The place of problem solving in contemporary mathematics curriculum documents

The place of problem solving in contemporary mathematics curriculum documents

Journal of Mathematical Behavior 24 (2005) 341–350 The place of problem solving in contemporary mathematics curriculum documents夽 Kaye Stacey Faculty...

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Journal of Mathematical Behavior 24 (2005) 341–350

The place of problem solving in contemporary mathematics curriculum documents夽 Kaye Stacey Faculty of Education, University of Melbourne, Melbourne 3010, Vic., Australia Available online 27 September 2005

Abstract This paper reviews the presentation of problem solving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problem solving in the documents is reviewed and contrasted, and illustrative problems from teachers’ support materials are used to demonstrate how problem solving is now more often treated as a teaching method, rather than a goal in itself. The paper also analyses how the curriculum documents describe the growth of students’ abilities in the process areas of mathematics, and assesses the guidance that this provides for teachers. At each stage, the paper suggests directions for research that would be useful in assisting curriculum documents to promote the fundamental but elusive goal of making students better problem solvers. © 2005 Elsevier Inc. All rights reserved. Keywords: Problem solving; Mathematics curriculum; Process; National curriculum; Learning outcomes; Research directions

The purpose of this paper is to review the role of problem solving in contemporary curriculum documents in the English-speaking world and to suggest some directions for research in problem solving that may help make curriculum specifications for problem solving increasingly useful in the future. Problem solving is one of the most fundamental goals of teaching mathematics, but also one of the most elusive. An ultimate goal of teaching is that students will be able to conduct mathematical investigations by themselves, and that they will be able to identify where the mathematics they have learned is applicable in real world situations. In the phrase of the mathematician Halmos (1980), problem solving is “the heart of mathematics”. However, while teachers around the world have had many successes in achieving this goal, especially with more able students, there is still a great need for improvement, so that more students can gain a deeper appreciation of what it means to think mathematically and to use mathematics to assist in their daily and working lives. Despite being a fundamental goal, problem solving may always be elusive. To get closer to the goal requires research directed to understanding the problem solving process for mathematics (in all its aspects), developing effective classroom processes, and designing excellent tasks. Moreover, the research needs to be closely intertwined with curriculum development and teacher development projects so that it can make an impact on practice. This paper will review the place of problem solving in school mathematics as portrayed in several official curriculum documents, and suggest some directions by which research on problem solving can assist educational systems to

夽 This paper is based on a presentation to the Topic Study Group 18 “Problem Solving in Mathematics” at ICME10 held in Copenhagen in 2004.

An earlier version of this paper is to appear in the Proceedings of the PROMATH2004 Conference, held in Lahti, Finland, July 2004. E-mail address: [email protected] 0732-3123/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2005.09.004

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promote mathematical problem solving in schools. I begin by describing how the teaching of mathematical problem solving is conceptualized in the official curriculum documents of Australia, UK, USA and Singapore. Problem solving has a place in each document, as the main goal itself or as a way of achieving a broader goal of achievement in school mathematics. Then, by examining some exemplary tasks from official curriculum advice, I will demonstrate that problem solving has become difficult to pinpoint in the official descriptions of mathematics curricula in some countries because it is enacted principally as a teaching methodology. In this approach, problem solving is, in part, regarded as something to learn, but even more as a process through which to learn, set within an amalgam of various attitudes toward and appreciations of mathematics. This view will be supported by considering examples of recommended teaching activities and curriculum documents. The way in which research on open problem solving can contribute in this climate is rather different from how it contributed in the past, when problem solving as an explicit goal was more evident. Some possible directions are given. In the final section, some examples are given of official descriptions of components of problem solving and of children’s progress. These descriptions highlight the need for research to make a stronger contribution to understanding how process aspects of mathematics develop, what should and can be taught, and how children progress. The data for this paper have been obtained from a document analysis of curriculum materials provided for teachers in books and on the internet by the UK National Curriculum, the National Council of Teachers of Mathematics (NCTM, USA), three of the states of Australia (Victoria, NSW, Western Australia) and Singapore. These jurisdictions were selected because they represent variations on what appears to be the current mainstream approach in English-speaking countries, with many paths of influence evident between them. The Australian and UK documents also share an “outcomes” based approach to curriculum documentation. Lying behind the issues discussed in this paper is the fact, documented over several decades of research, that successful mathematical problem solving depends upon many factors which have distinctly different characters. This is illustrated in Fig. 1. A deep knowledge of mathematics and a strong reasoning ability are essential. However, to be able to go beyond the routine problems, it is essential to have strategic knowledge (e.g., of common heuristic strategies) and to have a set of helpful beliefs and personal attributes to organize and direct the problem solving process. Dealing with large scale problems also requires attributes such as the ability to work with others and the ability to communicate what one has done. It follows that teachers who wish to make their students better problem solvers will always have to work on many different levels. Consequently there is an inevitable complexity of teaching problem solving and of specifying it in a curriculum document.

