**ADVANCED MATHEMATICAL THINKING**

Advanced mathematical thinking (AMT) is concerned with the introduction of formal definitions and logical deduction. Of particular interest is the transition from elementary school mathematics (geometry, arithmetic, algebra) to advanced mathematical thinking (axiomatic proof) at university.

My ideas have developed over my life, starting from my early textbooks for undergraduates, the edited book on * Advanced Mathematical Thinking* and the development of a theoretical framework of three distinct worlds of mathematics, embodied, (proceptual) symbolic, and formal. (eg Tall 2004, 2005).

My initial interest began as a mathematician in the early 70s, when I began to write textbooks on undergraduate mathematics with Ian Stewart, including** Foundations of Mathematics, Complex Analysis, Algebraic Number Theory and Fermat's Last Theorem. **There followed my own research in the late 70s with studies of students moving in transition from calculus to analysis, concepts of limits and infinity, and the concept images constructed by students to give meaning to formal mathematical concepts. Such images may or (more usually) may not be consonant with the formal theories. Research developed by Marcia Pinto in her PhD thesis under my supervision revealed that some individuals have a

Even though AMT is signalled by the existence of formal proof, the creation of new mathematical theories requires a sense of global relationships that suggest suitable definitions and proof. Tall 2001 suggests that not only do thought experiments suggest possible new theorems (which then require appropriate formal proof), but formal proofs can also lead to *structure theorems* that enable the prover to use the structure to imagine a more subtle embodiment. Thus thought experiment and formal proof can complement each other through creative thought processes that move from one to the other and back again.

More recent papers are leading me to the realisation that embodiment and formalism are complementary and that both proceptual symbolism and formal proof from verbally formulated definitions both use cognitive structure that no longer links simultaneously to embodiment. We manipulate numbers mentally without thinking of groups of objects. We do algebra by manipulating symbols that have meaning which need no longer be linked to an embodied meaning. Formal proof may work for many different embodiments and a link to a single embodiment can be misleading (Tall & Chin 2000). Later research has led me to realise that embodiment, proceptual thinking and formal proof essentially inhabit quite different mental worlds, with different forms of proof (embodied proof is by thought experiment or physical experiment to confirm a prediction, proceptual proof is through calculation or symbol manipulation, formal proof is through logical deduction.) Latest developments are found in a presentation on three different worlds of mathematics in calculus/analysis given in Rio de Janiero and the Overheads for a recent talk in Bogota. These ideas are currently being developed into a book available in draft form here and are in my most recent publication on the topic here.

*Selected papers on Advanced Mathematical Thinking:*

1981d The mutual relationship between higher mathematics and a complete cognitive theory of mathematical education, *Actes du Cinquième Colloque du Groupe Internationale P.M.E.*, Grenoble, 316-321.

[The ‘call to arms’, seeking an extension of school mathematics education to university level.]

In 1985, Gontran Ervynck (who originated the term 'advanced mathematical thinking') and I founded the Advanced Mathematical Thinking Working Group of PME, with later discussions such as:

1988i The Nature of Advanced Mathematical Thinking, *Papers of the Working Group of AMT*.

An early Working Group discussion paper on Advanced Mathematical Thinking.

This led to:

*Advanced Mathematical Thinking* (ed. David Tall), published by Kluwer, 1991, including the following chapters:

1991k The Psychology of Advanced Mathematical Thinking, in Tall D. O. (ed.) Advanced Mathematical Thinking, Kluwer: Holland, 3-21.

1991l (with Ed Dubinsky), Advanced Mathematical Thinking and the Computer, in Tall D. O. (ed.), Advanced Mathematical Thinking, Kluwer: Holland, 231-248.

1991m Reflections, in Tall D. O. (ed.), Advanced Mathematical Thinking, Kluwer: Holland, 251-259.

Other papers followed, for example:

1992c (with Md Nor Bakar) Students' Mental Prototypes for Functions and Graphs, Int. J. Math Ed Sci & Techn., 23 1, 39-50.

1991b (with Guershon Harel) The General, the Abstract, and the Generic in Advanced Mathematics, *For the Learning of Mathematics*, 11 1, 38-42.

1992e The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity, and Proof, in Grouws D.A. (ed.) Handbook of Research on MathematicsTeaching and Learning, Macmillan, New York, 495-511. [This is a review of some of the contemporary research about the transition from school to university].

1992m Construction of Objects through Definition and Proof, *PME Working Group on AMT*, Durham, NH.

1992f Current difficulties in the teaching of mathematical analysis at university: an essay review of Victor Bryant Yet another introduction to analysis, *Zentralblatt für Didaktik der Mathematik*, 92/2, 37-42.

1993a Mathematicians Thinking about Students Thinking about Mathematics, *Newsletter of the London Mathematical Society*, 202, 12-13 (full version available as pre-print).

1993d (with Mohamad Rashidi Razali), Diagnosing Students' Difficulties in Learning Mathematics, *Int J. Math Ed, Sci & Techn.*, 24 2, 209-202.

1994e The Psychology of Advanced Mathematical Thinking: Biological Brain and Mathematical Mind, *Abstracts of the Working group on Advanced Mathematical Thinking (A.M.T.)*, PME 18, Lisbon, 33-39.

1994g Understanding the Processes of Advanced Mathematical Thinking, *Abstracts of Invited Talks, International Congress of Mathematicians*, Zurich, August 1994, 182-183.

1995b Mathematical Growth in Elementary and Advanced Mathematical Thinking, plenary address. In L. Meira & D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, I, 61-75. [The first paper I wrote covering the full span of mathematical development from child to mathematician.]

