Selected Lectures

The information on this page relates to lectures and seminars given at various conferences.They represent various stages of thinking as my work develops, building on preceding work and presentations in other places. On occasion a lecture that is given for one audience (say mathematicians) is modified for another (say teachers). So there is overlap between some of them. The later versions usually represent the latest thinking.

All of the following are keynote talks in which I have chosen to speak on the overall framework of three worlds of mathematics in different parts of the curriculum.

The three worlds of mathematics and the brain. Overheads of plenary relating the long-term framework of 3 worlds and the neurophysiology of the brain. April 2016.
The three worlds of mathematics focused on the calculus. Latest development of ideas focusing on the calculus, clarifying the focus in the three worlds on perception of, operations on and properties of mathematical objects relating dynamic change and growth, building from the coherence of practical ideas to the consequence of theoretical formulations to the set-theoretic logical definition and deduction in analysis. April 2016.
The Historical & Individual Development of Mathematical Thinking: Ideas that are set-before and met-before. Plenary at HTEM conference UFRJ, Rio, May 5th 2008. Overheads. The first presentation of my hypothesis that three set-befores (recognition, repetition and language) underpin mathematical thinking and lead to the three worlds of mathematics.
Learning to think flexibly in mathematics using Japanese Lesson Study. Overheads of talks given to Masters students at URFJ and at UNIBAN Saõ Paulo.
The way in which the introduction of algorithms in Japanese lesson study relates to the theory of compression of operations into knowledge structures relating embodiment and symbolism.What we have met before: why individual students, mathematicians & math educators see things differently. Plenary at UNIBAN, Saõ Paulo May 13 2008. Overheads. Also presented to Masters students at Severino Sombra University, Vassouras and at URFJ.
The first presentation of the manner in which individual met-befores cause differences between the ways mathematicians, mathematics educators and students conceive of mathematical concepts.Setting Lesson Study within a long-term framework of learning. Presented at APEC Conference on Lesson Study in Thailand, August 2007. The first presentation of my ideas linking the three worlds of mathematics to the processes of Japanese Lesson Study.
Embodiment, Symbolism, Argumentation and Proof, Keynote presented at the Conference on Reading, Writing and Argumentation at National Changhua Normal University, Taiwan, May 2007. The development of argumentation and proof through the three worlds of mathematics, (including the first mention of blending knowledge structures).
 Teachers as Mentors to encourage both power and simplicity in active mathematical learning. 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.

Embodiment, Symbolism and Formalism in Undergraduate Mathematics Education
Research in Undergraduate Mathematics Education, San Diego, USA, February 2007.
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.
[The overheads for the presentation are here.]

Encouraging Mathematical Thinking that has both power and simplicity.
APEC-Tsukuba International Conference, December 3–7, 2006, Tokyo, Japan, December, 2006.
A presentation the overall framework of three worlds of mathematics to an audience interested in elementary school teaching, concentrating on the relationship between embodiment and symbolism.
The transition from embodied thought experiment and symbolic manipulation to formal proof. Fifth Southern Hemisphere Symposium on Undergraduate Mathematics and Statistics Teaching and Learning Fraser Island, Australia 2005.
A conference for university mathematicians and mathematics educators.This plenary was accidentally reviewed as a research report and rejected. It was well received as a plenary ... Does this say my paper is poor, or that those who review it didn't understand what I was talking about? Read it and see...
A Theory of Mathematical Growth through Embodiment, Symbolism and Proof. 
L'enseignement des mathématiques de la prime enfance à l'age adulte, Mons, Belgium, July 2005, later published in Annales de Didactique et de Sciences Cognitives, Irem de Strasbourg. 11, 195–215.
Prepared for a conference on the full development of mathematical growth from infancy to adult.


Seminars, Taipei, Taiwan, 2002

The Theory of Procepts, October 14, 2002. Three Worlds of Mathematics, October 21, 2002 (National Taiwan Normal University Mathematics Education Department).

How the Biological Brain affects the Mathematical Mind using Visual and Symbolic Ideas, (version for Science Education Students), October 31, 2002, The relationship between physical embodiment and mathematical symbolism: the concepts of force and vector, November 7th 2002 (in Science Education Department).

Information and overheads for downloading

Four seminars, Bogota, Columbia, 2002
Overheads for Biological Brain, Mathematical Mind and Technological Tools
Presented in Bogota, July 2–5, 2002.

Plenary Lecture, Universidade do Estado do Rio de Janiero February, 2002
Download Overheads for Using Technology to Support an Embodied Approach to Learning Concepts in Mathematics
Presented at the First Workshop in History and Technology in the Teaching of Mathematics. Rio De Janiero, February 21-23, 2002.

Seminar, Universidade Federal Rio de Janeiro February, 2002
How do we think about axioms and proof in mathematics?
Overheads and list of papers

Seminar, Birmingham October 2001
What research into mathematical thinking tells us at university level
Overheads and list of papers

Seminars for USA, September 2001
Overheads for Visualization and Proof in Calculus and Mathematical Analysis, outlines for Technology and Calculus, The Theory of Procepts
Plan of seminars and list of papers 

Seminars for Brazil, September 2000
five three-hour seminars on cognitive development in arithmetic, algebra, calculus, proof and an overall Theory of Cognitive Development
Plan of seminars and list of papers

Other selected Overheads for direct download

2002 How do we think about axioms and proof in mathematics?

2001 Biological Brain, Mathematical Mind and Computational Computers

2000 Transition from EMT to AMT

2000 Problem-solving

2000 National Curriculum

1998 Plenary at Mathematical Association Annual Conference

last modified: Wednesday, May 14, 2008