A thirty-year reflection on constructivism in mathematics education in PME

Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future

Volumn , Issue , 2006, Pages 305-345

  Confrey, Jere ^a   Kazak, Sibel ^a  

^a WASHINGTON UNIVERSITY   (United States)

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EID: 85028604872     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (189)

1. Artigue, M. (1987). Ingénierie didactique à propos d'équations differentielles. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 3, 236-242.
2. Artigue, M. (1990). Ingénierie didactique. Recherches en Didactique des Mathématiques, 9, 283-307.
3. Artigue, M. (1992). The importance and limits of epistemological work in didactics. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th PME International Conference, 3, 195-216.
4. Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in cabri and modalities of transition from conjectures to proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 2, pp. 32-39.
5. Bachelard, G. (1938). La formation de l'esprit scientifique. Paris, France: Vrin.
6. Bakker, A., & Gravemeijer, K. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147-168). Dordrecht, The Netherlands: Kluwer.
7. Balacheff, N. (1990). Towards a problematique for research on mathematics teaching. Journal for Research in Mathematics Education, 21(4), 258-272.
8. Balacheff, N. (1993). Artificial intelligence and real teaching. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 131-158). Berlin, Germany: Springer.
9. Ball, D. L. (1993). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157-196). Hillsdale, NJ, USA: Lawrence Erlbaum.
10. Bartolini Bussi, M. (1993). Geometrical proofs and mathematical machines: An exploratory study. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F.-L. Lin (Eds.), Proceedings of the 17th PME International Conference, 2, pp. 97-104.
11. Bass, H., & Ball, D. L. (2005). Mathematical knowledge for teaching. Presentation at the MET Summit II Follow-up Conference, Atlanta, GA, USA.
12. Bauersfeld, H. (1995). The structuring of the structures: Development and function of mathematizing as a social practice. In L. P. Steffe & G. Gale (Eds.), Constructivism in education (pp. 137-158). Hillsdale, NJ, USA: Lawrence Erlbaum.
13. Bauersfeld, H. (1998). Remarks on the education of elementray teachers. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 195-212). New York, NY, USA: Cambridge University Press.
14. Baulac, Y., Bellemain, F., & Laborde, J. M. (1988). Cabri-géomètre (v. 1.7) [Computer program]. Paris, France: Cedic-Nathan.
15. Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 9-61). New York, USA: Academic Press.
16. Bell, A., Swan, M., Onslow, B., Pratt, K., Purdy, D., et al. (1985). Diagnostic teaching for long term learning (ESRC Project No. HR8491/1). Nottingham, UK: Shell Centre for Mathematical Education, University of Nottingham.
17. Ben-Zvi, D. (2000). Toward understanding the role of technological tools in statistical learning. Mathematical Thinking and Learning, 2, 127-155.
18. Ben-Zvi, D., & Garfield, J. (2004). Statistical literacy, reasoning, and thinking: Goals, definitions, and challenges. In D. Ben-Zvi & J. Garfield (Eds.), Challenges in developing statistical reasoning, thinking, and literacy (pp. 3-16). Dordrecht, The Netherlands: Kluwer.
19. Biehler, R. (2001). Developing and assessing students' reasoning in comparing statistical distributions in computer-supported statistics courses. In C. Reading (Ed.), Proceedings of the 2nd International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRLT-2). Armidale, Australia: University of New England.
20. Blakemore, C. (1973). Environmental constraints on development in the visual system. In R. A. Hinde & J. Stevenson-Hinde (Eds.), Constraints on learning. New York, USA: Academic Press.
21. Blum, W. (1993). Mathematical modelling in mathematics education and instruction. In T. Breiteig, I. Huntley, & G. Kaiser-Messmer (Eds.), Teaching and learning mathematics in context (pp. 3-14). New York, USA: Ellis Horwood.
