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Symbols and the bifurcation between procedural and conceptual thinking
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Tall, David, Gray, Edward Martin, Bin Ali, Maselan, Crowley, Lillie, DeMarois, Phil, McGowen, Mercedes, Pitta, Demetra, Pinto, Marcia, Thomas, Michael and Yusof, Yudariah. (2001) Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, Vol.1 . pp. 81104. ISSN 14926156

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Abstract
Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to d o mathematical problems and to think about mathematical relationships. In this presentation we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra and calculus, then on to the formalism of axiomatic mathematics. This is taken from a number of research studies recently performed for doctoral dissertations at the University of Warwick by students from the USA, Malaysia, Cyprus and Brazil, with data collected in the USA, Malaysia and the United Kingdom. All the studies form part of a broad investigation into why some students succeed yet others fail.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics L Education > L Education (General) 
Divisions:  Faculty of Social Sciences > Institute of Education 
Library of Congress Subject Headings (LCSH):  Signs and symbols, Mathematics  Research, Mathematical ability, Mathematics  Study and teaching 
Journal or Publication Title:  Canadian Journal of Science, Mathematics and Technology Education 
Publisher:  Routledge 
ISSN:  14926156 
Date:  2001 
Volume:  Vol.1 
Page Range:  pp. 81104 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Open Access 
References:  Ali, M. b. (1996). Symbolic Manipulation Related to Certain Aspects Such as Interpretations of Graphs, PhD Thesis, University of Warwick. Ali, M. b. & Tall, D. O., (1996). Procedural and Conceptual Aspects of Standard Algorithms in Calculus, Proceedings of PME 20, Valencia, 2, 19–26. Anderson, C. (1997). Persistent errors in indices: a cognitive perspective, PhD Thesis, University of New England, Armidale, Australia. Cornu, B. (1991). Limits. In D. O. Tall (Ed.), Advanced Mathematical Thinking, (pp. 153–166). Dordrecht: Kluwer. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process schema, Journal of Mathematical Behavior, 15, 167–192. Crowley, L. & Tall, D. O. (1999). The Roles of Cognitive Units, Connections and Procedures in achieving Goals in College Algebra. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 2, 225–232. DeMarois, P. (1998). Aspects and Layers of the Function Concept, PhD Thesis, University of Warwick. DeMarois, P. & Tall, D. O. (1999). Function: Organizing Principle or Cognitive Root? In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 2, 257–264. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced Mathematical Thinking, (pp. 95–123). Dordrecht: Kluwer. Dubinsky, E., Elterman, F. & Gong, C.(1988). The student’s construction of quantification, For the Learning of Mathematics, 8, 44–51. Gray, E. M. & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal of Research in Mathematics Education, 26 (2), 115–141. Gray, E. M., & Tall, D. O. (1991). Duality, Ambiguity and Flexibility in Successful Mathematical Thinking, Proceedings of PME XIII, Assisi, Vol. II, 7279. Hiebert, J. & Lefevre, P. (1986). Procedural and Conceptual Knowledge. In J. Hiebert, (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 127). Hillsdale, NJ: Erlbaum. McGowen, M. (1998). Cognitive Units, Concept Images and Cognitive Collages, PhD Thesis, University of Warwick. McGowen, M. & Tall, D. O. (1999). Concept Maps & Schematic Diagrams as Devices for Documenting the Growth of Mathematical Knowledge. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 3, 281–288. Pinto, M. M. F. (1998). Students’ Understanding of Mathematical Analysis, PhD Thesis, University of Warwick. Pinto, M. M. F, & Tall, D. O. (1999). Student constructions of formal theory: giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 4, 65–73. Pitta, D. (1998). In the mind. Internal representations and elementary arithmetic, Unpublished Doctoral Thesis, Mathematics Education Research Centre, University of Warwick, UK. Pitta, D. & Gray, E. (1997). ‘In the Mind. What can imagery tell us about success and failure in arithmetic?’ In G. A. Makrides (Ed.), Proceedings of the First Mediterranean Conference on Mathematics, Nicosia: Cyprus, pp. 29–41. 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. Tall, D. O. (1995). Mathematical Growth in Elementary and Advanced Mathematical Thinking, Proceedings of the Nineteenth International Conference for the Psychology of Mathematics Education, Recife, Brazil, I, 61–75. Tall, D. O. & Thomas, M. O. J. (1991). Encouraging Versatile Thinking in Algebra using the Computer, Educational Studies in Mathematics, 22 2, 125–147. Van Hiele, P. (1986). Structure and Insight. Orlando: Academic Press. Yusof, Y. (1995) Thinking Mathematically: A Framework for Developing Positive Attitudes Amongst Undergraduates, PhD thesis, University of Warwick. Yusof, Y. & Tall, D. O. (1996), Conceptual and Procedural Approaches to Problem Solving, Proceedings of PME 20, Valencia, 4, 3–10. Yusof, Y, & Tall, D. O. (1999). Changing Attitudes to University Mathematics through Problemsolving, Educational Studies in Mathematics, 37, 67–82. 
URI:  http://wrap.warwick.ac.uk/id/eprint/473 
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