PROBLEM SOLVING: ITS ASSIMILATION TO THE TEACHER'S PERSPECTIVE

Paul Ernest

University of Exeter

p.ernest @ ex.ac.uk

INTRODUCTION

It could be said that there is a problem-solving bandwagon rolling. Many influential reports published in the USA, UK and elsewhere, such as the Agenda for Action (NCTM, 1980), the Cockcroft Report (DES, 1982), the Standards (NCTM, 1989) and the National Curriculum in Mathematics (NCC, 1989), all strongly endorse a problem solving approach to school mathematics. These recommendations have been made for over a decade -- three decades, if we include the 'discovery learning' of the 1960s. There are powerful arguments behind these endorsements, which I will not rehearse here. But the fact is that most mathematics teaching remains routine and 'instrumental'. More often than not, children are given a method for carrying out a type of task, and then many graded exercises to practice and reinforce the method. Each task has a unique correct answer. Of course such procedures do generate some 'relational' and well as 'instrumental' understanding on the part of the learner, and perhaps some strategic skills. But the primary focus of such exercises is the successful acquisition and deployment of procedures, and the acquisition of relational understanding or strategic skills is incidental.

So what is going on? Why the disparity between the problem solving recommendations and the routine and convergent nature of much of school mathematics? There are many reasons, including institutional resistance to change, vested interests behind the status quo, individuals' resistance to change, be they teachers, learners, parents or others, and so on. In this paper I want to pick out one strand to explore, which in my view has received insufficient attention. That is the assimilation of problem solving to the teacher's perspective, by which I mean the teacher's mathematics-related belief-system. Teachers have different beliefs about the nature of mathematics and its teaching and learning, which powerfully affect their classroom practices. In some form or other, this relationship is now widely accepted, and I elaborate on it below. But one unremarked consequence of this, I wish to claim, is that problem solving is understood differently by different teachers, in accordance with their beliefs. A demonstration of the lack of unique meaning and the diversity of interpretations given to the term 'problem-solving' by teachers and others might go some way towards explaining how widespread espousal of problem-solving can co-exist with its widespread rejection in practice.

THE MEANINGS OF PROBLEM SOLVING

Problem solving has been a widespread part of the rhetoric of British mathematics education since Cockcroft (1982). Worldwide, problem solving can be traced further back, at least to Brownell (1942) and Polya (1945), and probably earlier. (In fact Descartes' Rules for the direction of the mind of 1628 constitute a remarkably complete set of problem solving heuristics.) One of the difficulties in discussing problem solving is that the concept is ill defined and understood differently by different authors. However, there is agreement that it relates to mathematical inquiry. Thus, there are a number of preliminary distinctions which can be usefully applied, for it is possible to distinguish the object or focus of inquiry, the process of inquiry, and an inquiry based pedagogy. In this section, I shall explore these different aspects of problem solving.

1.	THE OBJECT OF INQUIRY

The object or focus of inquiry is the problem itself. One definition of a problem is "a situation in which an individual or a group is called upon to perform a task for which there is no readily accessible algorithm which determines completely the method of solution...It should be added that this definition assumes a desire on the part of the individual or the group to perform the task." (Lester, 1980, page 287). This definition indicates the non-routine nature of problems as tasks which require creativity for their completion. This must be relativized to the solver, for what is routine for one person may require a novel approach from another. It is also relative to a mathematics curriculum, which specifies a set of routines and algorithms. For a problem is only non-routine if it belongs to the complement of the scope of a given set of routines. The definition also involves the imposition of a task on an individual or a group, and willingness or compliance in performing the task. The relationship between an individual (or group), the social context, their goals, and a task, is complex, and the subject of the Soviet Activity Theory which has developed from the work of Vygotsky.

2.	THE PROCESS OF INQUIRY

In contrast with the object of inquiry is the process of inquiry itself. If a problem is identified with a question, the process of mathematical problem solving is the activity of seeking a path to the answer. However this process cannot presuppose a unique answer, for a question may have multiple solutions, or none at all, and demonstrating this fact represents a higher order solution to the problem.