Fig. 1. Factors contributing to successful problem solving.

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Fig. 2. Different curriculum subdivisions of process aspects of mathematics. (* Retained in 2000 revision.)

1. How problem solving is conceptualized in curriculum documents In Australia and the UK, since the 1990s, the curriculum has been officially described in terms of outcomes: what students should know and be able to do at the end of various stages of schooling. In Australia, the rewriting of the mathematics curriculum in terms of outcomes began with the “National Profile” (Australian Education Council, 1994), which has been the basis for the curriculum standards and frameworks used in the schools in every state of Australia. The National Profile organized the goals of mathematics education into six major strands; five “content” strands plus a “process” strand entitled Working Mathematically. (The word “strand” will be used in this paper for consistency, although it is not used in all jurisdictions.) The five content strands (number, algebra, space, measurement, chance and data) specify in broad terms the mathematical concepts and skills to be acquired, and set outcomes to be assessed. The “Working Mathematically” strand also aims to set learning outcomes to be assessed in the process areas of mathematics. The “Working Mathematically” strand organized the process of doing mathematics into the following six substrands (also shown in Fig. 2 with variations from elsewhere): • • • • • •

Investigating. Conjecturing. Using problem solving strategies. Applying and verifying. Using mathematical language. Working in context.

These substrands were a first attempt to define and structure what it means to work mathematically, and were intended to be used to plan both teaching and assessment. The National Profile and associated documents also recognized the importance of attitudes and appreciations (e.g., confidence to apply mathematics, persistence, creativity and ability to work cooperatively and independently) although they were not listed as assessable outcomes. During the 1990s, the various Australian states, which have independent education systems, modified the National Profile to their individual needs for curriculum planning and as a guide to assessment. Our interest here is to see how the process strand was reorganized. In Victoria, the Curriculum and Standards Framework (Board of Studies, 1995) organized mathematics into the five content strands of the national profile, and a sixth “process” strand (Tools and Procedures) which had five substrands (see Fig. 2), some of which are clearly derived from the National Profile. Since 2000, the successor to the Board of Studies, the Victorian Curriculum and Assessment Authority, having found that teachers were over-burdened by requirements to assess according to the framework, simplified the structure leaving just two substrands: Reasoning and Strategies (VCAA, 2000). Another state of Australia, New South Wales, in its 2002 syllabus (Board of Studies NSW, 2002), conceptualizes mathematics as things “to learn about” and things “to learn to do”. Thus students will “learn about” the material

