ZDM Mathematics Education (2007) 39:475–490 DOI 10.1007/s11858-007-0045-0
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 practice, 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 construction of the curriculum and in related policy documents have subsumed problem solving within the broader category of Working Mathematically. 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 assessed.
D. Clarke (&) International Centre for Classroom Research, University of Melbourne, 109 Barry Street, Carlton, VIC 3053, Australia e-mail:
[email protected] 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
1 Introduction Problem solving in mathematics in Australia has undergone 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 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 the broader notion of mathematical thinking, so too has research on teachers’ problem solving practices begun to draw on more general theoretical perspectives to investigate classroom processes and cultures that promote mathematical thinking. Prescription of mathematics curricula in Australia remains the responsibility of the state jurisdictions, although collaboration between states at various times has produced position statements representing a form of national curricular consensus. Contemporary curriculum documents in Australia have variously interpreted the problem solving agenda in terms of applications, heuristics or problembased 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, Working 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
123
476
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 jurisdictions. 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.
2 How has research on problem solving evolved in Australia? Although problem solving was a major focus of mathematics 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 research 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. However, 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 1
Around 80% of MERGA conference papers are presented by Australian researchers, and most of the remaining papers are presented by researchers from New Zealand. Similar themes are evident in mathematics 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.
123
D. Clarke et al.
style or methodological approach changed from the taskbased studies that typified early research on problem solving (such as analyses of mathematical tasks or students’ 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 problem solving internationally is no longer a clearly identifiable 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.
2.
3.
Obliteration. Problem solving research has been overtaken by other research and policy agendas (such as those stimulated by debates related to numeracy education). 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 problem solving). 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 Mathematically’’).
The first theme suggests that problem solving research has diminished because other emerging issues have required 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 structure 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
Problem solving and Working Mathematically
477
Table 1 Classification of current research on problem solving in Australia Theme
Research domain Students’ problem solving performance
1. Move towards ‘‘applied’’ problem solving research (Maturation)
Teachers’ instructional practices
1.1 How can students be assisted to form appropriate 1.2 What type of problem solving tasks should visual representations of problems? teachers choose for use in the classroom? Technology (Lowrie)
Problem posing (Lowrie)
Diagrams (Diezmann)
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 2.1 How can a problem solving approach promote explore more general theoretical mathematical thinking? concepts and perspectives Using modelling to connect mathematics with real (Generalisation) world contexts (English; Galbraith & Stillman)
2.2 What classroom processes promote a culture of inquiry to support problem solving? Communities of inquiry (Goos; Groves et al.) Collaborative learning (Barnes)
Using investigations to develop mathematical reasoning (Diezmann et al.) Developing creativity in mathematical thinking (Williams) Teaching for abstraction (Mitchelmore & White)
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 orientation 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 scaffolding that identified metacognition with self-scaffolding. Goos (Goos, 2002; Goos, Galbraith, & Renshaw, 2002) took metacognitive theorising in a new direction by analysing 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 students’ problem solving performance by using visual representations, 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 environments. 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 simulations, 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 regarded as an important adjunct to problem solving as the ability to pose problems requires metacognitive abilities
123
478
in recognising different problem structures and goals. Lowrie (2002b) has found that young children can generate open-ended problems with varying levels of complexity, especially when supported by a teacher in a structured problem posing environment. Sweller and his co-workers have conducted a long-term program of research 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-ofschool contexts (Lowrie, 2004, 2005) as well as characteristics 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 students’ 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 teachers’ beliefs about problem solving. Anderson and colleagues 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 transmissive teaching strategies and those with more contemporary 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 resource materials, as well as clear evidence that problem solving approaches improved student learning (Anderson, 2005).
