This page gives information about recent changes on the site.

August, 2016. After a period of ill-health, which has restricted my ability to travel, I have continued to work at home. New editions of Foundations of Mathematics and Algebraic Number Theory have been published, work has continued on developing the framework of long-term development of mathematical thinking based on perception, operation and reason. This relates to the natural structure of mathematics, the neurphysiological development of mathematical thinking in the individual and differing communities of practice relating to different aspects of mathematics.

October, 2013. How Humans Learn to Think Mathematically published. Several more papers added to downloads page

February, 2013. Two papers accepted for publication on downloads page
Cauchy’s Conceptions of Function, Continuity, Limit, and Infinitesimal, with implications for teaching the calculus.


January, 2013. Drafts and new papers added to downloads page.

December, 2012. Drafts added to downloads page. CUP has confirmed that How Humans Learn to Think Mathematically is going into production and will appear in 2013.

April 24th, 2012. Page updated with information about the How Humans Learn to Think Mathematically (CUP, forthcoming), materials for my plenary on Making Sense of Mathematical Reasoning and Proof and various drafts of papers that are still under revision. See my downloads page.

January 24th, 2012, I added a paper drafted for PME with Kin Eng Chin on my downloads page. This has developments of ideas related to the development of mathematical concepts through perception, operation and reason with data that shows how student teachers conceptions of trigonometry are different from pupils in school, emphasizing the need for teachers to understand the ideas of supportive and problematic met-befores that may lead unintentionally to teaching procedurally and limiiting student learning.

December 20th, 2011, I have placed some drafts of papers written recently on my downloads page. These include analyses that develop the wider theory of mathematical growth. I have also updated the opening chapter on How Humans Learn to Think Mathematically. This gives the outline of the full theoretical framework.

May 19th, 2011, I put up a page of explanation about the development of the theory on How Humans Learn to Think Mathematically

On March 10th, I added a paper with Mikhail Katz on the tensions between intuitive infinitesimals and formal analysis. Available as a download.

On December 22nd, I added the final version of Crystalline Concepts to appear in For The Learning of Mathematics in January, and a paper with Nellie Verhoef on Lesson Study in Holland. Available as downloads.

On October 19th, I added Perceptions, Operations and Proof in Undergraduate Mathematics, which studies the three different approaches to proof in terms of embodiment, operation, and formal deduction. To appear in the Newsletter of Community for Undergraduale Learning in the Mathematical Sciences, University of Auckland, New Zealand.

On October 18th, I added Mathematical and emotional foundations for lesson study in mathematics, which studies the crystalline structures of increasingly sophisticated concepts in mathematics in terms of met-befores and the effects of the resulting emotional reactions. To be presented as a plenary at the APEC Conference on Lesson Study in Thailand and updated some earlier drafts that had since been published.

On September 13th, 2010 I put up more draft papers, including the ICMI chapter on the Cognitive Development of Proof, a paper on Cauchy’s conceptions of continuity, limit and infinitesimals with Mikhail Katz which has clear implications in modern teaching of calculus and a plenary to be given in Mexico on A Sensible Approach To The Calculus. All of these represent further developments using the framework of embodiment, symbolism and formalism. All are available as downloads.
On May 28th 2010 I put up several new papers that are in draft or being reviewed for publication. They contain some of my most recent ideas, including the notion of crystalline concept as the foundational concept of mathematical thinking and the role of met-before that plays an essential role in the long-term development of mathematical thinking. These are all available as downloads.
On March 2nd 2009, and earlier, I have added several new items. These include a scanned version of an earlier paper on the Blancmange Function (1982a) and recent papers on my downloads page, items 2008c, 2008d, 2008e, 2008f, 2009a and 2009x.
In December 2008, I completed the first draft of a book on How Humans Learn to Think Mathematically. E-mail me to find out more.
On May 5th 2008, I added my plenary talk on ‘The Historical & Individual Development of Mathematical Thinking: Ideas that are set-before and met-before’. HTEM conference, Rio, May 5th 2008. together with the overheads on the Downloads page above.
On May 1st 2008, I added a tribute to my late friend, James J Kaput (1942–2005) ‘Imagineer and Futurologist of Mathematics Education’, a paper with Masami Isoda on ‘Long-term development of Mathematical Thinking and Lesson Study’ and a new theme on Lesson Study.
On 17th August, I added a keynote on 'Setting Lesson Study within a long-term framework of learning'. Presented at APEC Conference on Lesson Study in Thailand, August 2007.

On May 6th 2007, I added a keynote talk on ‘Embodiment, Symbolism, Argumentation and Proof’, to be presented at the Conference on Reading, Writing and Argumentation at National Changhua Normal University, Taiwan, May 23, 2007.

On March 28th 2007, I added the paper ‘Procedural embodiment and magic in linear equations’ with Rosana Nogueira de Lima, accepted for publication in Educational Studies in Mathematics.

On March 12th 2007, I added a final(?) draft of a paper with Eddie Gray on ‘Abstraction as a natural process of mental compression’, submitted to the Mathematics Education Research Journal.

On February 20th 2007, I added a keynote talk on ‘Embodiment, Symbolism and Formalism in Undergraduate Mathematics Education’, presented at the 10th Conference on Research in Undergraduate Mathematics Education, Feb 22–27, 2007, San Diego, California, USA.