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specified in the five content strands (similar to those specified above) and students will “learn to do” what is specified in the Working Mathematically strand. The comment is made that Working Mathematically is not to be assessed separately from the content strands. In this document, Working Mathematically has five inter-related processes (as shown in Fig. 2). As is typical in moving from one education system to another, we see that the substructure of the “working mathematically” strand bears similarities to the earlier documents, but is not the same. This shows that there is not strong agreement on how process strand for mathematics is constituted, and consequently on what are the key aspects to teach and to assess. Western Australia (HREF1) has two process strands, and is the only jurisdiction reviewed to explicitly recognize the affective component in its framework. The first strand, “Appreciating Mathematics”, covers disposition to use mathematics, and appreciation of its cultural origins and significance in modern life. Note how this affective component is concerned with the intellectual appreciation of mathematics, rather than a purely emotional like or dislike; an important feature if students are to be fairly assessed. It is important for students’ progress that they feel positively towards their learning of mathematics, and it will, of course, be of interest to teachers to know if this is the case, but it is not a reasonable quality for assessment. The second strand, “Working Mathematically”, organizes the abilities to solve problems using strategies, technology and management skills; the ability to choose mathematical ideas to apply in practical situations and being able to investigate, generalize and reason about patterns. These contents are further organized into three substrands, of “Mathematical Strategies”, “Reason Mathematically” and “Apply and Verify”. In the USA, the “Principles and Standards for School Mathematics” (NCTM, 2000) has also attempted to describe what makes up a good mathematics education. They also have five content standards (broadly paralleling the Australian content strands) and five further standards for the process aspects, as shown in Fig. 2. Unlike the Australian documents, the NCTM Standards has problem solving as one of several process strands, which structurally puts problem solving on an equal footing with learning to work with representations, communicating results etc, rather than as the goal for these processes. In the National Curriculum of the United Kingdom (HREF2), there are four “Attainment Targets”, three of which describe content (Number and algebra; Shape, space and measures; Handling data) and a fourth entitled “Using and applying mathematics” which is further subdivided into the three components shown in Fig. 2. The somewhat simpler structure of the UK curriculum is also reflected in the process strand, with the three components being a subset of the components of the NCTM standards. As with the NCTM document, problem solving is a part of this strand, rather than the underlying goal of the strand. Fig. 3 illustrates the UK structure. The curriculum documents reviewed above have mostly organized the mathematics curriculum in terms of strands reflecting the content areas of mathematics together with one (usually) process strand which sits in the structure in parallel with the other strands (although there are various acknowledgements of its difference). The Singapore structure (see Fig. 4 and Ministry of Education, 2001) is an example of a curriculum structure which is distinctly different. It puts mathematical problem solving at the center of its curriculum framework, as a goal and purpose of mathematics education. “The primary aim of the mathematics program is to enable pupils to develop their ability in mathematical problem solving. . . . The attainment of this mathematical problem solving ability is dependent on five inter-related components — Concepts, Skills, Processes, Attitudes and Metacognition.” (Ministry of Education, 2001, p. 5). The Singapore curriculum therefore conceptualizes content as one factor that contributes to problem solving ability, along

Fig. 3. In the UK curriculum, problem solving is part of a good mathematics education.

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Fig. 4. Singapore curriculum has problem solving as the goal of school mathematics.

with helpful attitudes, metacognition, heuristic strategies, thinking skills including reasoning, generic skills (such as communication) and general mathematical skills (such as estimation). This is in line with the structure shown in Fig. 1, which reflects how research has generally been organized around problem solving as a goal, with many factors contributing to successful problem solving ability. The curriculum structure of other countries, such as the UK structure shown in Fig. 3, sees problem solving as one of the components of a successful mathematics education, rather than its overall goal. Research could examine whether and how these curriculum structures from different countries influence teachers’ understanding of the goals of teaching mathematics, and whether these different understandings make a real difference in the attention that teachers give to mathematical problem solving beyond the routine. The examples above mainly show variations on a theme in the process domain. This document analysis could be taken to show that there is not strong agreement on how the process strand for mathematics is constituted, and consequently on what are the key aspects to teach and to assess. On the other hand, while it is evident that agreement is not strong, it is not evident that there is real disagreement! It is unclear from the documents whether there is a fundamental difference or whether the variations are simply attempts by different people to communicate ideas to teachers more clearly and to provide a framework for assessment which individual writing groups judge to be more viable. An open question is whether the various structures reflect real differences or not, especially whether there are differences evident in practice. Do these various structures reflect the same goals, or do the differences result in attention being given to different goals? A further question for inter-country research is to establish which of these approaches have been more successful in communicating the process of working mathematically to teachers, and what has been the key to the success. 2. Exemplary assessment tasks involving problem solving From the beginning of the open problem solving movement for schools, which is often linked with the publication of the Agenda for Action (NCTM, 1980), distinctions have been made about the role and purpose of problem solving in the curriculum. One useful common distinction, which is used by Schroeder and Lester (1989) but which they acknowledge as having much earlier origins, is made among: • 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).