123
D. Clarke et al.
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 Mathematically’’). The studies we classify within the two research 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 significant focus of Australian research. Galbraith and Stillman’s research with secondary school students reflects a commitment 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 modelling perspective based on solving word problems (see Kaiser & Sriraman, 2006). Mathematical investigations have been proposed as another way of involving students in exploring meaningful real world problems. Following Jaworski (1986), Diezmann, 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 research 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. Williams’s (2002a, b, 2004) work in constructing a hierarchical framework for describing students’ mathematical thinking in terms of the processes of abstraction has proven useful for investigating the nature of spontaneity, autonomy and creativity in mathematical problem solving. Mitchelmore and White (2000) advocated a problem solving approach to teaching for abstraction, exemplified
Problem solving and Working Mathematically
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 research on instructional practices that engage students in problem solving begun to draw on more general theoretical perspectives to investigate classroom processes and cultures that promote mathematical thinking. Goos’s (2004) long-term study of a secondary school mathematics classroom 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 relationships 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 communities 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 participation 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 statements of the curriculum) in Australia, it is necessary to note some complexities that arise as a result of the separation of powers between the national and state governments. 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
479
national consistency in curriculum. These periods of collaboration 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 Mathematical 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 previously 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. Problem 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, Geometry, Measurement, Estimation and approximation, Alertness to the reasonableness of results, Reading, Interpreting and constructing tables and graphs, Using mathematics to predict, Applying mathematics to everyday situations, and Language.
123
480
D. Clarke et al.
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 intensive 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 significance to the emergence of the term ‘‘Working Mathematically’’ 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 curriculum 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 inquiry and its practical and theoretical application. This includes problem posing and solving, representation 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 others, are formulation, solution, interpretation and communication. The processes of Working Mathematically draw upon and make connections between the knowledge, skills and understandings acquired in Number, Algebra, function and pattern, Measurement, chance and data, and Space (pp. 3, 4).
Table 2 Working Mathematically in the Australian curriculum National Statement Attitudes & appreciations Attitudes
Appreciations
Mathematical inquiry Mathematical expression
Order and Justification arrangement
Choosing and using mathematics Problem-solving strategies
Applying mathematics
Mathematical modeling
National Profile n/a
123
n/a
Using mathematical Investigating language
Conjecturing Using problem-solving Applying and strategies verifying
Working in context
Problem solving and Working Mathematically Table 3 Comparing conceptions of problem solving strategies Problem solving strategies in the Problem solving strategies in the National Statement (p. 39) 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
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 mathematics.
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 problems 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
481
familiar contexts. The second and third alternatives represent more radical curricular innovations: the explicit teaching of problem solving heuristics and the development of new pedagogies such as problem-based learning (PBL). There has been an emerging emphasis in Australian mathematics curricula on ‘‘real world’’ contexts for mathematics, beginning in the 1980s (see, for example, Treilibs, 1986) and continuing until current times. Whilst curriculum documents including the national statement saw applications and modelling as distinct from problem solving, 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 applications 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 strategic ‘‘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
123
482
solving section, perhaps at the end of some or all chapters. These commonly bore little or no relationship to the content 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 mathematics as it has always been done’’, with ‘‘problem solving’’ added on as an interesting appendage, actively 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 heuristics 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 teaching 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 K10 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
123
D. Clarke et al.
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 geographically 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 assessment, 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 mathematics 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 Victoria, 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 highstakes, 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 chal4
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).
Problem solving and Working Mathematically
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 assessment practices in particular Australian states.
9 Problem solving and ‘‘the ripple effect’’ The Victorian Certificate of Education (VCE), implemented in 1990, assessed student performance in all subjects in the final 2 years of secondary schooling (11th and 12th grades). The VCE mathematics assessment acknowledged very different types of performance from which ‘‘mathematical competence’’ was constituted and employed a multi-component assessment instrument, which was intended to capture the major features of that competence 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 problems in unfamiliar situations, including real-life situations; Skills practice and standard applications: the study of aspects of the existing body of mathematical knowledge through learning and practising mathematical algorithms, routines and techniques, and using them to find solutions to standard problems; Projects: extended, independent investigations involving 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 international 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
483
learning, emphasizing that students learn mathematics best through interaction with other people, through investigation, and through the use of language to express mathematical 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, modeling 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 consequences of high-stakes assessment (Barnes, Clarke & Stephens, 2000). This study employed a combination of document analysis, questionnaires and interviews. Documents analysed included curriculum and policy documents, teacher planning and instructional materials, and teacherdevised 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 visibility 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 teachers 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 reflect 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 Victoria, such activities were endorsed explicitly by the way in which problem solving and investigation tasks with an outof-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.
