Mathematics as a constructive activity: Exploiting dimensions of possible variation

Making the Connection: Research and Teaching in Undergraduate Mathematics Education

Volumn , Issue , 2008, Pages 191-204

  Mason, John ^a   Watson, Anne ^b  

^a OPEN UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77953916375     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.5948/UPO9780883859759.016     Document Type: Chapter

References (57)

1. Alcock, L., & Weber, K. (2005). Referential and syntactic approaches to proof: Case studies from a transition course. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, 2, 33–40. Melbourne, Australia: PME.
2. Alcock, L. (2004). Uses of example objects in proving. In M. Høines & A. Fuglestad (Eds.) Proceedings of PME 28. Bergen, Norway: Bergen University College, 2, 17–24.
3. Anthony, G. (1994). The role of the worked example in learning mathematics. In A. Jones et al. (Eds.), SAME papers (pp. 129–143), Hamilton, New Zealand: University of Waikato.
4. Bills, L. (1996a). Shifting sands: students’ understanding of the roles of variables in “A’ level mathematics. Unpublished doctoral dissertation, Open University, Milton Keynes, United Kingdom.
5. Bills, L. (1996b). The use of examples in the teaching and learning of mathematics. In L. Puig and A. Gutierrez (Eds.) Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, 2.81–2.88, Valencia, Spain: Universitat de València.
6. Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning. Mahwah, NJ: Lawrence Erlbaum Associates.
7. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H. Steiner (Ed.) Theory of Mathematics Education, Paper 54 (110–119) Bielefeld, Germany: Institut fur Didaktik der Mathematik der Universitat Bielefeld.
8. Brousseau, G. (1997). Theory of Didactical Situations in Mathematics: Didactiques des Mathématiques, 1970–1990, (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Trans.). Dordrecht, The Netherlands: Kluwer.
9. Brown, L., & Coles, A. (2000). Same/different: a “natural’ way of learning mathematics. In T. Nakahara & M. Koyama (Eds.) Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, Hiroshima, Japan, 2–153.
10. Brown, S., & Walter, M. (1983). The Art of Problem Posing. Philadelphia: Franklin Press.
11. Bruner, J. (1986) Actual minds, possible worlds. Cambridge, MA: Harvard University Press.
12. Cardano, G. (1969). The rules of algebra: Ars magna (T. Witmer, Trans.). New York: Dover. (Original work published 1545)
13. Chi, M., & Bassok, M. (1989). Learning from examples via self-explanation. In L. Resnick (Ed.) Knowing, learning and instruction: essays in honour of Robert Glaser. Hillsdale, NJ: Lawrence Erlbaum Associates.
14. Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, G. Howson, & M. Otte (Eds.), Perspectives in mathematics education. (pp. 243–307). Dordrecht, The Netherlands: Reidel.
15. Coles, A., & Brown, L. (1999). Meta-commenting: Developing algebraic activity in a “community of inquirers’. In L. Bills (Ed.) Proceedings of the British Society for Research into Learning Mathematics, (pp. 1–6), Warwick University, Coventry, United Kingdom: MERC.
16. Courant, R. (1981). Reminiscences from Hilbert’s Göttingen. Math Intelligencer, 3(4) 154–164.
17. Dahlberg, R., & Housman, D. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics 33, 283–299.
18. Diophantos, (Heath, Trans.). (1964). Diophantus of Alexandria: a study in the history of Greek algebra. New York: Dover.
19. Dweck, C. (1999). Self-Theories: Their role in motivation, personality and development. Philadelphia, PA: Psychology Press.
20. Feynman, R., & Leighton, R. (1985). “Surely you’re joking, Mr. Feynman!”: Adventures of a curious character (E. Hutchings, Ed.). New York: Norton.
21. Floyd, A., Burton, L., James, N., & Mason, J. (1981). EM235: Developing Mathematical Thinking. Open University Course. Milton Keynes, United Kingdom: Open University.
22. Gattegno, C. (1987). The science of education part I: Theoretical considerations. New York: Educational Solutions.
23. Gillings, R. (1972 reprinted 1982). Mathematics in the time of the pharaohs. New York: Dover.
24. Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7) 519–524.
25. Halmos, P. (1983). Selecta: Expository writing. In D. Sarasen & L. Gillman (Eds.), New York: Springer-Verlag.
26. Halmos, P. (1985). I want to be a mathematician: An automathography. New York: Springer-Verlag.
27. Hewitt, D. (1994). The principle of economy in the learning and teaching of mathematics. Unpublished doctoral dissertation, Open University, Milton Keynes, United Kingdom.
28. Karp, A. (2004). Examining the interactions between mathematical content and pedagogical form: notes on the structure of the lesson. For the Learning of Mathematics, 24(1) 40–45.
29. Klymchuk, S. (2005). Counter-examples in calculus. Auckland, New Zealand: Maths Press.
30. Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children (J. Teller, Trans.). In J. Kilpatrick & I. Wirszup (Eds.). Chicago, IL: University of Chicago Press.
31. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, United Kingdom: Cambridge University Press.
32. Lakoff, G. (1987). Women, fire, and dangerous things: What categories reveal about the human mind. Chicago, IL: Chicago University Press.
33. Legrand, M. (1993). Débate scientifique en cour de mathématiques, Repères IREM, No. 10, Topiques Edition.
34. Love, E. (1996). Letting go: An approach to geometric problem solving. In L. Puig & A. Gutiérrez (Eds.). Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education. Valencia, Spain: Universitat de València.
35. MacHale, D. (1980). The predictability of counterexamples. American Mathematical Monthly, 87, 752.
36. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates.
37. Marton, F., & Trigwell, K. (2000). Variatio est mater studiorum. Higher Education Research & Development, 19(3) 381–395.
38. Marton, F., & Tsui, A. (2004). Classroom discourse and the space of learning. Mahwah, NJ: Lawrence Erlbaum Associates.
39. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London, United Kingdom: Addison Wesley.
40. Mason, J., & Johnston-Wilder, S. (2004a). Fundamental constructs in mathematics education. London, United Kingdom: Routledge Falmer.
41. Mason, J., & Johnston-Wilder, S. (2004b). Designing and using mathematical tasks. Milton Keynes, United Kingdom: Open University.
42. Mason, J. (2002). Mathematics teaching practice: A guide for university and college lecturers. Chichester, Sussex, United Kingdom: Horwood Publishing.
43. Mason, J. (2002a). Researching your own practice: The discipline of noticing. London: Routledge Falmer.
44. Michener, E. (1978). Understanding mathematics. Cognitive Science, 2, 361-383.
45. Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Cambridge, MA: Princeton University Press.
46. Pólya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined edition). New York: Wiley.
47. Runesson, U. (2005). Beyond discourse and interaction. Variation: A critical aspect for teaching and learning mathematics. Cambridge Journal of Education, 35(1) 69–88.
48. Schoenfeld, A. (1985). Mathematical problem solving. New York: Academic Press.
49. Senk, S., & Thompson, D. (Eds.). (2003). Standards-based school mathematics curricula: What are they? What do students learn? Mahwah, NJ: Lawrence Erlbaum.
50. Skemp, R. (1976). Relational and instrumental understanding. Mathematics Teaching, 77, 20–26.
51. Sweller, J., & Cooper, G. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition & Instruction, 2, 58–89.
52. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2) 151–169.
53. Watson, A., de Geest, E., & Prestage, S. (2003). Deep progress in mathematics: The improving attainment in mathematics project. Oxford, United Kingdom: Dept. of Educational Studies, Oxford University.
54. Watson A., & Mason, J. (1998). Questions and prompts for mathematical thinking. Derby: Association of Teachers of Mathematics.
55. Watson A., & Mason, J. (2002), Student-generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249.
56. Watson A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.
57. Whitehead, A. (1919) (12th impression, reprinted 1948). An introduction to mathematics. London, United Kingdom: Oxford University Press.