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This question is related to Is Euclid dead? or Should Euclidean geometry be taught to high school students?, but I am not asking about whether Euclidean geometry should be taught at all, but whether alternate systems should be taught first.

non-Euclidean geometries such as Taxicab geometry have always had a bit of a "cult following" among mainstream mathematics educators, since the notions of angles can be presented in a completely different fashion (though traditional Taxicab geometry uses Euclidean angles), and of course, the notion of a "square" circle is a bit of a paradigm shift for most people.

Since these generalizations may be helpful in creating a more abstract notion of space and definitions of metrics, etc., would it be effective to teach these concepts before the Euclidean ones, or has this been tried before and there have been deleterious effects on students' ability to learn the Euclidean principles?

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  • $\begingroup$ Better to learn the simpler thing first. If you pile on too many prerequisites, especially hard ones, you make learning anything very very difficult. Just look at how hard it is to get basic info out of any Wikipedia math article. If you want to go to deeper levels of understanding later on basic topics (with benefit of advanced math), fine. But do it later as a recursion...if it even makes sense in cost/benefit (won't for an engineer for instance). $\endgroup$ – guest May 29 '18 at 5:51
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When many people use the term "non-Euclidean" geometry they are often referring to the "classical" non-Euclidean geometries - elliptic geometry and Bolyai-Lobachevsky geometry. The question asks about intuition about shapes in the (Euclidean) plane. So my answer to the question if studying the classical non-Euclidean geometries helps with intuition about Euclidean shape is "no."

However, I think that notation of distances other than Euclidean distance should be looked at much earlier than has often been the case. Most students arrive in college with the idea that the only "distance" is Euclidean distance. In fact, in some ways learning about Euclidean distance is "harder" than learning about taxicab distance because for many students square roots are a rather subtle issue. Discussing in K-12 or college about the different ways to measure distance is very "liberating." The points that are unit distance from me in an open field depends on the distance function I use, and "shape" of circles depends on distance. It also would not do any harm to discuss Hamming distance and show how certain spell checkers use Hamming distance to help correct spelling errors.

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You can certainly show the geometry e.g. on a sphere. Going any futher would be very counterproductive: most (as in "almost all") everyday geometry is Euclidean. Trigonometry (and much of "initial calculus") requires/assumes Euclidean geometry. The list goes on.

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If you would like to look at a treatment of axiomatic geometry which begins with the abstract, featureless space then places structure (Euclidean, Projective, Hyperbolic) then you might want a copy of Fundamentals of MODERN GEOMETRY for College Students by Honorè P. Mavinga

I had the opportunity to sit in some of his lectures. His approach is really clean and, while it may require some patience your students do not possess, at the end of it you can clearly understand the distinctions between the different axiomatic geometries as they arise by adding structure to the same blank template: affine space.

My personal inclination, given students who care etc.. I very much like the idea of the text by McCleary.Geometry from a Differentiable Viewpoint by John McCleary. Part A is axiomatic classical geometry. Part B is classical differential geometry of curves and surfaces, the usual topics including Gauss Bonnet. I think it might be very nice to teach a course where a few weeks concisely covered the axiomatic problems then the remainder of the course shows how the geometry of curved surfaces gives not just one noneuclidean geometry, rather, boundless choices. There are various institutional pressures which make the birth of this course very unlikely, but, perhaps somewhere there exists an institution whose local conditions (students, needed prerequisites...) would encourage such a course...

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  • $\begingroup$ This is great, I wish I could select both answers. I will definitely check out those texts! $\endgroup$ – jonsca Jul 16 '14 at 8:09
  • $\begingroup$ No problem, I think the accepted answer is more an answer to your question anyway :) $\endgroup$ – James S. Cook Jul 16 '14 at 13:42

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