Seminars in Bogota, Columbia, July 2-5, 2002

Biological Brain, Mathematical Mind and Technological Tools.

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My presentations focus on the way that human beings use a biological brain to build a mathematical mind using technological tools. The group in Columbia base their work on the use of the TI-92. My perception is that this provides a part of a wider scenario in which we build up conceptual structures by perception, action and reflection.I put forward the theory that these three activities produce three fundamentally different worlds of mathematics which I term the EMBODIED, SYMBOLIC-PROCEPTUAL and FORMAL. The embodied world underpins all our activities. It begins with our perceptions and actions on the actual world and through the use of language for internal and inter-personal communication, it builds from perceptual representations to platonic representations. The proceptual world is the world of calculation in arithmetic and symbol-manipulation in algebra and symbolic calculus. It uses very special symbols (called ‘procepts’) which can function either as a process (such as addition) or as a concept (such as sum). This special type of symbol ideally suits the human need to do mathematics by carrying out calculations and manipulations and to think about the concepts of mathematics. Later maturity in embodied and proceptual mathematics leads to our recognising many properties of the concepts we study until we are able to take these properties and use them as fundamental axioms and definitions as a basis of a coherent formal structure of theorems and proof.

The four days each focus on a different aspect of the theory, beginning with an overview on the first day, then successive days on embodied aspects (with particular application to calculus), proceptual development of symbols, and the development of formal proof.

The overheads for the seminars are downloadable for private study.

Photographs taken at the Seminar can be found here.

1. Three worlds of Mathematics
How our biological sensory perception grows to develop EMBODIED, SYMBOLIC & FORMAL mathematical worlds.
The role of the computer as a tool in this development.
Download overheads (2.2mb)
Download draft paper on Three Worlds of Mathematics in the Calculus, as presented in Rio, February, 2002.

Workshop: from embodied to formal geometry using Cabri on the TI-92.

2. An Embodied approach to the Calculus
supported by technology, leading to symbolism & formal proof.
Download overheads (2.4mb), example from TI (36 kb),
FREE download of original Graphic Approach to Calculus (for MS-DOS and windows)
Latest version of Graphic Calculus (for windows).
Other reading:
David Tall (1997): Functions and Calculus. In A. J. Bishop et al (Eds.), International Handbook of Mathematics Education, 289–325, Dordrecht: Kluwer.
David Tall, David Smith and Cynthia Piez (in preparation): Technology and Calculus. To appear in Research in Technology in Teaching and Learning Mathematics.

Workshop: Visual Calculus using the TI-92.

3. Discontinuities in development of Symbols
Cognitive development in arithmetic, algebra, calculus.
Download overheads (1.6 mb)
Other reading:
Thomas, M. O. J. & Tall, D. O. (2001). The long-term cognitive development of symbolic algebra, International Congress of Mathematical Instruction (ICMI) Working Group Proceedings - The Future of the Teaching and Learning of Algebra, Melbourne, 2, 590-597.
David Tall, Eddie Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Michael Thomas, and Yudariah Yusof (2001): Symbols and the Bifurcation between Procedural and Conceptual Thinking, Canadian Journal of Science, Mathematics and Technology Education 1, 81–104.

Workshop: Programming in Calculus using the TI-92.

4. The Development of Proof
How proof develops cognitively through embodied, proceptual and formal stages.
Download overheads (708 kb)
Other reading:
David Tall, (1997) 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.

Workshop: Summarising the week’s ideas, and, if time permits, a final session on Problem-Solving and Proof.

last modified: Sunday, May 30, 2010