Learning via problem solving in mathematics education

Abstract

Three forms of mathematics education at school level are distinguished:

• direct expository teaching with an emphasis on procedures, with the expectation that learners will at some later stage make logical and functional sense of what they have learnt and practised (the prevalent form),
• mathematically rigorous teaching in terms of fundamental mathematical concepts, as in the so-called “modern mathematics” programmes of the sixties,
• teaching and learning in the context of engaging with meaningful problems and focused both on learning to become good problem solvers (teaching for problem solving) andutilising problems as vehicles for the development of mathematical knowledge andproficiency by learners (problem-centred learning), in conjunction with substantialteacher-led social interaction and mathematical discourse in classrooms.

Direct expository teaching of mathematical procedures dominated in school systems after World War II, and was augmented by the “modern mathematics” movement in the period 1960-1970. The latter was experienced as a major failure, and was soon abandoned. Persistent poor outcomes of direct expository procedural teaching of mathematics for the majority of learners, as are still being experienced in South Africa, triggered a world-wide movement promoting teaching mathematics for and via problem solving in the seventies and eighties of the previous century. This movement took the form of a variety of curriculum experiments in which problem solving was the dominant classroom activity, mainly in the USA, Netherlands, France and South Africa. While initially focusing on basic arithmetic (computation with whole numbers) and elementary calculus, the problem-solving movement started to address other mathematical topics (for example, elementary statistics, algebra, differential equations) around the turn of the century. The movement also spread rapidly to other countries, including Japan, Singapore and Australia. Parallel with the problem-solving movement, over the last twenty years, mathematics educators around the world started increasingly to appreciate the role of social interaction and mathematical discourse in classrooms, and to take into consideration the infl uence of the social, socio-mathematical and mathematical norms established in classrooms. This shift away from an emphasis on individualised instruction towards classroom practices characterised by rich and focused social interaction orchestrated by the teacher, became the second element, next to problem-solving, of what is now known as the “reform agenda”. Learning and teaching by means of problem-solving in a socially-interactive classroom, with a strong demand for conceptual understanding, is radically different from traditional expository teaching. However, contrary to commonly-held misunderstandings, it requires substantial teacher involvement. It also requires teachers to assume a much higher level of responsibility for the extent and quality of learning than that which teachers tended to assume traditionally. Over the last 10 years, teaching for and via problem solving has become entrenched in the national mathematics curriculum statements of many countries, and programs have been launched to induce and support teachers to implement it. Actual implementation of the “reform agenda” in classrooms is, however, still limited. The limited implementation is ascribed to a number of factors, including the failure of assessment practices to accommodate problem solving and higher levels of understanding that may be facilitated by teaching via problem solving, lack of clarity about what teaching for and via problem solving may actually mean in practice, and limited mathematical expertise of teachers. Some leading mathematics educators (for example, Schoenfeld, Stigler and Hiebert) believe that the reform agenda specifi es classroom practices that are fundamentally foreign to culturally embedded pedagogical traditions, and hence that adoption of the reform agenda will of necessity be slow and will require more substantial professional development and support programs than those currently provided to teachers in most countries.Notwithstanding the challenges posed by implementation, the movement towards infusing mathematics education with a pronounced emphasis on problem solving both as an outcome and as a vehicle for learning seems to be unabated. Substantial work on the development of more effective means for professional development and support of teachers is currently being done.