Trends in Researching and Teaching Problem Solving

in School Mathematics in Australia: 1997 – 2000.

Kaye Stacey

University of Melbourne

Overview

This paper presents a brief summary of recent trends in the role of problem solving in school mathematics in Australia. Australia has a federal system of government and there are 8 independent major education systems. Policies about mathematics education and their implementation vary among these systems, although there are many commonalties. Generalisation is difficult and the observations following are not uniformly applicable to all systems. However, there is substantial sharing of ideas and some national agreed policies, which makes a report on Australia feasible.

Making students better problem solvers was formally established as a goal of the mathematics curriculum in every state and territory in Australia during the 1980s (Stacey & Groves, 1990). In practice, this goal encompassed two somewhat distinct aspects. Firstly, there is the goal of improving the capacity of students to solve problems, in particular problems in a real world context. This goal has been addressed by ensuring that students have some genuine experience of substantial problem solving by being given the opportunity to solve real world problems at school, and also by giving some instructional attention to the processes of problem solving including helpful strategies for approaching problems. This aspect of the problem solving goal has received less attention in recent years. The second aspect of the problem solving goal is to teach mathematics through problem solving approaches. Nisbet and Putt (2000) observe that this aspect of the problem solving goal has been given impetus in recent years by the popularity of constructivist theories to guide teachers as facilitators of learning.

In recent years, government initiatives in mathematics education in most education systems of Australia have emphasized the importance of establishing basic educational foundations for all students, especially in literacy but also in numeracy. The emphasis on monitoring achievement on basic skills might have excluded any emphasis on problem solving. This, however, has not happened. Although not receiving the prominence that it had several years ago, problem solving has continued to be regarded at least as an essential competency in mathematics. Nisbet and Putt (2000) attribute this to two causes. Firstly, there was the identification of “using mathematical ideas and techniques” as one of a small number of “key competencies” for national vocational education (Mayer, 1992). These key competencies were intended to describe the broad outcomes of education for effective citizenship and for work, and so the emphasised applying knowledge across the curriculum and being able to solve problems. The second factor identified by Nisbet and Putt was the prevalence of constructivist philosophies, interpreted broadly, within the research community and also in professional development for teachers. Since students must ultimately construct their own mathematical knowledge, a crucial role for the teacher is to offer the problems and learning experiences which provide students with the raw material for constructing mathematical concepts.

Within the research community, problem solving has received further impetus from the study of small group learning processes. On-going research into the use of collaborative learning (of various types) in mathematics classrooms has continued to highlight the way in which students solve problems, how they come to construct knowledge through this process and the need for metacognitive awareness of problem solving processes.

The introduction of new technology, (especially graphics calculators) has been a major concern in recent years and there has been a great deal of exploration about the possibilities that this offers for new types of problems to be tackled, especially in senior secondary school mathematics (Stacey, 2000). Graphics calculators, spreadsheets and data logging equipment are widely used in some but not all states. They have opened new possibilities for gathering real data (e.g. about speed, light intensity or temperature), fitting a function to create a mathematical model and using the model to make predictions or explain phenomena. Widespread availability of statistics software (handheld and on school computers) has also stimulated interest in solving problems involving real data. Students conduct surveys, or gather data from published sources and increasingly the internet, to bring topical questions into the classroom. Although students are still taught to compute statistics by hand, the use of statistics software to analyse substantial data sets has further opened the classroom to the real world.

The remainder of this paper discusses two features of problem solving in Australia: the use of substantial real world modeling and problem solving investigations into high stakes assessment and the insight into problem solving from the study of small group processes. There is also substantial research being undertaken in Australia into the way that students solve problems in different content areas of mathematics, some of which is reviewed by Nisbet and Putt (2000), but is outside the scope of this paper. Nisbet and Putt (2000) and Nisbet, Putt and Taplin (1996) both provide extensive reviews of Australian research on problem solving for four year intervals. Keeves and Stacey (1999) report other aspects of problem solving, in the context of broad developments in research in mathematics education in Australia since 1965.