What’s important in learning mathematics? Regular readers won’t be surprised when I say that I’ve come to think of it using the shorthand of nouns, verbs and adjectives:
- Nouns: the mathematical facts, concepts, and skills to be learned
- Verbs: the practices involved in thinking and working mathematically
- Adjectives: the emotions and feelings experienced throughout
Thinking in terms of nouns, verbs and adjectives helps emphasise that learning mathematics goes beyond merely acquiring knowledge. Catherine Attard describes mathematical engagement in a similar way, seeing it as a blend of cognitive, affective, and operative aspects that together help students develop a deeper relationship with mathematics.
For a complete understanding of mathematics, all three elements—nouns, verbs, and adjectives—must be integrated. Yet, official curricula often emphasise only the cognitive aspects (the nouns), which could be criticised for promoting a view of mathematics as an inert ‘store of knowledge’ to be deposited by teachers into passive students—a concept Paulo Freire (1970) described as the ‘banking model’ of education. However, as Nick Covington and Michael Weingarth from the Human Restoration Project argue in their critique of The Science of Learning, students are not merely brains in jars. They go on to say that students “exist within bodies, as affective agents in the world, thinking purposefully about and with the tools they have.” In other words, the operative (verbs) and affective (adjectives) aspects of learning are just as important as the cognitive (nouns).
How might we rethink mathematics education with this in mind? Leone Burton’s research suggests a compelling alternative by examining how mathematicians themselves understand mathematics. Her model of ‘coming to know’ offers a holistic framework, acknowledging the cognitive, operative, and affective dimensions of learning. This post is the first in a series exploring Burton’s findings in greater depth, focusing on these three dimensions.
Burton’s phrase ‘coming to know mathematics’ is well chosen. It suggests a deep, meaningful, and perhaps emotional connection to mathematics—much like getting to know a person. We become attuned to their imperfections, peculiarities, and charms. Over time, a rapport grows, and the relationship becomes dynamic, engaging, and deeply rewarding. Similarly, ‘coming to know’ mathematics can be an immersive and profound experience. The phrase offers a compelling vision of what it truly means to understand mathematics.
To propose a model of what it means to come to know mathematics, Burton drew on literature in several fields, including mathematics, mathematics education, sociology of knowledge in mathematics and science, gender in education, and feminist science. She suggested that coming to know mathematics:
- is a product of people, cultures and societies, meaning there is no single, universal mathematics
- is linked to emotions and aesthetics
- relies on intuition and insight
- can be approached through different ways of thinking: visual, analytic, and conceptual (with the latter emerging from her interviews)
- depends on making connections and searching for mathematical coherence.
To test her model, Burton conducted a seminal study in 1997, interviewing 70 research mathematicians (35 female, 35 male)1 in the UK and Ireland. Through these in-depth interviews, she explored whether their experiences aligned with her theoretical framework. They largely confirmed her model but also brought surprises, such as the variety of ways in which mathematicians think, and the high value placed on collaboration. Burton reported her work in the book Mathematicians as Enquirers: Learning about Learning Mathematics and a series of papers.
The book vividly captures how mathematicians work, think, and feel, in their own words. It offers a rare glimpse into their private worlds. I was particularly captivated by the sections describing where the thinking happens, what mathematicians do when they get stuck, how it feels to be a mathematician, how they use intuition, and how aesthetics plays into mathematics.
Burton’s work not only highlights the multifaceted nature of mathematical inquiry but also challenges us to reconsider how we learn and, by extension, teach mathematics. In upcoming posts, we’ll take a closer look at what the mathematicians in Burton’s study had to say about how they think and how they feel, and explore how these insights can shape our approach to teaching mathematics.
- The scale of the endeavour is astonishing. Each interview averaged 1.5 hours. Annette Lareau suggests allowing 3 hours of writing up notes per interview hour. Without accounting for time spent organising and traveling, that’s at least 70 x 1.5 x 3 = 315 hours of data collection. As I said, astonishing. ↩︎
Photo by Katie Moum on Unsplash
Wonder in Mathematics 