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14 February 2019
Have you ever heard of Flatland, by Edwin A. Abott?
Some years ago I found this slim volume and it compelled me to examine geometry in a different way. I had heard of the book – ‘a political and romantic tale about shapes’ – and yet could not conceive of how it could be done. What could shapes, those cold unfeeling geometrical figures that I wearily constructed in mathematics, have to do with emotion, with drama, with intrigue and the human condition?
I loved the book. I have read it, or parts of it, to all my classes – from a bevy of tiny wondrous Year 2 faces to a sea of rolling eyes in Year 11 who tutted when I first did the ‘voices’ but then begged me to from there on in. Sometimes they cheered bits: ‘If our highly pointed Triangles of the Soldier class are formidable, it may be readily inferred that far more formidable are our Women’. At other times they booed: ‘Needless to say that henceforth the use of Colour was abolished, and its possession prohibited’. Some bits I always have to explain, like the use of ‘minutes’ to mean smaller parts of a degree here: ‘Some of our highest and ablest men, men of real genius, have during their earliest days laboured under deviations as great as, or even greater than, forty-five minutes’. But it never fails to delight me to behold the commentary, the arguments, the emotional reactions and the opinions that burst forth when we read this book. It’s as if geometry escapes from the page and becomes meaningful to students in a wholly unexpected way and on a wholly human scale.
Are you feeling sceptical? Well you might. It’s an important question as to whether such anthropomorphism is a good thing. Does it help students learn mathematics more effectively? To that we shall return.
Have a look at this video – it’s about two and half minutes long, and is an example of one of the earliest forms of stop-motion animation, a word which is particularly pertinent here as it not only means to make into a moving picture, but to bring to life, from the Latin anima = soul, animare = to instil with life. As you watch, try to make a note of your feelings about the action happening before you and your interpretation of it.
This was part of a research study done in 1944 called ‘An Experimental Study of Apparent Behavior’ by Fritz Heider and Marianne Simmel in the American Journal of Psychology. They found that their research subjects almost universally anthropomorphised the shapes in the video to an astonishing degree (they had 34 test subjects in the first condition – no guidance – and only one appeared to resist the temptation to do so). Here is an excerpt from one reaction:
The depth to which the test subjects explore the dramatic scene, ascribing emotions, motives, reactions and consequences is fascinating. Some more results:
It is easy to see (in 1944, of course) how the connections between circle, fearful, meek and feminine occur, although perhaps jarring to the modern reader. More surprising might be single-occurrence words such as beautiful or teasing in this context. This begs the question: are our students seeing shapes like this? Could or should we encourage them to?
Of course, this sort of thing happens all the time in children’s television – here is a particularly obnoxious example. Piagetian notions of young children’s animism (seeing agency or a soul in things) suggest that they routinely attribute intent and motivation to inanimate objects such as clouds or wind. However, later research has strongly argued with this idea, suggesting that four or five-year-olds have a very good sense of the distinction between animate and non-animate entities and that Piaget’s results have come about from a linguistic lag. In other words, the way that children can articulate these choices is not yet sophisticated enough to explain why or how they know (often called their ontological intuition). The question is raised as to whether the diet of the magical, the supernatural and the physics-defying that we routinely feed young children is hanging around as mental blubber long after we have debunked the childish notion of the tooth fairy. Are we, by starting with emphatically singing, wild-eyed shapes, only setting them up for later disappointment and confusion? (See Ganea et al., 2011, 2014 and Legare et al., 2013 for more on this.)
It would appear that most children intuitively subscribe to and then ‘grow out’ of anthropomorphising. While it may seem like perpetuating mental error, it happens selectively and may have benefits which include emotional connection, empathising and engagement on a narrative level with the subject. Given that pupils describe mathematics as ‘irrelevant to their lives, tedious, disconnected, elitist and depersonalised’ (Rogers in Leslie & Mendick, 2014) could this be a possible antidote?
I’d be tempted to play the Heider and Simmel video to pupils and ask them to note their reactions, looking especially at the way they assign motivations, actions and reactions to the shapes and whether it prompts thinking about mathematics in a different way for them – because, of course, all this psychological stuff has quite a lot to do with the learning of mathematics. The two fields collide in a particularly interesting way, and many people are doing important and fascinating work at the intersection.
In a keynote at the International Group for the Psychology of Mathematics Education last year, Professor Natalie Sinclair went even further, challenging the audience to consider what it means to be an agent ‘doing’ mathematics. She suggested not that we should anthropomorphise non-humans (i.e. aim to make all mathematical interactions human-focused) but that we should consider ‘a post-human aesthetic of school mathematics’. What might this look like? Could a non-human ‘do’ mathematics? Is a human lens the only possible way to ‘see’ mathematics (and what characterises this)?
These are important questions and not without serious impact, particularly in a rapidly emerging age of AI. Could we conceive of an intelligent robot that was able to grasp mathematical beauty? Can we assign mathematical agency to a rock formation growing in a clearly hexagonal structure? And as we ask our students to connect with mathematics, are we able to understand – and willing to acknowledge – the cocktail of types of interaction they may have with it, from the affective to the aesthetic?
Ganea, P. A., Canfield, C. F., Simons-Ghafari, K., and Chou, T. (2014). Do cavies talk? The effect of anthropomorphic picture books on children’s knowledge about animals. Front. Psychol. 5:283
Ganea, P. A., Ma, L., and Deloache, J. S. (2011). Young children’s learning and transfer of biological information from picture books to real animals. Child Dev. 82, 1421–1433
Legare, C. H., Lane, J. D., and Evans, E. M. (2013). Anthropomorphizing science: how does it affect the development of evolutionary concepts? Merrill Palmer Q. 59, 168–197
Rogers,L. (2014). Chapter 10: History of mathematics in and for the curriculum. In D. Leslie & H. Mendick (Eds.), Debates in mathematics education. London ; New York, NY: Routledge, Taylor & Francis Group
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