Problem solving in the Australian mathematics curriculum: what have we learnt from other countries?
Judy Anderson is Associate Professor, Mathematics Education at the University of Sydney. This article is an abridged version of a paper presented at the Australian Curriculum Studies Association's 2009 Biennial Conference.
As Australia continues the process of developing a national curriculum, it is important to learn from other countries about the best approach for including problem solving in the curriculum and for supporting implementation by teachers.
International approaches to problem solving in the curriculum
Many curriculum documents present the school mathematics curriculum as lists of topics or 'content' and a set of 'processes'. Typically content includes the fundamental ideas of mathematics, historically grouped into such topics as number, algebra, measurement, geometry and chance and data. Processes include the actions associated with using and applying mathematics to solve problems which may be routine or non-routine. In many state and territory mathematics curriculum documents the processes have been grouped together and labelled Working Mathematically (Clarke, Goos & Morony, 2007). The following section summarises the approach to problem solving in the mathematics curriculum and the support provided for teachers in a selection of countries, chosen to exemplify some of the approaches taken and to highlight issues involved in implementation.
In Singapore, the results of an early TIMSS study led to several changes in the curriculum: the content was reduced by about 30% (Kaur, 2001) and problem solving became the primary goal of learning mathematics. The framework of the mathematics curriculum represents problem solving as being dependent on five inter-related components: skills, concepts, processes, attitudes and metacognition. The content is presented as skills and processes; attitudes represents the affective dimensions of learning; metacognition highlights the importance of self-regulation; and processes includes acquiring and applying mathematical knowledge.
While problem solving has been a focus of the curriculum since 1992, Kaur and Yeap (2009) report limited implementation in classrooms, with textbooks typically containing closed, routine problems and instruction in mathematics lessons usually teacher-led. In response to the limited implementation of problem solving by teachers, examinations have recently contained novel, non-routine problems. Teachers are now being confronted with new challenges to design and use similar tasks in their lessons. In addition to this, two new initiatives, Thinking School, Learning Nation (TSLN) and Teach Less, Learn More (TLLM), have aimed to reduce the curriculum content further and engage students in more thinking and problem-solving tasks (Kaur & Yeap, 2009). As evidence of the government’s commitment to teachers and their growth as professionals, teachers are entitled to 100 hours of professional development every year (Kaur, 2001).
In his presentation at a forum organised by the National Curriculum Board, Wardlaw (2008) revealed that Hong Kong has undergone significant reform since 2000 with a focus on student learning through alignment of curriculum, pedagogy and assessment. Associated with this reform are the following fundamental principles:
The Hong Kong Curriculum Framework has three interconnected components: Key Learning Areas, Generic Skills, and Values and Attitudes. Mathematics is one of the Key Learning Areas and the Generic Skills include collaboration, communication, creativity, critical thinking, information technology, numeracy, problem solving, self-management and study skills. Interestingly, the Basic Education Curriculum Guide (Education Dept HKSAR, 2002) indicates the priorities for 2001–2006 were communication, critical thinking and creativity. While Hong Kong has a coherent curriculum with high expectations, which values learning and training in basic skills and fundamental concepts, and with teachers who have good pedagogical content knowledge, Wardlaw (2008) acknowledges students have low self-efficacy and poor attitudes, particularly in mathematics. Additionally, there is an examination orientation, the mathematics curriculum is dense and compact, and the teaching and learning is rushed.
Teachers in Hong Kong are more aware of problem-solving approaches to teaching mathematics, but there remains limited evidence of implementation. Those teachers who try to engage students in discussion, mathematical reasoning and problem solving continue to lead students on a predetermined solution pathway rather than allowing more open investigation and exploration of mathematical ideas (Mok, Cai & Fung, 2005). Observations in Year 1 classrooms were characterised by 'whole-class teacher-pupils interaction and highly structured group/pair work' (Mok & Morris, 2001). More recently, Mok and Lopez-Real (2006) noted little use of group work or open-ended questions suitable for exploratory problem solving in the lessons of Hong Kong secondary school teachers.
The latest mathematics curriculum documents in England for Key Stage 3 and Key Stage 4 (the first four years of secondary education) are less prescriptive, allowing more flexibility for teachers. They contain a framework of personal learning and thinking skills and have a focus on assessment for learning. Problem solving is described as 'lying at the heart of mathematics' (DCSF, 2008a, p. 5) and is represented as a cycle of processes including representing, analysing, interpreting and evaluation, and communicating and reflecting.