123
484 Fig. 1 Sample 12th grade problem solving task (Victorian Board of Studies, 1995a, b)
D. Clarke et al.
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.
Despite the interest among mathematics educators, the use of out-of-class work for the problem solving component 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
123
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 Victorian schools.
Problem solving and Working Mathematically
485
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 jurisdictions (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 Territory, 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 assessment 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 Fig. 2 Sample open-ended and rich assessment tasks
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 devolved 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 instruction. 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 mathematical 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, Territories and Commonwealth, jointly signed the Adelaide Declaration on National Goals for Schooling in the Twenty-First Century. In 2006, the Council for the Australian 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,
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).
What do you think this might be the graph of? Label the graph appropriately. What information is contained in your graph? (Sullivan & Clarke, 1991b)
123
486
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 15year-old perform well (on average) when it comes to careful reading, logical thinking, and the application of reading skills and mathematical and scientific understandings 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 solving’’ (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. Accountability 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 common patterns and structural features from mathematical 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
123
D. Clarke et al.
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 historical 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 enshrined 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 absence 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 correspondence, serves as a model for both. The importance of such alignment, as demonstrated in this study, should not be seen as support for an assessment-based accountability system. Systems that reward 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, ‘‘mathematical 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 Victorian Essential Learning Standards at http://vels.vcaa.vic.edu.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-
Problem solving and Working Mathematically
scale, systemically supported, sustained research and development initiatives5. These claim to be evidencebased, 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 absent 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 shallow 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 discourse’’ (p. 119). This finding is, as Stacey indicated, not able to be generalized 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 Australian schools. We have identified three key themes underlying observed 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 classroom 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 5 Examples include Count Me In Too (New South Wales) and First Steps in Mathematics (Western Australia).
487
Working Mathematically. As it is presently conceived within Australian mathematics curricula, Working 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 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 assessed.
References Anderson, J. (2005). Implementing problem-solving in mathematics classrooms: What support do teachers want? In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research and practice (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia, Melbourne, pp. 89–96). Sydney: MERGA. Anderson, J., & White, P. (2004). Problem solving in learning, teaching mathematics. In B. Perry, G. Anthony, & C. Diezmann (Eds.), Research in mathematics education in Australasia 2000– 2003 (pp. 127–150). Flaxton, QLD: PostPressed. Anderson, J., White, P., & Sullivan, P. (2004). Using a schematic model to represent influences on, and relationships between, teachers’ problem-solving beliefs and practices. Mathematics Education Research Journal, 17(2), 9–38. Anthony, G. (2004). MERGA: A community of practice. In I. Putt, R. Faragher & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville, pp. 2–15). Sydney: MERGA. Australian Education Council and Curriculum Corporation. (1991). A national statement on mathematics for Australian schools. Melbourne, Victoria: Curriculum Corporation. Barnes, M. (2001). Collaborative learning and student change: A case study. In J. Bobis, B. Perry & M. Mitchelmore (Eds.), Numeracy and beyond (Proceedings of the 24th annual conference of the Mathematics Education Research Group of Australasia, Sydney, pp. 82–89). Sydney: MERGA. Barnes, M. (2003). Patterns of participation in small-group collaborative work. In L. Bragg, C. Campbell, G. Herbert & J. Mousley (Eds.), Mathematics education research: Innovation, networking, opportunity (Proceedings of the 26th annual conference of the Mathematics Education Research Group of Australasia, Geelong, pp. 120–127). Sydney: MERGA. Barnes, M., Clarke, D. J., & Stephens, W. M. (2000). Assessment as the engine of systemic reform. Journal of Curriculum Studies, 32(5), 623–650.