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Fig. 5. Examples of pentominoes.

In this section, I examine how the official curriculum materials place problem solving within teaching and assessment activities. It appears that teaching through problem solving is now the dominant recommendation. Through the sample tasks which they provide, the curriculum documents reviewed can be seen to promote an open, less rigid approach to mathematical learning which has strong investigative elements. The assessment tasks assess process aspects of mathematics along with standard curriculum goals. In this way, the ideal is conveyed that problem solving is absorbed into normal teaching, as an attitude to learning and a process underpinning achievement in the normal curriculum. Process aspects of mathematics come across as equal amongst many goals of the school curriculum and problem solving is seen as a way to develop other skills. This view appears to be at least as prominent as that of seeing other skills as essential to improve problem solving, and perhaps more prominent. The evidence for this claim is in the document analysis of sample tasks on the official websites and in books of advice for teachers. This paper can only indicate a few examples, taken from the UK National Curriculum and from Victoria (Australia). The tasks are given to indicate the level of work expected by students at various stages, and to show teachers how assessment advice is to be interpreted. I assume that these are tasks which are selected as providing good models for classroom practice. A clear theme amongst the sample tasks is how a degree of openness is recommended to be added to routine work to replace other practice of skills, and also be used in the assessment of routine skills. A UK example at Key Stage 2 (students aged about 8–11) is a problem on area and perimeter. The sample work (HREF2) provides a drawing by Riyaz (not reproduced here due to copyright restrictions), which shows how he worked out perimeters and areas of pentominoes. Fig. 5 shows some examples of pentominoes, which are polygons made from five unit squares each adjacent to at least one other. Instead of setting a standard exercise on perimeter, the teacher asked the pupils to draw as many pentominoes as they could onto centimeter-square paper and to work out their perimeters. This task is made non-routine by the need to work out what pentominoes are possible, or at least to find some examples of pentominoes. Children do not need to find a complete set, or to find a set without repetition to meet the goal of practicing finding perimeters. To this extent, the task might be seen as a straightforward example of finding perimeters of rectilinear shapes, and perhaps also to show that there is no direct link between perimeter and area. The pentominoes can have perimeters of 12 or 10, but they all have area 5. However, the problem also has potential for exploring deeper ideas of transformation geometry (which pentominoes are really the same?) and to find out why the possible perimeters are only 12 or 10. Even without these deeper ideas, which would be beyond most students in the suggested age group, the problem has potential to stimulate class discussion about process aspects of mathematics. This might be at a simple level such as working systematically and keeping a good record of findings, or at a more advanced level of making conjectures (all the pentominoes have even perimeters) and proving them. An important question for research is whether the process goals are well provided for in the context of practicing new content ideas, or whether a different treatment is needed. Among schools, this remains an topic of debate. A recent government project conducted in Western Australia, for example, used action research to investigate these two alternatives (problem solving only within the teaching of other content or with special attention). The report presents it as a major dilemma which needs to be resolved (Gray, 2003). Systematic and formal research on this question is still required. In another UK example from Key Stage 2 (HREF2), students had to find the shortest length of a railway line to connect all the towns on a map. Here the teaching objectives were again a mix of content and process: • to use all four operations to solve simple word problems involving numbers and quantities, including time (key objective); • to present and interpret solutions to problems.

