I recently posted a paper at Research Gate titled, "How Math Can Be Taught Better."
http://www.researchgate.net/publication/282947903_How_Math_Can_Be_Taught_Better
My section on geometry includes an appalling example of a geometry textbook totally screwing up a proof that would have been easy work for anybody with even a passing familiarity with Book I of The Elements by Euclid.
Curious if this was an anomaly or if such mistakes are systemic, I visited a high school library and a used textbook store to look through their geometry books. What I found is that textbook authors acknowledge Euclid as the founder of their science but make no effort to use the axiomatic method.
Geometry is presented as a jumble of statements randomly labeled as axioms (postulates) or as theorems (propositions) and never the same from one textbook to the next. Generally, postulates are either missing entirely or are just propositions that the author did not want to (or know how to) prove. Except for Euclid's first postulate, usually mentioned within the first few pages and not labeled as a postulate, I found no mention of his other four postulates.
The index of one widely used textbook shows that the word "postulate" appears in exactly one passage, which I quote in its entirety:
"Both theorems and postulates are statements of geometric truth. The difference between postulates and theorems is that postulates are assumed to be true, but theorems must be proven to be true based on postulates and/or alrady-proven theorems. It's a fine distinction, and if I were you, I wouldn't sweat it."
The author, Mark Ryan of The Math Center and author of The 10 Habits of Highly Successful Math Students, does not make this distinction. I found about a quarter of the 48 propositions from Book I of The Elements scattered throughout Mark Ryan's textbook and all were just stated as facts with no attempt to prove them. The student exercises had two-column "proofs" where the right-hand column consisted of references to these statements/facts, or whatever you want to call them.
This is not how I learned geometry! Mathematics education today might help you win on Jeopardy, but it will not help you become a scientist. Educators must return math education to instruction in the axiomatic method.
Article How Math Can Be Taught Better
Attempts to improve the teaching of geometry began in the 19th century, if not before. An account of attempts in the UK in the early twentieth century can be found in this paper (see link below to the full version of the paper):
Fujita, T., & Jones, K. (2011). The process of re-designing the geometry curriculum: the case of the Mathematical Association in England in the early twentieth century. International Journal for the History of Mathematics Education, 6(1), 1-23.
Another relevant paper is:
Brock, W. H. (1975). Geometry and the Universities: Euclid and his Modem Rivals 1860-1901. History of education, 4(2), 21-35.
In his address at the Royaumont Seminar in 1959, the mathematician Jean Dieudonné famously called for the elimination of Euclidean geometry from the secondary school curriculum. In the published (English-language) version of his remarks ("New Thinking in School Mathematics", 1959), Dieudonné expressed this call with the slogan "Euclid must go!", and most subsequent references to this speech quote him with that. In Siobhan Roberts's book "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry", Dieudonné is quoted (on p. 157) as saying (in French) "A bas Euclide! Mort aux triangles!" ("Down with Euclid! Death to triangles!").
http://eprints.soton.ac.uk/183091/3/183091JONES25.pdf
Yes, I agree with you. Please look at my paper http://cheng-pu.hxwk.org/2018/03/27/how-to-define-a-flat-plane/. There is definitely a better way to teach math and to think about math
I will look at your paper. I asked this question three years ago and, since then, I have written a high school geometry textbook. The index is here at Research Gate and I would be happy to send you a review copy.
Article Volume One: Geometry without Multiplication
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