When I think back 4 years to my secondary school days studying A Level maths, I picture those chunky textbooks, thick as a brick in my backpack, which made a satisfying thump as you thumbed between questions at the front, answers at the back. My mum was always worried that I was damaging my back lugging these around, and I was always worried that I would forget to bring the right book (Mechanics, Pure, or Statistics) to class.

Print textbooks are functional: they do the job, in terms of containing relevant information and practice questions. But they are somewhat lifeless. You need a teacher to animate the written explanations with their words and gestures, and their pen on the whiteboard. You need someone to lift the material off the page and make it move. I think teachers are brilliant and essential, and their skills and dedication can’t be replaced.

But, at the same time, I wonder: can we design new learning materials which are better adapted to the imagination and intuition of maths students? With the rise of EdTech, we are no longer tied to print materials. Can visual, digital content help both teachers and students along the latter’s journey of discovery?

### Maths education in classrooms today

School curricula, in all subjects, not just maths, are largely print-based. Tandi Clausen-May argues that this disparages other ways of thinking and learning, which might be equally valid or useful. Definitions and proofs which can be easily printed in text are prioritised as the most “proper” for school maths, whilst those which depend on models or dynamic geometry are overlooked or considered as secondary supplements.^{1}

She writes: ‘“proper” school maths is defined as maths that can be printed in a book, and preferably in text.’ Thus, she gives the following examples:^{2}

Where visual representations such as those shown above are used in the classroom, they are likely to be treated as aids for those struggling to understand the symbolic representation, rather than as valid, and valuable, approaches in themselves. Clausen-May argues that this approach disadvantages pupils with visual and kinaesthetic learning styles.

School maths curricula are also focused on sequential teaching, whereby students first go through all the basics before building up to solve complex equations. This may sound intuitive, perhaps because it is the way things are done and what most of us are used to. Conversely, however, it may be that visual learners do not respond well to sequential maths learning: without a view of the bigger picture, they lack motivation and question why they are learning what they are learning.^{3}

### Visual learning style: a myth?

Arguments for visual learning methods are often based on the belief that students have different learning styles – visual, auditory, or kinaesthetic – and that they will learn more effectively when presented with information in the appropriate style.

In actual fact, evidence dispels this widespread misconception. Not only are there problems regarding the lack of an established framework of learning styles, and the fact that in many studies preferred learning styles are self-reported, but the evidence is tenuous on the hypothesis that adapting material to learning style helps students.^{4}

However, even if visual learners do not exist, per se, this is not to say that visual learning cannot be valuable.

In fact, rather than helping a portion of the population who self-define as preferring visual learning methods, it might even be a universally effective approach to maths education.

### Visual understanding of number: built into our biology?

PowerPoints, emojis, screenshare. Instagram, infographics, graphics tablets. Technology which allows us to transmit and receive visual information has become increasingly important in our modern lives. Teenagers and young adults who have grown up in a digital world can hardly imagine life without it.

But what if there is a deep-rooted explanation for our affinity with visual information? Visual technologies might be new, but the need or desire to see in order to understand may not be new at all. The very fact that in English we say ‘I see’ to mean ‘I understand’ suggests a link between the two.

Pierre Pica, a linguist who has spent time researching the Munduruku, an Amazonian indigenous group whose language has no words for numbers beyond five, believes that ‘understanding quantities approximately in terms of estimating ratios is a universal human intuition’, whilst ‘understanding quantities in terms of exact numbers is not a universal intuition; it is a product of culture’.^{5} That is to say, it was important for human survival to evaluate at a glance if a group of spear-wielding enemies was larger than our group, or which of two trees held more fruit. Exact quantities, tied to specific symbols and words, came later. And the value of this approximate number sense is not limited to our evolutionary past.

Surprisingly, a 2008 study by psychologists at John Hopkins University and the Kennedy Krieger Institute found that the ability to visually discern different numbers of dots correlated strongly with success in formal maths education. The better a student’s ability to visually estimate quantities, the higher their chance of getting good grades.^{6}

‘This might have serious consequences for education,’ writes Alex Bellos. ‘If a flair for estimation fosters mathematical aptitude, maybe maths classes should be less about times tables and more about honing skills at comparing sets of dots’.^{7}

Our intuitive visual understanding of number, that is, could be better recognised and nurtured to help students succeed with more complicated maths.

### So how can visual learning methods help maths education?

A visual learning process can help students leap from the concrete to the abstract: from observing something in front of them to visualising or imagining what happens next, or what might happen in a slightly different scenario.

Stuart Murphy outlines a set of five skills which constitute a mathematical visual learning process: observation, recognition, interpretation, perception, and self-expression.^{8}

Observation involves close and critical attention. It involves students asking questions such as ‘What is it?’ and ‘What makes that result look like that? Recognition includes visual recall and is about retaining the link between a visual representation and its meaning or interpretation. Knowing what a triangle is, for instance, or knowing a sine curve. Interpretation involves deepening understanding through questions such as ‘How does it work?’ and ‘What does that model tell me?’ Interactive graphs which students can manipulate to see how functions change according to inputs, for instance, can be a valuable tool for this. Perception is about extending current understanding to further applications. The enabling question for students here is: ‘What happens next?’ Finally, self-expression is the skill of employing visual techniques themselves to convey information. This requires asking ‘How can I get this idea across?’, understanding what kind of image, graph, chart, or diagram best fits the data at hand, and executing the presentation.

Words, numbers, and pictures must work together in order for students to comprehend complex mathematical concepts, argues Murphy. ‘The well thought-out and carefully developed visual/verbal/numerical co-expression of content’ is key.^{9} That is, visual learning materials are not just about jazzing up text with colours and images.

Visual learning materials should innovate in order to engage the imaginations and, crucially,

intuitions, of all maths learners.

### References

^{1} Tandi Clausen-May, Teaching Maths to Pupils with Different Learning Styles (London: Paul Chapman Publishing, 2005).

^{2} *ibid*.

^{3} ‘Visual Learning Helps Understanding Maths Concepts Easier,’ EdTech Review, 6 Apr 2017 <https://edtechreview.in/trends-insights/trends/2729-visual-math-learning>.

^{4} ‘No Evidence to Back Idea of Learning Styles’, letter signed by thirty academics, 12 Mar 2017, uardian <https://www.theguardian.com/education/2017/mar/12/no-evidence-to-back-idea-of-learning-styles>.

^{5} Alex Bellos, Alex’s Adventures in Numberland (London: Bloomsbury, 2010).

^{6} *ibid*.

^{7} *ibid*.

^{8} Stuart J. Murphy, ‘The Power of Visual Learning in Secondary Mathematics Education,’ Research into Practice: Mathematics (Pearson, 2011).

^{9} *ibid*.