The formulation of the process of problem solving in terms of finding a path to a solution, utilizes a geographical metaphor of trail-blazing to a desired location. Polya elaborates this metaphor. "To solve a problem is to find a way where no way is known off-hand, to find a way out of a difficulty, to find a way around an obstacle, to attain a desired end that is not immediately attainable, by appropriate means." (Krulik and Reys, 1980, page 1). This metaphor has been represented spatially (Ernest, 1988, Figure 8). Since Nilsson (1971) it has provided a basis for some of the research on mathematical problem solving, which utilizes the notion of a 'solution space' or "state-space representation of a problem [which] is a diagrammatic illustration of the set of all states reachable from the initial state. A state is the set of all expressions that have been obtained from the initial statement of the problem up to a given moment." (Lester, 1980, page 293). The strength of the metaphor is that stages in the process can be represented, and that alternative 'routes' are integral to the representation. However a weakness of the metaphor is the implicit mathematical realism. For the set of all moves toward a solution, including those as yet uncreated, and those that never will be created, are regarded as pre-existing, awaiting discovery. Thus the metaphor implies an absolutist, even platonist view of mathematical knowledge.

3.	INQUIRY BASED PEDAGOGY

A third sense of problem solving is as a pedagogical approach to mathematics. Cockcroft (1982) endorsed such approaches under the heading of 'teaching styles', although the terminology employed does not make the distinction between modes of teaching and learning. In considering problem solving as an inquiry based pedagogy,number of further distinctions can be made. A range of different approaches can be generated by distinguishing the roles of the teacher and the learner, as in Table 1 (below). In the literature a number of terms are used across the whole range of inquiry approaches to teaching mathematics, including 'discovery', and 'investigations' (or 'investigational approach'), as well as 'problem solving'. To fix the meanings of the terms more precisely I distinguish 'guided discovery', 'problem solving' and 'problem posing' pedagogies according to the openness of the approach.

Table 1: A comparison of inquiry methods for teaching mathematics

METHOD	Teacher's role	Student's role
GUIDED DISCOVERY	Poses problem, or chooses situation with goal in mind. Guides towards solution or goal	Follows guidance
PROBLEM SOLVING	Poses problem Facilitates solving.	Finds own way to solve problem
PROBLEM POSING	Frames initial context, (or sanctions student initiative) Facilitates posing and solving	Poses own problem Attempts to solve and extend in his or her own way.

Table 1 illustrates that the shift from a guided discovery, via problem solving, to a problem posing approach involves more than mathematical processes. It also involves a shift in power with the teacher relinquishing control over the answers, over the methods applied by the learners, and over the choice of content of the lesson. The learners gain control over the solution methods they apply, and then finally over the content itself. Thus the shift to a more open inquiry orientated approach involves increased learner autonomy and self-regulation. If the overall classroom ethos is to be consistent with this shift, a necessary accompaniment is increased learner self-regulation over classroom movement, over access to learning resources, and over initiating and participating in learner-learner and learner-teacher interactions (discussion).

Evidently problem solving and posing as teaching approaches force a consideration of the social context of the classroom, and its power relations. Problem solving allows the learner to apply his/her learning creatively, in a novel situation, but the teacher still maintains a great deal of control over the content and form of the instruction. Where the problem posing approach allows the learner to pose problems and questions for investigation relatively freely, it becomes empowering and emancipatory. However the characteristics that have been specified are necessary but not sufficient for such an outcome. What is also needed, I wish to claim, is the communication of a fallibilist philosophy of mathematics in the classroom experience, which de-emphasizes the uniqueness and correctness of answers and methods.

THEORIZING THE TEACHER'S MATHEMATICS-RELATED BELIEF-SYSTEM

In this paper I want to stress the importance of the teachers mathematics-related belief-system, their perspective on mathematics and its teaching and learning, which the title of the paper refers to, with regard to the implementation of problem solving in the classroom.

In the past few years there has been a growing recognition of the importance of teachers' beliefs and conceptions (Clark and Peterson, 1986), and in particular on their mathematics-related belief systems (Thompson, 1984; Cooney, 1988; Ernest, 1989). What underlies this new emphasis is the hypothesis, by now reasonably well confirmed, that teachers' belief systems are crucial determinants of their classroom practice. Teachers' conceptions of the nature of mathematics and their personal theories about its teaching and learning are very important factors in determining how they teach mathematics in the classroom, and in particular, the place given to problem solving in their teaching. This is not to deny the vital importance of many other factors, such as teachers' attitudes and confidence, the extent of their knowledge of mathematics, and of course the social context in which they work.

