Making Use of Children’s Powers to Produce Algebraic Thinking

Algebra in the Early Grades

Volumn , Issue , 2017, Pages 57-94

  Mason, John ^a  

^a OPEN UNIVERSITY   (United Kingdom)

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[No Author keywords available]

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EID: 85144349464     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.4324/9781315097435-4     Document Type: Chapter

References (89)

1. Ahmed, A., and Williams, H. (1992). Raising achievement in Mathematics Project report. London: WSIHE/HMSO.
2. Anghileri, J. (2000). Teaching number sense. London: Continuum.
3. Anghileri, J. (2001). Principles and practices in arithmetic teaching: Innovative approaches for the primary classroom. Buckingham, England: Open University Press.
4. Baruk, S. (1985). The captain’s age: On errors in mathematics. L’âge du capitaine: De l’erreur en mathématiques. Seuil: Seuil Point Sciences.
5. Bednarz, N., Kieran, C., and Lee, L. (Eds.). (1996). Approaches to algebra: Perspectives for research and teaching. Dordrecht, the Netherlands: Kluwer Academic.
6. Blanton, M., and Kaput, J. (in press). Design principles for instructional contexts that support students’ transition from arithmetic to algebraic reasoning: Elements of task and culture. In R. Nemirovsky, B. Warren, A. Rosebery, and J. Solomon (Eds.), Everyday matters in science and mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.
7. Boaler, J. (1997). Experiencing school mathematics: Teaching styles, sex and setting. Buckingham, England: Open University Press.
8. Booth, L. (1988). Children’s difficulties in beginning algebra. In A. Coxford and A. Schulte (Eds.), The ideas of algebra, K-12, NCTM 1988 yearbook (pp. 20-32). Reston, VA: National Council for Teachers of Mathematics.
9. Branford, B. (1908). A study of mathematical education. Oxford, England: Oxford University Press.
10. Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactiques des mathématiques, 1970-1990 (N. Balacheff, M. Cooper, R. Sutherland, and V. Warfield, Trans.). Dordrecht, the Netherlands: Kluwer Academic.
11. Brown, J., and van Lehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4, 379-426.
12. Brown, S., Collins, A., and Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-41.
13. Brown, S. I., and Walter, M. I. (1982). The art of problem posing. Philadelphia: Franklin Institute Press.
14. Bruner, J. (1991). The narrative construction of reality. In M. Ammanti and D. Stern (Eds.), Rappresentazioni e Narrazioni (pp. 17-42). Roma-Bari: Laterze.
15. Bruner, J. (1996). The culture of education. Cambridge, MA: Harvard University Press.
16. Buchdahl, G. (1961). The image of Newton and Locke in the age of reason (Newman History and Philosophy of Science Series). London: Sheed and Ward.
17. Carpenter, T. P., and Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform: The case of mathematics education in the U.S. [Special issue]. International Journal of Educational Research, 17(5), 457-470.
18. Carpenter, T. P., Franke, M. L., and Levi, L. (2001). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
19. Chamberlin, J. (2003). If this is your land, where are your stories?: Finding common ground. Toronto, Canada: Knopff Canada.
20. Davydov, V. (1972). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (Soviet Studies in Mathematics Education, Vol. 2). Reston, VA: National Council for Teachers of Mathematics.
21. de Morgan, A. (1865). A speech of Professor de Morgan, president, at the first meeting of the London Mathematical Society. In Proceedings of the London Mathematical Society (pp. 1-9). London: London Mathematical Society.
22. Dewey, J. (1897). My pedagogic creed by Dewey. The School Journal, 54(3), 77-80.
23. Donald, M. (1991). Origins of the modern mind: Three stages in the evolution of culture and cognition. Cambridge, MA: Harvard University Press.
24. Donaldson, M. (1978). Children’s minds. New York: Norton.
25. Dougherty, B. (2003). Voyaging from theory to practice: Measuring up. In N. Pateman, B. Dougherty, and J. Zilliox (Eds.), Proceedings of the 27th conference of the International Group for the Psychology of Mathematics Education held jointly with the 25th conference of the PME-NA (Vol. 1, pp. 17-30). Honolulu, HI: University of Hawaii.
