From Visual to Verbal, the Many Ways to Understand Mathematics

By Elizabeth Kimber – Programme Leader for Mathematics at Bishop Grosseteste University

The Conversation UK recently published a piece written by Vicky Neale (Whitehead Lecturer at Balliol College, University of Oxford) and myself about how there are many ways to think about mathematics. From visual to verbal – there’s more than one way to understand maths was written for a general audience and aimed to address the idea that being ‘mathematical’ is about thinking in one particular way. We used the example of ellipses because while they aren’t generally covered in school mathematics, they are all around us and are amenable to being looked at in various ways that may be familiar from school maths.

Encouraging students to think in different ways about mathematics is a big part of how we teach at BGU and we are not alone in this. At A level, many students have been encouraged to work algebraically and geometrically to get a fuller sense of functions and graphs, perhaps supported by dynamic geometry software. However, many students have developed a preferred approach to their school mathematics and expect to continue working only in this way during their undergraduate studies. If they do this, then they’ll be missing out on powerful ways of thinking. We need to build up students’ confidence in thinking in different ways and extend it undergraduate mathematics.

A big part of mathematics is about making sense of definitions and theorems. For example, what it means for a sequence of numbers to converge. These results can be expressed very formally using words and mathematical notation, as you can see here, and the pressures of assessment make it tempting to move quickly into helping students write in this formal style before the underlying ideas have taken shape. However, our workshop-style teaching sessions give students the opportunity to explore these underlying ideas. This might involve working visually or numerically, perhaps using technology, but the most crucial aspect is the opportunity for students to discuss their thinking with each other.

I’m struck by how much students use gestures to describe their mathematical ideas to each other, often before they are ready to commit these ideas to paper. The gestures and discussion help to reveal students’ different ways of thinking and provokes others to ask questions and refine their ideas. This frequently flags up related issues that I couldn’t possibly cover in a lecture. We do also spend time writing up solutions to problems to help develop students’ mathematical writing style, but we don’t rush into this at the expense of grappling with the underlying ideas.

“The way in which our lessons are styled as a workshop rather than a lecture has been extremely helpful in allowing me to gain an understanding of the course content. The informal style of the workshop allows me to feel much more comfortable when discussing a topic that I am not confident in.” 1^st Year student

“Since studying at BGU my confidence in and passion for mathematics has increased. The small group teaching creates the optimum environment for discussion – students no longer simply accept the maths but question what we are learning, to understand how and why the maths works. This has helped me to understand the maths in greater depth, compared to studying A-level.” 1^st Year student

Working on this piece with Vicky was an interesting exercise in itself, but it also prompted me to reflect on my teaching at BGU and the great advantage of our long workshop-style teaching sessions.

You can find the full article on The Conversation.

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