ORIGINAL ARTICLE

Problem solving and Working Mathematically:an Australian perspective

David Clarke Æ Merrilyn Goos Æ Will Morony

Accepted: 1 July 2007 / Published online: 28 July 2007

� FIZ Karlsruhe 2007

Abstract This article reviews ‘‘problem solving’’ in

mathematics in Australia and how it has evolved in recent

years. In particular, problem solving is examined from the

perspectives of research, curricula and instructional prac-

tice, and assessment. We identify three key themes

underlying observed changes in the research agenda in

Australia in relation to problem solving: Obliteration,

Maturation and Generalisation. Within state mathematics

curricula in Australia, changes in the language and con-

struction of the curriculum and in related policy documents

have subsumed problem solving within the broader cate-

gory of Working Mathematically. In relation to assessment,

research in Australia has demonstrated the need for align-

ment of curriculum, instruction and assessment, particu-

larly in the case of complex performances such as

mathematical problem solving. Within the category of

Working Mathematically, recent Australian curriculum

documents appear to accept an obligation to provide both

standards for mathematical problem solving and student

work samples that illustrate such complex performances

and how they might be assessed.

1 Introduction

Problem solving in mathematics in Australia has under-

gone significant change over recent decades. Research into

problem solving can be discussed in terms of three key

themes: Obliteration, Maturation and Generalisation. The

pursuit of the latter two themes has led to parallel initia-

tives related to the investigation of problem solving in

applied settings and in the development of more general

theoretical conceptions of problem solving as an activity.

While problem solving has been subsumed within the

broader notion of mathematical thinking, so too has re-

search on teachers’ problem solving practices begun to

draw on more general theoretical perspectives to investi-

gate classroom processes and cultures that promote math-

ematical thinking.

Prescription of mathematics curricula in Australia re-

mains the responsibility of the state jurisdictions, although

collaboration between states at various times has produced

position statements representing a form of national cur-

ricular consensus. Contemporary curriculum documents in

Australia have variously interpreted the problem solving

agenda in terms of applications, heuristics or problem-

based learning. These alternatives are encompassed within

the term ‘‘problem solving approaches’’ referring to any

instructional approach which gives explicit recognition to

mathematical problem solving as a curricular goal. Most

recently, such documents have subsumed problem solving

within the broader category of Working Mathematically.

As it is presently conceived within Australian mathematics

curriculum documents and instructional materials, Work-

ing Mathematically does have potential for informing and

enacting change that makes the doing of mathematics—and

therefore problem solving—central to mathematics in

schools. However, video studies of grade 8 mathematics

D. Clarke (&)

International Centre for Classroom Research,

University of Melbourne, 109 Barry Street,

Carlton, VIC 3053, Australia

e-mail: d.clarke@unimelb.edu.au

URL: http://extranet.edfac.unimelb.edu.au/DSME/lps/DC

M. Goos

University of Queensland, Brisbane, Australia

W. Morony

Australian Association of Mathematics Teachers, Inc.,

Adelaide, Australia

123

ZDM Mathematics Education (2007) 39:475–490

DOI 10.1007/s11858-007-0045-0

classrooms in Australia show little evidence of an active

culture of problem solving.

In relation to assessment, research in Australia has

demonstrated the need for alignment of curriculum,

instruction and assessment, particularly in the case of

complex performances such as mathematical problem

solving. The role of problem solving within high-stakes

assessment has varied significantly between state jurisdic-

tions. Within the category of Working Mathematically,

recent Australian curriculum documents appear to accept

an obligation to provide both standards for mathematical

problem solving and student work samples that illustrate

such complex performances and how they might be as-

sessed.

2 How has research on problem solving evolved

in Australia?

Although problem solving was a major focus of mathe-

matics education research in Australia throughout the

1990s (Anderson & White, 2004; Nisbet & Putt, 2000),

research priorities, styles and values began to change

during this time. Anthony (2004) considered these changes

in her analysis of the content, educational focus, and re-

search methodology of papers presented at the annual

conferences of the Mathematics Education Research Group

of Australasia (MERGA) in 1994 and 2003.1 Three trends

are relevant to the discussion of problem solving. First, in

both 1994 and 2003, just under half of the papers presented

had substantial mathematical content as their focus. How-

ever, the content categories most frequently investigated

shifted from problem solving and algebra in 1994 to

number and computation in 2003. Anthony noted that this

shift may have been related to the development and

implementation of large-scale numeracy programs by most

of Australia’s state-based education systems, and the

emerging need for research to support and evaluate these

programs.2 Secondly, although little change occurred in the

educational focus of MERGA conference papers, with

‘‘cognition’’ representing the largest category of papers in

both 1994 and 2003, the overall proportion of papers

focusing on cognition had declined. The decrease was

offset by increasing interest in technology, affect and

sociocultural issues. Finally, the most common research

style or methodological approach changed from the task-

based studies that typified early research on problem

solving (such as analyses of mathematical tasks or stu-

dents’ behaviour as they worked on tasks) to ethnographic

or case studies. Taken together, these trends are consistent

with Stacey’s (2005) claim that research on student prob-

lem solving internationally is no longer a clearly identifi-

able segment of the mathematics education research

literature.

In tracing the evolution of problem solving research in

Australia we can identify three themes that may explain the

trends outlined above.

1. Obliteration. Problem solving research has been

overtaken by other research and policy agendas (such

as those stimulated by debates related to numeracy

education).

2. Maturation. The focus of problem solving research has

moved from theory development into an ‘‘applied’’

phase in order to investigate the impact of curriculum

reform on classroom practice (teaching through prob-

lem solving).

3. Generalisation. The field of problem solving research

has broadened to explore more general theoretical

concepts and perspectives (problem solving as one

aspect of mathematical thinking or ‘‘Working Mathe-

matically’’).

The first theme suggests that problem solving research

has diminished because other emerging issues have re-

quired attention. While this is almost certainly true, it

would be a mistake to assume that research on problem

solving has disappeared entirely. Instead, problem solving

research has been transformed in the ways suggested by the

terms Maturation and Generalisation used to label the

second and third themes. These themes are used to struc-

ture the discussion of current research on problem solving

in Australia. Table 1 shows how the themes are addressed

within the two major research domains relating to students’

problem solving performance and teachers’ instructional

practice, and positions representative Australian studies

within this classification scheme.

