Commentary on Theories of Mathematics Education

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Preface to Part II Ernest’s Reflections on Theories of Learning Lakatos-Hersh-Ernest: Triangulating Philosophy-Mathematics-Mathematics Education Bharath Sriraman and Nick Haverhals Philosophy has always maintained an intricate relationship with mathematics. It was also implicitly accepted that the philosophical positions of a bearer influence his/her view on mathematics and its teaching (Törner and Sriraman 2007), which leads us into the domain of beliefs theory. However the centrality of philosophy and its intricate relationship to theory development in mathematics education only came about two decades ago when Paul Ernest and Hans-Georg Steiner (1987) each independently became aware of the importance of epistemological issues that impact the teaching and learning of mathematics. Sierpinska and Lerman (1996) state: Epistemology as a branch of philosophy concerned with scientific knowledge poses fundamental questions such as: ‘What are the origins of scientific knowledge?’ (Empirical? Rational?); ‘What are the criteria of validity of scientific knowledge?’ (Able to predict actual events? Logical consistency?); ‘What is the character of the process of development of scientific knowledge?’ (Accumulation and continuity? Periods of normal science, scientific revolutions and discontinuity? Shifts and refinement in scientific programs?). The question of what is mathematics, for teaching and learning considerations brings into relevance the need to develop a philosophy of mathematics compatible with mathematics education. In order to answer this question for mathematics education, several theorists have played a role directly or indirectly. In this preface, we briefly summarize the role that Lakatos, Hersh and Ernest have played. Reuben Hersh began to popularize Lakatos’ book Proofs and Refutations to the mathematics community in a paper titled, “Introducing Imre Lakatos” (1978) and called for the community of mathematicians to take an interest in re-examining the philosophy of mathematics. Hersh (1979) defined the “philosophy of mathematics” as the working philosophy of the professional mathematician, the philosophical attitude to his work that is assumed by the researcher, teacher, or user of mathematics and especially the B. Sriraman () · N. Haverhals Department of Mathematical Sciences, The University of Montana, Missoula, USA e-mail: sriramanb@mso.umt.edu N. Haverhals e-mail: nicolas.haverhals@umontana.edu B. Sriraman, L. English (eds.), Theories of Mathematics Education, Advances in Mathematics Education, DOI 10.1007/978-3-642-00742-2_3, © Springer-Verlag Berlin Heidelberg 2010 35
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Advances in Mathematics Education E
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Prof. Bharath Sriraman Dept of Math
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vi Series Preface a regular topic s
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Introduction A Synthesized and Forw
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Introduction xi Campbell, J. (Ed.)
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xiv Contents Reflections on Theorie
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xvi Contents Lack of Cumulativeness
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xviii Contents Part VIII Problem So
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xx Contents Sixth:NewQuestionsandNe
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xxii Contents Part XIV Feminist Ped
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xxiv Contents Concordance of Theori
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Contributors Miriam Amit Department
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Contributors xxix Scott Makeig Swar
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Preface to Part I Jeremy Kilpatrick
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Preface to Part I 5 I concluded,
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Preface to Part III Theoretical, Co
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On the Theoretical, Conceptual, and
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88 G. Harel that the ultimate purpo
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90 G. Harel enquiry system, one mus
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92 G. Harel (a) scholars in a broad
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94 G. Harel Harel, G. (2008). What
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98 N. Presmeg interview—a dramati
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100 S. Lerman A Language of Researc
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102 S. Lerman cognition. Adler and
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104 S. Lerman social linguistics, c
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106 S. Lerman tailed year by year t
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108 S. Lerman can be applied, but p
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122 L.D. English researchers draw o
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124 R. Lesh and B. Sriraman Arguabl
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126 R. Lesh and B. Sriraman new use
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128 R. Lesh and B. Sriraman many sc
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130 R. Lesh and B. Sriraman (philos
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132 R. Lesh and B. Sriraman less gr
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134 R. Lesh and B. Sriraman Fig. 1
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138 R. Lesh and B. Sriraman spite o
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140 R. Lesh and B. Sriraman assumpt
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142 R. Lesh and B. Sriraman solving
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146 R. Lesh and B. Sriraman Coray,
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148 M. Amit One reason why mathemat
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160 D.N. Boote generative metaphors
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162 D.N. Boote of knowledge, the hi
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164 D.N. Boote This suggests anothe
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166 D.N. Boote closest to describin
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168 D.N. Boote Sandoval, W. A., & B
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172 S.J. Hegedus learning environme
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174 J. Pegg and D. Tall growth of k
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176 J. Pegg and D. Tall Local Cycle
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178 J. Pegg and D. Tall imaging of
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180 J. Pegg and D. Tall This cycle
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182 J. Pegg and D. Tall Table 3 Loc
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184 J. Pegg and D. Tall The questio
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186 J. Pegg and D. Tall sessment in
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188 J. Pegg and D. Tall Local cycle
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190 J. Pegg and D. Tall The second
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192 J. Pegg and D. Tall Pegg, J. (1
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194 B. Dahl time and the existence
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196 B. Dahl Pegg and Tall have show
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198 B. Dahl Table 3 Overview of the
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200 B. Dahl 2005, 2006b; Hansen 200
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202 B. Dahl Can we reconcile this d
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204 B. Dahl Unified Theory and Trut
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206 B. Dahl mathematics education?