26. Eliot, G. (1990). Thomas Carlyle. In G. Eliot (Ed.), Selected essays, poems and other writings (pp. 343-348). Harmondsworth: Penguin. (Original work published 1855)
27. Fischbein, E. (1993). Figural concepts. Educational Studies in Mathematics, 24(2), 139-162.
28. Floyd, A., Burton, L., James, N., and Mason, J. (1981). Developing mathematical thinking in Algebra [Open University course]. Milton Keynes, England: Open University Press.
29. Freudenthal, H. (1974). Soviet research on teaching algebra at the lower grades of the elementary school. Educational Studies in Mathematics, 5, 391-412.
30. Freudenthal, H. (1978). Weeding and sowing: Preface to a science of mathematical education. Dordrecht, the Netherlands: Reidel.
31. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht, the Netherlands:Kluwer Academic.
32. Gattegno, C. (1970). What we owe children: The subordination of teaching to learning. London: Routledge and Kegan Paul.
33. Gattegno, C. (1973). In the beginning there were no words: The universe of babies. New York: Educational Solutions.
34. Gattegno, C. (1987). The science of education: Part I. Theoretical considerations. New York:Educational Solutions.
35. Groetsch, C. (1999). Inverse problems: Activities for undergraduates. Washington, DC:Mathematical Association of America.
36. Hewitt, D. (1996). Mathematical fluency: The nature of practice and the role of subordination. For the Learning of Mathematics, 16(2), 28-35.
37. Hewitt, D. (1998). Approaching arithmetic algebraically. Mathematics Teaching, 163, 19-29.
38. Johnson, M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. Chicago: University of Chicago Press.
39. Krutetskii, V. A. (1976). The psychology of mathematics abilities in school children (J. Teller, Trans.). Chicago: University of Chicago Press.
40. Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics 11-16 (pp. 102-119). London: Murray.
41. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, England: Cambridge University Press.
42. Legrand, M. (1993). Débate scientifique en cour de mathématiques (Repéres IREM, No. 10, Topiques ed.). Paris: Topiques.
43. Lewis, C. S. (1945). The great divorce: A dream. London: Geoffrey Bles.
44. Malara, N., and Navarra, G. (2003). ArAl: A project for an early approach to algebraic thinking. Retrieved August 2004 from http://www.math.unipa.it/~grim /SiMalara Navarra.PDF
45. Marton, F., and Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates.
46. Mason, J. (1988a). Doing and undoing [Project update]. Milton Keynes, UK: Open University Press.
47. Mason, J. (1988b). Expressing generality [Project update]. Milton Keynes, UK: Open University Press.
48. Mason, J. (1992). Doing and construing mathematics in screen space. In B. Southwell, B. Perry, and K. Owens (Eds.), Space-The first and final frontier: Proceedings of the 15th annual conference of the Mathematics Education Research Group of Australasia (MERGA-15, pp. 1-17). Retrieved from http://cme.open. ac.uk/article_jhm_merga.htm
49. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, and L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, the Netherlands: Kluwer Academic.
50. Mason J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1(3), 243-267.
51. Mason, J. (2001a). Mathematical teaching practices at tertiary level: Working group report. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 71-86). Dordrecht, the Netherlands: Kluwer Academic.
52. Mason, J. (2001b). Tunja sequences as examples of employing students’ powers to generalize. Mathematics Teacher, 94(3), 164-169.
53. Mason, J. (2006). Structured variation grids. Retrieved November 2006 from mcsmail/open.ac.uk/jhm3
54. Mason, J., Burton L., and Stacey K. (1982). Thinking mathematically. London: Addison-Wesley.
55. Mason, J., and Johnston-Wilder, S. (2004). Designing and using mathematical tasks. Milton Keynes, England: Open University Press.
56. Mason, J., with Johnston-Wilder, S., and Graham, A. (2005). Developing thinking in algebra. London: Sage
57. Mason, J., Graham, A., Pimm, D., and Gowar, N. (1985). Routes to, roots of algebra. Milton Keynes, England: Open University Press.
58. Melzak, Z. (1983). Bypasses: A simple approach to complexity. New York: Wiley.
59. Merseth, K. (1993, March). How old is the shepherd?: An essay about mathematics education. Phi Delta Kappan, 74, 548-554.