On February 17th 2007, I added a keynote talk on ‘Teachers as Mentors to encourage both power and simplicity in active mathematical learning’, presented at The Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, 17–19 March 2007, Abu Dhabi.

On November 6th 2006, I added a draft of a paper on ‘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.

On September 4th 2006, I have added the full festschrift: Retirement as Process and Concept consisting of 29 research papers presented at Charles University, Prague, July 15h 2006 by colleagues and friends from around the world, in honour of Eddie Gray and David Tall. All the papers are copyright the authors, so please contact the original authors given for any copyright issues.

On July 10th 2006, I have added six new papers. How remiss of me not to keep up to date! These include a celebration of those who have influenced my ideas throughout my academic life to be presented in Prague, July 15th, 2006.
On February 20th 2006, a paper with Rosana Nogueira on The Concept Of Equations: What Have Students Met Before? has been added to the drafts page.

On November 23rd 2005, a paper with John Pegg on the fundamental cycle of concept construction has been added.
On May 12th 2005, new drafts have been added for plenary talks to be given on the development of mathematics from childhood to adulthood, and the transition from worlds of embodiment and symbolism to the world of formal proof.
On October 31st, 2004, four papers have been added to the drafts page.
On Wednesday July 24, 2004, as the result of many requests, the Proceedings of MALT I (Mathematics And Lateral Thinking) have finally been released to the public. This includes the paper:
Davis, G.; di Giacomo, S.; Gray, E. M.; Hegedus, S.; McGowen, M.; Pinto, M. M. F.; Pitta, D.; Simpson, A. P.; Tall, D. O. (1997): The Object of the Encapsulation of a Distilled Spirit. Proceedings of Malt I, 22-100.
which we claim to be the shortest long paper ever published.
On May 29, 2004, Several papers have been moved to the downloads page, including:
Plenary for the Topic Subgroup on Calculus at ICME (with Juan Pablo Mejia Ramos)
Thinking things through three worlds of mathematics (the origins of the theory) to be presented at PME28
Reflections on research and teaching of equations and inequalities for PME28.
Photographs of recent visits to Rio De Janeiro and Sao Paulo added to my home site.

On January 15, 2004, several papers have been added to the downloads page and to (downloadable) drafts. These include two papers on ‘Three worlds of mathematics’ one outlining the origins of the theory, the other responding to comments made about the theory in For the Learning of Mathematics.

On November 3rd, 2003, some modifications to the draft of Mathematical Growth were added. These are not for citation, but are for comment. Should you read them and have any comment prior to publication, please contact me by e-mail.

On July 7th, 2003, draft chapters of the Book on Mathematical Growth were added here.
On January 27th, 2003, a new theme on ‘Three Worlds of Mathematics’ was added together with four new articles for downloading.

See books for information on Fermat's Last Theorem (with Ian Stewart) and Intelligence Learning and Understanding (A Tribute To Richard Skemp) with Michael Thomas.
October 3rd, 2002,: a new draft paper:
Anna Watson, Panayotis Spirou, David Tall, The Relationship between Physical Embodiment and Mathematical Symbolism: The Concept of Vector.
This includes further interpretations of the ‘Three Worlds of Mathematics’ (embodied, proceptual, formal), now applied to the case of vector. (see below)

September 24th 2002, some new draft papers added: (see drafts).

Victor Giraldo, Luiz Mariano Carvalho, David Tall: Theoretical-Computational Conflicts and the Concept Image of Derivative, submitted to BSRLM.
(In Portuguese): Victor Giraldo, Luiz Mariano Carvalho, David Tall, Conflitos Teorico-Computacionais e a Formacao da Imagem Conceitual de Derivada, to appear in the Annals of History and Technology in Mathematics Education, Rio de Janiero.

September 9th 2002, several new published works added (see downloads).

At present I am working on a new formulation of different kinds of mathematical thinking in different contexts referring to Three Worlds of Mathematics (Embodied, Proceptual, Formal). My first draft of the ideas of “the three worlds of mathematics” (applied to the case of calculus) is available in my presentation from the Rio Conference in February.

July 1st 2002, Materials added for my presentations in Columbia.
March 4th, 2002: Materials have been added from my presentations in Brazil.

All downloads are operational. However, the themes are only in a preliminary form and the glossary is not yet developed. Watch this space!

February 12th 2002. Added drafts:

Final version of:
Marcia Pinto & David Tall (2002). Building formal mathematics on visual imagery: a case study and a theory. For the Learning of Mathematics, 22(1) (in press).

January 15th Added drafts:

Several papers for PME26 were submitted. (See Drafts.) A new rule of PME limits co-authored papers to 2 for any individual. I have been informed that my name must be deleted from any papers exceeding the quota. Watch this space.

December 9th 2002. Added draft:

John Pegg & David Tall: Fundamental Cycles in Learning Algebra: An Analysis, ICMI Conference on Algebra, Melbourne, Dec 2001.

December 7th 2001. Several papers added in updated draft form, including:

David Tall and Tony Barnard, Cognitive Units, Connections and Compression in Mathematical Thinking. (Finalised and submitted Jan 31st 2002).
David Tall, David Smith and Cynthia Piez, Technology and Calculus. In preparation for Research in Technology in Teaching and Learning Mathematics.
Gary Davis and David Tall: What is a scheme? Prepared for Intelligence, Learning and Understanding: A Tribute to Richard Skemp.

last modified: Friday, May 20, 2011