3 Earlier traditions in Australian problem solving

research

Australian research on problem solving in the 1990s was

influenced by the pioneering work of US researchers such

as Schoenfeld (1985, 1987, 1992), Garofalo and Lester

(1985), and Silver (1985) in seeking to develop cognitive

and metacognitive models of students’ thinking as they

work on problem solving tasks. Representative of this work

are the studies of secondary school students’ problem

1 Around 80% of MERGA conference papers are presented by Aus-

tralian researchers, and most of the remaining papers are presented by

researchers from New Zealand. Similar themes are evident in math-

ematics education research in both countries.2 These numeracy programs were established to improve the teaching,

learning and assessment of foundational mathematical skills in the

primary school years (K-6/7), especially in the areas of number sense

and computation.

476 D. Clarke et al.

123

solving strategies and characteristics conducted by Goos

and Galbraith (1996) and Stillman and Galbraith (1998),

Stillman’s (1998, 2000) analyses of the cognitive demand

of problem solving tasks, and Lowrie’s investigation of

visual and nonvisual problem solving methods used by

elementary school students (Lowrie & Clements, 2001).

Some recent studies have maintained this theoretical ori-

entation towards studying thinking processes. Wilson and

Clarke (2004), for example, synthesised existing research

with their own empirical work to formulate an elaborated

model of mathematical metacognition, while Holton and

Clarke (2006) proposed an expanded conception of scaf-

folding that identified metacognition with self-scaffolding.

Goos (Goos, 2002; Goos, Galbraith, & Renshaw, 2002)

took metacognitive theorising in a new direction by anal-

ysing patterns of student–student social interaction that

mediated metacognitive activity during collaborative

problem solving.

4 Towards applied research on problem solving

Current Australian research on problem solving has a more

applied focus reflecting the curricular goal of ‘‘teaching

mathematics through problem solving’’. Some of the

studies we classify within the two research domains shown

in Table 1 were concerned with efforts to improve stu-

dents’ problem solving performance by using visual rep-

resentations, while other research centred on teachers’

instructional practices in problem solving classrooms.

4.1 Students’ problem solving performance

Developing an appropriate visual representation of the

information in a problem is crucial to successful problem

solving (e.g., Wheatley & Brown, 1997), and the increasing

availability of computer software has led to investigations

of the ways in which manipulation of computer images

might foster spatial visualisation skills that assist in solving

problems. Lowrie (2002a) has evaluated the effectiveness

of interactive computer programs in improving children’s

capacity to interpret and construct 3D-like images in

computer environments. He concluded that children may

need to develop understanding of perspective, orientation,

and depth via manipulation of 3D objects before engaging

with these concepts in computer-based virtual environ-

ments. In non-technology contexts, Diezmann (2000, 2005)

has shown that children also have difficulty in generating

or selecting appropriate diagrams to represent problem

structure. Australian studies of this type generate questions

about the type of teacher support needed to help students

move between visual-tactile activity, computer simula-

tions, and abstract diagrammatic representations.

4.2 Teachers’ instructional practices

Choice of problem solving tasks is one aspect of

instructional practice that has been studied from a

number of perspectives. Problem posing tasks are re-

garded as an important adjunct to problem solving as the

ability to pose problems requires metacognitive abilities

Table 1 Classification of current research on problem solving in Australia

Theme Research domain

Students’ problem solving performance Teachers’ instructional practices

1. Move towards ‘‘applied’’

problem solving research

(Maturation)

1.1 How can students be assisted to form appropriate

visual representations of problems?

Technology (Lowrie)

Diagrams (Diezmann)

1.2 What type of problem solving tasks should

teachers choose for use in the classroom?

Problem posing (Lowrie)

Cognitive engagement (Helme & Clarke; Williams)

Context (Clarke & Helme)

1.3 How do teachers’ beliefs about problem solving

influence their classroom practice?

Teachers’ problem solving beliefs (Anderson et al.)

2. Broadening of the field to

explore more general theoretical

concepts and perspectives

(Generalisation)

2.1 How can a problem solving approach promote

mathematical thinking?

Using modelling to connect mathematics with real

world contexts (English; Galbraith & Stillman)

Using investigations to develop mathematical

reasoning (Diezmann et al.)

Developing creativity in mathematical thinking

(Williams)

Teaching for abstraction (Mitchelmore & White)

2.2 What classroom processes promote a culture ofinquiry to support problem solving?

Communities of inquiry (Goos; Groves et al.)

Collaborative learning (Barnes)

Problem solving and Working Mathematically 477

123

in recognising different problem structures and goals.

Lowrie (2002b) has found that young children can gen-

erate open-ended problems with varying levels of com-

plexity, especially when supported by a teacher in a

structured problem posing environment. Sweller and his

co-workers have conducted a long-term program of re-

search into the cognitive consequences of some of the

instructional techniques integral to the various problem

solving approaches discussed in this article. By applying

the criterion of the minimisation of extraneous cognitive

load, this research has demonstrated and justified the

instructional value of both non-goal-specific tasks and

worked examples in mathematics (Sweller, 1992).

Drawing on the same theoretical rationale, this research

has problematised both the explicit teaching of heuristics

and those pedagogies that might be characterised as

‘‘problem-based learning’’ (Kirschner, Sweller, & Clark,

2006). Other Australian research aiming to provide

teachers with information on choosing appropriate tasks

has focused on the use of authentic artefacts or out-of-

school contexts (Lowrie, 2004, 2005) as well as char-

acteristics of tasks that increase cognitive engagement

(Helme & Clarke, 2001; Williams, 2000). Clarke and

Helme (1998) distinguished the social context in which

tasks were undertaken from the ‘‘figurative context’’

described in the task itself and related this to the stu-

dents’ capacity to find points of connection between their

own experience and what they are trying to understand

or to solve.