Figure 1 provides a simplified model of some of the relationships involved.

Figure 1: The relationship between beliefs, and their impact on practice

__________________

| |

| PHILOSOPHY OF |

| MATHEMATICS |

| |

___________________ ____________________

| | | |

| ESPOUSED MODEL | | ESPOUSED MODEL |

| OF LEARNING | <-------> | OF TEACHING |

| MATHEMATICS | | MATHEMATICS |

| | | |

----------------------------------------------

| CONSTRAINTS AND OPPORTUNITIES PROVIDED |

| BY THE SOCIAL CONTEXT OF TEACHING |

----------------------------------------------

__________________ ___________________ _________________

| | | | | |

| ENACTED MODEL | | ENACTED MODEL | | USE OF |

| OF LEARNING | <--> | OF TEACHING | <--> | MATHEMATICS |

| MATHEMATICS | | MATHEMATICS | | TEXTS |

| | | | | |

Figure 1 represents one central belief, the teacher's conception of the nature of mathematics, as underpinning two pedagogical components of the teacher's belief system, the theories of teaching and learning mathematics. These in turn have an impact on practice in the form of the models of learning and teaching mathematics which are enacted. One use of resources, namely that of mathematics texts, is singled out as important, for texts embody an epistemological approach to mathematics, and the extent to which their presentation and sequencing of school mathematics is followed is a crucial determinant of nature of the implemented curriculum (Cooney, 1988; Goffree, 1985). In particular, texts which embody an algorithmic approach to mathematics teaching can represent a major obstacle to the introduction of a problem solving pedagogy in the classroom. The downward arrows in the figure show the direction of influence on behaviour. The content of the upper positioned belief components is reflected in the nature of the lower positioned components. Because the enacted models are interrelated, as are the espoused theories of teaching and learning, this is represented in the figure as horizontal links drawn between them.

As figure 1 illustrates, the impact of the espoused theories on practice is mediated by the constraints and opportunities provided by the social context of teaching (Clark and Peterson, 1986). The social context has a powerful influence, as a result of a number of factors including the expectations of others, such as students, their parents, fellow teachers and superiors. It also results from the institutionalised curriculum: the adopted text or curricular scheme, the system of assessment, and the overall national system of schooling. The social context leads the teacher to internalise a powerful set of constraints affecting the enactment of the models of teaching and learning mathematics. According to Lacey (1977) the influence of the social context and significant others in it leads to the 'socialisation' of teachers. If the espoused beliefs of the teacher are at odds with those of an authority figure the teacher becomes 'strategically compliant', or in more extreme cases makes an internalized adjustment to the belief-system. Thus newly prepared mathematics teachers who are eager to employ a problem solving approach often need to adapt in their classroom practices to comply with the prevailing ethos of a traditional algorithmic approach. This may be a temporary 'strategic compliance' until they are able to influence and change the ethos. It may cause conflict and tension, resulting in the withdrawal of the teacher or in internal adjustments to the teacher's belief system.

Naturally the model illustrated in Figure 1 is greatly simplified, since the relationships are far more complex and far less mechanistic than they appear. Thus, for example, although the enacted beliefs in are shown separate from the social context, they are embedded in it. Furthermore, all the beliefs and practices are part of an interactive system, and pressures at any point, such as in classroom practices, or from authority figures such as heads of departments, as we have seen, will feed-back and may influence all the other components temporarily or permanently.

TEACHERS' PHILOSOPHIES OF MATHEMATICS

A central component in the model of mathematics-related belief-systems sketched above is the teacher's personal philosophy of mathematics. This is his or her conception of the nature of mathematics as a discipline. Three groupings or types of philosophies of mathematics can be distinguished: absolutist, progressive absolutist and conceptual change or social constructivist philosophies of mathematics (Confrey, 1981; Ernest, 1991).

