Creativity in the mathematics classroom
Creativity in the mathematics classroom: how the current debate misses the point
The current debate on the new National Curriculum revolves around the concept of creativity. Does the curriculum form a foundation for creativity or rather stifle creative processes? Michael Gove (Secretary of State for Education) contends that knowledge must come first before invention; Ken Robinson counters on the Guardian website that a student does not have to master all the necessary skills and concepts "before the creative work can begin." While the acknowledgement by both sides that creativity is possible in mathematics takes us beyond the tired claim that it is not a creative subject, the debate about when creativity can start misses the point completely. Neither formulation captures the relationship between knowledge construction and creativity that can occur in open activity.
Gove argues that unless students are competent in carrying out basic operations or have a stock of secure knowledge, they will not be able to use maths creatively. The draft National Curriculum for maths reinforces his point in its first aim:
All pupils will become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems.
Leaving aside the highly dubious claim that frequent practice leads to conceptual understanding, it certainly does little to uphold the draft’s description of mathematics as a creative discipline. Repetitive practice, which implies highly structured lessons based around textbook-style exercises, strangles any desire to be creative.
Robinson writes that any child has the potential to be creative within any subject as long as they are motivated and passionate. And motivation and passion derive from teachers’ creative solutions to each student’s individual needs. Creativity begets creativity. That’s nice! Unfortunately, Robinson does not go on to give any idea of what teachers could do in their classrooms to encourage the process of creativity. And more damaging to his case, he does not spell out how many of the necessary skills a student needs before the teacher could reasonably expect to see anything new.
Students’ creativity in mathematics classrooms comes in many forms. For Polya (in How to Solve It), the process of creativity involves bringing current knowledge to bear on a problem: by "varying the problem, we bring in new points, and so we create new contacts, new possibilities of contacting elements relevant to our problem." Polya includes going back to a definition, decomposing and recombining, introducing auxiliary elements, generalising or specialising and using analogies as the ways to vary a problem. Developing these skills in order to draw on existing knowledge is, I believe, in line with Gove’s approach – although, with Gove, students would be so disengaged from mathematics by the time they reached the problem, they would have no interest in solving it.
Watson and Mason (in Mathematics as a Constructive Activity) take creativity one step further by requiring students to make up their own examples in response to a task. The example spaces (collections of examples) that emerge can encourage "an appreciation of sameness and difference" through an "active comparison of special cases rather than causal pattern-spotting." Watson and Mason found that if tasks are too open, then students can become suspicious, feel insecure and play safe. Constraints in the task, paradoxically, lead to more creativity.
The generation of examples is, for me, in line with Robinson’s contention that creativity is possible before knowledge is fully formed. In Watson and Mason's first task (Think of some integers that have only two factors), a student would need to know the meanings of 'integer' and ‘factor’ to be able to create examples that lead to a basic understanding of the concept of a prime number.
However, is the generation of examples truly creative? Surely the teacher is leading students to discover knowledge that would make up a standard entry in a textbook. Is it possible to create new results that are not immediately available? Is it possible to encourage students to decide what that new knowledge might be and how they would arrive at it?
Answers to those questions are the key concerns of the Inquiry Maths teacher. The combined transformations inquiry, for example, starts with a statement about a particular case that leads, if not to standard knowledge, then to a neat and well-defined solution. By playing an active role in arriving at the result, students develop the confidence to pose new questions, decide how to develop the inquiry and seek out the conceptual knowledge they identify as necessary to make progress. They make general statements or propose conjectures before exploring further particular cases. The inquiries they create, each one of which is regulated in a different way from any other, often produce novel, yet legitimate, ideas.
Inquiry Maths reflects the process Vygotsky outlined in Imagination and Creativity of the Adolescent. Creativity in the secondary school is different from before because it is now based on thinking in abstract concepts – and conceptual thinking enriches students’ concrete activity:
Imagination is a creative transforming activity which moves from one form of concreteness to another. But the mere movement from a given concrete form to a newly created form of it and the very feasibility of a creative construction is only possible with the help of abstraction. So abstraction is incorporated into the process of imagination as an indispensable constituent part, but it does not form its centre. The movement from the concrete through the abstract to the construction of a new form of a concrete image, is the path which describes imagination in the adolescent age.
For Vygotsky, the intellectualisation of creativity constitutes a key development in the thinking of secondary school students. Now creativity relies on using concepts, and learning to use concepts requires creativity. It is not a case of content before creativity (Gove) or some content before creativity (Robinson); it is knowledge construction through creativity.
Andrew Blair, May 2013