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The commentary given on a student’s work on the teachers’ support website, indicates that the student initially adopted a random approach to solving this problem. The sample work shows how the student progressed to present the information in a clear and organized way, using color to distinguish the lines. The comment is made that the student shows mastery of the number operations she has used (in this case, only addition) but that to improve her performance she needs to become more systematic in the way she solves a problem. These are comments on both content and process. Similar use of problem solving to allow an element of openness into the teaching of content, add a strategic element and dilute repetitive work is evident in the sample work provided in support of the Curriculum and Standards Framework in Victoria, Australia (HREF3). For example, in a problem recommended for 15-year-old students, students have to calculate specified powers of 3, explain the repeating pattern in last digits, predict the last digits of 315 and 31997 and describe a general way of finding the last digit of any positive integral power of 3. This problem is provided in the support material to illustrate the student’s ability to justify conjectures, as well as also linking to the algebra substrand of expressing generality. An important point to note is that this problem is seen as linking to the algebra strand even though no algebraic manipulation of the symbols is required. The sample student work provides an example of achievement at the highest level where students construct a convincing argument to justify conjectures and assertions. At the other end of the age spectrum, expected achievement in both a content strand (Number) as well as the process strand is shown in response to the following situation: “Ten boys and girls are coming to our school. But we don’t know how many are boys and how many are girls. Show the possible combinations”. The sample student work (HREF 3) suggests that children would have concrete materials available to assist in solving the problem. One solution shows a child’s drawing of the possible combinations adding to 10, and includes correctly written numerals. Again, this open problem solving is implicitly recommended as an activity to replace having students simply write out number combinations to 10, or practice the writing of numerals. The examples above show clear recognition of problem solving as a teaching method, especially as a context for practicing skills. It is part of a more open approach to teaching which encourages the development of a mix of helpful attitudes and appreciations, general thinking processes and specific problem solving skills. These approaches are also evident in curriculum material, such as in Core-Plus Math Project (HREF4), funded by the National Science Foundation in the USA, where ideas are introduced and skills are practiced in a sustained problem context. There are many opportunities for research to examine the success of these teaching methods and to assist in the resolution of Gray’s dilemma (2003). These changes in recommended teaching approaches reflect a pervasive approach to teaching mathematics, where speaking about mathematics with peers, writing about mathematics, and developing ideas within a community of inquiry are highly valued. They also reflect a wide appreciation of children’s capacity for developing concepts through their own reasoning and mathematical activity (for example by inventing their own arithmetic algorithms). The blurring of problem solving as a teaching goal with a novel teaching methodology associated strongly with certain classroom organizations involving features such as group work, means that research specifically on student problem solving is no longer an identifiably separate part of the mathematics education research literature. 3. Challenges for specifying growth in problem solving abilities Despite the official support for open problem solving as a key component of teaching and learning, it is often lost. In both the research and professional literature, there are many reports of the rewards and challenges of teaching about, through, or for problem solving. There is a useful body of research which identifies and analyses the difficulty of maintaining a high level of engagement with challenging tasks in classrooms (see, for example, Henningsen & Stein, 1997). Teachers’ experiences often support this. The article by Schettino (2003) is just one example of a recent article of this type. In agreement with observations made by many teachers and researchers in the past, Schettino provides a good analysis of the difficulties encountered within the classroom and the challenges of overcoming them from a teacher’s point of view. Difficulty in implementing problem solving in the classroom is, however, not the only obstacle to making problem solving a classroom reality. The many variations in the curriculum structures used around the world demonstrate that a straightforward specification of the process area of mathematics is problematic and hence it is difficult to be precise about what is to be learned in the process area. In this section, we delve into the curriculum documents, to consider how they display what counts as progress in learning in the process area. This is important, so that teachers are given

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Fig. 6. Extracts from Victoria’s curriculum document (VCAA, 2000).

