KAZUHIKO NUNOKAWA

APPLYING LAKATOS' THEORY TO THE THEORY OF MATHEMATICAL PROBLEM SOLVING

ABSTRACT. In this paper, the relation between Lakatos' theory and issues about mathematics education - - especially issues about mathematical problem solving - - is reinvestigated by paying attention to Lakatos' methodology of a scientific research programme. By comparing the same findings about mathematical problem solving with the discussion in Lakatos' theory - - e.g. research programmes' hard cores, their negative and positive heuristics, and their goals - - we establish the correspondence between research programmes and solver's structures of a problem situation, i.e. structures given by a solver to a problem situation. After establishing this, the implications of Lakatos' theory, i.e. the nature of selection from competing programmes and the social origins of the cores of programmes, are applied to the discussion about mathematical problem-solving, with indications of the related evidence in the theory of mathematical problem solving which seems to support the application of those implications. Such an application leads to one view of mathematical problem solving, which reflects the irrational nature and social aspects of problem-solving activities, both in solving problems and in selecting better solutions.

INTRODUCTION

In recent years, Lakatos' theory has been introduced to the mathematics education community. Attention has been paid to his 'proofs and refutations' method (Lakatos, 1976) and his assertion of the quasi-empirical nature of mathematical knowledge (e.g. Lakatos, 1978b, p. 30). Many researches seem to be influenced by his theory: introduction of exploration and observation into mathematics class (Chazan, 1990; Schoenfeld, 1994); emphasizing student's creation of mathematical knowledge (Volmink, 1994); paying attention to informal mathematics (Schmittau, 1991); paying attention to fallibilism and social construction of knowledge (Ernest, 1991, 1994a); 1 paying attention to openness of problems (Zimmermann, 1991); paying attention to social aspects of mathematical proof (Balacheff, 1991; Hanna, 1991); emphasizing classroom discussion (Ball et al., 1990; Burton, 1994; Nickson, 1994). There is a viewpoint that students should adopt a view of mathematical knowledge compatible with fallibilism and the social construction of knowledge (Lampert, 1990). Although Lakatos (1976) employs the dialogue form of description, his method is the reflection of 'rationally reconstructed or 'distilled' history' (p. 5). Furthermore, according to Lakatos' theory (1978a), the rationally Educational Studies in Mathematics 31: 269-293, 1996. (~) 1996 Kluwer Academic Publishers. Printed in Belgium.

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reconstructed growth of knowledge occurs in Popper's third world (p. 92). Thus, it is not immediately possible to deduce from his theory the importance of social interactions or discussions in the construction of mathematical knowledge. This dialogue expresses the growth of mathematics as 'a living, growing organism, that acquires a certain autonomy from the [human] activity which has produced it' (Lakatos, 1976, p. 146). 2 Musgrave (1989) characterizes the Lakatosian school using the notion of 'deductive heuristics'. And Lakatos (1976) has Sigma say the following; [Y]ou will never 'induce' order from chaos. We started with long observation and lucky insight - - and failed. Now you propose to start again with longer observation and luckier insight. Even if we did arrive at a new naive conjecture - - which I doubt - - we shall only end up in the same mess. (p. 69)

Taking account of this, it becomes obscure whether Lakatos had in mind the simple induction based on the observation or not. When he mentions the quasi-empirical nature, he discusses the transmission of truth or falsity in the system; 3 The important logical flow in such quasi-empirical theories is not the transmission of truth

but rather the retransmissionof falsity -- from special theorems at the bottom ('basic statements')up towardsthe set of axioms.(p. 28) According to his definition, a theory which is quasi- empirical, in his sense, may be either empirical or non- empirical in the usual sense (p. 29). Thus, it is impossible to deduce the importance of experiments or observations in mathematics classrooms from Lakatos' assertions conceming the quasi-empirical nature of mathematical knowledge. Furthermore, concerning thought experiments, he calls Cauchy's proof for Euler's theorem a 'thought experiment'; I think that mathematicians would accept this [Cauchy's proof] as a proof, and some of them will even say that it is a beautiful one. It is certainly sweepingly convincing. But we did not prove anything in any however liberally interpreted logical sense. There are no postulates, no well- defined underlying logic, there does not seem to be any feasible way to formalize this reasoning. What we were doing was intuitively showing that the theorem was true. This is avery common way of establishing mathematicalfacts, as mathematicians now say. The Greeks called this process deikmyne and I shall call thought experiment. (Lakatos, 1978b, pp. 64-65; all italics are original ones)

This implies that we cannot simply discuss thought experiments in the same arena where we discuss usual experiments and observations. Thus his theory may not directly suggest the introduction of experiments into the mathematics classroom. As shown above, there is something vague concerning the relationship between Lakatos' theory and mathematics education, so in this paper we will reconsider it from a particular perspective. We will pay more attention to his idea of Methodology of Scientific Research Programme, which may

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be one of his most important ideas (cf. Worral, 1976), and try to apply it to mathematics education, especially the issue of mathematical problem solving. This will lead us to a new viewpoint for discussing mathematical problem solving.