Research on teacher beliefs has been a consistent theme

within mathematics education for many years, and this

theme is reflected in current Australian research on teach-

ers’ beliefs about problem solving. Anderson and col-

leagues have examined teachers’ support for problem

solving approaches by developing and evaluating a model

of factors that influence problem solving beliefs and

practices (Anderson, White, & Sullivan, 2004). While

teachers with ‘‘traditional’’ beliefs reported using trans-

missive teaching strategies and those with more contem-

porary beliefs favoured problem solving approaches in the

classroom, the model acknowledged that teachers’ early

experiences as learners of mathematics and perceived

constraints within the teaching context (e.g., students’ stage

of schooling and level of understanding, textbooks,

assessment pressures, parental expectations) were factors

moderating their plans for implementing problem solving

approaches. Teachers reported that they needed more

support for changing their practice, such as modelling and

demonstration of strategies and better access to good re-

source materials, as well as clear evidence that problem

solving approaches improved student learning (Anderson,

2005).

5 Towards more general theoretical concepts

and perspectives

The second theme we identify in current problem solving

research is a broadening of the field that places problem

solving within the realm of mathematical thinking (often

expressed in curriculum documents as ‘‘Working Mathe-

matically’’). The studies we classify within the two re-

search domains in Table 1 focus on approaches to

developing students’ mathematical thinking and classroom

processes that promote a culture of inquiry.

5.1 Students’ mathematical thinking

Research in this domain has followed two lines of inquiry

focusing on either contextualisation or abstraction as

mathematical thinking processes.

In the 1980s and 1990s, mathematics curriculum

development in Australia emphasised problem solving in

parallel with applications or modelling, and developing

students’ ability to use their mathematical knowledge to

address problems in real world contexts remains a signifi-

cant focus of Australian research. Galbraith and Stillman’s

research with secondary school students reflects a com-

mitment to teaching modelling processes (Galbraith, 2006;

Galbraith & Stillman, 2006), while English’s work with

younger children (Doerr & English, 2003; English &

Watters, 2004) is representative of the contextual model-

ling perspective based on solving word problems (see

Kaiser & Sriraman, 2006).

Mathematical investigations have been proposed as an-

other way of involving students in exploring meaningful

real world problems. Following Jaworski (1986), Diez-

mann, Watters and English (2001) describe mathematical

investigations as ‘‘contextualised problem solving tasks

through which students can speculate, test ideas and argue

with others to defend their solutions’’ (p. 170). This re-

search found that although young children could plan and

implement investigations, they faced a range of difficulties

in the process. Knowledge of these difficulties could enable

teachers to structure investigations and thus provide more

opportunities for success.

Contrasting with the emphasis on real world connections

in modelling and investigative approaches is research on

the development of abstract mathematical thinking. Wil-

liams’s (2002a, b, 2004) work in constructing a hierarchi-

cal framework for describing students’ mathematical

thinking in terms of the processes of abstraction has proven

useful for investigating the nature of spontaneity, auton-

omy and creativity in mathematical problem solving.

Mitchelmore and White (2000) advocated a problem

solving approach to teaching for abstraction, exemplified

478 D. Clarke et al.

123

through several successful teaching trials based on learning

angle concepts (White & Mitchelmore, 2003).

5.2 Promoting a culture of inquiry

Just as problem solving has been subsumed within the

broader notion of mathematical thinking, so too has re-

search on instructional practices that engage students in

problem solving begun to draw on more general theoretical

perspectives to investigate classroom processes and cul-

tures that promote mathematical thinking. Goos’s (2004)

long-term study of a secondary school mathematics class-

room is representative of this approach. Her research

developed a sociocultural framework for examining the

teacher’s specific actions in creating a culture of inquiry.

The analysis showed how the teacher established norms

and practices that emphasised mathematical sense making

and justification of ideas and arguments, and traced rela-

tionships between the teacher’s actions and students’

changing participation patterns. At the elementary school

level, Groves, Doig, and Splitter (2000) looked to cross

cultural studies of mathematics teaching in different

countries (e.g., Stigler & Hiebert, 1999) to inform their

research on mathematics classrooms functioning as com-

munities of inquiry.

Collaborative learning has been investigated in several

studies previously cited (e.g., Goos, Galbraith, & Renshaw,

2002; Williams, 2000) as a participation structure for

engaging students in problem solving. Barnes’s (2001,

2003) research in this area has identified factors that inhibit

or support productive peer interactions, such as the level of

challenge and interest generated by the task as well as the

positioning of students within these interactions. The par-

ticipation patterns she identified, such as ‘‘interactive

leaders’’ or attention seekers’’, highlighted the importance

for teachers of understanding social power relations in

small groups.

6 Problem solving and published curricula in Australia

Before outlining the evolution of the treatment of problem

solving in the intended curriculum (i.e., the official state-

ments of the curriculum) in Australia, it is necessary to

note some complexities that arise as a result of the sepa-

ration of powers between the national and state govern-

ments. Control of schools, including the curriculum, is the

constitutional responsibility of the states. As a result, there

is no ‘‘national’’ curriculum—there are, in fact, eight of

them. Hence, on the face of it, it is not possible to discuss

the ‘‘Australian curriculum’’.

There have, at times, been collaborative efforts by state

and the national governments to work together towards

national consistency in curriculum. These periods of col-

laboration have produced statements that were broadly

agreed upon by the governments. It is these statements that

form the basis for this brief historical survey. They indicate

points in time when all the jurisdictions more or less

agreed. Differences did emerge in the curricula from state

to state in the time between these nodes, but these were

reconciled in the next round of collaboration.

6.1 Australian Mathematics Education Program (1982)

The Australian Mathematics Education Program (AMEP)

was established by the Curriculum Development Centre

(CDC), an organization jointly owned by the state, and

national governments. Its ‘‘Statement of basic Mathemati-

cal Skills and Concepts’’ was ‘‘the first national statement

of basic mathematical skills and concepts’’ (CDC, 1982). It

was a brief document that identified ten domains of skills3

and concepts, of which the eighth was problem solving.

The CDC took the view that

‘‘Problem solving is the process of applying previ-

ously acquired knowledge in new and unfamiliar

situations. Being able to use mathematics to solve

problems is a major reason for studying mathematics

at school. Students should have adequate practice in

developing a variety of problem solving strategies so

they have confidence in their use’’ (p. 3).

This was a common and predominant view in curricula

around the country before the work of the AMEP and

through the 1980s. Two different views of problem solving

coexisted in the curriculum documents of the time. Prob-

lem solving was seen as the essence of doing mathematics

at school, while at the same time it was represented as a

series of strategies to be developed and then used on

mathematical problems within mathematics and in the

‘‘real world’’.