1. ABSOLUTISM

The most widespread philosophies of mathematics are absolutist, which view it as a body of fixed and certain, objective knowledge. Logicism, formalism and to a large extent platonism are absolutist (Benecerraf and Putnam, 1964). Most of these views are foundationist, regarding the truths of mathematics as being based on certain logical foundations. One consequence is that the structure of mathematical knowledge is seen to be hierarchical, building upwards from its logical foundations by chains of logic and definition. Another consequence is that mathematics is seen as objective, neutral, culture and value free.

2. PROGRESSIVE ABSOLUTISM

The progressive absolutist position also views mathematics as made up of certain, objective knowledge. But in addition, it accepts that new knowledge is continually being created and added by human creative activity. Confrey (1981) distinguishes this view, which underpins Popper's (1979) epistemology. It also describes the intuitionist philosophy of mathematics, which places human activity in as central in the creation of mathematics, and which argues that its logical foundations are never complete (Heyting, 1956). Thus a key feature of this view of this position is that it emphasizes the human processes of knowledge getting in mathematics, as much as their product.

Progressive absolutism can be identified as the philosophy of mathematics implicit in the ideology of progressive education (Ernest, 1991). From this perspective, the certainty of mathematical knowledge is not questioned, but the creative role of human activity in extending it is acknowledged. This is partly why this position emphasises the process and creative human aspects of school mathematics. (It also stems from the ideological model of childhood adopted by this position.)

3. FALLIBILISM

The fallibilist philosophy of mathematics is largely due to Lakatos (1976, 1978), but is also to be found in Davis and Hersh (1980) and Tymoczko (1986). This is a social view of mathematics which for all its rigour sees it at base as fallible and corrigible, the ever-changing product of social human creative activity. The claim of fallibilism is that the concepts and propositions of mathematics, as well as the logic on which its system of proof rests, are tentative human creations, rather than objective absolutes, and remain eternally open to revision and change. Fallibilism points to the history of mathematics in which theoretical structures rise and fall, like waves in the sea. The foundations of mathematics, that which is taken to be its firmest or most basic part, varies from generation to generation, and has included such things as geometry, arithmetic, logic, set theory, and category theory, in turn.

Given these three main groupings with regard to the philosophy of mathematics from the literature, there is the issue of their occurrence in teachers' of mathematics belief-systems. The claim I wish to make is:

Thesis 1

Absolutism, Progressive Absolutism and Fallibilism are the three main philosophies of mathematics held by school and college teachers of mathematics, although they may be implicit and unarticulated philosophies.

THE ASSIMILATION OF PROBLEM SOLVING TO THE TEACHER'S PERSPECTIVE

In this section I put forward the the main thesis of this paper, which is reflected in its title.

Thesis 2

Problem solving is assimilated to the teacher's mathematical perspective. In other words, what a teacher understands by problem solving in mathematics and in the teaching and learning of mathematics is largely a function of that teacher's personal philosophy of mathematics.

Relating this to the three different philosophies of mathematics distinguished above (Thesis 1), we have the following specific theses.

Thesis 2.1

A teacher with an absolutist view of mathematics will view problem solving as the carrying out of non-routine teacher imposed tasks with determinate right answers. Problem-solving is thus an activity which follows on from the transmission of mathematical content, and provides the means to apply previously learned knowledge and skills.

Associated with this perspective one might expect to find the view that there is a single best method for solving any problem. In addition, problems may be viewed as the means of motivating students through challenge or 'relevance'. The teacher's primary role is that of transmitter and communicator of knowledge, and problems are a secondary means of applying, reinforcing and motivating learning. Given that the emphasis of this perspective is on objective truth in mathematics, and hence on the determinate answer to any problem, there is a tendency to identify problem solving with the object of inquiry, the problem itself.

Thesis 2.2

A teacher with a progressive absolutist philosophy of mathematics will view problem-solving as the means to develop and employ the strategies and the processes of mathematics, and to uncover the truths and structures of mathematics.

This perspective can be expected to emphasise the role of human i.e. learner activity in the uncovering of mathematical knowledge, as well as valuing the process aspects of mathematics for themselves. Thus carefully chosen and contrived environments, contexts and problems are given to the learners to experience and explore, and they are guided to solve the problems implicitly or explicitly contained in them. Knowledge is expected to emerge from the learner's experiences, and the teacher's role is that of manager and facilitator. Given that the emphasis of this perspective is on human activity in creating or at least uncovering new mathematical knowledge, there is a tendency to identify problem solving with the process. However, strategic compliance with an ethos which is not conducive to an inquiry based classroom approach may lead to problem-solving being treated as another area of content to be added on to the mathematics curriculum. This may, in practice, result in the identification of problem solving with the object of inquiry, the problems themselves.