guidance on how well-taught children will typically develop, how to evaluate the skills of their students, and how to plan further instruction that moves children on in their abilities to work mathematically. Take, for example, the specification of children’s knowledge about problem solving strategies. General problem solving strategies are weak, widely applicable strategies, whereas most mathematical methods are powerful with narrow applicability. There is an inevitable trade-off between breadth of applicability and power: neither one nor the other type of strategy is “better”, but they are inherently different. For instance, the algebraic method of solving problems by naming the unknowns, writing equations and solving them using algebraic rules is extraordinarily powerful — in those situations where it can be applied. In contrast, the heuristic strategy of “draw a diagram” applies to problems of many different types, including those where algebraic equation solving strategies apply. It is often very helpful for getting an insight into a problem, but it is rarely sufficient in its own right. The process parts of mathematics are broad strategies which develop slowly, along with some helpful appreciations and attitudes, and they contrast markedly with the pace and specific nature of the rest of the mathematics curriculum. Figs. 6 and 7 give some extracts from the Victorian and NSW documents, which show how the detailed specification of the strands and substrands is handled there. These extracts demonstrate some of the strengths and difficulties of the curriculum specification. The obvious strength is the broad range of aspects of doing mathematics that appear in the curriculum documents. The division into the strands and substrands in Fig. 2 provide a place for at least an oblique reference to all the major processes likely to be relevant to investigating problems, to mathematising real situations, to solving mathematically formulated problems and to interpreting and communicating results. There are inevitable compromises of comprehensiveness, so that the brief summaries will always emphasize some things (e.g., guess-check-improve in Fig. 6) over other worthy contenders, but in general, there is a very rounded picture.

Fig. 7. Extract from NSW Mathematics Syllabus for years 7–10 (Board of Studies NSW, 2002).

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Specifying progress in the process aspects of mathematics is more difficult. One important issue, where more research could be very useful, concerns whether development occurs in the process itself, or principally in regard to the content to which the process is applied. An example is shown in Questioning in Fig. 7, which only advises that a student should be able to “ask[s] questions that could be explored in relation to Stage 3 content”. A statement like this might only remind teachers that they should encourage their students to ask questions. It does not give guidance to illustrate how questions might change as students progress, other than in the content of the questions, nor what teachers might do to encourage the growth of sophisticated questioning. These are areas where research can help. How, if at all, do students’ questions grow in sophistication? Can the key discriminators between more and less sophisticated questions be taught to students? What are reasonable expectations of students’ questioning at various age levels? Is this simply a question of the content of the question, or are there aspects of questioning that develop? The Victorian questioning descriptions (see 3.1 and 5.1 in Fig. 6) have addressed this to some extent by identifying as easiest the questions that are “from presented data and from familiar contexts” (level 3 and level 5) and, by implication seeing questions arising from “inquiries in relation to previous tasks” as being more sophisticated (level 5 only). Whether this is a genuine difference and whether, or how, it can be taught or learned are questions which could be researched by those interested in improving practice. Many of the other entries in both Fig. 6, Fig. 7 and the source documents provide similarly useful starting points for research: what are the characteristics of student growth in the process areas? Another way in which growth is hard to see in the curriculum specification is when there is little obvious link between statements at different levels. As we noted above, there are many different components to working mathematically, and a curriculum specification will, of necessity, highlight some over others. However, highlighting different aspects at different levels makes growth hard to see. The NSW entries for Reasoning (Fig. 7) provide an example. The two statements have some commonalities and differences, but it is hard to see whether the differences are key to the growth to be expected or merely a different phrasing of the same sentiment. The descriptions of “applying strategies” have a similar characteristic. Both are reasonable short descriptions of good things to do, but they do not highlight the development or teaching that might take place. Research on the nature of reasoning and strategies that can be expected at various levels, such as the work of Ishida (1997, 2002) could be applied here. These curriculum documents intend to describe what can be expected of well taught students at each level, and the description is then to be used for assessment and for planning teaching. An important research agenda is to provide this information to curriculum authorities. 4. Research directions The sections above have described the way in which the “process” aspects of mathematics are now handled in official curriculum documents and accompanying advice to teachers in several English-speaking educational jurisdictions. First, I highlighted the different ways in which problem solving, and “working mathematically” are positioned within the mathematics curriculum, and how these process strands are constituted. My examples come from the systems with which I am most familiar, and are all from English speaking countries. The variation within even this limited group shows that specifying the process aspects of mathematics is a difficult task. There are opportunities for researchers to conduct empirical or theoretical enquiries into the nature of this part of a mathematics curriculum and into the ways in which teachers understand what is meant by these specifications. Research on whether these different specifications affect how teachers perceive their work could provide stronger foundations for future curriculum development. In the second section, I discussed the way in which the pervasive approach to teaching mathematics has recast problem solving primarily as “teaching through problem solving”. Problem solving activities are prominent in the recommended tasks only to the extent that they support the learning of other parts of the curriculum. In this vision, open problem solving is absorbed into normal teaching, as an attitude to learning and a process underpinning achievement in the normal curriculum. The goal of the tasks highlighting problem solving is to promote good learning of routine content and to develop useful strategic and metacognitive skills, rather than explicitly to strengthen students’ ability to tackle unfamiliar problems. This vision follows a welcome and widespread appreciation of children’s capacity for developing concepts through their own reasoning, mathematical activity and invention. There is also a widespread acknowledgement that routine skills can be practiced in an open environment, thereby encouraging creativity and investigative strength at the same time. With a pervasive approach to learning mathematics which values speaking, writing, and doing mathematics within communities of inquiry, open problem solving is not easily pinned down but is more an amalgam of skills, attitudes and appreciations across a broad front of doing mathematics. In this context, problem solving research therefore blends with research on many other aspects of teaching and learning; a trend which