22. Borba, M. (1993). Students' understanding of transformations of functions using multi-representational software. Unpublished PhD dissertation, Cornell University, Ithaca, NY, USA.
23. Borba, M. (1994). A model for students' understanding in a multi-representational environment. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 2, pp. 104-111.
24. Borba, M. (1995). Overcoming limits of software tools: A student's solution for a problem involving transformation of functions. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 248-255.
25. Borba, M., & Confrey, J. (1996). A student's construction of transformations of functions in a multiple representational environment. Educational Studies in Mathematics, 31, 319-337.
26. Borba, M., & Scheffer, N. (2004). Coordination of multiple representations and body awareness [videopaper]. Educational Studies in Mathematics, 57(3) [on CD-ROM], 16 min. video (132 Kb), 126 text pages, 118 images (1156 Kb).
27. Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking information and communication technologies, modeling, experimentation and visualization. New York, USA: Springer.
28. Brousseau, G. (1983). Les obstacles épistémologique et les problèmes en mathématiques. Revue Internationale de Philosophie Recherches en Didactique des Mathématiques, 4, 165-198.
29. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. G. Steiner (Ed.), Theory of mathematics education (ICME-5 Topic Area and miniconference) (pp. 110-119). Bielefeld, Germany: Institut für Didactik der Mathematik, Universität Bielefeld.
30. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht, The Netherlands: Kluwer.
31. Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Research, 18, 32-42.
32. Burkhardt, H. (1981). The real world and mathematics. Glasgow, UK: Blackie and Son.
33. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH, USA: Heinemann.
34. Carpenter, T. P., Moser, J. M., & Romberg, T. (1982). Addition and subtraction: A cognitive approach. Hillsdale, NJ, USA: Lawrence Erlbaum.
35. Carraher, T. N., & Schliemann, A. D. (1988). Culture, arithmetic and mathematical models. Cultural Dynamics, 1, 180-194.
36. Chevallard, Y. (1988). Sur l'analyse didactique: Deux études sur els notions de contrat et de situation. Marseille, France: IREM 14.
37. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York, USA: Macmillan.
38. Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307-333). Mahwah, NJ, USA: Lawrence Erlbaum.
39. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Shauble, L. (2003). Design experiments in educational research. Educational Researcher, 32, 9-13.
40. Cobb, P., Gravemeijer, K. P. E., Bowers, J., & McClain, K. (2001). Statistical minitools (Version Revised). Utrecht, The Netherlands: The Freudenthal Institute, Utrecht University.
41. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Joumal for Research in Mathematics Education, 14, 83-94.
42. Cobb, P., Yackel, E., & Wood, T. (1990). Classrooms as learning environments for teachers and researchers. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 125-146). Reston, VA: NCTM.
43. Confrey, J. (1984). Towards a framework for constructivist instruction. In J. M. Moser (Ed.), Proceedings of the 6th PME-NA Annual Meeting, 1, 477-483.
44. Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. In C. Cazden (Ed.), Review of research in education (16, pp. 3-56). Washington, DC, USA: American Educational Research Association (AERA).
45. Confrey, J. (1991). Learning to listen: A student's understanding of powers of ten. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 111-138). Dordrecht, The Netherlands: Kluwer.
46. Confrey, J. (1995). How compatible are radical constructivism, sociocultural approaches, and social constructivism? In L. P. Steffe & G. Gale (Eds.), Constructivism in education (pp. 185-225). Hillsdale, NJ, USA: Lawrence Erlbaum.
47. Confrey, J. (1998). Voice and perspective: Hearing epistemological innovation in students' words. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 104-120). New York, NY, USA: Cambridge University Press.
48. Confrey, J. (in press). The evolution of design studies as methodology. In K. Sawyer (Ed.), Cambridge handbook of the learning sciences.
49. Confrey, J., & Costa, S. (1996). A critique of the selection of "mathematical objects" as a central metaphor for advanced mathematical thinking. International Journal of Computers for Mathematical Learning, 2, 139-168.
50. Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 231-266). Mahwah, NJ, USA: Lawrence Erlbaum.
51. Confrey, J., & Maloney, A. (1991). Function Probe [Computer program]. Santa Barbara, CA, USA.
52. Confrey, J., & Maloney, A. (2005). A theory of mathematical modeling in technological settings. In ICMI (Ed.), Applications and modelling in mathematics education. Berlin, Germany: Springer.
53. Confrey, J., & Smith, E. (1989). Alternative representations of ratio: The Greek concept of anthyphairesis and modern decimal notation. In D. E. Herget (Ed.), The history and philosophy of science in science education: Proceedings of the 1st International Conference (pp. 71-82). Tallahassee, FL, USA: Science Education & Department of Philosophy, Florida State University.
54. D'Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5, 44-18.
55. De Lange, J. (1987). Mathematics, insight, and meaning. Utrecht, The Netherlands: OW & OC.
56. De Lange, J. (2001). The P in PME: Progress and problems in mathematics education, In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 1, 3-4.
57. Dennis, D. (1995). Historical perspectives for the reform of mathematics curriculum: Geometric curve drawing devices and their role in the transition to an algebraic description of functions. Unpublished PhD dissertation, Cornell University, Ithaca, NY, USA.
58. Désautels, J. (1998). Constructivism-in-action: students examine their idea of science. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 121-138). New York, USA: Cambridge University Press.
59. diSessa, A., & Cobb, P. (2004). Ontological innovation and the role of theory in design experiments. The Journal of the Learning Sciences, 13(1), 77-103.
60. Douady, R. (1986). Jeu des cadres et dialectique outil-objet. Recherches en Didactiques des Mathématiques, 7, 5-31.
61. Dreyfus, T. (1993). Didactic design of computer-based learning environments. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics, education, and technology (pp. 101-130). New York, NY: Springer.
62. Driver, R., & Easley, J. (1978). Pupils and paradigms: A review of literature related to concept development in adolescent science students. Studies in Science Education, 5, 61-84.
63. Dubinsky, E., & McDonald, M. (2001). Apos: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 273-280). Dordrecht, The Netherlands: Kluwer.
64. Ellerton, N. F. (Ed.). (1999). Mathematics teacher development: International perspectives. Perth, Australia: Meridian Press.
65. Ernest, P. (1991a). Constructivism, the psychology of learning, and the nature of mathematics. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 2, pp. 25-32.
66. Ernest, P. (1991b). The philosophy of mathematics education. Bristol, PA, USA: The Falmer Press, Taylor and Francis Inc.
67. Finzer, W. (2001). Fathom dynamic statistics (v. 1.16) [Computer program]. Emeryville, CA, USA: KCP Technologies.
68. Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht, The Netherlands: Kluwer.
69. Fuson, K. C. (1988). Children's counting and concepts of number. New York, USA: Springer.
70. Gagné, R. M. (1965). The conditions of learning. New York, USA: Holt, Rinehart & Winston.
71. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16, 163-176.
72. Ginsburg, H. P. (1989). Children's arithmetic. Austin, TX, USA: Pro-Ed.
73. Ginsburg, H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. New York, USA: Cambridge University Press.
74. Goldin, G. A. (1989). Constructivist epistemology and discovery learning in mathematics. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13th PME International Conference, 2, pp. 15-22.
75. Goldin, G. A. (2002). Representation in mathematical learning and problem solving. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 197-218). Mahwah, NJ, USA: Lawrence Erlbaum.
76. Goldin, G. A., & Gennain, Y. (1983). The analysis of a heuristic process: "Think of a simpler problem". In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th PME-NA Annual Meeting, 2, 121-128.
77. Gravemeijer, K. (1994). Developing realistic mathematics education. PhD dissertation, CD-ß Press / Freudenthal Institute, Utrecht, The Netherlands.
78. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155-177.
79. Gravemeijer, K. (2002). Preamble: From models to modeling. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7-24). Dordrecht, The Netherlands: Kluwer.
80. Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 146-169). Dordrecht, The Netherlands: Kluwer.
81. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 25, 115-141.
82. Greeno, J. (1989). Situations, mental models, and generative knowledge. In D. Klahr & K. Kotovsky (Eds.), Complex information processing. Hillsdale, NJ, USA: Lawrence Earlbaum.
83. Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 361-367.
84. Greer, B. (1987). Nonconservation of multiplication and division involving decimals. Journal for Research in Mathematics Education, 18, 37-45.
85. Gutiérrez, A. (1995). Learning dynamic geometry: Implementing rotations. In A. A. diSessa, C. Hoyles, & R. Noss (Eds.), Computers and exploratory learning (pp. 275-288). Berlin, Germany: Springer.
86. Gutiérrez, A., & Jaime, A. (1993). An analysis of the students' use of mental images when making or imagining movements of polyhedra. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F.-L. Lin (Eds.), Proceedings of the 17th PME International Conference, 2, 153-160.
87. Hall, R. (1995). Exploring design-oriented mathematical practices in school and work settings. Communications of the ACM, 62.
88. Hall, R. (1998). A case study of making and using representations of quantity in architectural design. Unpublished manuscript, University of California, Berkeley, USA.
89. Harel, G., & Confrey, J. (1994). Development of multiplicative reasoning in the learning of mathematics. Albany, NY, USA: State University of New York Press.
90. Harel, I., & Papert, S. (1991). Constructionism. Norwood, NJ, USA: Ablex.
91. Hart, K. A. (1984). Ratio: Children 's strategies and errors. Windsor, Berkshire, UK: NFER-Nelson.
92. Hawkins, D. (1978). Critical barriers to science learning. Outlook, 3, 3-25.
93. Hollebrands, K. (2002). The role of a dynamic software program for geometry in high school students' developing understandings of geometric transformations. In D. Mewborn (Ed.), Proceedings of the 24th PME-NA Annual Meeting, 695-706.
94. Hoyles, C., & Noss, R. (1988). Formalising intuitive descriptions in a parallelogram logo microworld. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 2, 417-424.
95. Jackiw, N. (1991). The Geometer's Sketchpad [Computer program]. Berkeley, CA, USA: Key Curriculum Press.
96. James, W. (1907). Pragmatism: A new name for some old ways of thinking. New York, USA: Longman Green and Co.
97. Janvier, C. (1987). Problems of representation in the learning of mathematics. Hillsdale, NJ, USA: Lawrence Erlbaum.
98. Jaworski, B. (1991). Some implications of a constructivist philosophy for the teacher of mathematics. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 2, 213-220.
99. Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry. London, UK: The Falmer Press.
100. Jaworski, B., Wood, T., & Dawson, A. J. (1999). Mathematics teacher wducation: Critical international perspectives. London: Falmer Press.
101. Kamii, C. K. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York, USA: Columbia University, Teachers College Press.
102. Kaput, J. (1987). Representation and mathematics. In C. Janvier (Ed.), Problems of representation in the learning of mathematics (pp. 19-26). Hillsdale, NJ, USA: Lawrence Erlbaum.
103. Kaput, J. (2001). New activity structures exploiting wirelessly connected graphing calculators. In R. Speiser, C. Maher, & C. Walter (Eds.), The Proceedings of the 23rd PME-NA Annual Meeting, 2, 1017-1018.
104. Kieren, T. E. (1992). Rational and fractional numbers as mathematical and personal knowledge: Implications for curriculum and instruction. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323-372). Hillsdale, NJ, USA: Lawrence Erlbaum.
105. Kilpatrick, J. (1987). What constructivism might be in mathematics education. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, pp. 3-27.
