This short note is devoted to the role played by intuitive explanations in mathematical education. We provide a few examples of such explanations. They are related to: verbal commentaries, perception, physical models. We recall also some examples of internal explanations, inside mathematics itself.
We discuss the role of a heuristic principle known as the Principle of Permanence of Forms in the development of mathematics, especially in abstract algebra. We try to find some analogies in the development of modern formal logic. Finally, we add a few remarks on the use of the principle in question in mathematical education.
This paper contains a few critical remarks concerning some fundamental assumptions and claims propagated by Lakoff and Núñez in their monograph Lakoff, Núñez (2000). Our attitude is skeptical (cf. also Pogonowski,2011). We agree with the idea that conceptual metaphors may play some role in the formation of elementary mathematical notions. However, we disagree with the authors’ claim that such metaphors provide the main mechanism in the emergence of new notions in advanced mathematics.
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We discuss the creative role of objects called pathologies by mathematicians.Pathologies may become “domesticated” and give rise to newmathematical domains. Thus they influence changes in mathematical intuition.
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We introduce the concept of the context of transmission. It coversthe ways in which mathematical knowledge and mathematical abilities aretransmitted in education and popularization of mathematics. We stress therole of intuitive explanations in these processes. Several examples of suchexplanations are presented, related to: linguistic explanations, perception,empirical models, and internal explanations inside mathematics itself.
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