The role and uses of technologies for the teaching of algebra and calculus

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

Volumn , Issue , 2006, Pages 237-273

  Ferrara, Francesca ^a   Pratt, Dave ^b   Robutti, Ornella ^a  

^a UNIVERSITY OF TURIN   (Italy)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 76249092210     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (102)

1. Aczel, J. (1998). Learning algebraic strategies using a computerised balance model. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 2, 1-8.
2. Ainley, J. (1994). Building on children's intuitions about line graphs. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 2, 1-8.
3. Ainley, J. (1995). Reasons to be formal: Contextualising formal notation in a spreadsheet environment. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 26-33.
4. Ainley, J. (2000). Exploring the transparency of graphs and graphing. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 2, 9-17.
5. Ainley, J., Nardi, E., & Pratt, D. (1998). Graphing as a computer-mediated tool. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 1, 243-250.
6. Ainley, J., & Pratt, D. (2002). Purpose and utility in pedagogic task design. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 2, 17-24.
7. Anichini, G., Arzarello, F., Ciarrapico, L., & Robutti, O. (Eds.). (2004). New mathematical standards for the school from 5 through 18 years (Edition for ICME 10, on behalf of UMI-CIIM, MIUR). Bologna, Italy: Unione Matematica Italiana.
8. Artigue, M. (1991). Analysis. In D. Tall (Ed.), Advanced mathematical thinking (pp. 167-198). Dordrecht, The Netherlands: Kluwer.
9. Arzarello, F. (2005). Mathematical landscapes and their inhabitants: Perceptions, languages, theories (Plenary lecture). In M. Niss et al. (Eds.), Proceedings of the 10th International Congress of Mathematics Education.
10. Arzarello, F., Bazzini, L., & Chiappini, G. (1995). The construction of algebraic knowledge: Towards a socio-cultural theory and practice. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 1, 199-206.
11. Arzarello, F., & Robutti, O. (2003). Approaching algebra through motion experiences. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 1, 111-115.
12. Ball, L., Pierce, R., & Stacey, K. (2003). Recognising equivalent algebraic expressions: An important component of algebraic expectation for working with CAS. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 4, 15-22.
13. 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, 104-111.
14. Borba, M. C. (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.
15. Borba, M., & Scheffer, N. (2003). Sensors, body, technology and multiple representations. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 1, 121-128.
16. Borba, M. C., & Villareal, M. E. (1998). Graphing calculators and the reorganisation of thinking: The transition from functions to derivative. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 2, 136-143.
17. Cavanagh, M., & Mitchelmore, M. (2000). Student misconceptions in interpreting basic graphic calculator displays. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 2, 161-168.
18. Cedillo, T. E. A. (1997). Algebra as a language in use: A study with 11-12 year olds using graphing calculators. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 2, 137-144.
19. Confrey, J. (1994) Six approaches to transformation of functions using multi-representational software. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 2, 217-224.
20. Confrey, J., & Smith, E. (1992). Revised accounts of the function concept using multi-representational software, contextual problems and student paths. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th PME International Conference, 1, 153-160.
21. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Dortrecht, The Netherlands: Kluwer.
22. Dagher, A., & Artigue, M. (1993). The use of computers in learning to correlate algebraic and graphic representation of functions. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F.-L. Lin (Eds.), Proceedings of the 17th PME International Conference, 2, 1-8.
23. Doerr, H. M., McClain, K., & Bowers, J. (1997). Creating a shared context: The use of a multimedia case in a teacher development course. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 2, 217-224.
24. Doerr, H. M., & Zangor, R. (1999). Creating a tool: An analysis of the role of the graphing calculator in a pre calculus classroom. In O. Zaslavsky (Ed.), Proceedings of the 23rd PME International Conference, 2, 265-272.
25. Drijvers, P. (2001). Reaction to "A meta study on IC technologies in education. Towards a multidimensional framework to tackle their integration". In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 1, 123-130.
26. Euler, L. (1988), Introduction to Analysis of the Infinite (1, pp. 50-55). New York, USA: Springer Verlag. (translated by John Blanton)
27. Friedlander, A., & Stein, H. (2001). Students' choice of tools in solving equations in a technological learning environment. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 2, 441-448.
28. Gitirana, V. (1998). Conceptions as articulated in different microworlds exploring functions. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 3, 9-16.
29. Goldenberg, P. (1997). Believing is seeing: How preconceptions influence the perception of graphs. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 197-203.
30. Gómez, P., & Fernández, F. (1997). Graphing calculators use in pre-calculus and achievement in calculus. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 3, 1-8.
31. Guin, D., & Trouche, L. (2000). Thinking of new devices to make viable symbolic calculators in the classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 3, 9-16.
32. Guttenberger, E. W. (1992) The learning of trigonometric functions in a graphical computer environment. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th PME International Conference, 3, 106-113.