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208 B. Dahl Marshall, I., & Zohar,
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212 S.J. Hegedus The authors then b
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214 L. Moreno-Armella and B. Sriram
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234 G.A. Goldin In connection with
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236 G.A. Goldin of particular mathe
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Problem Solving Heuristics, Affect,
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Preface to Part IX Jinfa Cai As the
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Problem Solving for the 21 st Centu
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298 A. Zollman Even though all nati
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300 A. Zollman vancing problem-solv
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Preface to Part X Layne Kalbfleisch
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Preface to Part X 307 Barth, H., La
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310 S.R. Campbell ignated as “the
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312 S.R. Campbell typically loath t
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314 S.R. Campbell Various manners i
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316 S.R. Campbell educational pract
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318 S.R. Campbell and so on, have t
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320 S.R. Campbell Fig. 1 Interdisci
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322 S.R. Campbell they can work jus
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324 S.R. Campbell however, while ce
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326 S.R. Campbell mentation (such a
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328 S.R. Campbell Campbell, S. R. (
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330 S.R. Campbell Lee, K. (2003). N
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342 L. Moreno-Armella complex consi
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344 G. Harel teaching, and learning
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346 G. Harel development spanning a
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348 G. Harel Offer two values for t
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350 G. Harel We are looking for pos
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352 G. Harel 2. We found that the a
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354 G. Harel stance on the nature o
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356 G. Harel As I engaged deeply in
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358 G. Harel Through such repeated
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360 G. Harel reasoning that reinfor
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362 G. Harel (Fragment 13). Still f
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364 G. Harel Fig. 9 Quantitative ap
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366 G. Harel References Anderson, J
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Appreciating Scientificity in Quali
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Preface to Part XIII Studying Goals
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Preface to Part XIII 399 Davis, R.
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402 G. Törner et al. analysis. In
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404 G. Törner et al. also focused
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406 G. Törner et al. structure of
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408 G. Törner et al. turn in the t
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410 G. Törner et al. some open tas
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412 G. Törner et al. This belief n
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414 G. Törner et al. Bundle (C): B
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416 G. Törner et al. instance char
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418 G. Törner et al. Ball, D. L. (
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420 G. Törner et al. Törner, G. (
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422 D. Tirosh and P. Tsamir The tea
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424 D. Tirosh and P. Tsamir that sh
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426 D. Tirosh and P. Tsamir our com
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Preface to Part XIV Feminist Pedago
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Preface to Part XIV 433 more import
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436 J.E. Jacobs of the school curri
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438 J.E. Jacobs Chart 1: Women’s
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440 J.E. Jacobs through induction w
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442 J.E. Jacobs ways of knowing. Th
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444 J.E. Jacobs impeccable reasonin
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446 J.E. Jacobs Hanson, K. (1992).
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448 G.C. Leder Different Faces of F
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450 G.C. Leder sources, Kaiser and
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452 G.C. Leder have witnessed vario
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454 G.C. Leder Klein, S. S. (Ed.) (
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456 S. Bulut et al. or class, resea
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458 S. Bulut et al. Table 2 Higher
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460 S. Bulut et al. Table 4 Studies
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462 S. Bulut et al. Table 8 Number
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464 S. Bulut et al. Conclusion Alth
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466 S. Bulut et al. Ubuz, B. (1999)
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468 G. Pálsdóttir and B. Sriraman
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Preface to Part XV Tommy Dreyfus As
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Preface to Part XV 481 one’s own
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484 A. Bikner-Ahsbahs and S. Predig
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508 F. Arzarello 3. Strategies for
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510 F. Arzarello Since mathematics
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512 F. Arzarello different lenses i
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516 S. Prediger and A. Bikner-Ahsba
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On Networking Strategies and Theori
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538 U. Gellert open towards the com
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540 U. Gellert Explicitness in Math
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542 U. Gellert no attempt of checki
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544 U. Gellert race to 20 is not ma
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546 U. Gellert It is quite clear fr
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548 U. Gellert classroom practice,
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550 U. Gellert Gravemeijer, K. (199
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552 U. Gellert It is interesting th
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554 U. Gellert research phenomenon
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556 T. Wedege on action—when work
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558 T. Wedege From Bricoleur to Ref
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Preface to Part XVII The Importance
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The Importance of Complex Systems i
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Complexity Theories and Theories of
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594 B. Sriraman of the eminent arti
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596 N. Sinclair collections of sign
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598 N. Sinclair Fig. 1 The frozen g
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600 N. Sinclair gesture. While it i
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602 N. Sinclair that the values of
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604 N. Sinclair students in France.
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606 N. Sinclair was responsible for
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608 N. Sinclair and elation felt by
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610 N. Sinclair in taking Walkerdin
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612 N. Sinclair Núñez, R. (2006).
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614 D. Pimm In this chapter, I will
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616 D. Pimm fact that the ancient E
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618 D. Pimm Mathematics tries to de
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622 B. Sriraman et al. mathematics
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624 B. Sriraman et al. politics is
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626 B. Sriraman et al. • a homoge
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628 B. Sriraman et al. and promotio
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630 B. Sriraman et al. “abstract
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632 B. Sriraman et al. carried out
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634 B. Sriraman et al. can then be
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636 B. Sriraman et al. References A
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638 B. Sriraman et al. Sriraman, B.
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640 K. Yasukawa This opens up the q
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642 K. Yasukawa treatments that the
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Author Index A Abelson, R., 413, 41
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Author Index 647 Bryant, P., 268, 2
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Author Index 649 Elliott, P., 279,
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Author Index 651 Henn, H.-W., 319,
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Author Index 653 Lenoir, T., 9, 30
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Author Index 655 Pappas, S., 281, 2
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Author Index 657 Sica, A., 164, 167
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Author Index 659 Watters, J. J., 27
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662 Subject Index 234, 316, 324, 33
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664 Subject Index 200, 203, 205-206
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666 Subject Index 299, 300, 341, 38
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668 Subject Index learning environm
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