Thesis 2.3

A teacher with a fallibilist view of mathematics will view problem solving as the appropriate pedagogy to employ in the classroom. In particular, it is seen as a socially mediated process of problem posing and solution construction, requiring discussion for the negotiation of meanings, strategies and proofs.

In an appropriate environment, this perspective can be expected to emphasise the autonomy of the learner in choosing and posing problems for exploration and solution. These problems can be expected to arise from mathematical puzzles and relationships, from issues in the learners' cultural and social environment, as well as engagement with the kinds of tasks that will be used (usually externally imposed) for assessment and certification. The learners can be expected to question the choice and nature of classroom activity, pedagogy, classroom organisation and the teacher's authority. By challenging some of the authority structures and status quo in education, such an approach can be expected to be particularly vulnerable to social pressures and expectations. Thus there may be a tendency to adapt a problem solving rather than problem posing pedagogy, with learner control over methods, but not over content. However this can still admit substantial discussion and learner negotiation over meanings and answers. Thus strategic compliance may lead to problem posing being treated as classroom based inquiry, and to a problem-solving pedagogy. (Really powerful dissonance may lead to the type of compliance discussed under Thesis 2.2.)

The claimed links between philosophies of mathematics, theories of teaching mathematics and theories of learning mathematics are summarised in Table 2.

Table 2: Three contrasting teacher mathematics-related belief-systems

PHILOSOPHY OF MATHEMATICS	ABSOLUTIST A fixed structured body of applicable pure knowledge	PROGRESSIVE ABSOLUTIST An absolute body of knowledge uncovered by human activity	FALLIBILIST A social construction: challenged, changing and reformulated
THEORY OF TEACHING MATHEMATICS	TRANSMISSION Clear explanation Illustration Motivation	LEARNER-CENTRED Facilitation of exploration Management of inquiry Process emphasis	NEGOTIATION Discussion Negotiation Overt discussion of teacher's role
THEORY OF LEARNING MATHEMATICS	UNDERSTANDING Acquisition of meaning Application	DISCOVERY Active search for pattern Creativity Experience	EMPOWERMENT Problem-posing Discussion collaboration Development of Autonomy

EMPIRICAL EVIDENCE SUPPORTING THESIS 2

A number of studies offer partial confirmation of Thesis 2 and its sub-theses. A growing number of these are referred to in the literature, including recent surveys of research on teacher beliefs in mathematics, such as those of Thompson (1990) and Brown, Cooney and Jones (1988). The empirical studies quoted below are representative selections, chosen on the basis of their explicit treatment of conceptions of mathematics and problem solving in the classroom.

Lerman (1989) describes how open-ended problem posing and solving work has been introduced into school mathematics in Britain as a consequence of the new GCSE examination for 16 year olds with coursework assessment of 'investigations'. He reports how the intentions of this innovation have been subverted by some teachers' view that there is a unique correct outcome to these tasks, betraying an underlying absolutist philosophy of mathematics. This supports Thesis 2.1.

Marks (1987) provides a detailed case study of a single high school mathematics teacher Sandy. Sandy views problem solving, by which he means non-routine inquiry into received problems, as central to mathematics. "It is not...Problem Solving, a topic to be squeezed in somewhere between Fractions and Quadratic Equations. It is problem solving with a small "p"...woven into the fabric of mathematics rather than stamped on top" (page 2). He also states that it is central to the teaching of mathematics; high school algebra in his case. Sandy's classroom practice bears this out, with his frequent use of high-level questioning and his emphasis on heuristic methods. The case study gives every indication of that Sandy has a progressive absolutist conception of mathematics. Consequently, in view of his pedagogical beliefs and practices, it supports Thesis 2.2.