To assist teachers, a wide range of support material has been prepared for school and district-based professional development with examples of problems and rich tasks for each of the content strands. Teachers are encouraged to analyse tasks to identify the processes. This support is vital if teachers are to embed the processes in lessons and provide regular problem-solving opportunities for students. It is too soon to determine the impact of the changes, but assessment items will also be changed to include more open-ended questions.
For at least 30 years, researchers from the Freudenthal Institute in the Netherlands have been developing a mathematics curriculum and a pedagogical approach known as Realistic Mathematics Education (RME). The framework is based on the notion that mathematics is a human activity and that students need to experience 're-inventing' the mathematics for themselves or 'mathematising' during lessons. Problems based on imaginable contexts (those which make sense to students) are used to develop mathematical skills and processes. Rather than using a more traditional teaching approach of demonstration of formal mathematics followed by skills practice and then applications to problems, this approach uses realistic problems as a starting point for learning and applying new mathematical ideas. However, for some students a more formal problem may be appropriate since the focus is on problem contexts that are 'imaginable' or 'realisable' for the learner (Van den Heuvel-Panhuizen, 2003). The theoretical approach developed in the Netherlands has been adapted in several other countries including the United States and England (see for example Romberg, 2001).
Teachers have freedom in determining the curriculum, although textbooks represent the main source of guidance, followed by Key Goals and domain descriptions provided by the Dutch government – interestingly, 'problem solving' is not listed explicitly as one of the goals. Given this flexibility, what is taught in most schools is very similar (Van den Heuvel-Panhuizen, 2000). More recently learning trajectories for particular content topics have been developed to assist teachers, but these are not meant as a 'recipe' for what and how to teach. While RME aimed to support the implementation of a problem-oriented curriculum, there is little evidence of non-routine problem solving in Dutch classrooms (Doorman et al., 2007). A lack of such problems in both textbooks and examinations is cited as the main reason for limited implementation.
The new Australian Curriculum approach
The draft Australian Curriculum: Mathematics will be available for consultation from mid-February to May 2010, prior to implementation from 2011. The guiding statement for writers, the Shape of the Australian Curriculum: Mathematics (NCB, 2009), presents the structure as three content strands – Number and algebra, Measurement and geometry, and Statistics and probability – as well as four proficiency strands – Understanding, Fluency, Problem solving and Reasoning (informed by Kilpatrick, Swafford & Findell, 2001). Problem solving is described as 'the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively' (NCB, 2009, p. 6). Expectations for problem solving will be elaborated to support teaching and assessment: this is critical since teachers will need models of practice to support effective implementation.
Kilpatrick et al. (2001, p. 5) included a fifth proficiency, productive disposition, described as 'habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy'. It is disappointing that this proficiency was not included in the Mathematics Shape Paper since it is critical that we focus on developing positive dispositions in mathematics, particularly given recent reports highlighting negative views about both the content and teaching of mathematics, especially in the early secondary years (McPhan, et al., 2008). The decision to exclude this proficiency was probably based on the need to develop achievement standards which would be difficult to write. However as Kilpatrick et al. (2001) note, the five proficiencies are 'interwoven and interdependent' which will make the writing of separate achievement standards for each of the proficiencies a challenging task.
In a summary of international trends in mathematics curriculum development, Wu and Zhang (2006) noted an increased focus on problem solving and mathematical modelling in countries from the West as well as the East. Curriculum developers recognise that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways. It is through problem solving that students develop a deeper understanding of mathematical ideas, become more engaged and enthused in lessons, and appreciate the relevance and usefulness of mathematics.
Given the efforts to date by many countries (including Australia) to include problem solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable resources and more time are important steps, it is possible that problem solving in the mathematics curriculum will only become valued when it is included in high-stakes assessment. In addition, teachers need readily available examples of useful non-routine problems, particularly in textbooks.
Australian Education Council (1991). A national statement on mathematics for Australian schools. Carlton: Curriculum Corporation.
Anderson, J. A., & Bobis, J. (2005). 'Reform-oriented teaching practices: A survey of primary school teachers.' In H. L. Chick, & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol 2, pp. 65–72), Melbourne: PME.
Board of Studies NSW (2002). Mathematics Years 7–10 Syllabus. Sydney: BOS NSW.
Cai, J. (2003). 'What research tells us about teaching mathematics through problem solving.' In F. K. Lester (Ed.), Teaching mathematics through problem solving: Prekindergarten–Grade 6 (pp. 241–253). Reston, VA: NCTM.
Clarke, D., Goos, M., & Morony, W. (2007). Problem solving and working mathematically: An Australian perspective. ZDM Mathematics Education, 39, 475–490.
