This page gives information about recent changes on the site.

**August**, 2018. I still continue to work at home and welcome visitors and internet contacts. The second edition of *Complex Analysis* with Ian Stewart has a new chapter that revolutionises how we can think about the calculus, both in history and in teaching and research in analysis. This is part of a wider framework that includes neurophysiological ideas concerning how the human brain perceives the world, processes information internally, and communicates ideas through gestures, writing, drawing and speaking. It also focuses on how the inner brain enhances or suppresses neuronal links in a primitive 'fight or flight' reaction that affects attitudes, gives pleasure, or causes anxiety. This extends the framework of 'three worlds of mathematics'.

This has fundamentally enhanced and profoundly changed the way in which I view research and publications on the teaching and learning of mathematics. Drafts will appear on my **downloads page** and on ResearchGate as the work progresses.

**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 Mathematic*s 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, 2017.* *Continuing to update downloads page

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

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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.

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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.

which we claim to be the shortest long paper ever published.

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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.

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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

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!**

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).

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.

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

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*.

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