'PROOFS AND REFUTATIONS' METHOD AND RESEARCH PROGRAMME

Among Lakatos' works, it may be 'Proofs and Refutations' (Lakatos, 1976) that is most directly related to mathematics. Here, we will discuss the relation between the 'proofs and refutations' method and the research programme which is used in this paper. Lakatos' (1976) Proofs and Refutations can be considered an attempt to apply Popper's (1963) idea to the field of mathematics and to describe the rational growth of mathematical knowledge (p. 5). His central idea and the essence of the 'proofs and refutations' method can be found most clearly in the appendix of his book, rather than in the main text which is written in the dialogue form. According to that appendix, the fourth stage of 'proofs and refutations method' is like the following: (4) Proof re-examined:the 'guilty lemma' to whichthe global counterexamplesis a 'local' counterexample is spotted. This guilty lemma may have previously remained 'hidden' or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - - the improved conjecture- - supersedes the primitive conjecturewith the new proof-generatedconcept as its paramountnew feature. (Lakatos, 1976, p. 127) As shown in this stage, the 'proofs and refutations' method seems to be a work of analyzing the proof associated with the statement when a counterexample of that statement appears, and making explicit the hidden assumption underlying that proof. That is, proof-analysis is an important component of the 'proofs and refutations' method. Theorems can be considered the things to be explained, and proofs explain theorems in mathematics as theories explain facts in science (Marchi, 1976). This interpretation of the 'proofs and refutations' method allows us to establish the correspondence between the 'proofs and refutations' method and Popper's (1963) 'conjectures and refutations.' We can consider that a statement at a certain moment and a proof associated with it may correspond to a conjecture in Popper (1963). They are tested by searching for counterexamples, and if some counterexamples can refute the statement or the proof, then we try to make them better by taking account of the refutation, especially taking into account hidden assumptions. If we notice the fact that the Lakatos' (1976) tittle is not Conjectures and Refutations, but Proofs and Refutations, what occurs in the process should be the dialectic

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between 'proofs' and 'refutations.' Unlike natural science, in mathematics conjectures can be 'proved' logically. This may make it difficult to apply the relation between conjectures and refutations to the domain of mathematics. By indicating that proofs are refutable, Lakatos (1976) reinterprets that relation as the relation between proofs and refutations in mathematics.4 In science, presenting bold conjectures can be a starting point of the growth of knowledge; in mathematics, presenting tentative proofs can be the starting point of the growth of knowledge, even if they contain hidden assumptions or lemmas that have not been proved at that moment (Lakatos, 1978b, p. 41). After that, proof-analysis may generate proof- generated concepts, let ideas take on different meaning (Fisher, 1966, p. 141) and elicit the growth of knowledge in mathematics4. For Lakatos, proof is a thought-experiment or a quasi- experiment 'which suggests a decomposition of the original conjectures into subconjectures or lemmas' (Lakatos, 1976, p. 9). In other words, richer proofs are always liable to some uncertainty on account of hitherto unthoughtof possibilities, and have unlimited possibilities for introducing more and more terms, more and more axioms, more and more rules in the form of new so-called 'obvious' insights (Lakatos, 1978b, pp. 61- 69). So, they cannot be complete. On the contrary, formalized ones are reliable; however, in them imagination is tied down to a poor recursive set of axioms and some scanty rules. So, it is not clear what they are reliable about. Here, the question 'what does a mathematical proof prove?' (Lakatos, 1978b, p. 61) occurs, and mathematical truth cannot be verified by proofs. This may be the nature of fallibilism of mathematics and the premise for using the 'proofs and refutations' method. When we interpret the 'proofs and refutations' method as the above and consider statements and proofs appearing during a sequence of 'proofs and refutations' to be theories, we can draw its correspondence to research programme methodology. As Lakatos (1976) shows, a mathematical problem or statement concerns many concepts and ideas in mathematics, and they are closely related to one another (Consider, for example, the notion of 'proof-generated concept.') 5 Thus, a problem or a statement constructs a certain system or a theory. Indeed, Ernest (1994a) incorporates 'informal theory' into the components of logic of mathematical discovery. And when Yuxin (1990) establishes the correspondence between Lakatos' (1976) logic of mathematical discovery and Lakatos' (1976a) methodology of a scientific research programme, he considers lemmas and a conclusion corresponding to protective belt and hard core of a theory respectively, and thinks that the proof-analysis is a heuristic method (pp. 395-396). If the monster-barring can be seen to be the state where the demarcation

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among the research communities is relatively clear (Bloor, 1982), we can find some competing research programmes in the discussion in Lakatos (1976). When taking a system consisting of problems, statements, proofs, lemmas as a programme, we can make a correspondence between a 'proofs and refutations' method and the methodology of research programmes. Thus, an application of the research programme methodology to the theory of mathematical problem solving may also lead to reconsidering the relation between the 'proofs and refutations' method and mathematics education.