Department of Children, Schools and Families [DCSF] (2008a). Secondary mathematics guidance papers. Nottingham, UK: DCSF.
Department of Children, Schools and Families [DCSF] (2008b). Renewing the framework for secondary mathematics: Spring 2008 subject leader development meeting. Nottingham, UK: DCSF.
Doorman, M., Drijvers, P., Dekker, T., Van den Heuvel-Panhuizen, M., de Lange, J., & Vijers, M. (2007). 'Problem solving as a challenge for mathematics education in The Netherlands'. ZDM Mathematics Education, 39(5/6), 405–418.
Education Department HKSAR (2002). Basic education curriculum guide: Building on strengths (Primary 1–Secondary 3). Hong Kong: The Curriculum Development Council.
Kaur, B. (2001). 'TIMSS & TIMSS-R – Performance of grade eight Singaporean students'. In C. Vale, J. Horwood, & J. Roumeliotis (Eds), 2001 a mathematical odyssey (pp. 132–144). Proceedings of the 38th annual conference of the Mathematical Association of Victoria, Brunswick, Vic: MAV.
Kaur, B., & Yeap, B. H. (2009). Mathematical problem solving in Singapore schools. In B. Kaur, B. H. Yeap & Kapur, M., Mathematical problem solving: Yearbook 2009 (pp. 3–13). Singapore: Association of Mathematics Education and World Scientific.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Lovitt, C., & Clarke, D. (1988). Mathematics curriculum and teaching program, Activity Banks 1 and 2. Carlton, Vic.: Curriculum Corporation.
McPhan, G., Morony, W., Pegg, P., Cooksey, R., & Lynch, T. (2008). Maths? Why Not? Canberra: Department of Education, Employment and Workplace Relations.
Ministry of Education Singapore (2006). Mathematics syllabus: Secondary. Singapore: Curriculum Planning and Development Division.
Mok, I. A., Cai, J., & Fung, A. (2005). Teaching mathematics through problem solving: Struggles a Hong Kong teacher faces. Paper presented at Earcome3 conference, Shanghai, China.
Mok, I. A., & Lopez-Real, F. (2006). 'A tale of two cities: A comparison of six teachers in Hong Kong and Shanghai'. In D. Clarke, C. Keitel & Y. Shimizu, Mathematics Classrooms in Twelve Countries: The ACSA Conference 2009 Insider's Perspective (pp. 237-246). Rotterdam: Sense Publishers.
Mok, I.A.C., & Morris, P. (2001). 'The metamorphosis of the 'Virtuoso': Pedagogic patterns in Hong Kong primary mathematics classrooms'. Teaching and Teacher Education, 17(4), 455–468.
National Curriculum Board (2009). Shape of the Australian curriculum: Mathematics. Barton, ACT: NCB.
Peter-Koop, A. (2005). 'Fermi problems in primary mathematics classrooms: Fostering children's mathematical modeling processes'. Australian Primary Mathematics Classroom, 10(1), 4–8.
Romberg, T. A. (2001). Standards-based mathematics assessment in middle school: Rethinking classroom practice. New York: Teachers College Press.
Stacey, K. (2003). 'The need to increase attention to mathematical reasoning'. In H. Hollingsworth, J. Lokan, & B. McCrae, Teaching Mathematics in Australia: Results from the TIMSS 1999 Video Study (pp. 119–122), Camberwell, Vic.: Australian Council of Educational Research.
Stacey, K. (2005). 'The place of problem solving in contemporary mathematics curriculum documents'. Journal of Mathematical Behaviour, 24, 341–350.
Van den Heuvel-Panhuizen, M. (2000). Mathematics education in the Netherlands: A guided tour. Freudenthal Institute CD Rom for ICME9. Utrecht: Utrecht University.
Van den Heuvel-Panhuizen, M. (2003). 'The didactical use of models in Realistic Mathematics Education: An example from a longitudinal trajectory on percentage'. Educational Studies in Mathematics, 54, 9–35.
Vincent, J., & Stacey, K. (2008). 'Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS Video Study criteria to Australian Eighth-grade mathematics textbooks'. Mathematics Education Research Journal, 20(1), 82–107.
Wardlaw, C. (2008). Presentation to the National Curriculum Forum held in Melbourne, 2008.
Wu, M., & Zhang, D. (2006). 'An overview of the mathematics curricula in the West and the East'. In F. K. S. Leung, K, Graf & F. J. Lopez-Real (Eds.), Mathematics Education in Different Cultural Traditions – A Comparative Study of East Asia and the West (pp. 181–194).
Key Learning AreasMathematics
Subject HeadingsInquiry based learning