**PROOF**

What is proof? Mathematicians think they know, and they share a culture in which they consider proof to be central. However, their concept of mathematical proof is meaningless to a young child. So how do children come to know about the meaning of proof? My research has looked at both the cognitive development of proof and also the cognitive nature of formal proof. Below are papers I have written over the years which show a common thread building from early experiences, intuitions and thought experiments using visualisation and on to symbolism and proof.

As children grow in experience and maturity, different forms of 'proof' become available. In ‘te cognitive development of proof’ (1999j below) I relate possible forms of proof to the child's cognitive development, including physical enactive proof through arithmetic and algebraic manipulation, thought experiments imagining prototypical instances, and on to formal proof in university mathematics. Later these ideas have been refined and integrated into the development of mathematical thinking through the embodied and symbolic worlds of elementary mathematics and the formal-axiomatic world of advanced mathematics (**2002k** below, **2005c **below).

At university level, I began to realise the enormous change that students encounter in moving from school mathematics to formal proof and how they operate in different ways to attempt to make sense of it. The research of Marcia Pinto revealed how individuals develop the notion of formal proof in different ways. *Natural learners* develop their concept imagery to embrace and illuminate the given concept definition; this allows them to imagine thought experiments based on imagery that suggest possible theorems. *Formal learners* use the concept definition to construct the mathematics directly through formal deduction. The distinction between thought experiments (involving conceptual embodiment) and formal proof (involving formal deduction) has led to further theoretical developments. The formal approach developed by Hilbert builds on embodied concepts which are axiomatised and used as a basis for formal theory. Conversely, structure theorems give more sophisticated embodiments to formal theory. Abe Chin found that students embody their definitions by relating them to earlier experiences and this embodiment can act as an obstacle to the development of formal theory.

Now Pablo Mejia-Ramos is working on public and private aspects of proof and relating this to individuals earlier experiences of embodiment and symbolism. The quest continues to see mathematical proof as part of a natural human development from embodied perceptions and symbolized actions to increasingly sophisticated properties that themselves become axioms as foundations for formal theories and formal proofs.

**1979d Cognitive aspects of proof, with special reference to the irrationality of the square root of 2**, *Proceedings of the Third International Conference for the Psychology of Mathematics Education*, Warwick, 206-207.

A preliminary study of first year university students understanding of the irrationality of root 2. Students on the whole prefer a thought experiment that shows that squaring a number doubles the number of occurrences of its prime factors. The paper discusses the conflict between making the idea meaningful in terms of prototypical examples andthe formulation of proof by contradiction. The former has a short-term payoff, but the latter is essential in the long-term development of axiomatic proof.

1989a The nature of mathematical proof, *Mathematics Teaching*, 127, 28-32.

A humorous reflection on the nature of proof written for teachers and teenagers, with some deeper considerations.

1991e To prove or not to prove, *Mathematics Review* 1 3, 29-32.

A rewrite of the previous paper specifically for sixth-form students (aged 16-18).

1988h (with John Mills) From the Visual to the Logical, *Bulletin of the I.M.A.* 24 11/12 Nov–Dec, 176–183.

Thinking about the relationship between visual intuition (supported by computers) and logical proof.

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.

A review of the literature concerning the transition to formal mathematics, including a consideration of the transition to formal proof.

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

Preliminary ideas concerning the way in which mathematical concepts are *constructed* from definitions using formal proof.

1995f Cognitive Development, Representations & Proof. In *Justifying and Proving in School Mathematic*s, Institute of Education, London, 27-38.

A first consideration of the different types of proof available depending on the representations that are meaningful for the learner.

1997d (with Tony Barnard), Cognitive Units, Connections and Mathematical Proof, *Proceedings of PME 21*, Finland, 2, 41-48.

My first attempt to develop Tony Barnard's notion of 'cognitive unit' relating it to the proof that root 2 is irrational.

**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.

A paper which I regard as being seminally important, (although I have not followed it up as carefully as it deserves). This paper investigates the development of students’ abilitiy to use a definition in an operable manner to deduce theorems as they follow a lecture course introducing a definition and then using it to build a formal theory. Of the five students interviewed over two 10 week terms, none were able to make the definitions operable. One struggled with the definition, one learned it because of the interviews, the other three simply built intuitively on their concept imagery.