123
488 Beesey, C., Clarke, B. A., Clarke, D. M., Stephens, W. M., & Sullivan, P. (1998). Exemplary assessment materials: Mathematics. Carlton: Longman/Board of Studies. Burton, L. (1996) Assessment of mathematics—what is the agenda? In M. Birenbaum, & F. Dochy (Eds.), Alternatives in assessment of achievements, learning processes and prior knowledge (pp. 31–62). Dordrecht: Kluwer. Clarke, D. J. (1988) Assessment alternatives in mathematics. Canberra: Curriculum Development Centre. Clarke, D. J. (1995). Quality mathematics: How can we tell? The Mathematics Teacher, 88(4), 326–328. Clarke, D. J. (1996). Assessment. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 327–370). Dordrecht: Kluwer. Clarke, D.J. (1997). Constructive assessment in mathematics: practical steps for classroom teachers. Berkeley: Key Curriculum Press. Clarke, D. J., Clarke, D. M., & Lovitt, C. J. (1990). Changes in mathematics teaching call for assessment alternatives. In T. J. Cooney, & C. R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (pp. 118–129). Reston: NCTM. Clarke, D. J., Emanuelsson, J., Jablonka, E., & Mok, I. A. C. (Eds.) (2006). Making connections: comparing mathematics classrooms around the world. Rotterdam: Sense Publishers. Clarke, D. J., & Helme, S. (1998). Context as construction. In: O. Bjorkqvist (Ed.), Mathematics teaching from a constructivist point of view (pp. 129–147). Vasa, Finland: Faculty of Education, Abo Akademi University. Clarke, D. J., Keitel, C., & Shimizu, Y. (Eds.) (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publishers. Clarke, D. J., & Stephens, W. M. (1996) The ripple effect: the instructional impact of the systemic introduction of performance assessment in mathematics. In: M. Birenbaum, F. Dochy (Eds.), Alternatives in assessment of achievements, learning processes and prior knowledge (pp. 63–92). Dordrecht: Kluwer. Close, G. S., Boaler, J., Burt, A., Coan, E., & Symons, K. (1992). Materials to support the assessment of Ma1: using and applying mathematics. London: School Examinations and Assessment Council. Curriculum Corporation (1994). Mathematics—a curriculum profile for Australian schools. Melbourne: Curriculum Corporation. Curriculum Development Centre (1982). Australian Mathematics education program: a statement of basic mathematical skills and concepts. Canberra: CDC. Curtis, D., & Denton, R. (2003). The authentic performance-based assessment of problem-solving. Station Arcade, South Australia: NCVER. Dawkins, P. (2006). Federalist paper 2—the future of schooling in Australia: a report by the states and territories. Melbourne: Council for the Australian Federation. Diezmann, C. (2000). The difficulties students experience in generating diagrams for novel problems. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th annual conference of the International Group for the Psychology of Mathematics Education (vol 2, pp. 241–248). Hiroshima: PME. Diezmann, C. (2005). Primary students’ knowledge of the properties of spatially oriented diagrams. In H. Chick, & J. Vincent (Eds.), Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (vol 2, pp. 281–288). Melbourne: PME. Diezmann, C., Watters, J., & English, L. (2001). Implementing mathematical investigations with young children. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.), Numeracy and beyond (Proceedings of the 24th annual conference of the Mathematics
123
D. Clarke et al. Education Research Group of Australasia, Sydney, pp. 170– 177). Sydney: MERGA. Doerr, H., & English, L. (2003). A modelling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34, 110–137. Downton, A., Knight, R., Clarke, D., Lewis, G. (2006). Mathematics assessment for learning: rich tasks & work samples. Fitzroy: Australian Catholic University. English, L., & Watters, J. (2004). Mathematical modelling with young children. In M. Hoines, & B. Fugelsted (Eds.), Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 335–342). Bergen: PME. Galbraith, P. (2006). Real world problems: Developing principles of design. In P. Grootenboer, R. Zevenbergen, & M. Chinnapan (Eds.), Identities, cultures and learning spaces (Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia, Canberra, pp. 229–236). Adelaide: MERGA. Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. Zentralblatt fu¨r Didaktik der Mathematik, 38(2), 143–162. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring and mathematical performance. Journal for Research in Mathematics Education, 25(4), 36–54. Goos, M. (2002). Understanding metacognitive failure. Journal of Mathematical Behavior, 21, 283–302. Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35, 258–291. Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30, 229–260. Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49, 193–223. Groves, S., Doig, B., & Splitter, L. (2000). Mathematics classrooms functioning as communities of inquiry: possibilities and constraints for changing practice. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1–8). Hiroshima: PME. Helme, S., & Clarke, D. J. (2001). Identifying cognitive engagement in the mathematics classroom. Mathematics Education Research Journal, 13(2), 133–153. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, A., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeck, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: results from the TIMSS 1999 video study. Washington: US Department of Education, National Center for Education Statistics. Holton, D., & Clarke, D. J. (2006). Scaffolding and metacognition. International Journal of Mathematical Education in Science and Technology, 37(2), 127–143. Jaworski, B. (1986). An investigative approach to teaching and learning mathematics. Milton Keynes, UK: Open University Press. Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. Zentralblatt fu¨r Didaktik der Mathematik, 38(3), 302–310. Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.