33. Healy, L., & Hoyles, C. (1996). Seeing, doing and expressing: An evaluation of task sequences for supporting algebraic thinking. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 3, 67-74.
34. Hegedus, S. J., & Kaput, J. (2003). The effect of SimCalc connected classrooms on students' algebraic thinking. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 3, 47-54.
35. Hershkowitz, R., & Kieran, C. (2001). Algorithmic and meaningful ways of joining together representatives within the same mathematical activity: An experience with graphing calculators. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 1, 96-107.
36. Hong, Y. Y., & Thomas, M. (1997). Using the computer improve conceptual thinking integration. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 3, 81-88.
37. Kaput, J. (1987). PME XI algebra papers: A representational framework. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 345-354.
38. Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York, USA: Macmillan.
39. Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 77-156). Hillsdale, NJ, USA: Lawrence Erlbaum.
40. Kaput, J., & Hegedus, S. J. (2002). Exploiting classroom connectivity by aggregating student constructions to create new learning opportunities. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 3, 177-184.
41. Kaput, J., Noss, R., & Hoyles, C. (2002). Developing new notations for a learnable mathematics in the computational era. In L. English (Ed.), Handbook of international research in mathematics education (pp. 51-75). Hillsdale, NJ, USA: Lawrence Erlbaum.
42. Kendal, M., & Stacey, K. (1999). CAS, calculus and classrooms. In O. Zaslavsky (Ed.), Proceedings of the 23rd PME International Conference, 3, 129-136.
43. Kendal, M., & Stacey, K. (2000). Acquiring the concept of derivative: Teaching and learning with multiple representations and CAS. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 3, 127-134.
44. Kendal, M., & Stacey, K. (2001). Influences on and factors changing technology privileging. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 3, 217-224.
45. Kidron, I. (2002). Concept definition, concept image, and the notion of infinite sum in old and new environments. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 3, 209-216.
46. Kidron, I., Zehavi, N., & Openhaim, E. (2001). Teaching the limit concept in a CAS environment: Students' dynamic perceptions and reasoning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 3, 241-248.
47. Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York, USA: Macmillan.
48. Kissane, B. (2001). Algebra and technology: Emerging issues. Presentation at the Biennial Conference of the Australian Association of Mathematics Teachers, Canberra, Australia.
49. Lagrange, J. B. (1999). Learning pre-calculus with complex calculators: Mediation and instrumental genesis. In O. Zaslavsky (Ed.), Proceedings of the 23rd PME International Conference, 3, 193-200.
50. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, USA: Basic Books.
51. Lesh, R., & Herre, J. (1987). Dienes revisited: Multiple embodiments in computer environments. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 211-220.
52. Li, L., & Tall, D. (1993). Constructing different concept images of sequences & limits by programming. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F.-L. Lin (Eds.), Proceedings of the 17th PME International Conference, 2, 41-48.
53. Magidson, S. (1992). What's in a problem? Exploring slope using computer graphing software. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th PME International Conference, 2, 64-71.
54. Mariotti, M. A., & Cerulli, M. (2001). Semiotic mediation for algebra teaching and learning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 3, 343-349.
55. Mesa, V. M., & Gómez, P. (1996). Graphing calculators and pre-calculus: An exploration of some aspects of students' understanding. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 3, 391-398.
56. Milani, R., & Baldino, R. (2002). The theory of limits as an obstacle to infinitesimal analysis. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 3, 345-352.
57. Monaghan, J., Sun, S., & Tall, D. (1994). Construction of the limit concept with a computer algebra system. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 3, 279-286.
58. Moreira, L. (2002). Mathematics education and critical consciousness. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 3, 369-376.
59. Nathan, M. J. (1992). Interactive depictions of mathematical constraints can increase students' levels of competence for word algebra problem solving. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th PME International Conference, 2, 160-169.
60. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA, USA: NCTM.
61. Nemirovsky, R. (1994). Slope, steepness, and school math. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 3, 344-351.
62. Nemirovsky, R. (1995). Panel Presentation. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 1, 79-86.
63. Nemirovsky, R. (1998). Symbol-Use, fusion and logical necessity: On the significance of children's graphing: A reaction paper to Ainley et al. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 1, 259-267.
64. Nemirovsky, R. (2003). Three conjectures concerning the relationship between body activity and understanding mathematics. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 1, 105-113.
65. Nemirovsky, R., Kaput, J., & Roschelle, J. (1998). Enlarging mathematical activity from modelling phenomena to generating phenomena. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd PME International Conference, 3, 287-294.
66. Noble, T., & Nemirovsky, R. (1995). Graphs that go backwards. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 256-265
67. Norton, S., & Cooper, T. J. (2001). Students' responses to a new generation ILS algebra tutor. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 3, 439-446.
68. Pratt, D. (1994). Active graphing in a computer-rich environment. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 4, 57-64.