The first (essence of mathematics) view was somewhat

idealistic, and something of a given in commonly held

views of mathematics, e.g., Polya (1957). As a result it did

not have practical impact in classrooms. The second view

owes much to the heuristics described by Polya (1957) and

has had a continuing presence in school mathematics in

Australia. It gave rise to specifications in documents about

expectations for the teaching and learning of problem

solving. For example, in Victoria ‘‘The Mathematics

Framework: P-10’’ (1988) had a sequence of learning for

problem solving for the compulsory years of schooling that

3 The others were Number Skills and Computational Skills, Geome-

try, Measurement, Estimation and approximation, Alertness to the

reasonableness of results, Reading, Interpreting and constructing ta-

bles and graphs, Using mathematics to predict, Applying mathematics

to everyday situations, and Language.

Problem solving and Working Mathematically 479

123

largely emphasized strategies. It is also noteworthy that

problem solving was the only ‘‘process’’ aspect included in

the scope and sequence for Victorian schools at that time.

All the others related to mathematical content areas.

6.2 National statement and National Profile

(c. 1991–1993)

The early 1990s saw Australia come close to adopting a

truly national curriculum for school mathematics. Much of

the groundwork was done, but at a pivotal time the states

agreed to maintain autonomy in curriculum.

Two mathematics documents were the result of inten-

sive collaborative curriculum development and extensive

consultation in the late 1980s and early 1990s. Despite the

lack of formal and agreed adoption, both have had a major

impact on mathematics curriculum in Australia.

A National statement on mathematics for Australian

schools (Australian Education Council and Curriculum

Corporation, 1991) defined the broad scope and content of

the school mathematics curriculum. It mirrored the duality

of the previous AMEP work in that solving problems was

assumed as key to the mathematical enterprise, but that the

sole embodiment of this was the development of strategies.

The ‘‘Mathematical Inquiry’’ strand had ‘‘problem solving

strategies’’ as one of its four sub-strands, along with

‘‘Mathematical expression’’, ‘‘Order and arrangement’’ and

‘‘Justification’’. This strand was intended to address

‘‘communication skills, ways of thinking and habits of

thought which are explicitly, although not exclusively,

mathematical’’ (p. 37). Through the arrangement of the

national statement, problem solving was dissociated from

the use of mathematics in real world and applied contexts

(the ‘‘choosing and using mathematics’’ strand).

The publication Mathematics—a curriculum profile for

Australian schools in 1994 ‘‘describe(d) the progression of

learning typically achieved by students’’ (Curriculum

Corporation, 1994, p. 1; our emphasis). It described, for the

first time, the agreed set of intended learning outcomes for

students, and this was a big shift in thinking that continues

to have ramifications in Australian education.

Whilst these two documents were described as ‘‘linked’’,

the structure of the National Profile departed from that of

the national statement. The five content strands were

identical, but the National Profile used the term ‘‘Working

Mathematically’’ to capture all of the process aspects of

learning mathematics. Stacey (2005) also attached signifi-

cance to the emergence of the term ‘‘Working Mathemat-

ically’’ in this key curriculum document. The Working

Mathematically strand consisted of sequences of outcomes

in the sub-strands shown in Table 2.

Table 3 lists the strategies specified in each document.

Those in the National Profile begin with outcomes for

young children at the top, ending with those expected of

students at the end of schooling. There is no ‘‘hierarchy’’ in

the list for the national statement.

6.3 Statements of learning for mathematics

(2005–2006)

The most recent national project involving the states and

national government working together to develop curricu-

lum has been the National Consistency in Curriculum

Outcomes Project that sought, for mathematics among

several subject areas, to identify ‘‘knowledge, skills,

understandings and capacities that students in Australia

should have the opportunity to learn and develop in the

mathematics domain’’. These have been expressed as

‘‘opportunities to learn’’ that ‘‘education jurisdictions have

agreed to implement in their own curriculum documents’’

(MCEETYA, 2006; p. 1).

From the statements of learning for mathematics:

Working Mathematically involves mathematical in-

quiry and its practical and theoretical application.

This includes problem posing and solving, represen-

tation and modelling, investigating, conjecturing,

reasoning and proof and estimating and checking the

reasonableness of results or outcomes. Key aspects of

Working Mathematically, individually and with oth-

ers, are formulation, solution, interpretation and

communication. The processes of Working Mathe-

matically draw upon and make connections between

the knowledge, skills and understandings acquired in

Number, Algebra, function and pattern, Measure-

ment, chance and data, and Space (pp. 3, 4).

Table 2 Working Mathematically in the Australian curriculum

National Statement

Attitudes & appreciations Mathematical inquiry Choosing and using mathematics

Attitudes Appreciations Mathematical

expression

Order and

arrangement

Justification Problem-solving

strategies

Applying

mathematics

Mathematical

modeling

National Profile

n/a n/a Using mathematical

language

Investigating Conjecturing Using problem-solving

strategies

Applying and

verifying

Working

in context

480 D. Clarke et al.

123

The detailed descriptions provided in the Statements of

Learning identify problem solving strategies in a manner

similar to that of previous documents. The strategies are

not viewed in isolation, but as part of the whole. Moreover,

the Statements of Learning for Mathematics and the state

curricula to which they are connected represent another

opportunity to put the doing of mathematics, in the form of

Working Mathematically, at the centre of school mathe-

matics.

7 Curricular alternatives: applications, heuristics

and problem-based learning

In relation to the role and purpose of problem solving in

mathematics curricula in Australia, it is useful to consider

the distinctions drawn by Schroeder and Lester (1989, and

cited in Stacey, 2005) between:

• teaching for problem solving (teaching mathematical

content for later use in solving mathematical problems);

• teaching about problem solving (teaching heuristic

strategies to improve generic ability to solve problems);

• teaching through problem solving (teaching standard

mathematical content by presenting non-routine prob-

lems involving this content) (Stacey, 2005, p. 345).

These three categories succinctly summarise the three

major approaches employed by Australian curriculum

developers. The first can be seen as a simple elaboration of

the traditional curriculum to include the ‘‘application’’ of

conventional mathematical content in more complex or less

familiar contexts. The second and third alternatives repre-

sent more radical curricular innovations: the explicit

teaching of problem solving heuristics and the develop-

ment of new pedagogies such as problem-based learning

(PBL).