SOLVER'S STRUCTURES OF A PROBLEM SITUATIONS AND RESEARCH PROGRAMME

Lakatos did not discuss the notion of methodology of a scientific research programme in the context of mathematical knowledge. There is a view, however, that he made an attempt to apply it to mathematics (Koetsier, 1991, p. 66) and there are some others' attempts to construct the discussion on the research programme methodology in the domain of mathematics (Hallett, 1979; Hawson, 1979). These suggest the possibility of an application of the discussion of the research programme to the domain of mathematics. Of course, it may be suspect whether we can apply Lakatos' theory, the discussion about communities of scientists or mathematicians, to the issues about individual solvers, or to the more elementary level. Related to this, there is a researcher who says that Lakatos' research programmes are more individual entities than Kuhn's paradigms (Giorello, 1992, p. 159). According to Confrey, (1990), one of the resources of the research on children's conceptions is the results of philosophy of science, and there, children's concept acquisition is discussed by analogy with them. Wood et aL (1993) mention the event which has happened in the classroom as the analogue of a scientific revolution. A similar analogy conceming theoryladenness can be found in Nunokawa (in press, a). Furthermore, if the term 'methodology' Lakatos uses has a sense akin to Prlya's 'heuristic' (Lakatos, 1976, p. 3), and if we remember that we have also discussed his heuristic on the individual and rather elementary level, then it is not so unreasonable to apply Lakatos' theory to the issues on such a level In fact, the example of I.Akatos (1976) can be mterpreted as a solving process of a proof problem (see also the example mentioned in Lakatos, 1976, pp. 101-102). Since Lakatos' theory reflects the continuous aspect of the growth of mathematical knowledge as well as the discontinuous one (Yuxin, 1990), so

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it can be assumed that it may be applied more easily to the discussion about mathematical problem solving reflecting its continuous aspects, than to one based on the discrete stages in the solving process. Here we shall consider the correspondence between research programmes and the discussion about mathematical problem solving based on the notion of solver's structures of a problem situation (Nunokawa, 1994a, c), which reflects the continuous aspects of the solving process. Here, a solver's structures of a problem situation are structures given by a solver to a problem situation, which consists of elements the solver recognizes in the situation, relationships he/she establishes among the elements, and senses he/she gives to the elements or the relationships. According to Lakatos (1978a), each research programme has: (i) hard core, (ii) positive heuristics. It also has a protective belt of auxiliary hypotheses to protect the hard core (pp. 110-111). Unlike the case of the naive falsification, falsification does not necessarily imply rejection, and people can keep from following it. From the view of the research programme, one programme must be abandoned only when it has been defeated by another more fertile programme (p. 112). Thus, consequent 'fruits' (Orton, 1988, p. 40) might decide whether the initial cores and heuristics should be retained or abandoned. Here we can see the quasi-empirical nature of knowledge.

Cores in mathematical problem solving According to Lakatos (1978a), the core of the research programme is a fundamental premise for the researchers participating in that programme, and it cannot be refuted. The attempt to avoid the line of thought rejecting the core may produce negative heuristics. The consideration which seems to refute the core at a glance can be removed taking advantage of auxiliary hypotheses of the protective belt. In Newton's programme, for example, the three law of dynamics and the law of gravitation are included in the core. Even if, at the earlier stages, there are some counterexamples to such a core, the programme does not need to be abandoned. Lakatos (1978a) says that sometimes these counterexamples could be turned into corroborating instances (p. 48). When we go back to the discussion of mathematical problem solving, what may occur to us first as a candidate of such cores is a problem statement which explains the content of a problem to solvers. Nunokawa (1992) shows, however, that solvers can make various interpretations of the problem statement, none of which are contradictory to that statement. This makes it difficult to simply see the problem statement as the stable core in mathematical problem solving. Furthermore, to consider a problemsolving process from the viewpoint of 'solver's structures of a problem

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Figure 1. Structure intended by teachers (this shows the case of 10 houses for simplicity)

situation' means to see it as changes of ways a solver tries to make sense of the problem situation. Thus, what will be taken as the core must be the kernel which might appear and be retained among the solver's changing structures, and give basic assumptions for constructing those structures. The existence of such a kernel in solver's structures is suggested by one student's solution mentioned in Nunokawa (1992) (This student is called 'student D' in that article). The student produced this solution to the following 'telephone line problem' (Miwa, 1991): We connect a house to another house by a direct telephone line. We put just one telephone line between each house. How many telephone line are there for twenty houses? (p. 82)

In this problem, it is intended that all of the pairs of houses are connected by a direct telephone line (Figure 1). The student D, however, drew many diagrams showing various ways of connecting, and gave different answers (numbers of the lines) to each of these diagrams (see Figure 2). Although they seem rather strange to us, they seem to have the common assumption for constructing them - - for each house, to ensure a certain route to each of the other 19 houses. This is represented most primitively in the right-middle diagram. The lines are drawn like a net to make a route from one house to another. In the other diagrams, routes are constructed more elegantly. After houses are divided into some groups, which consists