69. Pratt, D. (1995). Passive and active graphing: A study of two learning sequences. In L. Meira. & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 210-217.
70. Radford, L., Cerulli, M., Demers, S., & Guzmán, J. (2004). The sensual and the conceptual: Artefact-mediated kinesthetic actions and semiotic activity. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th PME International Conference, 4, 73-80.
71. Radford, L., Demers, S., Guzmán, J., & Cerulli, M. (2003). Calculators, graphs, gestures, and the production meaning. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 4, 55-62.
72. Rasmussen, C., & Nemirovsky, R. (2003). Becoming friends with acceleration: The role of tools and bodily activity in mathematical learning. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th PME International Conference, 1, 127-134.
73. Reid, D. A. (2002). Describing young children's deductive reasoning. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 4, 105-112.
74. Resnick, Z., Schwarz, B., & Hershkowitz, R. (1994). Global thinking "between and within" function representations in a dynamic interactive medium. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 4, 225-232.
75. Robutti, O., & Ferrara, F. (2002). Approaching algebra through motion experiences. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 1, 111-118.
76. Samurçay, R. (1985). Learning programming: Constructing the concept of variable by beginning students. In L. Streefland (Ed.), Proceedings of the 9th PME International Conference, 1, 77-82.
77. Schwarz, B., & Bruckheimer, M. (1988). Representations of functions and analogies, In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 2, 552-559.
78. Schwarz, B. B., & Hershkowitz, R. (1996). Effects of computerized tools on prototypes of the function concept. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 4, 259-266.
79. Smart, T. (1995). Visualising quadratic functions: A study of thirteen-year-old girls learning mathematics with graphic calculators. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 272-279.
80. Sutherland, R. (1987). A study of the use and understanding of algebra related concepts within a Logo environment. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 241-247.
81. Sutherland, R., & Rojano, T. (1996). Mathematical modelling in the sciences through the eyes of Marina and Adam. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 4, 291-298.
82. Tabach, M., & Hershkowitz, R. (2002). Construction of knowledge and its consolidation: A case study from the early-algebra classroom. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th PME International Conference, 4, 265-272.
83. Tall, D. (1985). Using computer graphics programs as generic organisers for the concept image of differentiation. In L. Streefland (Ed.), Proceedings of the 9th PME International Conference, 1, 105-110.
84. Tall, D. (1986a). Building and testing a cognitive approach to the calculus using interactive computer graphics. Unpublished PhD. dissertation, University of Warwick, UK.
85. Tall, D. (1986b). A graphical approach to integration and the Fundamental Theorem. Mathematics Teaching, 113, 48-51.
86. Tall, D. (1987a). Algebra in a computer environment. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 262-271.
87. Tall, D. (1987b). Constructing the concept image of a tangent. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International, 3, 69-75.
88. Tall, D. (1989). Concept images, generic organizers, computers & curriculum change. For the Learning of Mathematics, 9(3), 37-12.
89. Tall, D., & Sheath, G. (1983). Visualizing higher level mathematical concepts using computer graphics. In R. Hershkowitz (Ed.), Proceedings of the 7th PME International Conference, 357-362.
90. Thomas, M., & Tall, D. (1988). Longer-term conceptual benefits from using a computer in algebra teaching. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 2, 601-608.
91. Thompson, P. W., & Thompson, A. G. (1987). Computer presentations of structure in algebra. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 1, 248-254.
92. Trouche, L., & Guin, D. (1996). Seeing is reality: How graphing calculators may influence the conceptualisation of limits. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 4, 323-330.
93. Tsamir, P., & Bazzini, L. (2001). Can x=3 be the solution of an inequality? A study of Italian and Israeli students. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th PME International Conference, 4, 303-310.
94. Ubuz, B., & Kirkpinar, B. (2000). Factors contributing to learning of calculus. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th PME International Conference, 4, 241-248.
95. Valero, P., & Gómez, P. (1996). Precalculus and graphic calculators: The influence on teacher's beliefs. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th PME International Conference, 4, 363-370.
96. Verillon, P., & Rabarbel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, X (1), 77-101.
97. Vinner, S. (1987): Continuous functions - Images and reasoning in college students, In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th PME International Conference, 3, 177-183.
98. Yerushalmy, M. (1997). Emergence for new schemes for solving algebra word problems: The impact of technology and the function approach. In E. Pehkonen (Ed.), Proceedings of the 21st PME International Conference, 1, 165-172.
99. Yerushalmy, M., & Bohr, M. (1995). Between equations and solutions: An odyssey in 3D. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th PME International Conference, 2, 218-225.
100. Yerushalmy, M., & Shternberg, B. (1994). Symbolic awareness of algebra beginners. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 18th PME International Conference, 4, 393-400.
101. Yerushalmy, M., Shternberg, B., & Gilead, S. (1999). Visualisation as a vehicle for meaning of problem solving in algebra. In O. Zaslavsky (Ed.), Proceedings of the 2nd PME International Conference, 1, 197-206.
102. Zehavi, N. (1988). Substitutions leading to reasoning. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference, 2, 665-672.