There has been an emerging emphasis in Australian

mathematics curricula on ‘‘real world’’ contexts for math-

ematics, beginning in the 1980s (see, for example, Treilibs,

1986) and continuing until current times. Whilst curricu-

lum documents including the national statement saw

applications and modelling as distinct from problem solv-

ing, teachers and students have increasingly been involved

in solving problems that involved using mathematics in the

‘‘real world’’. Many textbooks and other support materials

have tried to adopt this orientation, and there have been

assessment-driven changes to promote the use of applica-

tions for teaching and learning mathematics. The advent of

the Internet in recent years has made real data much more

available to teachers than ever before.

Many contemporary Australian textbooks have separate

sections for applications, although these are less frequently

referred to as ‘‘problem solving’’ since the term itself

seems to have become less popular. For example, ICE-EM

Mathematics (2006) is a text series designed for national

use. It has a ‘‘challenge section’’ at the end of each chapter.

These are linked to the content of the chapter, but there is

no explicit discussion or instruction about problem solving

strategies.

Materials that supported the development of problem

solving strategies were prevalent from the early 1980s.

Stacey and Groves (1985) provided detailed lesson notes to

support the teaching and learning of problem solving in

junior secondary classrooms (grades 7/8–10). Their thesis

was that ‘‘problem solving can be improved by:

• practising solving non-routine problems;

• developing good problem solving habits;

• learning to use problem solving strategies; and

• thinking about and discussing these experiences’’

(inside front cover).

The focus on non-routine problems was pronounced.

The problems were selected to exemplify particular stra-

tegic ‘‘themes’’ in problem solving. Although clearly

mathematical, the activities were not directly linked to the

rest of the curriculum. This led to instructional practices

that treated problem solving as a distinct and separate

component of school mathematics.

The book was something of a landmark publication in

Australian mathematics education. It provided teachers

with practical guidance on the teaching of problem solving.

Indeed, the approach exemplified by Stacey and Groves’

work had—and arguably still has significant impact. Since

1985, Australian textbooks have often had a problem

Table 3 Comparing conceptions of problem solving strategies

Problem solving strategies in the

National Statement (p. 39)

Problem solving strategies in the

National Profile (p. 4)

Guessing, checking and

improving

Answer questions by acting out

a story, showing with objects

or pictures

Looking for patterns Trial and error

Making a model or drawing

a picture

Selecting key information

Making an organised list

or table

Representing information in

models, diagrams and lists

Restating the problem (Strategies) based on selecting

and organizing key information

and being systematic

Separating out irrelevant

information

Identifying and working on related

problems or sub-problems

Identifying and attempting

sub-tasks

Generalizing from one problem

situation to another

Solving a simpler version

of the problem

Rethinking problem conditions

and constraints

Eliminating possibilities

Problem solving and Working Mathematically 481

123

solving section, perhaps at the end of some or all chapters.

These commonly bore little or no relationship to the con-

tent of the rest of the chapter. Typically they were

‘‘something extra’’ and perhaps ‘‘a bit of fun’’ when the

real work of the chapter (fractions or algebra or whatever)

was completed. Often this section was reserved only for

those students who were quick with their other work, with

the implication that it was not core mathematical learning

for all the students (even those who might be struggling

with other work).

Siemon (1986) has criticised such an ‘‘appendage

mentality’’ in relation to problem solving:

‘To spend the majority of one’s time ‘‘doing mathe-

matics as it has always been done’’, with ‘‘problem

solving’’ added on as an interesting appendage, ac-

tively acts against encouraging a problem-solving

approach (to mathematics)’ (p. 35).

In other words, whilst curriculum planners had viewed the

introduction and emphasis on problem solving as part of

making school mathematics more relevant and engaging,

problem solving risked being constructed in classrooms

and in the minds of students according to the existing

paradigms of views of mathematics and approaches to its

teaching and learning. Both the applications and the heu-

ristics alternatives were open to (mis)interpretation as

being disconnected from the central and more conventional

content of the curriculum.

Lovitt and Clarke (1988) in their influential mathematics

curriculum and teaching program (MCTP) added an

important new slant on problem solving in mathematics.

They promoted ‘‘using problem solving as the most

effective way to teach’’ (p. 469). Problem solving was seen

by these authors as a teaching methodology, and the MCTP

materials exemplified this approach. This involved teach-

ing through applications and modelling, an approach that

became prevalent in some courses of study in grades 11

and 12, and in which students learned by grappling with

‘‘real world problems’’. The generic term ‘‘problem-based

learning’’ (PBL) captures these approaches and has been

growing in currency, particularly in the secondary years.

Efforts to move in this direction have been reinforced

through their connection to broader curriculum directions

being adopted by state and territory curriculum authorities.

For example, in New South Wales, the term Working

Mathematically has been strongly embraced in the new K-

10 mathematics syllabus. Anderson (2005) noted that the

elements of Working Mathematically were easily and

strongly linked to the elements and dimensions of ‘‘quality

teaching’’ as described in the education department’s

generic instructions to all teachers ‘‘quality teaching in

NSW public schools’’ (NSWDET, 2003). In other words,

by implementing the Working Mathematically elements of

the mathematics syllabus, teachers will also be able to meet

the other requirements in the broader curriculum. This

convergence of purpose has the potential to encourage and

enhance the efforts of teachers of mathematics to work in

ways that emphasise and develop Australian students’

capacities to work mathematically, and, incidentally, to

develop as mathematical problem solvers.

8 The assessment of mathematical problem solving

in Australia

During the 1990s, a consistency could be seen in the trends

in mathematics assessment in communities as geographi-

cally dispersed as Australia (Victorian Board of Studies,

1995a, b), the Netherlands (Van den Heuvel-Panhuizen,

1996), the Pacific region (Pacific Resources for Education

and Learning, 1997), Portugal (Leal and Abrantes, 1993),

Sweden (National Agency for Education, 1995), the UK

(Close et al., 1992), and the USA (National Council of

Teachers of Mathematics, 1995). The common elements of

these assessment initiatives included the use of open-ended

tasks, the use of contextualized settings for many tasks, the

use of technology in instruction and its presence in assess-

ment, and the expansion of the means of assessment beyond

time-restricted examinations. This consistency derived

from a new conception of the mathematics curriculum and

the consequent demands for forms of assessment that were

sensitive to new standards in mathematics. These various

national trends have been drawn together in significant

international documents (e.g. OECD: PISA, 2003) that have

recommended a broader framework for assessing mathe-

matics than that found in traditional tests.4 The assessment

of problem solving provided one of the key challenges for

mathematics educators in Australia during the 1990s.