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Figure 2. Student D's solution

of one to five houses, each group is connected to the central point. A line from one house can go to the other group's house through this central point. To these student's structures based on this assumption, of course, you can make the following objection: 'A communication from one house has to go along several small lines, and this does not satisfy the given condition that we put just one line between each house.' To this objection, however, the student D can re-object like this: 'Those lines are connected to one another and make one route from a house to another, and this is o n e line as a telephone line.' This re-objection directs our attention to the definition of the notion of 'one telephone line' to protect the central idea for constructing the above structures. This suggests the possibility of protecting the central idea by adding an auxiliary hypothesis about its definition. If you object to them again saying that those ways of connecting may cause the mixedness, student D can re-object saying that there is a special exchanger at the center, or some kind of magic lines are used and so on. In this case, the central idea can be protected by operating on auxiliary hypotheses, which play a role as the protective belt.

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A

c._ Figure 3. A diagram drawn by the solver while solving the pentagon problem Nunokawa (1994c) shows another example where a solver transferred even some of the conditions given in the problem statement into auxiliary hypotheses in order to protect the central idea for constructing his structures. The solver, one graduate student, tackled the following problem; A given convex pentagon ABCDE has the property that the area of each of the five triangles ABC, BCD, CDE, DEA and EAB is unity. Show that all pentagons with the above property

have the same area, and calculate that area. Show, furthermore,that there are infinitely many non-congruentpentagonshavingthe abovearea property.(Klamkin, 1988, p. 2) In the latter half of his solving process, he had the idea that a structure of the problem situation could be based on the parallel relations between sides and diagonals. When he tried to show that there were many non-congruent pentagons according to that structure, however, he got confronted with the situation where using this idea about the parallel relations could not make two of the five triangles remain to be unit-area ones. At this moment, his refutation was directed not to the parallel relations, but to the given condition that the area of each of the five triangles is unit triangle, which is given in the problem statement. He shelved this condition tentatively, and made a decision that, after thinking about the problem taking advantage of the parallel relations, he would make an adjustment to satisfy the given

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condition. Although the central idea about the parallel relations somewhat contradicted the given condition at this moment, it was not abandoned. Rather, the auxiliary hypothesis, that a later adjustment could recover the given condition, was added to the periphery of this condition. These central ideas for creating solver's structures can be protected from objections by accommodating other parts (e.g. interpretations of some concepts), even when they are different from the standard one, as shown in the first example. Furthermore, as shown in the second example, even conditions given in the problem statement can become a object of this accommodation if it is needed to protect central ideas. From this discussion, we can assume the existence of a kind of central core concerning solver's structures of a problem situation, which is protected by a belt of auxiliary hypotheses.

Positive heuristics in mathematical problem solving Positive heuristics of the research programme are considered to show longterm order and policies of the research. Based on these policies, researchers build up their own methods. In doing that, they are allowed to ignore some of the actual data or counterexamples (Lakatos, 1978a, p. 50). First, we would consider two examples, mentioned in section 3.1. Student D who tackled the 'telephone line problem' could make the various ways of connecting houses, if she applied the policy that the houses are divided into some groups and those groups are connected to each other at the central point. Changing the grouping can lead to various ways of connecting. Indeed, all but one structure she created (see Figure 2) can be built up in this manner. This idea also suggests a way of counting the lines. The number of the lines within a group is the same as that of the houses in that group, and the number of the lines used to connect the groups to the centre is the same as the number of the groups. Furthermore, this policy can play a role to create a structure of the problem situation which fits the central idea of ensuring for each house a route to each of the other houses. The policy mentioned here gives the solver the order and direction of thinking alongside of the core discussed in the previous subsection. In the solving process mentioned in Nunokawa (1994c) there occurred 'global restructuring of the solver's structures' based on the parallel relations between the sides and diagonals of the pentagon. That is, the parallel relations, which had been first the inner characteristics of the problem situation, were turned into a basic idea for constructing his own structures of the problem situation, and his whole structure of the problem situation was constituted based on that idea. The attempt to apply these parallel relations to the broader part of the situation as possible can be found in this process.

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This implies that it was his policy for making sense of the problem situation to apply the parallel relations to the broader part of the situation as possible, and following this policy can suggest a certain order of thinking alongside the core discussed in the previous subsection. In other words, this policy also contains the order and direction of thought, and functions as a positive heuristic. As stated above, researchers construct their own models according to the positive heuristics, and are allowed to ignore some of the actual data or counterexamples. This means that, even when the researcher notices what cannot be explained within his research programme, he is allowed to proceed following the order and policy shown in that programme. 6 Such phenomena are also found in mathematical problem solving, relating to 'prospective structures' of the problem situation. The subject in Nunokawa (1993 a), who was the same person as that in Nunokawa (1994c), tackled the following problem: If A and B are fixed points on a given circle and XY is a variable diameter of the same circle, determine the locus of the point of intersection of lines AX and BY. You may assume that AB is not a diameter. (Klamkin, 1988, p. 5)

