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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