The assessment of mathematical problem solving in

Australia has had a colourful and even controversial history.

In 1990, the Department of Education in the state of Vic-

toria, piloted and subsequently implemented an innovative

assessment regime at grades 11 and 12, in which the explicit

assessment of problem solving was a key component. In

Victoria, as in most Australian states, the 12th grade

examination system is state-mandated and extremely high-

stakes, in that it mediates access to subsequent university

and other tertiary studies. Given this, the attempt to assess

mathematical problem solving within such a high-stakes

context, provided significant insight into the practical,

conceptual, philosophical, political and educational chal-

4 In several countries, developments in assessment can be linked to

specific national projects or initiatives. Some of these are illustrated in

Clarke (1996) and Burton (1996). Other related issues are discussed in

Leder (1992) and Stephens and Izard (1992).

482 D. Clarke et al.

123

lenges associated with such an initiative. It also provided an

opportunity to investigate the instructional consequences of

such an innovation and to carry out comparative research

into the extent to which problem solving was manifest in

curriculum documents, classroom instruction, and assess-

ment practices in particular Australian states.

9 Problem solving and ‘‘the ripple effect’’

The Victorian Certificate of Education (VCE), imple-

mented in 1990, assessed student performance in all sub-

jects in the final 2 years of secondary schooling (11th and

12th grades). The VCE mathematics assessment acknowl-

edged very different types of performance from which

‘‘mathematical competence’’ was constituted and em-

ployed a multi-component assessment instrument, which

was intended to capture the major features of that compe-

tence through the use of very different instrument types. An

underlying principle of VCE mathematics was that all

students engage in the following mathematical activities:

• Problem-solving and modeling: the creative application

of mathematical knowledge and skills to solve prob-

lems in unfamiliar situations, including real-life situa-

tions;

• Skills practice and standard applications: the study of

aspects of the existing body of mathematical knowl-

edge through learning and practising mathematical

algorithms, routines and techniques, and using them to

find solutions to standard problems;

• Projects: extended, independent investigations involv-

ing the use of mathematics.

These three learning activities were incorporated into all

courses for VCE mathematics in Grades 11 and 12 as

formal work requirements. These work requirements were

intended to promote key aspects of mathematical behaviour

and to guide the work of teachers and students. The three

work requirements were directly linked to the ways in

which mathematical performance was assessed. They were

intended to be used in an integrated way to develop

understanding of concepts, communication skills, and a

capacity to justify mathematical claims. The Victorian

multi-component assessment scheme attracted interna-

tional interest and was featured prominently in the NCTM

Assessment Standards (NCTM, 1995, 61–63). Figure 1

shows a typical VCE problem solving task.

In contrast to the situation in Victoria, teachers in the

state of New South Wales (NSW) received contradictory

messages about what the system expected of them in

mathematics. On the one hand, a ‘‘Statement of Principles’’

was incorporated into all curriculum documents which

discussed, among other issues, the nature of mathematics

learning, emphasizing that students learn mathematics best

through interaction with other people, through investiga-

tion, and through the use of language to express mathe-

matical ideas. The syllabi for Grades 7 and 8 were

well-aligned with this statement, emphasizing problem

solving, investigative approaches, and communication. On

the other hand, there was no requirement at any level to

incorporate specific investigative, problem solving, mod-

eling or communication tasks into school assessments.

There was a clear implication in the various curriculum

documents that assessment solely by means of examination

was perfectly acceptable. Barnes, Clarke and Stephens

exploited the difference in alignment between curriculum

and assessment in the two most populous Australian states

to conduct a major investigation of the instructional con-

sequences of high-stakes assessment (Barnes, Clarke &

Stephens, 2000). This study employed a combination of

document analysis, questionnaires and interviews. Docu-

ments analysed included curriculum and policy documents,

teacher planning and instructional materials, and teacher-

devised assessment materials. Theoretical sampling of

schools and teachers in both states included rural and

metropolitan schools, government and non-government

schools, and a variety of social demographic characteristics

(including ethnicity and language). The classroom visibil-

ity of problem solving activities and assessment emerged

as the key difference between the two states.

The greatest difference between NSW and Victorian

teachers (according to Barnes, Clarke & Stephens, 2000)

was the importance they attached to students developing

report-writing skills. Fifty-five percent of Victorian teach-

ers regarded it as highly important as compared with only

ten percent of NSW teachers. NSW teachers also gave very

much less support than Victorian teachers to students

developing investigative skills, the item supported most

strongly by Victorian teachers. These two statements re-

flect aspects of doing mathematics which were emphasized

in VCE assessment procedures, but which were of little

importance in preparing students for the NSW 12th grade

examinations. The same applied to students undertaking

extended and open-ended mathematical activities. In Vic-

toria, such activities were endorsed explicitly by the way in

which problem solving and investigation tasks with an out-

of-class component were built into and assessed in the

VCE. Teachers in NSW did not attach comparable

importance to these activities, most probably because they

could not be tested by means of traditional examinations.

Most striking in this analysis, was the evidence in Victoria

of the ‘‘ripple effect’’ (Clarke & Stephens, 1996), whereby

the language and format of teacher-devised assessment

tasks employed in grades 7 to 10 in Victorian schools

echoed their officially mandated correlates in the 12th

grade VCE to an extraordinary level of detail.

Problem solving and Working Mathematically 483

123

Despite the interest among mathematics educators, the

use of out-of-class work for the problem solving compo-

nent of the high-stakes VCE mathematics assessment was

not viewed particularly favourably by the general public or

by some university recipients of the graduates of this

assessment scheme. After some experimentation with

methods by which the assessment of out-of-class work

could be calibrated against more traditional examination

performances, the use of problem solving activities in 12th

grade assessment was relegated to the status of an option

within a school-based assessment component rather than

being a mandated and externally set assessment. Given the

choice and little encouragement, schools favoured the use

of more easily administered conventional examinations,

and by the end of the 1990s, the assessment of problem

solving as a significant element in high-stakes mathematics

assessment had largely fallen from common use in Victo-

rian schools.