A number of further studies offer evidence supporting both sub-theses 2.1 and 2.2. Thompson (1984) provides a case study of three teachers which support these theses. In only two of the cases (Lynn and Kay) is problem-solving explicitly mentioned. Lynn has an absolutist conception of mathematics described in terms such as: "cut and dried", and "exact, predictable absolute and fixed" (page 116). She sees classroom problem solving as "recall[ing] the appropriate method or procedure...correctly apply...and obtain the right answer." (page 117) This fits perfectly with Thesis 2.1. In contrast Kay sees mathematics as certain but "continuously expanding...and undergoing changes to accommodate new developments" (page 113), a progressive absolutist view. Her conception of problem solving in the classroom and her observed practice was to "encourage the students to guess and conjecture...allow them to reason on their own rather than show them how to reach a solution or an answer" (page 114). During teaching Kay made explicit reference to the heuristic processes of mathematics. All in all, she exemplifies Thesis 2.2 very well.

Dougherty (1990) studied 11 teachers of grades 4-6. In brief, ten of these were found to have some sort of absolutist conception of mathematics (emphasising mathematics as rules and procedures, as a tool, or as step by step methods). Each of these teachers viewed problem solving as the more or less routine application of learned methods and procedures. Thus these ten teachers confirm Thesis 2.1. The remaining teacher in the sample viewed mathematics as "experiential and not a static body of knowledge" (page 123), indicative of progressive absolutism. This teacher also viewed classroom problem solving as 'solving thinking problems', that is as a creative activity. Classroom observation also showed an appreciation of divergent student thinking, and students taking responsibility for contributing meaningful explanations and presentations. This teacher is confirmatory of Thesis 2.2.

It is harder to find empirical evidence in favour of Thesis 2.3, because of the infrequent realisation of a problem-posing pedagogy in practice. Mellin-Olsen (1987) implicitly endorses a fallibilist, socially based view of mathematics and gives convincing examples of a problem-posing mathematics curriculum as realised in Norway. This fits well with Thesis 2.3, but is a carefully argued academic case with practical illustrations, rather than an empirical case study. Elsewhere I have referred to the combination of a fallibilist philosophy of mathematics with a problem-posing pedagogy as indicative of a 'public educator' ideology, and illustrated its adherents, support and implementations, from the literature (Ernest, 1991).

One example of classroom practice based on a fallibilist view of mathematics is the constructivist teaching experiments carried out by Paul Cobb, Terry Wood and Erna Yackel (see for example Cobb, 1987; Yackel, 1987). These experiments strategically complied with the expectations of the school context, such as teaching the normal Grade 2 curriculum. They adopted a small-group problem-solving pedagogical approach in which children discussed and contrasted their own solutions and methods, rather than relying on the authority of the teacher for arbitration. Thus this example is similar to the strategic compliance discussed under Thesis 2.3.

I should close this section by admitting that having pieced together some confirmatory evidence, as I have done, does not constitute a scientific test of my theses. For I have not sought to falsify them. However, what I offer is a plausible but conjectured explanation, with some theoretical and empirical support.

CONCLUSION

The key point I have tried to make in this paper is that the terms 'problem' and 'problem-solving' vary in meaning according to the perspective of the speaker. In particular, I have offered a theory based on the assumption that the teacher's personal philosophy of mathematics is the major determinant of what the teacher means by problem solving in school mathematics. I would like to suggest this as an area of research that could be fruitful in the future. What do teachers (and student teachers) mean by problem solving? How do their meanings relate to the mathematics-related belief-systems, especially their philosophies of mathematics? If my conjectures and theories are borne out by further research, there are serious implications for problem solving in schools. Namely, that if we wish to promote a problem solving pedagogy in schools, it is not enough to gain sincere agreement from teachers and others. For the concepts and understandings underpinning our agreements will vary greatly, according to the perspective of the person involved. One teacher's agreement to implement problem solving may involve word problems, another may mean the occasional non-routine exercise, a third may carry some practical application. None of these reflect a process-orientated classroom, let alone a problem-posing pedagogy.

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This paper was first published as P. Ernest (1992) ‘Problem Solving: Its Assimilation to the Teacher’s Perspective’ in J. P. da Ponte, J. F. Matos, J. M. Matos and D. Fernandes, Eds. Mathematical Problem Solving and New Information Technologies, Berlin: Springer-Verlag, 1992, 287-300.