“I’ve never seen a question on pyramids like that before. We’ve not been taught that,” said one of my star pupils as she left a mock exam. Did I detect in her eyes the look of one who had been betrayed? A pang of guilt struck. But then, I can’t teach them every possible problem that might come up – but am I supposed to? Do my pupils assume I do, and think that there’ll be no surprises, no “now what do I do now?” moments, in the exam?
How to teach the unexpected? Or how to teach how to handle the unexpected. Problem solving, with its 30% element in the reformed GCSE, tackles the issue of dealing with “complex and unfamiliar problems” (Watson 2014) where “the methods to use are not obvious or there may be a choice of different methods” (Schoenfeld 1992). But for pupils to learn how to problem solve requires them to do the exploring, the thinking, the discovery. A transmission mode of teaching, where learning “is an individual activity based on watching, listening and imitating until fluency is attained” (Swan and Swain 2010) may not provide the experience the pupils need to develop their problem solving skills. The teaching needs to become pupil-centred – and, according to Ofsted (2008, 2012), for many of us this requires a change of teaching style. But teachers’ professional development itself “has been described as an ‘unsolved problem’, particularly where there is an expectation to change teaching practices from teacher-centred orthodoxies to more pupil-centred approaches” (Watson 2014). A couple of research papers attempt to shed some light on what works.
Swan and Swain (2010) implemented a programme “designed to challenge existing practices and beliefs by investigating how teachers might incorporate the following pedagogical principles into their teaching”, including:
- exposing common misconceptions
- promoting explanation, application and synthesis rather than mere recall
- encouraging reasoning rather than ‘getting answers’
- creating connections between topics, so that pupils do not see mathematics as a set of unrelated tricks and techniques to be memorised
all in the context of collaborative work with an emphasis on discussion. Crucially, rather than informing teachers of the theories they would need to embrace, instead
- their existing beliefs and practices were recognised
- contrasting practices viewed and conflicts discussed
- different approaches experimented with and supported
- time for self-reflection and discussion provided.
The researchers witnessed practices becoming less transmission-oriented, supporting their view that “changes in beliefs are more likely to follow changes in practice, after the implementation of well-engineered, innovative methods, as processes and outcomes are discussed and reflected upon.”
Watson (2014) viewed professional development through the idea that learning is dependent on three key factors: (1) direct observation of the desired behaviour, (2) an individual’s belief in their potential success, and (3) the coherence between an individual’s beliefs, the social context and the individual’s behaviour. Watson highlights the importance of appropriate models, suggested-lesson plans, video examples, and suitable classroom activities, as well as the individual’s self-belief, in implementing the suggested approach. He suggests that moving from a teacher-centred approach is constrained by the demands of day-to-day teaching as well as the attraction that it provides “routines that students, parents and teachers have familiarity with”.
There are many courses and resources (see below) now available for developing the skills for facilitating problem solving. The ideas in these research papers may help make this a reality in the classroom. And perhaps we can also make our pupils happier: "I can't remember the numbers, but the one about Hannah's sweets in particular made me want to cry." None of us wants this to be our pupils’ abiding memory of GCSE maths.
What helps a teacher create a class full of confident problem solvers? How have you managed to achieve this? Let us know, by email to email@example.com or Twitter @NCETMsecondary.
Many resources are available for Professional Development, a selection is listed here:
- NCETM Departmental Workshops
- Bowland Maths
- PRIMAS, a European project to promote inquiry-based learning in mathematics and science
- FRESH courses from MEI
- NRICH - A Guide to Problem Solving
- For another angle on “problem-solving” see the Inquiry maths website.
Department for Education (2013) Subject content and assessment objectives for GCSE in mathematics for teaching from 2015.
Department for Education (2014) Mathematics: Programme of study for Key Stage 4.
Ofsted (2008) Mathematics: Understanding the score. London: Office for Standards in Education.
Ofsted (2012) Mathematics: Made to measure. London: Office for Standards in Education.
Schoenfeld, A.H. (1992) Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In Grouws, D. (Ed.) Handbook of research on mathematics teaching and learning (pp. 334–370). New York: MacMillan.
Swan, M. and Swain, J. (2010) The impact of a professional development programme on the practices and beliefs of numeracy teachers, Journal of Further and Higher Education, 34:2, 165-177
Watson, S. (2104) The impact of professional development on the teaching of problem solving in mathematics: A Social Learning Theory perspective in Pope, S. (Ed.) Proceedings of the 8th British Congress of Mathematics Education, 351- 358.