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has been evident in the research literature for some years. However, there are also important questions still to be answered about the ways in which this blended approach can achieve both its content and process goals. In the third section, I noted that the detailed specification of curriculum objectives also provides opportunities for research. Research is needed to identify more definitively what is to be learned, and at what age and what are the signs of progress. To encourage teachers to work on problem solving in spite of the well-documented obstacles of classroom implementation, they need clear ideas of what characterizes the progress of children and which aspects can be taught. From the earliest problem solving research, it became clear that even a little teaching about problem solving pays dividends in making students’ better problem solvers. However, there is much more research to be done to get a good description of what can be achieved for students of various ages and abilities. With these suggestions, we see that research on open problem solving needs to regularly change its character to meet the new visions of the ideal classroom and to provide information that is useful to contemporary curriculum advice. References Australian Education Council. (1994). Mathematics — A curriculum profile for Australian schools. Melbourne: Curriculum Corporation. Board of Studies NSW. (2002). Mathematics years 7–10 syllabus. Sydney: Author. Board of Studies, Victoria. (1995). Curriculum and standards framework. Melbourne: Author. Gray, R. (2003). Introduction — The dilemma. In M. O’Brien (Ed.), Working mathematically in the middle years. Perth: Mathematical Association of Western Australia. Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519–524. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549. HREF1 Curriculum Council (WA). (n.d.). Curriculum framework. Mathematics. Retrieved March 26, 2005, from http://www.curriculum.wa.edu.au/ pages/framework/framework08. HREF2. (n.d.). National curriculum in action. Retrieved October 15, 2004. Retrieved from http://www.ncaction.org.uk. HREF3 Victorian Curriculum and Assessment Authority. (n.d.). Curriculum and standards framework II. Retrieved October 15, 2004. Retrieved from http://csf.vcaa.vic.edu.au/home.htm. HREF4. (n.d.). Core-plus mathematics project. Retrieved April 15, 2005. Retrieved from http://www.wmich.edu/cpmp/index.html. Ishida, J. (1997). The teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process. School Science and Mathematics, 97(3), 155–162. Ishida, J. (2002). Students’ evaluation of their strategies when they find several solution methods. The Journal of Mathematical Behavior, 21, 49–56. Ministry of Education, Singapore. (2001). Lower secondary mathematics syllabus. Singapore: Author. National Council for Teaching Mathematics. (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: Author. Schettino, C. (2003, November). Transition to a problem solving curriculum. Mathematics Teacher, 534–537. Schroeder, T., & Lester, F. (1989). Developing understanding in mathematics via problem solving. In P. Trafton & A. Shulte (Eds.), New directions for elementary school mathematics (1989 Yearbook). Reston, VA: NCTM. Victorian Curriculum Assessment Authority (VCAA). (2000). Curriculum and standards framework II. Melbourne: Author.