Applying Lakatos' theory to the theory of mathematical problem solving

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.

[1]  Herbert Breger.  Tacit Knowledge in Mathematical Theory , 1992 .

[2]  Kazuhiko Nunokawa,et al.  Problem solving as modelling: a case of augmented‐quotient division problem , 1995 .

[3]  Tatsuro Miwa.  A Comparative Study on Classroom Practices of Mathematical Problem Solving Between Japan and the U.S. , 1991 .

[4]  Peter Urbach,et al.  The Bayesian Alternative to the Methodology of Scientific Research Programmes , 1989 .

[5]  Imre Lakatos,et al.  The methodology of scientific research programmes: Contents , 1978 .

[6]  H. Putnam.  What is mathematical truth , 1975 .

[7]  P. Feyerabend,et al.  Against Method: Outline of an Anarchistic Theory of Knowledge , 1976 .

[8]  Philip Kitcher,et al.  The nature of mathematical knowledge , 1985 .

[9]  Larry Sowder,et al.  Drawings for Story Problems: Practices in Japan and the United States. , 1994 .

[10]  D. Bloor,et al.  Knowledge and Social Imagery , 1977 .

[11]  C. S. Fisher,et al.  The death of a mathematical theory: a study in the sociology of knowledge , 1966 .

[12]  Michael Hallett.  Towards a Theory of Mathematical Research Programmes (II) , 1979, The British Journal for the Philosophy of Science.

[13]  J. Confrey.  A Review of the Research on Student Conceptions in Mathematics, Science, and Programming , 1990 .

[14]  Daniel Chazan,et al.  Quasi-empirical views of mathematics and mathematics teaching , 1990 .

[15]  Zheng Yuxin,et al.  From the Logic of Mathematical Discovery to the Methodology of Scientific Research Programmes* , 1990, The British Journal for the Philosophy of Science.

[16]  I. Lakatos.  PROOFS AND REFUTATIONS (I)*† , 1963, The British Journal for the Philosophy of Science.

[17]  M. Crowe.  Ten misconceptions about mathematics and its history , 1988 .

[18]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[19]  Aris Koutougos,et al.  Research Programmes and Paradigms as Dialogue Structures , 1989 .

[20]  D. A. Anapolitanos,et al.  Proofs and Refutations: A Reassessment , 1989 .

[21]  Magdalene Lampert,et al.  When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching , 1990 .

[22]  Imre Lakatos,et al.  Mathematics, science and epistemology: Indexes , 1978 .

[23]  Murray S. Klamkin.  USA Mathematical Olympiads 1972-1986 , 1988 .

[24]  I. Lakatos,et al.  Proofs and Refutations: Frontmatter , 1976 .

[25]  P. Feyerabend.  Against Method: Outline of an Anarchistic Theory of Knowledge , 1976 .

[26]  Methodology in Non-Empirical Disciplines , 1979 .

[27]  K. Nunokawa.  Solver's structures of a problem situation and their global restructing , 1994 .

[28]  M. Nickson.  The Culture of the Mathematics Classroom: An Unknown Quantity? , 1994 .

[29]  Gad Yair,et al.  Tutorial relations — The dynamics of social contract and personal trust , 1989 .

[30]  Jere Confrey,et al.  Learning to Listen: A Student’s Understanding of Powers of Ten , 1991 .

[31]  J. Worrall.  Imre Lakatos (1922–1974): Philosopher of Mathematics and Philosopher of Science , 1976 .

[32]  Kazuhiko Nunokawa,et al.  Improving Diagrams Gradually: One Approach to Using Diagrams in Problem Solving. , 1994 .

[33]  E. Corte,et al.  Teaching word problems in the primary school: what research has to say to the teacher , 1989 .

[34]  Leone Burton,et al.  Whose Culture Includes Mathematics , 1994 .

[35]  J. Smart.  Methodology and Ontology , 1989 .

[36]  K. Popper,et al.  Conjectures and refutations;: The growth of scientific knowledge , 1972 .

[37]  T. Wood.  Rethinking Elementary School Mathematics: Insights and Issues , 1993 .

[38]  R Zahler.  Errors in mathematical proofs. , 1976, Science.

[39]  J. Volmink.  Mathematics by All , 1994 .

[40]  R. Orton.  Two Theories of "Theory" in Mathematics Education: Using Kuhn and Lakatos to Examine Four Foundational Issues. , 1988 .

[41]  Imre Lakatos,et al.  The Methodology of Scientific Research Programmes , 1978 .

[42]  K. Popper,et al.  Conjectures and refutations;: The growth of scientific knowledge , 1972 .

[43]  T. Koetsier.  Lakatos' Philosophy of Mathematics: A Historical Approach , 1991 .

[44]  Kurt Reusser.  Problem solving beyond the logic of things: contextual effects on understanding and solving word problems , 1988 .

[45]  Nicolas Balacheff.  Treatment of Refutations: Aspects of the Complexity of a Constructivist Approach to Mathematics Learning , 1991 .

[46]  Jean Schmittau.  Mathematics Education in the 1990s: Can It Afford to Ignore Its Historical and Philosophical Foundations?. , 1991 .