Location of Repository

Encouraging versatile thinking in algebra using the computer

By David Tall and Michael Thomas


In this article we formulate and analyse some of the obstacles to understanding the notion of a variable, and the use and meaning of algebraic notation, and report empirical evidence to support the hypothesis that an approach using the computer will be more successful in overcoming these obstacles. The computer approach is formulated within a wider framework ofversatile thinking in which global, holistic processing complements local, sequential processing. This is done through a combination of programming in BASIC, physical activities which simulate computer storage and manipulation of variables, and specific software which evaluates expressions in standard mathematical notation. The software is designed to enable the user to explore examples and non-examples of a concept, in this case equivalent and non-equivalent expressions. We call such a piece of software ageneric organizer because if offers examples and non-examples which may be seen not just in specific terms, but as typical, or generic, examples of the algebraic processes, assisting the pupil in the difficult task of abstracting the more general concept which they represent. Empirical evidence from several related studies shows that such an approach significantly improves the understanding of higher order concepts in algebra, and that any initial loss in manipulative facility through lack of practice is more than made up at a later stage.\u

Topics: LB, QA
Publisher: Springer Netherlands
Year: 1991
OAI identifier: oai:wrap.warwick.ac.uk:508

Suggested articles



  1. (1934). 1628: Regulae ad Directionem Ingenii, (Published posthumously and also translated into English as “Rules for the Direction of the Mind”, doi
  2. 1989a: “Dynamic Algebra”,
  3. 1989b: ““Verbal Evidence for Versatile Understanding of Variables in a Computer Environment”,
  4. (1988). A Conceptual Approach to the Early Learning of Algebra Using a Computer, unpublished Ph.D. thesis,
  5. (1984). Algebra Learning Project -
  6. (1981). Algebra”,
  7. (1986). Building and Testing a Cognitive Approach to the Calculus Using Interactive Computer Graphics,
  8. (1974). Cerebral Dominance Viewed as a Decision System”,
  9. (1990). Concept Images, Generic Organizers,
  10. (1981). Conservation of equation and function under transformations of variable”, doi
  11. (1977). Conservation of Equation, Conservation of Function and Their Relationship to Formal Operational Thinking, unpublished doctoral dissertation,
  12. (1982). Consistent differences in cognitive styles shown for qualitative biological problem-solving”, doi
  13. (1980). Constructing Meaning for the Concept of Equation”,
  14. (1988). Establishing fundamental concepts in algebra through numerical problem-solving”,
  15. (1979). Intelligence, Learning and Action, doi
  16. (1969). Interhemispheric Relationships: The Neocortical Commissures: Syndromes of Hemispheric Deconnection”,
  17. (1903). La Science et l’hypothèse, included in The Foundations of Science (translated by Halsted G.B.,
  18. (1980). Learning Without Understanding : The Effect of Tutorial Strategies on Algebra Misconceptions”,
  19. (1988). Longer-Term Conceptual Benefits From Using A Computer in Algebra Teaching”,
  20. (1981). Pre-algebraic notions among 12 and 13 year olds”,
  21. (1976). Relational understanding and instrumental understanding” doi
  22. (1988). Resequencing Skills and Concepts in Applied Calculus using the Computer as a Tool”, doi
  23. (1985). The Effects of BASIC Computer Programming on the Understanding of the Use of Letters as Variables in Algebra, unpublished M.Sc. thesis,
  24. (1969). The Other Side of the Brain: 2. An Appositional Mind”,
  25. (1977). The Self and Its Brain, doi
  26. (1985). The Social Brain, Basic Books,
  27. (1980). Towards a Computational Theory of Algebraic Competence”,
  28. (1988). WIll computers really count?”,

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