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ACSA Conference 2009

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Mathematics Curriculum Development and the Role of Problem Solving

Judy Anderson

The University of Sydney

<j.anderson@edfac.usyd.edu.au>

… a fundamental aim of the mathematics curriculum is to educate students to be active, thinking

citizens, interpreting the world mathematically, and using mathematics to help form their

predictions and decisions about personal and financial priorities. (NCB, 2009, p. 5).

Problem solving is recognised as an important life skill involving a range of processes

including analysing, interpreting, reasoning, predicting, evaluating and reflecting. It is either

an overarching goal or a fundamental component of the school mathematics curriculum in

many countries. However, developing successful problem solvers is a complex task requiring

a range of skills and dispositions (Stacey, 2005). Students need deep mathematical knowledge

and general reasoning ability as well as heuristic strategies for solving non-routine problems.

It is also necessary to have helpful beliefs and personal attributes for organizing and directing

their efforts. Coupled with this, students require good communication skills and the ability to

work in cooperative groups (Figure 1).

Figure 1. Factors contributing to successful problem solving (Stacey, 2005, p. 342)

Teachers have had many opportunities to build knowledge about teaching problem solving

and using problems as a focus of learning in mathematics (Cai, 2003). In Australia advice to

teachers has been provided in a range of publications including books (e.g., Lovitt & Clarke,

1988) and professional journals (e.g., Peter-Koop, 2005), in national curriculum statements

(e.g., Australian Education Council, 1991) as well as in state and territory curriculum

documents (e.g. BOS NSW, 2002). Such advice has been accompanied by pre-service and in-

service programs to change teaching practices from more traditional approaches to

contemporary or reform methods where teachers use non-routine problems and problem-

centred tasks (Anderson & Bobis, 2005). Given the amount of policy advice and resource

development, there are concerns about the limited opportunities for Australian students to

solve problems other than those of low procedural complexity (Stacey, 2003). It is possible

that the main constraints on implementation are the types of questions including in

Personal attributes

e.g. confidence,

persistence,

organisation

Communication

skills

Solving problems

successfully requires

a wide range of skills

General

reasoning

Deep mathematical

knowledge

Heuristic

strategies

Abilities to work

with others

effectivel

y

Helpful beliefs e.g.

orientation to ask

questions

ACSA Conference 2009

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examinations and in textbooks (Doorman et al., 2007; Kaur & Yeap, 2009; Vincent & Stacey,

2008).

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

supporting teachers are varied with some countries developing realistic tasks (e.g. Holland),

and others reducing the content in the curriculum to allow teachers more time for problem

solving (e.g. Singapore). Examining the efforts of other countries and considering the

constraints and affordances for teaching problem solving will inform the efforts required for

successful national curriculum development and implementation in Australia.

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. While processes includes 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 Singapore, Hong Kong, England and the Netherlands – this selection

of countries has been chosen to exemplify some of the approaches taken and to highlight

issues involved in implementation.

Singapore

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. Figure 2 represents the framework of the mathematics

curriculum with problem solving dependent on five inter-related components – skills,

concepts, processes, attitudes and metacognition. The content is presented as skills and

processes while attitudes represents the affective dimensions of learning, metacognition

highlights the importance of self-regulation, and processes includes acquiring and applying

mathematical knowledge.

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Figure 2. Mathematics framework from the Singapore mathematics curriculum (Ministry of

Education, Singapore, 2006, p. 2)

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).

Hong Kong

In his presentation at a forum organised by the National Curriculum Board Wardlaw (2008)

revealed Hong Kong has undergone significant reform since 2000 with a focus on student

learning through alignment of curriculum, pedagogy and assessment (Figure 2). Associated

with this reform are the following fundamental principles:

• all students have opportunities to learn and should not be screened out early;

• life-long learning capabilities are needed for a contemporary and future world;

• whole person development for enhancing quality of life in society, culture, economy;

• conceptions of knowledge changing – cross disciplinary, personal, co-constructed; and

• structural changes to facilitate opportunities and pathways for all young people (Wardlaw,

2008, slide 5).

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Figure 3. Aligning curriculum, pedagogy and assessment in Hong Kong (Wardlaw, 2008)

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 priority for 2001-2006 was 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. For those teachers who

try to engage students in discussion, mathematical reasoning and problem solving, they

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.

England

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. The

explanation for the relationships depicted in Figure 4 is “the diagram represents the dual

nature of mathematics: it is both a tool for solving problems in a wide range of contexts and a

discipline with a distinctive and rigorous structure” (DCSF, 2008a, p. 19).

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Figure 4. A representation of the processes involved in problem solving (DCSF, 2008a, p. 6).

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, e.g., a task

involving finding patterns and relationships in a hundreds chart is accompanied by the

template presented in Figure 5. This support is vital if teachers are to embed the processes in

lessons and provide regular problem-solving opportunities for students. However, it is too

soon to determine the impact of the changes but assessment items will also be changed to

include more open-ended questions.

Figure 5. Template to aid teacher identification of processes in a task involving a hundreds

chart (DCSF, 2008b, p. 4).

The Netherlands

For at least 30 year, 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 ‘mathematizing’ 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

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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 they 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.

Considering the mathematics curriculum and problem solving initiatives of Singapore, Hong

Kong, England and The Netherlands reveals some similarities and some differences.

Singapore made a significant change by reducing mathematics content, the RME approach in

the Netherlands was designed to build mathematics learning from relevant problem contexts,

and in England the latest curriculum provides increased flexibility and examples of rich

problem-solving tasks. Hong Kong is yet to develop the same levels of support for teachers.

All recognise that for teachers to include more problem solving opportunities in lessons,

textbooks will need to include more examples of problems and examinations need to assess

problem solving.

The New Australian Curriculum Approach

The Australian Curriculum: Mathematics is currently being written with opportunities for

consultation early in 2010 and implementation in 2011. The guiding statement for writers, the

Shape of the Australian Curriculum: Mathematics (National Curriculum Board [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, particularly 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.

Concluding Comments

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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 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.

References

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.

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Education Department HKSAR (2002). Basic education curriculum guide: Building on strengths (Primary 1 –

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