Problem 1 – The art gallery

Question 1

A room in an art gallery contains a picture which you are interested in viewing.

The picture is two metres high and is hanging so that the bottom of the picture is

one metre above your eye level. How far from the wall on which the picture is

hanging should you stand so that the angle of vision occupied by the picture is at

a maximum? What is this maximum angle?

Question 2

On the opposite wall there is another equally interesting picture which is only

one metre high and which is also hanging with its base one metre above eye

level, directly opposite the first picture. Where should you stand to maximise

your angle of vision of this second picture?

Question 3

How much advantage would a person 20 centimetres taller than you have in

viewing these two pictures?

Question 4

This particular art gallery room is six metres wide. A gallery guide of the same

height as you wishes to place a viewing stand one metre high in a fixed position

which provides the best opportunity for viewing both pictures simply by turning

around. The guide decides that this could best be done by finding the position

where the sum of the two angles of vision is the greatest. Show that the

maximum value which can be obtained by this approach does not give a suitable

position for the viewing stand.

Question 5

The gallery guide then decides to adopt an alternative approach which makes the

difference between the angles of vision of the two pictures, when viewed from

the viewing stand, as small as possible. Where should the viewing stand be

placed using this approach? Comment on your answer.

Fig. 1 Sample 12th grade

problem solving task (Victorian

Board of Studies, 1995a, b)

484 D. Clarke et al.

123

A recent national study, Year 12 Curriculum Content

and Achievement Standards (Matters & Masters, 2007),

examined curriculum documents in the form of mandated

courses of study and assessment schemes in use at 12th

grade level in all seven Australian states and territories.

Problem solving was present in the curriculum documents

related to 12th grade mathematics in all seven state juris-

dictions (p. 24). However, it was only evident in the

statements of assessment standards in use in four of the

seven jurisdictions: not appearing at all in the 12th grade

assessment conducted in South Australia/Northern Terri-

tory, Tasmania, and Western Australia (p. 79). This

inconsistent valuing of problem solving in statements of

assessment standards on a state by state basis may indicate

continuing practical difficulties in the assessment of

mathematical problem solving in high-stakes contexts, or it

may just reflect the difference between the curricular

rhetoric of policy documents and assessment practice.

10 Mathematics assessment trends in Australia

In 1988, Assessment Alternatives in Mathematics was

published in Australia as part of the national mathematics

curriculum and teaching program (Clarke, 1988). This

teacher resource publication included a section titled,

‘‘Assessment of Problem Solving and Investigative Work’’

(Clarke, 1992, pp. 35–42). The publication reflected an

increasing contemporary Australian interest in the assess-

ment of complex mathematical performances and in the use

of open-ended mathematics tasks for assessment as well as

instruction (eg Clarke, Clarke & Lovitt, 1990; Sullivan &

Clarke, 1991a and b; Sullivan & Clarke, 1992; Clarke,

1995). Since the early 1990s, the progressive subordination

of problem solving to the broader curriculum component

‘‘Working Mathematically’’ has been matched by an

increasing interest among curriculum developers and

teachers in ‘‘rich assessment tasks’’ (Beesey, et al., 1998;

Downton, et al., 2006). Three such tasks are shown in

Fig. 2. The common characteristic of such assessment tasks

was the requirement that significant responsibility be de-

volved to the student for the construction of the response.

Open-ended tasks and complex, non-routine mathematical

problems offered suitable vehicles for this devolution of

responsibility and control and it was argued that, as a

consequence, the student’s response was more reflective of

the student’s own mathematical understandings and more

likely to usefully inform the teacher’s subsequent instruc-

tion. The term ‘‘constructive assessment’’ was coined to

combine the prioritising of a constructed response with the

commitment to constructive action as a consequence of

assessment (Clarke, 1997). The use of complex mathe-

matical tasks was a key component of this approach and

remained a central feature, while the original emphasis on

problem solving was progressively subsumed within the

more inclusive ‘‘Working Mathematically’’ (and somewhat

subordinated to the increasing interest in numeracy).

11 The contemporary assessment of mathematical

problem solving in Australia

In 1999, the governments of the Australian States, Terri-

tories and Commonwealth, jointly signed the Adelaide

Declaration on National Goals for Schooling in the

Twenty-First Century. In 2006, the Council for the Aus-

tralian Federation undertook a review of ‘‘the achievements

of cooperative federalism in the area of school policy since

the Adelaide Declaration’’ (Dawkins, 2006). The review

reported a new level of federal collaboration that included

the development of a national statement of learning for

mathematics, setting out key learning goals for grades 3, 5,

What do you think this might be the

graph of?

Label the graph appropriately.

What information is contained in your

graph?

(Sullivan & Clarke, 1991b)

Fred’s apartment has five rooms and a total area of 60 square metres. Draw a possible plan of

Fred’s apartment. Label all rooms and show the dimensions – length and width – of each room

(Clarke, 1995).

In my backyard I have some chooks [chickens] and some dogs. Altogether I can count 25 heads

and 78 legs. How many dogs do I have? (Downton et al., 2006).

Fig. 2 Sample open-ended and

rich assessment tasks

Problem solving and Working Mathematically 485

123

7 and 9. This federal collaboration included The National

Assessment Program, intended to provide ‘‘full cohort

literacy and numeracy testing in Years 3, 5, 7 and 9’’

(Dawkins, 2006, p. 8). While problem solving could be

seen as a constituent element in both ‘‘numeracy’’ and in

‘‘Working Mathematically,’’ the term ‘‘problem solving’’ is

inconsistently evident in contemporary curricular and

assessment documents in Australia, most commonly in

reference to the situated application of specific skills. In the

same review, the performance of Australian students in the

OECD’s Programme for International Student Assessment

(PISA) in reading, mathematics, science, and in problem

solving was celebrated as indicating that ‘‘Australian 15-

year-old perform well (on average) when it comes to

careful reading, logical thinking, and the application of

reading skills and mathematical and scientific understand-

ings to everyday problems’’ (Dawkins, 2006, p. 7).