He tried to show that the locus was a circle in his solving process. In order to do that, he introduced the angle appearing at the intersection point as the new element of his structure of the problem situation, and got the idea that the problem situation basically consisted of the sliding angle, which would be the inscribed angle of the circle made by the locus. This required verifying that all the angles appearing at different intersection points were congruent. This verification was realized by relating each angle to one particular angle showing that each angle was congruent to that particular angle. The above idea about the problem situation suggested the order and direction of thought, and relating the angles to one another became a positive heuristic in this case. In doing this, however, he got confronted with the problem - - what would happen if the position of the moving points X and Y relative to the fixed point A and B was reversed? At first, he tried to apply the same idea about the sliding angle to this new case. He then noticed that applying the same idea might be impossible. The failure of this application seemed to him as a counterexample to the idea, and he began to suspect it. In fact, the solver said, 'Should I do it more generally?' at this moment, and this utterance implied the negation of his previous effort. After all, though, he did not give it up. He shifted to reconsidering the idea to relate it to the new case. Consequently, he showed that adding the angles at the intersection points in the new case to the angles in the past case made always 180 ~ He deduced from this

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T' Figure 4. A diagramdrawnby the solverwhile solving the circle problem

fact that, in the new case, all the angles at the intersection points were congruent. That new case was naturally found during this process, but it was at first assumed implicity that the new case could be treated just the same way as the past case. In other words, it was assumed that the problem situation was homogeneous which was unaffected by the positions of the points. His structure of the problem situation at this stage was a simple and prospective one, in a sense that it contained the implicit assumption which made the problem situation simpler (cf. Nunokawa, 1993a). Although the new case appeared afterwards as a counterexample to the central idea, it was finally turned into a corroborative example in the sense that it got placed within the framework based on the central idea through the attempt based on that idea. The implicit assumption, which supported the simple prospective structure, facilitated his solving process by keeping him tentatively from the complex procedure of identifying cases and checking each case. This might support implementing the positive heuristic indicated by the central idea. Indeed, Mason (1985) says the following concerning one conjecture:

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I have recorded it as a conjecture because I do not want to spend time now justifying it. It feels right and, since it is written down clearly, I will check it in my Review [phase]. My mathematical experience reinforces my observation with a strong sense of its verifiability and I am confident enough to proceed without diverting my concentration from the main thread of the investigation. (p. 71)

The ambiguity contained in a prospective structure seems related to Agassi's (1980) view that Lakatos (1976) succeeded in showing the ambiguity contained in a mathematicians' logical evidence. In fact, the process of elaborating prospective structures by making hidden assumptions explicit and taking account of them, is similar to the process of 'Proofs and Refutations' method which makes hidden assumptions in proofs explicit by proof-analysis. Turning new cases into one's own framework, in any case, may remind us of Lakatos' (1976) 'monster-adjustment,' where new cases and possibilities can appear because of ambiguities of our arguments. These seem related furthermore to Lakatos' (1978a) statement that, even if new cases or possibilities seem counterexamples at first, we have no need to give up our ideas and we can sometimes expect to 'digest anomalies and even turn them into positive evidence' (p. 4) and that 'the first versions may even 'apply' only to non-existing 'ideal' cases' (p. 65). As shown in the above, in the discussion about mathematical problem solving we can find the correspondents of positive heuristics and the treatment of counterexamples in the research programme.

Goals in mathematical problem solving Gillies (1992) mentions positive heuristics and goals as the essential components of research programmes. In Frege's research programme, according to him, its goal is to show that arithmetic can be reduced to logic, and its positive heuristic is to try to prove arithmetic theorems using only general logical rules. As shown in this example, goals of research programmes are considered to be what they would want to achieve by following their positive heuristics. Nunokawa's (1993c) analysis implies a hint for considering a goal of the solving process to the 'pentagon problem' discussed in Nunokawa (1994c). Global restructuring based on the parallel relations occurred in that solving process, as mentioned above. By comparing that process with the other process which was not accompanied by global restructuring, he finds the following characteristic of that process: in the former process, to change the solver's structures of the problem situation, the solver tried to give new sense to the elements of the literally created structure of the situation, rather than to introduce new elements which were not stated in

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the problem statement, and such an attempt seemed to lead to the global restructuring. The positive heuristic of that solving process was applying the parallel relations to as broad parts of the problem situation as possible. Nunokawa (1993c) shows, however, that making parallel relations in the situation by adding new elements was not intended in the solving activities following this positive heuristics. What was intended here was to make sense of the initial literally-created structure of the problem situation by using the parallel relations. In other words, applying the parallel relations was implemented with a certain kind of intention. This intention can be considered a goal of this solving process which fits its positive heuristic. And such intentions relating to positive heuristics, instead of goals of the problems themselves, can be taken to correspond to goals as a component of research programmes, when we take a viewpoint based on solver's structures of a problem situation.