106. Konold, C., & Miller, C. D. (2005). Tinkerplots dynamic data exploration (v. 1.0) [Computer program]. Emeryville, CA, USA: Key Curriculum Press.
107. Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. Chicago, IL, USA: The University of Chicago Press.
108. Laborde, C. (1993). The computer as part of the learning environment: The case of geometry. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 48-67). Berlin, Germany: Springer.
109. Lakatos, I. (1976). Proofs and refutations. Cambridge, UK: Cambridge University Press.
110. Larochelle, M., & Bednarz, N. (1998). Constructivism and education: Beyond epistemological correctness. In M. Larochelle, N. Bednarz, & J. Garrison (Eds.), Constructivism and education (pp. 3-20). New York, NY, USA: Cambridge University Press.
111. Lehrer, R., & Pritchard, C. (2002). Symbolizing space into being. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 59-86). Dordrecht, The Netherlands: Kluwer.
112. Lehrer, R., & Schauble, L. (2000). Modeling in mathematics and science. In R. Glaser (Ed.), Advances in instructional psychology, Vol. 5: Educational design and cognitive science (pp. 101-159). Mahwah, NJ, USA: Lawrence Erlbaum.
113. Lesh, R., & Doerr, H. (2003). Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ, USA: Lawrence Erlbaum.
114. Lesh, R., & Kelly, A. E. (2000). Multitiered teaching experiments. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 197-230). Mahwah, NJ, USA: Lawrence Erlbaum.
115. Ma, L. (1999). Knowing and teaching mathematics: Teachers understanding of fundamental mathematics in China and the United States. Mahwah, NJ, USA: Lawrence Eribaum.
116. Maher, C. A., Speiser, R., Friel, S., & Konold, C. (1998). Learning to reason probabilistically. In S. Berenson, K. Dawkins, M. Blanton, W. Columbe, J. Kolb, & K. Norwood (Eds.), Proceedings of the 20th PME-NA Annual Meeting, 1, pp. 82-87.
117. Marrades, R., & Gutiérrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87-125.
118. Mathematical Sciences Education Board, N. R. C. (2004). On evaluating curricular effectiveness: Judging the quality of k-12 mathematics evaluations. Washington, DC, USA: The National Academies Press.
119. McClain, K., Cobb, P., & Gravemeijer, K. (2000). Supporting students' ways of reasoning about data. In M. J. Burke & F. R. Curico (Eds.), Learning mathematics for a new century (pp. 174-187). Reston, VA, USA: NCTM.
120. McDermott, J. J. (1981). The philosophy of John Dewey. Chicago, IL, USA: University of Chicago Press.
121. Meira, L. (1995). Mediation by tools in the mathematics classroom. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 1, 102-111.
122. Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education. I (pp. 139-168). Providence, RI, USA: American Mathematical Society.
123. Nakahara, T. (1997). Study of the constructive approach in mathematics education: Types of constructive interactions and requirements for the realization of effective interactions. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 3, 272-279.
124. Nemirovsky, R. (2002). On guessing the essential thing. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing and modeling in mathematics education (pp. 233-256). Dordrecht, The Netherlands: Kluwer.
125. Nemirovsky, R., & Borba, M. (2004). Bodily activity and imagination in mathematics learning. Educational Studies in Mathematics, 57(3) [on CD-ROM].
126. Nesher, P. (1988a). Multiplicative school word problems: Theoretical approaches and empirical findings. In M. Behr & J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp. 19-40). Reston, VA, USA: Lawrance Erlbaum and NCTM.
127. Nesher, P. (1988b). Beyond constructivism: Learning mathematics at school. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 1, 54-74.
128. Niss, M. (1992). Applications and modelling in school mathematics - directions for future development. In I. Wirzup & R. Streit (Eds.), Development in school mathematics education around the world (Vol. 31). Reston, VA, USA: NCTM.
129. Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part ii - problem structure at successive stages: Problem solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11(3), 331-363.