Problem solving and Working Mathematically Leal, L. C., & Abrantes, P. (1993) Assessment in an innovative curriculum project for mathematics in grades 7–9 in Portugal. In M. Niss (Ed.), Cases of assessment in mathematics education (pp. 173–182). Dordrecht: Kluwer. Leder, G. C. (Ed.) (1992). Assessment and learning of mathematics. Melbourne: Australian Council for Educational Research. Lovitt, C., & Clarke, D. M. (1988). Mathematics curriculum and teaching program professional development package: activity bank vol 2. Canberra: Curriculum Development Centre. Lowrie, T. (2002a). The influence of visual and spatial reasoning in interpreting simulated 3D worlds. International Journal of Computers for Mathematics Learning, 7, 301–318. Lowrie, T. (2002b). Young children posing problems: the influence of teacher intervention on the type of problems children pose. Mathematics Education Research Journal, 14(2), 87–98. Lowrie, T. (2004). Authentic artefacts: influencing practice and supporting problem solving in the mathematics classroom. In I. Putt, R. Faragher, & M. McLean (Eds.), Mathematics education for the third millennium: Towards 2010 (Proceedings of the 27th annual conference of the Mathematics Education Research Group of Australasia, Townsville, pp. 351–358). Sydney: MERGA. Lowrie, T. (2005). Problem solving in technology rich contexts: Mathematics sense making in out-of-school environments. Journal of Mathematical Behavior, 24, 275–286. Lowrie, T., & Clements, K. M. (2001). Visual and nonvisual processes in grade 6 students’ mathematical problem solving. Journal of Research in Childhood Education, 16(1), 77–93. Matters, G., & Masters, G. (2007). Year 12 curriculum content and achievement standards. Barton: Department of Education, Science and Technology. Ministerial Council on Education, Employment, Training and Youth Affairs (MCEETYA) (2006). Statements of learning for mathematics. Downloaded from (http://www.curriculum.edu.au) on 30 March 2007. Mitchelmore, M., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41, 209–238. National Agency for Education (1995). National test in mathematics course A. Stockholm: Swedish National Agency for Education. National Council of Teachers of Mathematics (NCTM) (1995). Assessment standards for school mathematics. Reston: NCTM. New South Wales Department of Education and Training (NSWDET) (2003). Quality teaching in NSW public schools: a discussion paper. Ryde: NSWDET. Nisbet, S., & Putt, I. (2000). Research in problem solving in mathematics. In K. Owens, & J. Mousley (Eds.), Research in mathematics education in Australasia 1996–1999 (pp. 97–121). Sydney: MERGA. Organisation for Economic Cooperation and Development: Program for International Student Assessment (OECD: PISA) (2003). Assessing scientific, reading and mathematical literacy: a framework for PISA 2006. Paris: OECD. Pacific Resources for Education, Learning (1997). Standards for Pacific education. Pacific Education Updates, 9(2), 16. Polya, G. (1957). How to solve it. 2nd edn. Princeton: Princeton University Press. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic. Schoenfeld, A. H. (1987). What’s all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 334–370). Hillsdale: Erlbaum. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition and sense making in mathematics. In D. Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 189–215). New York: Macmillan.