Mathematical problem solving and research programmes To sum up this section's discussion, we can find the correspondents in mathematical problem solving to each of the basic components of research programmes. We can conclude that the process in which a solver gives his own structures to the problem situation and tries to make sense of it by using his knowledge, which is problem solving, may be one (even small-scale) research programme, containing its hard core, protective belt of auxiliary hypothese, positive heuristics, and goals relating to those heuristics. Such a correspondence, in turn, suggests that the solving activities mentioned in this section may be non-capricious ones, but activities reflecting important aspects of mathematical problem solving.

SELECTING OUT OF COMPETING SOLUTIONS

Employing the viewpoint of research programmes, we can assume that a programme constituting a progressive problemshift is successful and one showing degenerating problemshift is unsuccessful (Lakatos, 1978a, pp. 33-34). When we consider mathematical problem solving on the analogy of research programmes, the issue of selecting 'good' solutions should reflect a certain kind of progressive problemshifts concerning solver's structures or solving processes. It is on this level which Lakatos (1978a) discusses the necessity of the empirical methods (Smart, 1989). That issue is related to whether a research programme can help researchers produce new facts. In the domain of

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mathematics, however, such fertility is not limited to the form of foreseeing phenomena and verifying them. It can take the form of presenting new problems and does not necessarily need empirical verification (Howson, 1979, p. 263). Here we can mention at least two types of problems produced from solver's structures or solving processes of the original problem. The first are problems which can be produced in extending the original one or making new problems based on the original one. Fujii (1992) shows an important finding relating to this type. He explores a relation between the types of problems newly made by students based on the original one, and whether students could notice a certain characteristic of the original problem situation. And he finds that 'there was the relation between students' understanding of the original problem and the types of the newly-made problems' (p. 78; my translation). This implies that what kind of structures a solver constructs to the original problem situation would affect what kind of problems he will make in extending the original one. In the case of above mentioned 'telephone line' problem student D tackled, new problems can easily be made by changing the number of houses. Solvers can foresee solutions or lines of solving processes to such new problems: 'I would be able to solve it like this because the old one could be solved like that.' The degree to which such foreseeing would be possible and foreseen solutions would be verified afterwards may be an index of a progressive problemshift. Student D's structures of the problem situation will be able to help a solver foresee solutions to the new problems. To the same problem, it is possible to construct other structures which don't contradict the statement of 'telephone line' problem. For example, a structure in which 20 houses are divided into 10 pairs and only the two houses within the same pair are connected to each other can be noncontradictory to the statement under a certain interpretation of it (Nunokawa, 1992, p. 2 6 ; such variety reminds us of many interpretations of 'polyhedora' in Lakatos (1976)). When the solver foresees a solution to a new problem treating the odd number of houses, he will be confronted with the problem of pairing of houses and forced to introduce ad hoc assumptions (e.g. 'This case should be treated differently'). 7 The solution based on the latter structure of the problem situation may show the degenerating problemshift with respect to this point. The second instance is where problems appear as new problems while articulating the ideas used in the solution of the original problem. Let us consider as an example the seventh grade classroom's treating 'telephone line' problems reported in Miwa (1990). In this lesson, some students

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Figure 5. Student B's chain-like connecting

presented the solution in which the number of lines was obtained by 19 + 18 +...+ 2 +1. Through the discussion of this, a new problem about the sum of the arithmetic sequence whose common difference is 1 and the first term is 1 was settled, this problem was interesting and challenging enough for the seventh graders and elicited new discussion among them (pp. 134-162). Solutions to this new problem would affect the degree to which the 'telephone line' problem can be extended, and solutions to those extension can be foreseen based on the solution using the expression 19+ 18 +...+2+ 1. For example, if a student finds a simple procedure for obtaining the sum of such sequence which can be easily implemented even w h e n the number of terms is rather large, then he can foresee the solutions to the 'telephone line' problem including a very large number of houses. If a student finds it difficult to implement his own procedure for the large number of terms, then he may be forced to seek the other solutions different from the original problem. In such a sense, the problem about the sum of arithmetic sequences is closely related to the 'telephone line' problem. And generating such an interesting and challenging problem can be considered an index of the progressive problemshift of the solver's structure which supported the solution based on the expression 19 + 18 +...+ 2 +1. On the other hand, the chain-like way of connecting lines (Figure 5) is not contrary to the problem statement under a certain interpretation (see the discussion about the student B's solution in Nunokawa (1992). The solution supported by this structure of the problem situation may lead to a so-called 'planting problem' as a new problem. In one Japanese textbook, this type of problem is treated at the third grade. When this type of problem can be considered too easy for the sixth graders, to which the above- mentioned student B belonged, generating such a 'planting problem' cannot be seen as an index of progressive problemshift. It may be possible that, based on this fact, the structure showing the chain-like way of connecting and the solutions supported by such a structure are not employed in the sixth-grade class. When we apply the discussion about research programmes to the issue about mathematical problem solving, it may be possible for us to select

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some solvers' structures and the solution supported by them among the others by resorting to the standards with respect to what they can newly produce. Such a selection requires us to assess solvers' structures and their solutions referring not only to those structures and solutions themselves, but also to the extended line of thought supported by a common basic idea, including new problems generated during or after the solving process. This is an implication of the Lakatos' theory. It often happens that what seems critical for the flow of research can be recognized as critical only afterwards (Fisher, 1966; Lakatos, 1978a).