130. Opper, S. (1977). Piaget's clinical method. Journal of Children's Mathematical Behavior, 5, 90-107.
131. Papert, S. (1993). The children's machine: Rethinking school in the age of the computer. New York, NY, USA: Basic Books.
132. Perry, B., & Dockett, S. (2002). Yound children's access to powerful mathematical ideas. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 81-111). Mahwah, NJ, USA: Lawrence Erlbaum.
133. Pfannkuch, M., Budgett, S., Parsonage, R., & Horring, J. (2004). Comparison of data plots: Building a pedagogical framework. Paper presented at ICME-10, Copenhagen, Denmark.
134. Piaget, J. (1954). The construction of reality in the child. New York, USA: Basic Books.
135. Piaget, J. (1976). The child's conception of the world. Totowa, NJ, USA: Littlefield.
136. Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26, 165-190.
137. Polanyi, M. (1958). Personal knowledge: Towards a post-critical philosophy. Chicago, IL, USA: University of Chicago Press.
138. Polya, G. (1957). How to solve it. Princeton, NJ, USA: Princeton University Press.
139. Reading, C., & Shaughnessy, J. M. (2000). Students' perceptions of variation in a sampling situation. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 4, pp. 89-96.
140. Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 827.
141. Resnick, M. (1994). Turtles, termites, and traffic jams, explorations in massively parallel microworlds. Cambridge, MA, USA: MIT Press.
142. Resnick, M., & Ocko, S. (1991). Lego/Logo: Learning through and about design. In I. Harel & S. Papert (Eds.), Constructionism (Vol. 2). Norwood, NJ, USA: Ablex Publishing.
143. Sawyer, K. (in press). Cambridge handbook of the learning sciences.
144. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL, USA: Academic Press.
145. Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Educational Research, 4, 1-94.
146. Schwartz, J., & Yerushalmy, M. (1985). The Geometric Supposer [Computer program]. Pleasantville, NY, USA: Sunburst Communications.
147. Schwartz, J. L., & Yerushalmy, M. (1988). The Function Supposer: Explorations in algebra [Computer program]. Pleasantville, NY, USA: Sunburst Communications.
148. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.
149. Shapin, S., & Schaffer, S. (1989). Leviathan and the air-pump: Hobbes, Boyle, and the experimental life. Princeton, USA: Princeton University Press.
150. Shaughnessy, M. (2004). Students' attention to variability when comparing distributions. Paper Presented at the Research Presession of the 82nd Annual Meeting of the NCTM, Philadelphia, PA.
151. Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10, 24-36.
152. Sierpinska, A. (1992). On understanding the notion of function. In G. Harel & E. Dubinsky (Eds.), The concept of function. Aspects of epistemology and pedagogy (pp. 25-58). Washington, DC, USA: Mathematical Association of America.
153. Simon, M. A. (1988). Formative evaluation of a constructivist mathematics teacher inservice program. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 2, 576-583.
154. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.
155. Simon, M. A. (2000). Research on the development of mathematics teachers: The teacher development experiment. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 335-359). Mahwah, NJ, USA: Lawrence Erlbaum.
156. Simon, M. A., Tzur, R., Heinz, K., & Kinzel, M. (2004). Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35, 305-329.
157. Sinclair, H. (1987). Constructivism and the psychology of mathematics. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, pp. 28-41.
158. Skemp, R. (1978). The psychology of learning mathematics. Hillsdale, NJ, USA: Lawrence Erlbaum.
159. Smith, E., Dennis, D., & Confrey, J. (1992). Rethinking functions: Cartesian construction. In S. Hills (Ed.), Proceedings of the 2nd International Conference on the History and Philosophy of Science in Science Teaching (2, pp. 449-466). Kingston, Ontario, Canada: Queens University.
160. Steffe, L. P. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). Albany, NY, USA: State University of New York Press.
