writing code on a computer to try and check something.” He reflects on this experience in a short video and this ABC News article — both of which are worth checking out.

Brazillian Fields Medallist Artur Avila, in an interview with Chalkdust, revealed that he often does mathematics without writing anything down, making breakthroughs on planes and trains. He describes the restrictions of travel as forcing him to structure the question in a ‘smarter way’, leading to a better understanding.

What do these experiences tell us? Mathematical thinking often thrives in unconventional places—whether while walking, showering, or travelling—because these moments offer freedom from distraction and allow ideas to evolve and connect naturally. For educators, this serves as a reminder that fostering mathematical thought doesn’t always require a formal setting. And it encourages us to give students thought-provoking problems that linger in their minds, helping break the notion that mathematics is something that only happens within the four walls of the classroom.

Yet, beyond where mathematical thinking happens, another interesting question is how it takes shape in the mind. What mental images do mathematicians form as they work through these ideas in their heads? Burton’s research offers insights into this inner world of mathematical thought.

What mental images are formed?

Burton initially hypothesised that the research mathematicians in her study would predominantly think in two ways — visually and analytically — and that they would move flexibly between these styles. To explore this, she asked them to describe what appeared on their mental screen as they worked through mathematical problems.

However, rather than confirming her hypotheses, Burton uncovered a third style of thinking, which she termed ‘conceptual’. She also discovered that many of the mathematicians tended to rely on one thinking style instead of shifting between multiple approaches.

• Visual thinkers described thinking in pictures and dynamic mental images. Some saw vivid, sometimes colourful, pictures, while others imagined motion in their minds.
• Analytic thinkers described thinking symbolically, visualising symbols, formulae, and equations, often manipulating them mentally.
• Conceptual thinkers described working with broader ideas and structures, organising ‘blobs’ of concepts and making connections by re-positioning these ideas in their mind.

Early in my research career, I was amazed by colleagues who discussed mathematics without writing anything down. I grew to become very accustomed to it. We often talked conceptually, using words to describe visual images, turning to symbolic language only when precision was needed. This mirrors my approach to thinking: when working through details, I often picture an imaginary whiteboard and manipulate symbols on it. But when thinking at a higher level, I see blobs and connections. What about you?

Among the 64 mathematicians in this part of the study, visual thinking tended to dominate, particularly in pure mathematics, while applied mathematicians and statisticians showed a more even spread across the three styles. Sixty percent of the mathematicians used a combination of styles depending on the problem. However, relatively few used all three thinking styles. Only three participants used all three, 36 relied on a combination of two, and 25 preferred just one (15 were visual thinkers, 3 analytic, and 7 conceptual).

For educators, this highlights the importance of encouraging and valuing diverse approaches to understanding, exploring, and communicating mathematical thinking. By doing so, we can better support students with different thinking preferences while helping them build flexibility and depth in their mathematical understanding.

How are mathematical ideas explored?

While Burton’s research focuses on where and how mathematicians think, another crucial aspect is how they explore and record their ideas. Mathematical thinking is not limited to abstract mental processes—it often involves physical tools, visual aids, and creative manipulation of objects. This hands-on exploration is key to developing mathematical intuition.

In primary school classrooms, students engage with mathematics through tangible, hands-on experiences like manipulatives, blocks, and games. However, as students progress through their education, these tools are often phased out, leaving behind only pen-and-paper tasks in higher education. This shift diminishes an essential mode of understanding, and it’s no wonder that some students come to feel that mathematics must be done entirely in the mind. But as mathematician Reuben Hersh notes: “Intuition…is the effect in the mind/brain of manipulating concrete objects—at a later stage, of making marks on paper, and still later, manipulating mental images. This experience leaves a trace, an effect, in the mind.”

It’s also perhaps because mathematics is largely seen as a highly abstract, intellectual activity that students become to feel that the only acceptable way is to do it in their heads. Primary school classrooms are rich in hands-on mathematical experiences for students. For reasons I don’t understand, we gradually withdraw this powerful experience from our classrooms until in universities predominantly only pen-and-paper remain.

Mathematicians often use physical objects and visual representations to push the boundaries of abstract thinking. One of my favourite anecdotes comes from Australian Fields Medallist, Terry Tao in this Sydney Morning Herald article:

Terry Tao recalls the day his aunt found him rolling around her living room floor in Melbourne with his eyes closed. He was about 23. He was trying to visualise a “mathematical transform”. “I was pretending I was the thing being transformed; it did work actually, I got some intuition from doing that.” His aunt is likely still puzzled. “Sometimes to understand something you just use whatever tools you have available.”

Terry is known to use his children’s toys to explore tiling configurations, demonstrating that even seemingly simple objects can aid complex problem-solving. This playful approach underscores the value of physical interaction with ideas, where abstract concepts can take on tangible forms. Similarly, Iranian Fields Medallist Maryam Mirzakhani preferred doodling on large sheets of paper, scribbling formulas on the periphery of her drawings. Her daughter described her work as “painting”, which beautifully captures the artistry of her approach to mathematics.

Chalkboards, too, serve as a powerful medium for mathematical exploration, often becoming canvases for intricate and beautiful representations of mathematical thought. Julia Collins’ website, What’s on My Blackboard? showcases examples of chalkboard work from mathematicians, revealing how these spaces foster creativity and problem solving. Similarly, Jessica Wynne’s beautiful book Do Not Erase highlights the visual artistry of mathematicians’ chalkboards. (Check out the ‘Press’ section of her website for glorious images.)

These examples illustrate that mathematics is not solely an abstract pursuit of the mind—it is also an art, a hands-on process that involves visualising, manipulating, and representing ideas in diverse, creative ways. As educators, we should encourage students to experiment with different tools, from physical objects to visual representations, to develop a deeper, more intuitive understanding of mathematics.

In future posts, we’ll dive into another aspect of thinking, inspired by Burton’s research—what happens when mathematicians get stuck, and how aha! moments happen.

Photo by Enis Can Ceyhan on Unsplash