A SOCIAL ASPECT OF MATHEMATICAL PROBLEM SOLVING

Although it may be a problem to make conclusion about the importance of social interactions in mathematics classrooms directly from the dialogic form of description in Lakatos (1976), there are some researchers who would find the other forms of social aspects in the Lakatos' theory. Bloor (1982), for example, thinks that various ways of responding to refutations in Lakatos (1976) are influenced by the social character of the research communities. 8 In doing so, there is no 'correct' response. 'It isn't a case of 'discovering' the right status' but 'a case of deciding' (Bloor, 1994, p. 26). This decision is influenced by '[t]he existing state of culture, the context around the new result, and the interest that inform our practices' (idem, p. 27). Dunmore (1992) considers the core of a research programme to be 'the community's shared background' (p. 224) and on the metamathematics level. Referring to Kitcher's (1984) notion of 'mathematical practice', language to be used, problems to be selected as important, accepted reasonings, and accepted views on mathematics are included in the components on such a level different from the level of mathematics itself. If we recall that Kitcher (1984) considers that they can change historically and socially, we can assume that the core may reflect the community's social character. Indeed Dunmore (1992) says that there occur revolutions in mathematics on this level. The core of a research programme contains various explicit or implicit assumptions about doing research, including what were mentioned in Section 3.1, and the above discussion implies that some of them have a social nature or social origins. 9 Now our question is whether such social aspects of doing research can be also found in mathematical problem solving. One type of appeamce of social aspects in mathematical problem solving is implicit restrictions on ways of understanding and responding to problems which are not given in the problem statements. Understanding of problems or lines of thought which can be seen to be viable under a

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certain interpretation of the problem statements is sometimes rejected. In such cases, it is required to solve the problems 'mathematically', and that understanding of those lines of thought are rejected because they are considered inappropriate under the 'mathematical' value, although they are not contradictory to the problem statements (Nunokawa, 1992; Reusser, 1988). Shigematsu and Sowder (1994) report on an American girl who considers use of diagrams in problem solving to be a kind of 'cheating' and not to be legitimized. This restriction on her lines of thought can be seen to reflect the character of the US mathematics classrooms (p. 544) - - that is, it is one of their social aspects. Furthermore, according to DeCorte and Verschaffel (1989), similar phenomena can be found in ways of responding to problems. Even the same problem may be answered differently depending on the context in which it is presented; whether it is presented in the everyday-life context, or in the mathematical-problem-solving context. These suggest the existence of the taken-shared (or expected to be taken-shared) recognition about solving problems 'mathematically' in the communities of learning peers and teachers, and it can, at least to some extent, determine the ways of understanding and responding to problems or the appropriate lines of thought. Another instance of social aspects is the solver's control over his own solving activities by taking account of expected levels of problem difficulty (Nunokawa, 1993b). That expected levels of problems' difficulty influence solver's solving process is shown experimentally (Reusser, 1988). While the information about expected difficulty was externally presented by the experimenter in Reusser's (1988) study, the control based on it occurred internally in the case of Nunokawa's (1993b) subject. The latter implies that the solver might feel the expectation about the appropriate level of difficulty in the 'mathematical' problem solving context he was placed in, and tried to adapt his solution to that expectation. Social aspects of mathematical problem solving can also be found in the selections of solutions. Whether the solution is logically correct is not the only criterion for selection. As Nunokawa (1992) shows, the solutions that may be generally considered incorrect can be logically coherent under a certain construal of the words or concepts in the problem statement, especially when we see them in non-mathematical contexts. If extra production of the solutions or solver's structures of the problem situation is employed as a criterion for selecting solutions, it becomes an important factor what kind of problems or facts can be accepted as interesting ones. The solutions or solver's structures Which have a potentiality to produce 'mathematically interesting' further problems or facts are selected. Referring to Kitcher's (1984) idea that accepted problems can change historically and socially,