1996a Advanced Mathematical Thinking and the Computer. Proceedings of the 20th University Mathematics Teaching Conference, Shell Centre, Nottingham.

1996i Understanding the Processes of Advanced Mathematical Thinking, *L'Enseignement des Mathématiques*, 42, 395-415. [The text of a talk I gave at the International Congress of Mathematicians, Zurich, 1994.]

1997 Making Research in Mathematics Education Relevant to Research Mathematicians, Paper presented to MER Session of AMS/MAA Conference, San Diego, January 1997. Published on the World Wide Web: http://www.math.uic.edu/MER/sps/Tall/paper.html

1997 Cognitive Difficulties in Learning Analysis, for *Mathematical Association Committee on Teaching Undergraduate Mathematics*. Published on the world wide web http://www.bham.ac.uk/ctimath/talum/austin/analysis/tall/tall.htm

1997f Metaphorical objects in Advanced Mathematical Thinking, *International Journal of Computers for Mathematical Learning* 1, 61–65.

1998a (with Liz Bills), Operable Definitions in Advanced Mathematics: The case of the Least Upper Bound, *Proceedings of PME 22*, Stellenbosch, South Africa, 2, 104-111. [Tracing the development of students' ideas of least upper bound to see if and when they become 'operable' in the sense that the student can operate with the definition to deduce theorems.]

1997g From School to University: the effects of learning styles in the transition from elementary to advanced mathematical thinking. In Thomas, M. O. J. (Ed.) *Proceedings of The Seventh Annual Australasian Bridging Network Mathematics Conference*, University of Auckland, 9-26.

1999c Reflections on APOS theory in Elementary and Advanced Mathematical Thinking. In O. Zaslavsky (Ed.), *Proceedings of the 23rd Conference of PME*, Haifa, Israel, 1, 111-118.

1999k (with Eddie Gray, Demetra Pitta, Marcia Pinto), Knowledge Construction and diverging thinking in elementary and advanced mathematics, E*ducational Studies in Mathematics*, 38 (1-3), 111–133. [A review of the bifurcation between success and failure through cognitive development of the growing individual.]

2000h Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). In Wei-Chi Yang, Sung-Chi Chu, Jen-Chung Chuan (Eds), *Proceedings of the Fifth Asian Technology Conference in Mathematics*, Chiang Mai, Thailand (pp. 3–20). ATCM Inc, Blackwood VA. ISBN 974-657-362-4. [A summary of my work on cognitive development with special reference to the calculus.]

2001a David Tall, Eddie Gray, Maselan Bin Ali, Lillie Crowley, Phil DeMarois, Mercedes McGowen, Demetra Pitta, Marcia Pinto, Michael Thomas, and Yudariah Yusof (2001). Symbols and the Bifurcation between Procedural and Conceptual Thinking, *Canadian Journal of Science, Mathematics and Technology Education 1*, 81–104. [a summary of several research studies looking at the success and failure in thinking of mathematical concepts at various stages of the curriculum]

2001b David Tall (2001). Cognitive Development in Advanced Mathematics Using Technology. *Mathematics Education Research Journal*. 12 (3), 196–218. [An overview of my work.]

2001c Chae, S. D., & Tall, D. O. (2001). Aspects of the construction of conceptual knowledge in the case of computer aided exploration of period doubling. In C. Morgan, T. Rowlands (Eds), *Research in Mathematics Education, Volume 3*. BSRLM Publications, Graduate School of Education, University of Bristol, England.

2001d Tony (A. D.) Barnard & David Tall (2001). A Comparative Study of Cognitive Units in Mathematical Thinking. In Marja van den Heuvel-Panhuizen (Ed.) *Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education* 2, 89–96. Utrecht, The Netherlands.

2001g (Abe) Ehr-Tsung Chin & David Tall (2001). Developing Formal Mathematical Concepts over Time. In Marja van den Heuvel-Panhuizen (Ed.) *Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education 2*, 241-248. Utrecht, The Netherlands. [Students developing through a sequence of stages: informal, definition-based, theorem-based, compressed concept.]

2001j Marcia Maria Fusaro Pinto & David Tall (2001). Following student’s development in a traditional university classroom. In Marja van den Heuvel-Panhuizen (Ed.) *Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 4*, 57-64. Utrecht, The Netherlands.

2001m Chae, S. D. and Tall, D. (2001). ‘Students’ concept images for period doublings as embodied objects in chaos theory’. Proceedings of the International Conference on Computers in Education, Vol. 3, 1470-1475, Seoul: Korea. [The subtleties of student thought in experimenting with visual images of recursion and chaos.]

2001o David Tall and Dina Tirosh (2001). Infinity – The never-ending struggle, *Educational Studies in Mathematics, 48* (2&3), 129–136. [an introduction to a collection of papers on infinity.]

2001p David Tall (2002). Natural and Formal Infinities, *Educational Studies in Mathematics*, 48 (2&3), 199–238.[This paper contrasts natural ideas that are built by extending images of finite concepts and formal ideas developed through definition and formal proof.]

2002a Marcia Pinto and David Tall (2002). Building formal mathematics on visual imagery: a theory and a case study. *For the Learning of Mathematics* . [The case of a gifted 'natural' learner who builds his conceptions of formal mathematics by thought experiment, reconstructing his images as he goes.]

2002i David Tall and Erh-Tsung Chin, (2002). *University students embodiment of quantifier*. In Anne D. Cockburn & Elena Nardi (Eds), *Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4*, 273–280. Norwich: UK. [How some students use prototypical embodiments to handle general statements involving the universal quantifier.]

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