Unexpectedly, perhaps the most explicit evidence that

problem solving remains a priority within some Australian

education circles can be found in a document titled, ‘‘The

authentic performance-based assessment of problem solv-

ing’’ (Curtis & Denton, 2003). This very thorough and

interesting report was produced by the national centre for

vocational education research (NCVER) and it may be that

this document embodies most clearly the apparent national

trend to interpret problem solving in applied (in this case,

vocational) terms.

The other evident trend to emerge in the last decade in

Australia has been an increasing demand for educational

accountability of systems, schools and teachers. Account-

ability has largely taken the form of increased demands for

a variety of forms of assessment: from the introduction of

state-mandated tests of mathematics content at prescribed

grade levels to the requirement of new levels of detail in

the evidence of student performance that classroom

teachers are required to collect. Documents such as the

Victorian Essential Learning Standards (VELS) (VCAA,

2005) partition the curriculum into domains (such as

mathematics) and dimensions within those domains

(number, space, measurement, chance and data, structure,

and Working Mathematically). ‘‘Standards’’ are prescribed

for one or more dimensions within each domain. The

standard for Working Mathematically at level 6 (Grades 9

and 10) includes the following:

In Working Mathematically students abstract com-

mon patterns and structural features from mathe-

matical situations and formulate conjectures,

generalisations and arguments in natural language

and symbolic form . . . Students choose, use and

develop mathematical models and procedures with

attention to assumptions and constraints. They collect

relevant data, represent relationships in mathematical

terms, and test the suitability of the results obtained . .

. Students engage in investigative tasks and problems

set in a wide range of practical, theoretical and his-

torical contexts (VCAA, 2005, pp. 35, 36).

The aspirations of the problem solving agenda of the 1990s

are clearly evident in the above Standard. A challenge for

classroom teachers is to interpret the above standard in

assessable terms. Downloadable examples are available of

student work samples, illustrating how the standard for

Working Mathematically might be evidenced in a student’s

written response to a suitable mathematical task.

Barnes, Clarke and Stephens (2000) drew attention to

the need for congruence between the performances en-

shrined in the curriculum, those practiced in the classroom,

and those required for assessment purposes. Without this

alignment, more complex mathematical performances

(such as mathematical problem solving) can vanish from

the taught curriculum as a direct consequence of their ab-

sence from the assessed curriculum.

The emphasis in this paper is on a ‘‘curricular

alignment’’ by which assessment matches curricular

goals and instructional practice and, by this corre-

spondence, serves as a model for both. The impor-

tance of such alignment, as demonstrated in this

study, should not be seen as support for an assess-

ment-based accountability system. Systems that re-

ward or punish teachers on the basis of the

assessment of their students’ performance appeal to a

philosophical framework (and a model of teacher

professionalism) entirely different from the rationale

of congruence between curricular policy, instruction

and assessment (Barnes, Clarke & Stephens, 2000, p.

645).

It appears that forms of mathematical problem solving can

be discerned within the more generic contemporary label of

Working Mathematically (and, for example, ‘‘mathemati-

cal inquiry’’ in the Australian Capital Territory). There are

also encouraging signs that curriculum developers and

those responsible for assessment design recognize the need

to provide teachers with models of the types of student

performances commensurate with mathematical problem

solving (e. g., assessment advice provided with the Victo-

rian Essential Learning Standards at http://vels.vcaa.vic.e-

du.au/assessment/).

12 Some summary comments on trends in relation to

mathematical problem solving in Australia

Like many other countries, Australia has recently focused

on ‘‘numeracy’’ in the early years, with a number of large-

486 D. Clarke et al.

123

scale, systemically supported, sustained research and

development initiatives5. These claim to be evidence-

based, and to some extent there are elements that promote

the learning of problem solving behaviours, while others

have some affinity with problem-based learning (i.e.,

problem solving as pedagogy). It would be inaccurate,

however, to characterize these as substantially about

‘‘problem solving’’.

The video study undertaken by the third international

mathematics and science study provided evidence that

there was not a large presence of problem solving in any of

its guises in the Australian Grade 8 classes that were

studied. In the Learner’s Perspective Study (Clarke, Keitel,

& Shimizu, 2006; Clarke, Emanuelsson, Jablonka, & Mok,

2006), problem solving activities were almost entirely ab-

sent from the classrooms of the ‘‘competent’’ mathematics

teachers videotaped in Australia. Stacey (1999) indicated

that

‘‘(T)he average lesson in Australia reveals a cluster of

features that together constitute a syndrome of shal-

low teaching, where students are asked to follow

procedures without reasons. The evidence for this

syndrome lies in the low complexity of problems

undertaken with excessive repetition, and an absence

of mathematical reasoning in the classroom dis-

course’’ (p. 119).

This finding is, as Stacey indicated, not able to be gener-

alized in any scientific way. It is, however, a finding that is

challenging after more than a quarter of a century of

emphasis on problem solving in curriculum, advice and

resources to support the teaching of mathematics in Aus-

tralian schools.

We have identified three key themes underlying ob-

served changes in the research agenda in Australia in

relation to problem solving: Obliteration, maturation and

generalisation. The pursuit of the latter two themes has led

to parallel initiatives related to the investigation of problem

solving in applied settings and in the development of more

general theoretical conceptions of problem solving as an

activity. While problem solving has been subsumed within

broader notions of mathematical thinking, mathematical

inquiry and Working Mathematically, so too has research

on teachers’ problem solving practices begun to draw on

more general theoretical perspectives to investigate class-

room processes and cultures that promote mathematical

thinking.

Within state mathematics curricula in Australia, changes

in the language and construction of the curriculum have

subsumed problem solving within the broader category of

Working Mathematically. As it is presently conceived

within Australian mathematics curricula, Working Mathe-

matically does have potential for informing and enacting

change that makes the ‘‘doing of mathematics’’—and

therefore problem solving—central to mathematics in

schools. However, video studies of classroom practice in

middle school mathematics classes in Australia (Clarke,

Keitel, & Shimizu, 2006; Hiebert et al., 2003), show little

evidence of an active culture of problem solving.

In relation to assessment, research in Australia has

demonstrated the need for alignment of curriculum,

instruction and assessment, particularly in the case of

complex performances such as mathematical problem

solving. Within the category of Working Mathematically,

recent Australian curriculum documents appear to accept

an obligation to provide both standards for mathematical

problem solving and student work samples that illustrate

such complex performances and how they might be as-

sessed.

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