Location of Repository

Symbols and the bifurcation between procedural and conceptual thinking

By David Tall, Edward Martin Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Michael Thomas and Yudariah Yusof


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.\ud 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\ud 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\ud 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

Topics: QA, L1
Publisher: Routledge
Year: 2001
OAI identifier: oai:wrap.warwick.ac.uk:473

Suggested articles



  1. (1998). Aspects and Layers of the Function Concept, PhD Thesis,
  2. (1998). Cognitive Units, Concept Images and Cognitive Collages, PhD Thesis,
  3. (1999). Concept Maps & Schematic Diagrams as Devices for Documenting the Growth of Mathematical Knowledge.
  4. (1996). Conceptual and Procedural Approaches to Problem Solving,
  5. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. doi
  6. (1991). Encouraging Versatile Thinking in Algebra using the Computer, doi
  7. (1999). Function: Organizing Principle or Cognitive Root?
  8. (1998). In the mind. Internal representations and elementary arithmetic, Unpublished Doctoral Thesis,
  9. (1997). In the Mind. What can imagery tell us about success and failure in arithmetic?’
  10. (1991). Limits. In doi
  11. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on processes and objects as different sides of the same coin, doi
  12. (1997). Persistent errors in indices: a cognitive perspective,
  13. (1986). Procedural and Conceptual Knowledge. In doi
  14. (1991). Reflective abstraction in advanced mathematical thinking. In doi
  15. (1986). Structure and Insight.
  16. (1999). Student constructions of formal theory: giving and extracting meaning.
  17. (1998). Students’ Understanding of Mathematical Analysis,
  18. (1996). Symbolic Manipulation Related to Certain Aspects Such as Interpretations of Graphs, PhD Thesis,
  19. (1999). The Roles of Cognitive Units, Connections and Procedures in achieving Goals in College Algebra.
  20. The student’s construction of quantification,
  21. (1995). Thinking Mathematically: A Framework for Developing Positive Attitudes Amongst Undergraduates, PhD thesis,
  22. (1996). Understanding the limit concept: Beginning with a co-ordinated process schema, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.