60. Newman, J. R. (Ed.). (1956). The world of mathematics. New York: Simon and Schuster.
61. Noticeboard. (2004). Mathematics teaching 186. Accessed from www.atm.org.uk/mt/archive/met186files/TM-MT186-20-21.pdf
62. O’Connor, M. (1998). Language socialization in the mathematics classroom:Discourse practices and mathematical thinking. In M. Lampert and M. Blonk (Eds.), Talking mathematics in school: Studies of teaching and learning (pp.17-55). Cambridge, England: Cambridge University Press.
63. Ollerton, M., and Watson, A. (2001). Inclusive mathematics: 11-18. London: Sage.
64. Open University. (1978). M101: Mathematics foundation course. Milton Keynes, England: Author.
65. Peirce, C. S. (1956). The essence of mathematics. In J. Newman (Ed.), The world of mathematics (Vol. 3, pp. 1773-1783). London: Allen and Unwin. (Original work published 1902)
66. Piaget, J. (1971). Biology and knowledge. Chicago: University of Chicago Press.
67. Pirie, S., and Kieren, T. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(4), 7-11.
68. Polya, G. (1957). How to solve it. New York: Anchor.
69. Polya, G. (1962). Mathematical discovery: On understanding, learning, and teaching problem solving (combined ed.). New York: Wiley.
70. Rhadakrishnan S. (1953). The principal upanishads. London: Allen and Unwin.
71. Rojano, T., and Sutherland, R. (1993). Towards and algebraic approach: The role of spreadsheets. In I. Hirayabashi, N. Nohoda, K. Shigematsu, and Fou-Lai Lin (Eds.), Proceedings of the 17th International Conference for the Psychology of Mathematics Education (pp. 189-196). Japan: University of Tsukuba.
72. SMP 11-16. (1984). Doing and undoing. Cambridge, England: Cambridge University Press.
73. South Notts Project. (n.d.). Material for secondary mathematics. Nottingham, England: Shell Centre, University of Nottingham.
74. Stacey, K., and MacGregor, M. (2000). Curriculum reform and approaches to algebra. In R. Sutherland, T. Rojano, A. Bell, and R. Lins (Eds.), Perspectives on school algebra, (pp. 141-153). Dordrecht, the Netherlands: Kluwer Academic.
75. Tahta, D. (1972). A Boolean anthology: Selected writings of Mary Boole on mathematics education. Derby: Association of Teachers of Mathematics.
76. Tahta, D. (1980). About geometry. For the Learning of Mathematics, 1(1), 2-9.
77. Tahta, D. (1981). Some thoughts arising from the new Nicolet film. Mathematics Teaching, 94, 25-29.
78. Tahta, D. (1989). Is there a geometrical imperative? Mathematics Teaching, 129, 20-29.
79. Torretti, R. F. (1978). Philosophy of geometry from Riemann to Poincare. Dordrecht, the Netherlands: Reidel.
80. van Hiele, P. (1986). Structure and insight: A theory of mathematics education (Developmental Psychology Series). London: Academic Press.
81. Vergnaud, G. (1981). Quelques orientations théoriques et méthodologiques des recherches françaises en didactique des mathématiques. In Some theoretical orientations and French research methodologies in teaching mathematics (Vol. 2, pp. 7-17). Grenoble: Edition IMAG.
82. Vygotsky, L. (1965). Thought and language (E. Hanfmann and G. Vakar, Trans.). Cambridge, MA: MIT Press.
83. Ward, J. (1713). The young mathematician’s guide: Being a plain and easie introduction to the mathematicks in five parts (2nd ed.). London: Tho. Horne.
84. Watson, A. (2000). Going across the grain: Mathematical generalization in a group of low attainers. Nordisk Matematikk Didaktikk (Nordic Studies in Mathematics Education), 8(1), 7-22.
85. Watson, A., De Geest, E., and Prestage, S. (2004). Deep progress in mathematics: The improving attainment in mathematics project. Oxford, England: Oxford University Press.
86. Watson, A., and Mason, J. (2004). Mathematics as a constructive activity: The role of learner-generated examples. Mahwah, NJ: Lawrence Erlbaum Associates.
87. Whitehead, A. (1948). An introduction to mathematics. London: Oxford University Press. (Original work published 1911)
88. Whiteside, D. (Ed.). (1972). The mathematical papers of Isaac Newton: Vol. 5. 1683-1684. Cambridge, England: Cambridge University Press.
89. Zazkis, R. (2001). From arithmetic to algebra via big numbers. In H. Chick, K.Stacey, J. Vincent, and J. Vincent (Eds.), Proceedings of the 12th ICMI study conference:The future of the teaching and learning of algebra (pp. 676-681). Melbourne, Australia: University of Melbourne Press.