489 Schroeder, T., & Lester, F. (1989). Developing understanding in mathematics via problem solving. In P. Traafton, & A. Shulte (Eds.), New directions for elementary school mathematics (1989 Yearbook). Reston: NCTM (cited in Stacey, 2005). Siemon, D. (1986). Problem solving—a challenge for change or a passing fad? In K. Swinson (Ed.), Mathematics teaching: challenges for change (Proceedings of the Eleventh Biennial Conference of the Australian Association of Mathematics Teachers Inc.). Brisbane: AAMT. Silver, E. (1985). The teaching and assessing of mathematical problem solving. Reston: NCTM. Stacey, K. (1999). 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. TIMSS Australia Monograph No. 5. Camberwell: Australian Council for Educational Research Ltd. Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behavior, 24, 341–350. Stacey, K., & Groves, S. (1985). Strategies for problem solving. Burwood: VICTRACC. Stephens, W. M., & Izard, J. (Eds) (1992). Reshaping assessment practices: assessment in the mathematical sciences under challenge: proceedings from the first national conference on assessment in the mathematical sciences. Melbourne: Australian Council for Educational Research. Stigler, J., & Hiebert, J. (1999). The teaching gap: best ideas from the world’s teachers for improving education in the classroom. New York: The Free Press. Stillman, G. (1998). Task context and applications at the senior secondary level. In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching mathematics in new times (Proceedings of the 21st annual conference of the Mathematics Education Research Group of Australasia, Gold Coast, pp. 454–461). Gold Coast: MERGA. Stillman, G. (2000). Impact of prior knowledge of task context on approaches to applications tasks. Journal of Mathematical Behavior, 19(3), 333–361. Stillman, G., & Galbraith, P. (1998). Applying mathematics with real world connections: Metacognitive characteristics of secondary school students. Educational Studies in Mathematics, 36, 157– 195. Sullivan, P., & Clarke, D. J. (1991a). Communication in the classroom: the importance of good questioning. Geelong: Deakin University Press. Sullivan, P., & Clarke, D. J. (1991b). Catering to all abilities through ‘good’ questions. Arithmetic Teacher 39 (October issue, 14–18). Sullivan, P., & Clarke, D. J. (1992). Problem solving with conventional mathematics content: responses of pupils to open mathematical tasks. Mathematics Education Research Journal, 4(1), 42–60. Sweller, J. (1992). Cognitive theories and their implications for mathematics instruction. Chapter 3. In G. Leder (Ed.), Assessment and learning of mathematics (pp. 46–62). Hawthorn: Australian Council for Educational Research. Treilibs, V. (1986). Applications—a partial solution to an impossible problem? In K. Swinson (Ed.), Mathematics teaching: challenges for change (Proceedings of the Eleventh Biennial Conference of the Australian Association of Mathematics Teachers Inc.). Brisbane: AAMT. Van Den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht: CD-b Press. Victorian Board of Studies (1995a). Mathematics assessment activities using number, tools and procedures. Melbourne: Victorian Board of Studies.
123
490 Victorian Board of Studies (1995b). VCE mathematics: specialist—official sample CATs. Blackburn: HarperSchools. Victorian Curriculum, Assessment Authority (VCAA) (2005). Victorian essential learning standards. Discipline-based learning strand: Mathematics. East Melbourne: VCAA. Wheatley, G., & Brown, D. (1997). Components of imagery and mathematical understanding. Focus on Learning Problems in Mathematics, 19(1), 45–70. White, P., & Mitchelmore, M. (2003). Teaching angle by abstraction from physical activities with concrete materials. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th annual conference of the International Group for the Psychology of Mathematics Education and 25th annual conference of PMENA (Vol. 4, pp. 403–410). Honolulu: PME. Williams, G. (2000). Collaborative problem solving and discovered complexity. In J. Bana, & A. Chapman (Eds.), Mathematics education beyond 2000 (Proceedings of the 23rd annual conference of the Mathematics Education Research Group of Australasia, Fremantle, pp. 656–663). Sydney: MERGA.
123
D. Clarke et al. Williams, G. (2002a). Associations between mathematically insightful collaborative behaviour and positive affect. In A. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 402–409). Norwich: PME. Williams, G. (2002b). Identifying tasks that promote creative mathematical thinking: a tool. In B. Barton, K. Irwin, M. Pfannkuch, & M. Thomas (Eds.), Mathematics education in the South Pacific (Proceedings of the 25th annual conference of the Mathematics Education Research Group of Australasia, Auckland, pp. 698–705). Sydney: MERGA. Williams, G. (2004). The nature of spontaneity in high quality learning situations. In M. Hoines, & B. Fugelsted (Eds.), Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 433–440). Bergen: PME. Wilson, J., & Clarke, D. J. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25–49.
