This was actually designed to show that dealing with limits in terms of finite decimals has inherent difficulties. 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.ġ980d Mathematical intuition, with special reference to limiting processes, Proceedings of the Fourth International Congress on Mathematical Education, Berkeley, 170-176.Įarly empirical studies on students cognitive perceptions of the limit concept.ġ981c Intuitions of infinity, Mathematics in School, 10, 3 30-33.Ī 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.ġ981e Infinitesimals constructed algebraically and interpreted geometrically, Mathematical Education for Teaching, 4, 1 34-53.Ī theoretical construction of infinitesimals algebraically, viewed pictorially.ġ993e (with Lan Li), Constructing Different Concept Images of Sequences and Limits by Programming, Proceedings of PME 17, Japan, 2, 41-48.Ī study of an introduction to limits using programming. ![]() Measuring infinity gives an alternative theory of 'points' which is more consonant with intuitive measuring concepts than formal cardinal concepts.ġ980c Intuitive infinitesimals in the calculus, Abstracts of short communications, Fourth International Congress on Mathematical Education, Berkeley, page C5. 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. Schwarzenberger) Conflicts in the learning of real numbers and limits, Mathematics Teaching, 82, 44-49.Ī 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.ġ980b The notion of infinite measuring number and its relevance in the intuition of infinity, Educational Studies in Mathematics, 11 271-284.Īt this time, the concept of infinity' was often conceived in terms of cardinal infinity. My later papers look at this broader theoretical development.ġ978c (with R. A moving value tending to zero is seen as an arbitrarily small number, a cognitive infinitesimal. This gives rise to cognitive conflict in terms of cognitive images that conflict with the formal definition. 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). It is therefore amenable to an analysis in terms of the theory of procepts. ![]() The limit concept is conceived first as a process, then as a concept. 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. ![]() (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. My earliest research began with calculus and limits, leading to the discovery of differences between mathematical theories and cognitive beliefs in many individuals.
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