#### Onion Eater

##### Well-known member

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I recently posted a paper at Research Gate titled, “How Math Can Be Taught Better.”

https://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:

The author, Mark Ryan of The Math Center and author of

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.

https://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:

Mark Ryan said: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 already-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 his 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.

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