HOW HUMANS LEARN TO THINK MATHEMATICALLY

 

My book explaining my overall framework to explain and predict how humans learn to think mathematically is due to be published by Cambridge University Press (USA) on June 30 2013. I have been allowed by the publishers to place the first chapter of the book in draft copy here. The book can be ordered in advance on the web at £24 paperback, £60 hardback. The best deal I have found is £18 including post and package worldwide from The Book Depository. The book is written for a wide range of readers in a manner that is both theoretically sound and practically helpful, not only for theorists, but for teachers and learners. It begins with the new born child and studies how children, students and mathematicians develop ideas at successive levels of sophistication. It offers a foundation of human mathematical thinking in terms of perception, action and language and how this develops in two complementary ways in school (through human perception, action and thought experiment on the one hand and the operational use of symbols that translate operations such as counting and measuring to mathematical concepts such as number). At university level the ideas are reorganised into a third way of working based on set-theoretic definitions and deductions.

This leads to three fundamentally different ways of building mathematical concepts: through a focus on:

• the structural properties of objects (as in geometry, or in graphical representations)
• the operational properties of actions on objects (as in counting, sharing, arithmetic, algebra)
• the formal properties of mathematical objects given by formal definition and proof.

The overall linking concept is the notion of crystalline concept. This is a concept with strong connections that are implicit in a given concept, for instance the notion of 'isosceles triangle' in geometry which not only has two equal sides, it must have two equal angles and other properties, such as symmetry about the line bisecting the vertex, or 'five' in arithmetic which not only equals 3+2 or 2+3 or 7–2, but also, if 3 is taken from 5 then 2 must be left, and so on. This idea develops in appropriate ways in each of the three worlds of mathematics.

The transition to new areas of mathematics is built on experiences that were met before and in a new context may be supportive (e.g. 2+2=4) or problematic (e.g. take away makes smaller in the context of negative quantities). Supportive met-befores allow appropriate generalization and give pleasure in operating in new contexts. For a confident individual a problematic met-before may cause frustration and a determination to conquer the problem. For a less-confident individual, a problematic met-before may cause anxiety and the desire to avoid the problem. In the latter case, the individual may seek to learn procedures by rote to be able to pass tests. If successful, this gives a new kind of pleasure in passing examinations, but it may not promote a flexible way of thinking and lead to longer-term difficulties.

Earlier papers that eventually led to the full framework are given below. Important papers are 2010b on met-befores, 2011a on crystalline concepts, and the keynote talk 2010c on the effects of emotion in mathematical thinking.
2004a Introducing Three Worlds of Mathematics. For the Learning of Mathematics. The first paper on three worlds written as a response to published comments on 'the three worlds', at that time under development, including a discussion on the building of theories.
2004b Thinking Through Three Worlds of Mathematics.Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Bergen, Norway, 4, 281–288. An introduction to the origins and ideas in ‘the three worlds‘.
2005e David Tall (2005). A Theory of Mathematical Growth through Embodiment, Symbolism and Proof. Plenary Lecture for the International Colloquium on Mathematical Learning from Early Childhood to Adulthood, Belgium, 5-7 July 2005. A description of the framework of development from child to adult, starting from foundational principles.
2005f David Tall (2005). The transition from embodied thought experiment and symbolic manipulation to formal proof. Plenary Lecture for the Delta Conference, Frazer Island, Australia, November 2005. [An analysis of the transition from conceptual embodiment and proceptual symbolism to formal proof.]
2006b David Tall (2006). A life-time’s journey from definition and deduction to ambiguity and insight. Retirement as Process and Concept: A Festschrift for Eddie Gray and David Tall, Prague. 275-288, ISBN 80-7290-255-5. [A celebration of those who have taught me almost everything I know.]
2006h David Tall & Juan Pablo Mejia-Ramos (2006). The Long-Term Cognitive Development of Different Types of Reasoning and Proof, presented at the Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany. (pre-publication draft).
2007f  Eddie Gray & David Tall (2007). Abstraction as a natural process of mental compression. Mathematics Education Research Journal, 19 ( 2), 23–40.
2008e David Tall (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 2008, 20 (2), 5-24
2009x David Tall (2009). Cognitive and social development of proof through embodiment, symbolism & formalism. ICMI Conference on Proof.
2010b Mercedes McGowen & David Tall (2010). Metaphor or Met-before? The effects of previous experience on the practice and theory of learning mathematics. Journal of Mathematical Behavior 29, 169–179.
2010d David Tall (2010). Perceptions, Operations and Proof in Undergraduate Mathematics, CULMS Newsletter (Community for Undergraduale Learning in the Mathematical Sciences), University of Auckland, New Zealand, 2, November 2010, 21-28.
2011a David Tall (2011) Crystalline concepts in long-term mathematical invention and discovery. For the Learning of Mathematics. January 2011.

For more recent papers, look at the latest downloads.

Some recent keynote talks focusing of different aspects of the theory:

2006h David Tall (2006). Encouraging Mathematical Thinking that has both power and simplicity. Plenary presented at the APEC-Tsukuba International Conference, December 3–7, 2006, at the JICA Institute for International Cooperation (Ichigaya, Tokyo). [The overall framework of three worlds of mathematics for an audience interested in elementary school teaching, concentrating on the relationship between embodiment and symbolism.]
2007b David Tall (2007). Embodiment, Symbolism and Formalism in Undergraduate Mathematics Education, Plenary at 10th Conference of the Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, Feb 22–27, 2007, San Diego, California, USA. [A presentation to an audience interested in undergraduate mathematics education, concentrating on the relationship between embodiment and symbolism in school and the formalism of definition-theorem-proof.] [Overheads]
2007c  David Tall (2007). Teachers as Mentors to encourage both power and simplicity in active mathematical learning. Plenary at The Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, 17–19 March 2007, Abu Dhabi. [A presentation to secondary mathematics teachers, focusing on the relationship between embodiment and symbolism and the need for teachers to take into account ideas of compression of knowledge and what students bring to their studies.]
2007d Embodiment, Symbolism, Argumentation and Proof, Keynote presented at the Conference on Reading, Writing and Argumentation at National Changhua Normal University, Taiwan, May 2007.
2007e Setting Lesson Study within a long-term framework of learning. Presented at APEC Conference on Lesson Study in Thailand, August 2007.
2010c Mathematical and emotional foundations for lesson study in mathematics. Plenary presented at the APEC Lesson Study Conference, Chiang Mai, Thailand, November 2010.
2012c Making Sense of Mathematical Reasoning and Proof. Plenary at Mathematics & Mathematics Education: Searching for Common Ground: A Symposium in Honor of Ted Eisenberg, April 29-May 3, 2012, Ben-Gurion University of the Negev, Beer Sheva, Israel.