This book formulating an overall framework to explain and predict how humans learn to think mathematically has now been published by Cambridge University Press (USA) in September 2013. I was allowed by the publishers to place the first chapter of the book in draft copy here. The price quoted for the book has varied over the last few months. It is now confirmed as paperback $39.99, Hardback $99.99. The best deal I have found, including post and package worldwide is from The Book Depository. (A few errors in the first printing are listed here.)
    It 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. In particular, it contains many examples at all levels to illustrate the ideas for the general reader. 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.
The framework is consistent with other general theories in mathematics education and is designed encourage all interested in mathematics to grasp how mathematical thinking develops in the long-term, both in the development of the child and in the historical evolution of ideas. The development of individual mathematical thinking builds from the child's perceptions and operations through increasingly sophisticated ways of reasoning. At each stage, the learner builds on previous experiences that may work in one situation but then may become supportive or problematic in new contexts. This has cognitive and emotional consequences that offer insight into the bifurcation between pleasurable sense-making and the challenge required to overcome difficulties that may lead either to greater pleasure or to increasing disaffection and even anxiety.

It identifies three essentially different long-term developments of mathematical thinking involving:

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 and leads to connections between mathematical ideas that gives mathematics its coherent structure.

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 later lead to longer-term difficulties.

The framework focuses on cognitive and emotional development of mathematical thinking from the viewpoint of the learner while remaining aware of the crystalline nature of mathematics at all levels. It blends with other theories to reveal how approaches suitable in one context may be problematic in others. It encourages theorists to become aware of their own met-befores that colour their opinions and offers an analysis into current controversies in mathematics education that arise from the views of different communities of practice.


The book 'How Humans Learn to Think Mathematically' is the reflective summation of forty years of research and development. The papers that led to its development may be found in the Downloads page of my website, including the first chapter of the book which can be found either on the Downloads page, or directly from here:
Chapter 1 of How Humans Learn to think Mathematically

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. 31 (1) 3–8.
2013e Mercedes McGowen & David Tall (2013). Flexible Thinking and Met-befores: Impact on learning mathematics, With Particular Reference to the Minus sign. Journal of Mathematical Behavior 32, 527–537.

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.
2013 Integrating History, Technology and Education in Mathematics. Plenary Presentation: História e Tecnologia no Ensino da Matemática, July 15, 2013, Universidade Federal de São Carlos, Brazil.

last modified: Friday September 13 2013.