**LIMITS, INFINITESIMALS AND INFINITIES**

My earliest research began with calculus and limits, leading to the discovery of differences between mathematical theories and cognitive beliefs in many individuals. (For instance, the limit ‘nought point nine repeating’ has mathematical limit equal to one, but cognitively there is a tendency to view the concept as getting closer and closer to one, without actually ever reaching it.) Over the years the reason behind this distinction has become clearer. The primitive brain notices *movement*. Hence the mental notion of a sequence of points tending to a limit is more likely to focus on the *moving* *points* than on the *limit point*. The limit concept is conceived first as a process, then as a concept. It is therefore amenable to an analysis in terms of the theory of *procepts*. In the case of the limit, the process of tending to a limit is a *potential* process that may *never* reach its limit (it may not even have an explicit finite procedure to carry out the limit process). This gives rise to *cognitive conflict* in terms of cognitive images that conflict with the formal definition. A ‘moving value’ tending to zero is seen as an arbitrarily small number, *a cognitive infinitesimal*. My later papers look at this broader theoretical development.

1978c (with R. L. E. Schwarzenberger) Conflicts in the learning of real numbers and limits, *Mathematics Teaching*, 82, 44-49.

A paper which notes the existence of cognitive conflict in the learning of limits, written by two mathematicians who were beginning to think about cognitive problems.

1980b The notion of infinite measuring number and its relevance in the intuition of infinity, *Educational Studies in Mathematics*, 11 271-284.

At this time, the ‘concept of infinity' was often conceived in terms of cardinal infinity. This paper considers the manner in which experiences with measuring can give different intuitions about infinity which conflict with cardinal infinity, showing that infinite concepts conceived in different contexts can have incompatible properties. Measuring infinity gives an alternative theory of 'points' which is more consonant with intuitive measuring concepts than formal cardinal concepts.

1980c Intuitive infinitesimals in the calculus, Abstracts of short communications, *Fourth International Congress on Mathematical Education*, Berkeley, page C5.

The full length poster related to the abstract reports empirical research into the cognitive development of mathematics students being introduced to a logical form of infinitesimal calculus.

1980d Mathematical intuition, with special reference to limiting processes, *Proceedings of the Fourth International Congress on Mathematical Education*, Berkeley, 170-176.

Early empirical studies on students cognitive perceptions of the limit concept.

1981c Intuitions of infinity, *Mathematics in School*, 10, 3 30-33.

A discussion of various views of infinity which reveals how children's intuitive ideas of infinity are often closer to measuring infinity with infinitesimals as inverses rather than cardinal numbers which have no inverses. [Not yet in PDF format].

1981e Infinitesimals constructed algebraically and interpreted geometrically, *Mathematical Education for Teaching*, 4, 1 34-53.

A theoretical construction of infinitesimals algebraically, viewed pictorially. [Not yet in PDF format].

1993e (with Lan Li), Constructing Different Concept Images of Sequences and Limits by Programming, *Proceedings of PME 17*, Japan, 2, 41-48.

A study of an introduction to limits using programming. This was actually designed to show that dealing with limits in terms of finite decimals has inherent difficulties. In this matter it succeeded.

1993h Real Mathematics, Rational Computers and Complex People, *Proceedings of the Fifth Annual International Conference on Technology in College Mathematics Teaching*, 243–258.

This paper is mainly about computers and calculus but it does include an interesting idea of plotting graphical models of functions which are different on the rationals and irrationals that has relevance for Lebesgue integration and for intuitions about the relative 'sizes' of the set of rationals (repeating decimals) and irrationals (non-repeating decimals).

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 study following several students learning a first analysis course and how they handle the definition of a least upper bound. Some students go through the course without learning the definition and without even being able to reproduce it. A student may be able to ‘explain' the definition in a personal manner with its own concept image, but not have a definition that is ‘operable' in the sense that it can be used, by the student, to prove theorems and solve problems.

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 broad review of some of my theories about learning in mathematics, relating conceptions that arise in a biological brain to formal mathematical conceptions and how the computer may be used to support different kinds of mathematical thinking.

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 a range of studies that consider the development of symbolism in mathematics. In particular it considers 'operational procepts' in arithmetic, that have built in operations of calculation, such as 3+2, 'potential procepts' in algebra such as 3+2x (which cannot be evaluated until x is known) and 'potentially infinite procepts' (limits) in calculus and analysis.

2001b David Tall (2001). Cognitive Development in Advanced Mathematics Using Technology. *Mathematics Education Research Journal*. 12 (3), 196–218.

Another discussion on the development of mathematical concepts.

2001l A Child Thinking about Infinity. *Journal of Mathematical Behavior*.

A study of a remarkable conversation with a six-year old about infinity, revealing the natural belief in infinity as a very large number that can be operated on as in arithmetic, then a conflicting view that there is only one infinity that cannot be reached or passed (though if you could, you would reach the minus numbers again!) Also includes a discussion on cardinal infinity.

2001p Natural and Formal Infinities. to appear in *Educational Studies in Mathematics*, 48 (2&3), 199–238.

An in-depth discussion of the conflicts between 'natural infinities' that arise as a result of extending everyday finite experience and different formal infinities that arise by selecting different axioms as foundations for infinite concepts.

This is an extension of the ideas of natural and formal approaches to analysis, developed by Marcia Pinto and explained in:

2001j Marcia Maria Fusaro Pinto & David Tall: 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.

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