Intuitive explanations of mathematical ideas

Jerzy Pogonowski

Abstract


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.

Keywords


mathematical intuition, understanding in mathematics, intuitive explanation

References


Adams, C. C.: 2004, The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, Providence, Rhode Island.

Ajdukiewicz, K.: 1974, Pragmatic logic, D. Dordrecht Boston: D. Reidel Publishing Co., Warszawa: Polish Scientific Publishers.

Ben-Zeev, T., Star, J.: 2001, Intuitive Mathematics: Theoretical and Educational Implications, in: B. Torff, R. J. Sternberg (ed.), Understanding and Teaching the Intuitive Mind: Student and Teacher Learning, Lawrence Erlbaum Associates Publishers, Mahwah, New Jersey, London, 29–56.

Cohen, P.: 1966, Set theory and the continuum hypothesis, W. A. Benjamin, New York.

Conway, J. H., Guy, R.: 1996, The Book of Numbers, Springer Verlag, New York.

Davis, P. J., Hersh, R.: 1981, Mathematical experience, Birkhäuser, Boston.

Fischbein, E.: 1987, Intuition in Science and Mathematics: an educational approach, Kluwer Academic Publishers, New York / Boston / Dordrecht / London / Moscow.

Fraenkel, A. A., Bar-Hillel, Y., Levy, A.: 1973, Foundations of set theory, North-Holland Publishing Company, Amsterdam, London.

Galperin, G.: 2003, Playing Pool with pi (The Number pi from a Billiard Point of View), Regular and Chaotic Dynamics 8(4), 375–394.

Ghrist, R.: 2014, Elementary Applied Topology, Createspace.

Hanna, G., Jahnke, H. N., Pulte, H.: 2010, Explanation and Proof in Mathematics. Philosophical and Educational Perspectives, Springer, New York, Dordrecht, Heidelberg, London.

Heath, T. L.: 2002, The Works of Archimedes, Dover Publications, Inc., Mineola, New York.

Kitcher, P.: 1984, The Nature of Mathematical Knowledge, Oxford University Press, Oxford.

Koepke, P.: 2013, Models of set theory. Lecture notes available at: www.math.uni-bonn.de/ag/logik/teaching/2013SS/models_of_set_theory_1/Skript.pdf.

Lakatos, I.: 1976, Proofs and Refutations. The Logic of Mathematical Discovery, Cambridge University Press, Cambridge.

Lakoff, G., Núñez, R. E.: 2000, Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.

Lange, M.: 2014, Depth and Explanation in Mathematics, Philosophia Mathematica. Advance Access published September 12, 2014: http://philmat.oxfordjournals.org/.

Laraudogoitia, J. P.: 1996, A Beautiful Supertask, Mind 105, 81–83.

Lénárt, I.: 1996, Non-Euclidean Adventures on the Lénárt Sphere, Key Curriculum Press, USA.

Levi, M.: 2009, The Mathematical Mechanic. Using Physical Reasoning to Solve Problems, Princeton University Press, Princeton and Oxford.

Mancosu, P.: 2015, Explanation in mathematics, in: E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Summer 2015 edn, Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/sum2015/entries/mathematics-explanation/.

Needham, T.: 1997, Visual complex analysis, Clarendon Press, Oxford.

Pogonowski, J.: 2016, Kontekst przekazu w matematyce (The context of sharing mathematical knowledge, skills and abilities), Annales Universitatis Paedagogicae Cracoviensis Studia ad Didacticam Mathematicae Pertinentia 8, 119–137.

Polya, G.: 2009, Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Ishi Press International, New York, Tokyo.

Polya, G.: 2014, Mathematics and Plausible Reasoning. Vol. I: Induction and Analogy in Mathematics, Vol. II: Patterns of Plausible Inference, Martino Publishing, Mansfield Centre, CT.

Prasolov, V. V.: 2011, Intuitive topology, Providence, Rhode Island: Series Mathematical World, vol. 4, American Mathematical Society.

Romero, G. E.: 2014, The collapse of supertasks, Foundation of science 19(2), 209–216.

Schoenfeld, A. H.: 1985, Mathematical Problem Solving, Academic Press, Inc., Orlando.

Sierpinska, A.: 1994, Understanding in Mathematics, The Falmer Press, London.

Steiner, M.: 1978, Mathematical Explanation, Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition 34(2), 135–151.

Tall, D.: 2013, How Humans Learn to Think Mathematically. Exploring the Three Worlds of Mathematics, Cambridge University Press, Cambridge.


Full Text: PDF

e-ISSN: 2450-341X, ISSN: 2080-9751

AUPC SDMP is on the List of the Ministry’s scored journals (part B) with 5 points for 2016