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the above- mentioned way of selection may be an instance of social aspects of mathematical problem solving. Indeed, social factors can be also found in selecting research programmes. Coherence of the theories in a programme is necessary for it to become a candidate for selection, but it is not the final criterion (Anapolitanos, 1989). Some kinds of social value affect that selection. They can make a case for attention to the aesthetic aspect, to the community's favor of continuity of research, fruitfulness of theory, all of which the 'Proofs and Refutations' method lacks (Anapolitanos, 1989). Mori (1990) says that Lakatos' idea of research programmes did not satisfy Popper and was beyond his expectation. And Feyerabend (1975), a well-known anarchist, mentions the similarity between his anarchism and Lakatos' methodology of research programme (p. 198). There are views that Kuhn's influence can be found in Lakatos' theory (Yuxin, 1990), and that dialogue structures can be found among competing research programmes (Koutougos, 1989). These researchers' statement shows that rational aspects are weakened in the methodology of research programmes and social factors play a role in selecting programmes even in Lakatos' theory. 10 Selection of one programme means selection of a particular core, positive heuristics, and background assumptions supporting them. So, such core or assumptions are socially chosen (this fits with Bloor's (1991) statement; see note 9). In turn, these socially-selected cores or assumptions influence research activities. If we take account of the analogy between research programmes and mathematical problem solving, we can expect that what solutions are selected may influence the solver's later creation of their structures of problem situations. This suggests the following: (i) we should explore social origins of the components (correspondents of cores, heuristics, goals) of mathematical problem solving discussed in the third section; (ii) we should explore the selection of solutions in mathematics lessons as the process which may construct those components socially, and in turn how that selection is influenced by 'mathematical' value existing outside of the classroom. Concerning these issues, the research on the social origins of what is considered mathematical in the classroom (Kumagai, 1994; Wood et al., 1993) is directly related to mathematical problemsolving research.

CODA

In this paper, we have tried to establish a correspondence between Lakatos' methodology of the research programme and the discussion about mathematical problem solving. This gives us a view of problem solving: A

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solver goes on making sense of the problem situation as far as possible by following his central ideas, and tentative solutions can be a starting point of the solving process even if they contain hidden assumptions: A solver does not need to give up the central ideas at once even if, during the solving process, he finds something which seems not to fit the ideas, and he is allowed to put it aside temporarily and come back to it afterwards: A solver should produce further problems or facts from his own solutions and his structures of the problem situation, and demonstrate the fruitfulness of his 'programme.' This view can become easily compatible with the view which takes problem solving as mathematical modelling (Nunokawa, in press, b). Selection should be done by referring to the fruitfulness in order to avoid the chaotic state of various solutions. Although Lakatos considers that this selection can be implemented rationally, succeeding researchers find some social factors involved. What mathematics teachers use to select solutions in mathematics lessons is also a kind of social value, i.e. mathematical value. As other societies' values are not taken for granted by us, mathematical value may not seem natural for students, who will become members of (the school's) mathematics community from now. So, in some cases, we as mathematics teachers should persuade students to come with us by showing the fruitfulness of our approach. 11 During this persuasion, we should expect that, when students solve problems, they can create their own structures of problem situations or their own lines of thought which have mathematically competitive power. Lakatos' theory also suggests the importance of the fruitfulness and the persuasion based on it.

NOTES * This research was partially supported by Grant-in-Aid for ScientificResearch (No. 05 780 149), Ministry of Education, Science and Culture of Japan. Tymoczko(1986) uses the term 'quasi-empiricism' followingLakatos, and he says that 'quasi- empiricism views the constructions of mathematicians more as social products' (p. xvi). 2 In fact, Confrey (1991) cites the same part of Lakatos (1976), and points out Lakatos' (1976) incompleteness in linking between mathematics and human activity (p. 116). And Ernest (1994b) and Welch (1985) also mention the less-social nature of Lakatos' theory. 3 This standpoint differs from that of Putnam (1986) that quasi-empirical methods are analogous to the methods of the physical sciences. The latter seems to match better with the popular interpretation, although it takes account of the importance of fertileness of mathematical consequences.

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4 According to Zahler (1976), 'many published mathematical articles undoubtly contain serious undiscovered errors.' (p. 98) 5 Crowe (1988) says, '[M]athematical assertions are usually not tested in isolation but in conjunction with other elements in the system.' This is mentioned in relation to the Duhem-Quine Thesis, which is also related to the discussion in the third section of this paper. 6 In some cases, counterexamples can be turned to corroborative examples. Thus, ignoring counterexamples may mean not only persisting in one programme, but also postponing the attempt to convert the counterexamples into corroborative ones and explain them within his programme. 7 Balacheff (1991) reports that students introduced such assumptions or excluded the cases of odd numbers as exceptions when they solved the problem of counting the number of diagonals of polygons. In a closed system with a leadership whose authority derives from the discovery of a theorem, counterexamples become the basis for a revolution. This social character tends to provoke monster-barring. In a large and diverse system, many interests have been accommodated and complex relations have emerged between the different segments of society. This character tends to provoke monster-adjustment or exception-barring (Bloor, 1982, pp. 199-200). 9 Bloor, (1991) says, 'At any given time mathematics proceeds by, and is grounded in, what its practitioners take for granted. There are no foundations other than social ones' (p. 153). m According to Urbach (1989), Lakatos wished all occurrences of the word 'rational' to be expunged from his writings in 1973 (p. 400). Urbach (1989) himself considers that a subjective element is ineliminable in a method of evaluating the hard core hypotheses of a research programme (p. 411). 11 This is rather different from rejecting the students' solutions by showing their deficiencies. The latter approach has a kind of difficulty that students do not feel conflict even when those deficiencies are shown (Vinner, 1990). This difficulty is, however, compatible with Lakatos' theory.

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Department of Mathematics Joetsu University of Education Joetsu, Niigata, 943 Japan