1998b (with Adrian Simpson), Computers and the Link between Intuition and Formalism. In *Proceedings of the Tenth Annual International Conference on Technology in Collegiate Mathematics*. Addison-Wesley Longman. pp. 417-421.

A further development of earlier ideas on visual intuition and formal logic.

**1999j The Cognitive Development of Proof: Is Mathematical Proof For All or For Some?** In Z. Usiskin (Ed.), *Developments in School Mathematics Education Around the World*, vol, 4, 117-136. Reston, Virginia: NCTM.

A development of 1995f (above) considering how child development leads to a sequence of different stages of proof development related to the development of enactive, visual, symbolic, formal modes of thought.

**1999a The Chasm between Thought Experiment and Mathematical Proof**. In G. Kadunz, G. Ossimitz. W. Peschek, E. Schneider, B. Winkelmann (Eds.), Mathematische Bildung und neue Technologien, Teubner, Stuttgart, 319-343.

A plenary lecture expanding on the transition from elementary to advanced mathematical thinking, including a discussion of the work of Marcia Pinto. (Published before 1999j, but given after.)

**1999g (with Márcia Maria Fusaro Pinto), Student constructions of formal theory: giving and extracting meaning**. In O. Zaslavsky (Ed.), *Proceedings of the 23rd Conference of PME*, Haifa, Israel, 4, 65-73.

The emergence of the ideas of 'natural' and 'formal' thinking, based on a grounded theory built from observations of first year university students coping with an anaylsis course.

**2001j (with Marcia Maria Fusaro Pinto) 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.

A development of 1999g (above) from the PhD thesis of Marcia Pinto, based on the distinction between natural and formal thinking.

2000c (with Ehr-Tsung Chin), Making, Having and Compressing Formal Mathematical Concepts. In T. Nakahara and M. Koyama (eds) *Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education 2*, 177-184. Hiroshima, Japan.

A preliminary study of the development of equivalence relation and partition, from intuitive ideas through definitions, deductions and theorems

2001d Barnard, A. D. & Tall, D. O. (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.

A preliminary version of a major paper on Cognitive Units, Connections and Compression in Mathematical Thinking. This considers the cognitive constructions and links involved in mathematical thinking and proof.

**2001g (with (Abe) Ehr-Tsung Chin) 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.

An analysis of the relationship between the formal definitions of equivalence relation and partition, with the equivalence relation being more related to a formal definition and the partition (which is logically equivalent) being cognitively associated with intuitive visual ideas.

**2001p Natural and Formal Infinities.** *Educational Studies in Mathematics*.

This paper considers the difference between cognitive and formal conceptions of infinity in great detail, with particular reference to how formal thinking can lead to structure theorems that have natural interpretations in visual form. It therefore has wider implications for the development of formal proof.

**2002a** (with Marcia Pinto), **Building formal mathematics on visual imagery: a case study and a theory**. *For the Learning of Mathematics *. 22 (1), 2–10.

A focus on a natural learner building formalism from coherently organised visual intuition.

**2002k** David Tall, (2002). **Differing Modes of Proof and Belief in Mathematics**, *International Conference on Mathematics: Understanding Proving and Proving to Understand*, 91–107. National Taiwan Normal University, Taipei, Taiwan.

This is the first paper describing the cognitive development of proof in terms of the *three worlds of mathematics*, relating to embodiment, symbolism and formalism.

**2002l** Ehr-Tsung Chin, David Tall (2002), Proof as a Formal Procept in Advanced Mathematical Thinking, *International Conference on Mathematics: Understanding Proving and Proving to Understand*, 212-221. National Taiwan Normal University, Taipei, Taiwan.

A formulation of proof as process and concept.

2004d David Tall (2004). 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 outline of the *three worlds of mathematics*, relating to embodiment, symbolism and formalism, which includes the development of proof.

2005c Juan Pablo Mejia-Ramos & David Tall (2005). **Personal and Public Aspects of formal Proof: A Theory and a Single-Case Study.** To be presented at the *British Colloquium of Mathematics Education*, Warwick.

A study of a mathematics graduate responding to the proof of 'the derivative of a differentialble even function is odd' from embodied, symbolic and formal viewpoints, contrasting the private aspects of belief and the public aspect of formal proof.

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