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‘rien n'est plus difficile a comprendre pour un psychologue que ce que les mathematiciens entendent par intuition’.
Piaget (1962, pp. 223-241)
‘Nothing is more difficult for a psychologist to understand than what mathematicians mean by ‘intuition’’
When teaching mathematics I regularly asked my students to explain their reasoning, to convince me of their argument, or simply kept asking them ‘Why?’ until I felt satisfied they had thought about something to the level of depth I intended. Getting them to communicate their ideas, solution methods, and workings gave me an insight into how they constructed their answers, the mistakes they made and misconceptions they held. I was able then to support them and add to my own understanding of how students learnt.
On a number of occasions, when working on particularly challenging problems – especially when introducing those that were in novel contexts – I would be surprised when a moment of enlightenment hit. Yet so often the student wouldn’t be able to explain where the idea came from:
‘It just does’, they would say, ‘Just because,’ or ‘I just know it is’.
This ‘gut feeling’ for the content, an instinct that guides towards a solution to a mathematical problem – almost a sixth sense in being able to spot an underlying structure – came up in conversation with Martin Hyland, professor of mathematical logic at the University of Cambridge, with respect to some of his recent research. Professor Hyland explained that he had spent the morning trying to use some equations to solve a particular problem. He had an intuition that these were the right equations, but no one had used them in this way. He was convinced they would work but wasn’t at all convinced why he thought they would.
What is this ‘ability’? How do we suddenly spot a route through a challenging problem? Is there such a thing as mathematical intuition? If it exists, can we learn it? If so, how?
In a Huffington Post article from 2014 Steve Jobs is quoted as saying that intuition is ‘more powerful than intellect’. The article continues by quoting Sophy Burnham, bestselling author of The Art of Intuition, who states that ‘It’s different from thinking, it’s different from logic or analysis … It’s knowing without knowing’.
I met with Professor Hyland last year to get his opinion on the debate about the approaches to geometry and potential content for the Framework. I was at a point where I was recognising the magnitude of the task ahead of me: mapping geometry for all learners up to 19…where do you stop? What approach do you take? How do you blend Euclid, transformations, vectors? Should you?
The response he gave took me somewhere quite different. What Professor Hyland explained was that the specific content of a curriculum wasn’t as important as developing geometric intuition. The aim of a curriculum should be to establish such intuition in whatever content we feel important, although he admitted he didn’t necessarily know how one might do this. My interest was sparked.
Geometric intuition prepares and guides our mental and practical activity (Jones, 1993); it is the ability to see obvious features within a configuration and the creative skill of spotting critical hidden features. It involves making productive conjectures and identifying useful constructions (French, 2004). Intuition doesn’t require extrinsic justification (Fischbein, 1982), yet has its own intrinsic certainty (Fischbein, 1987). It’s an interpretation of a solution, yet is essential to move forward:
…intuition is able to organize information, to synthesize previously acquired experiences, to select efficient attitudes, to generalize verified reactions, to guess, by extrapolation, beyond the facts at hand…. intuition offers a global perspective of a possible way of solving a problem and, thus, inspires and directs the steps of seeking and building the solution…the role of intuition is to prepare action…
p.12 Fischbein, E., 1982
Intuition is developed through drawing (Schoenfeld, 1985), previous experience, and visual imagery (Fishbein, 1987 in Jones, 1993).
Importantly, as stated by Jones (1994), geometric intuition is not necessarily a ‘basis for working out a solid solution’ but does provide conjectures worthy of further consideration using analytical tools.
So what does this mean for the Framework? Is developing a mathematical intuition equivalent to developing the ability to recognise mathematical structures? If not how is it different? Either way, how do we develop it in the Framework? How do we recognise it in the classroom – or even in ourselves?
Fischbein, E. (1982). Intuition and proof. For the learning of mathematics, 3(2), 9-24.
Fischbein, E. (1987). Intuition in mathematics and science: An educational approach.
French, D. (2004). Teaching and learning geometry. A&C Black.
Gregoire, C. (2014). 10 Things Highly Intuitive People Do Differently. The Huffington Post [online] https://www.huffingtonpost.co.uk/2014/03/19/the-habits-of-highly-intu_n_4958778.html
Jones, K. (1993). Researching Geometrical Intuition. Proceedings of the British Society for Research into Learning Mathematics, 13(3), 15–19.
Jones, K. (1994). On the Nature and Role of Mathematical Intuition. Proceedings of the British Society for Research into Learning Mathematics, 14(2), 59–64.
Beth, E. W. & Piaget, J. (1962). Épistémologie mathématique et psychologie. Essai sur les relations entre la logique formelle et la pensée réelle. Étude d'épistémologie génétique, XIV. Les Etudes Philosophiques 17(2), 248-249.
Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando, Florida, Academic Press.
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