161. Steffe, L. P. (1995). Alternative epistemologies: An educators' perspective. In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 489-523). Hillsdale, NJ, USA: Lawrence Erlbaum.
162. Steffe, L. P., Cobb, P., & Von Glasersfeld, E. (1988). Young children's construction of arithmetical meanings and strategies. New York, USA: Springer.
163. Steffe, L., & Gale, J. (1995). Constructivism in education. Hillsdale, NJ, USA: Lawrence Erlbaum.
164. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307). Hillsdale, NJ, USA: Lawrence Erlbaum.
165. Steffe, L. P., Von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory, and application. New York, USA: Praeger.
166. Streefland, L. (1991). Fractions in realistic mathematics education. A paradigm of developmental research. Dordrecht, The Netherlands: Kluwer.
167. Sutherland, R., & Rojano, T. (1993). A spreadsheet approach to solving algebra problems. Journal of Mathematical Behavior, 12, 353-382.
168. Taimina, D. (2005). Historical mechanisms for drawing curves. Retrieved June 3, 2005, from http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.htmm/2004-9
169. Tall, D. O. (Ed.). (1991). Advanced mathematical thinking. Dordrecht, The Netherlands: Kluwer.
170. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
171. Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing and modeling in mathematics education (pp. 197-220). Dordrechht, The Netherlands: Kluwer.
172. Thorndike, E. (1922). The psychology of arithmetic. New York, USA: Macmillan.
173. Van Hiele, P. M., & Van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary schools. In H. Freudenthal (Ed.), Report on methods of initiation into geometry (pp. 67-80). Groningen, The Netherlands: J. B. Wolters.
174. Vergnaud, G. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In T. P. Carpenter, M. J. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 39-58). Hillsdale, NJ, USA: Lawrence Erlbaum.
175. Vergnaud, G. (1987). About constructivism. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 42-54.
176. Vergnaud, G. (1996). The theory of conceptual fields. In L. Steffe & P. Nesher (Eds.), Theories of mathematical learning (pp. 219-239). Mahwah, NJ, USA: Lawrence Erlbaum.
177. Verschaffel, L., De Corte, E., & Vierstraete, H. (1999). Upper elementary school pupils' difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, 30(3), 265-285.
178. Vinner, S., & Hershkowitz, R. (1980). Concept images and some common cognitive paths in the development of some simple geometric concepts. In R. Karplus (Ed.), Proceedings of the 4th PME International Conference, 177-184.
179. Voigt, J. (1985). Patterns and routines in classroom interaction. Researches en Didactique de Mathématiques, 6(1), 69-118.
180. Von Foerster, H. (1984). On constructing a reality. In P. Watzlawick (Ed.), Invented reality: How do we know what we believe we know? (pp. 41-61). New York, USA: Norton.
181. Von Glasersfeld, E. (1982). An interpretation of Piaget's constructivism. Revue Internationale de Philosophie, 36(142-143), 612-635.
182. Von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In R. D. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (JRME Monograph 4) (pp. 19-29). Reston, VA, USA: NCTM.
183. Von Glasersfeld, E. (1991). Radical constructivism in mathematics education. Dordrecht, The Netherlands: Kluwer.
184. Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London, UK: Falmer Press.
185. Watson, J. M., & Moritz, J. B. (1999). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.
186. Wheeler, D. (1987). The world of mathematics: Dream, myth or reality? In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, pp. 55-66.
187. Wood, T., Cobb, P., & Yackel, E. (1995). Reflections on learning and teaching mathematics in elementary school. In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 401-422). Hillsdale, NJ, USA: Lawrence Erlbaum.
188. Yoshida, M. (1999). Lesson study: A case study of a japanese approach to improving instruction through school-based teacher development. Unpublished PhD dissertation, University of Chicago, USA.
189. Zack, V., Mousley, J., & Breen, C. (1997). Developing practice: Teachers inquiry and educational change. Melbourne, Australia: Deakin University Press.