Recent Research in Mathematics Education at Warwick University

The seminars on “Recent Research at Warwick University” are specified in terms of five three hour sessions. Each session is based on a particular topic, and papers for that session consist of one ideas paper and one or more research papers. The ideas papers cover the broad area in a popular style, the research papers involve empirical research related to the ideas. At the moment it is my plan to present materials from the ideas papers, referring to some of the specified research articles. However, I will respond to the audience should a different manner of presentation be appropriate.

The files here are in PDF (portable document format) which may be read using Adobe Acrobat Reader on any PC, Unix Machine, or Macintosh computer.


Click here to download a copy of Acrobat Reader for your computer.
If you have any difficulties, please e-mail me at

1. Cognitive development in ARITHMETIC

Ideas: How young children encapsulate counting processes as number concepts - the notion of PROCEPT and how this underlies success and failure in arithmetic.

1993b, Eddie Gray and David Tall: Success and Failure in Mathematics: The Flexible Meaning of Symbols as Process and Concept, Mathematics Teaching, 142, 6—10.

Research: Data on children doing arithmetic age 7-11, and how the use of an advanced calculator can help a child move from procedural counting to a conceptual approach using number concepts.

1994a, Eddie Gray and David Tall: Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic, The Journal for Research in Mathematics Education, 26 (2), 115—141.

1997, Gray, E.M. & Pitta, D. Changing Emily’s Images, Mathematics Teaching, 161, 38–51

2. Cognitive development in ALGEBRA

Ideas: Algebraic symbols as process and object, how to use a computer to improve flexible thinking.

1993f, David Tall: The Transition from Arithmetic to Algebra: Number Patterns or Proceptual Programming?, New Directions in Algebra Education, Queensland University of Technology, Brisbane, 213—231.Research: Selections from several pieces of research into variables and straight line equations

1991c, David Tall & Michael Thomas: Encouraging Versatile Thinking in Algebra using the Computer, Educational Studies in Mathematics, 22 2, 125—147.

1999d, Lillie Crowley & David Tall, 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.

1999f, Mercedes McGowen & David Tall: 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.

1999e, Phil DeMarois & David Tall: Function: Organizing Principle or Cognitive Root? In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel, 2, 257—264.

2000e, Mercedes McGowen, Phil DeMarois and David Tall: The Function Machine as a Cognitive Root for the Function Concept, Proceedings of PME-NA (in press).

3. Cognitive development in CALCULUS

Ideas: A locally straight approach to the calculus using visual software.

1985a, David Tall: Understanding the calculus, Mathematics Teaching 110 49—53.

Research: A review of research into recent developments in the teaching of calculus.

1997a,, David Tall: Functions and Calculus. In A. J. Bishop et al (Eds.), International Handbook of Mathematics Education, 289-325, Dordrecht: Kluwer.

4. Cognitive development in PROOF

Ideas: Showing how the notion of proof is dependent on the child's development.

1999j, David Tall: The Cognitive Development of Proof: Is Mathematical Proof For All or For Some? In Z. Usiskin (Ed.), Developments in School Mathematics Education Around the World, vol, 4, 117-136. Reston, Virginia: NCTM

Research: Two papers on very different approaches to proof, one showing how university students build proofs in different ways, the other showing how some students cannot cope with proof and have idiosyncratic images of the meaning of rational number.

1999g, Márcia Maria Fusaro Pinto & David Tall, 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.

1996h, Márcia Maria Fusaro Pinto & David Tall: Student Teachers’ Conceptions of the Rational Numbers, Proceedings of PME 20, Valencia, 4, 139—146.


Ideas: A summary of the cognitive development of symbolism and proof from a range of sources.

2001a, David Tall, Eddie Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Yudariah Yusof). Symbols and the Bifurcation between Procedural and Conceptual Thinking, Canadian Journal of Science, Mathematics and Technology Education (in press).

Research: Another view of cognitive development, this time with a balance between the visual and the symbolic.

1995b, David Tall: Mathematical Growth in Elementary and Advanced Mathematical Thinking, plenary address. In L. Meira & D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, I, 61—75.

last modified: Friday, October 4, 2002