The Importance of Geometry in one's Mathematical Foundation

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Chandller
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Hi PF,
What would you say is the level of importance of Geometry, as opposed to say, Algebra I or II, in a future Mathematician's mathematical foundation? Would not covering it thoroughly leave holes that would show up later in higher mathematics? Any thoughts?


Thank you,
Chandller
 

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Geometry and Algebra are the bedrock of any mathematician. While its true, one will specialize in a particular field where one or the other is more important, it's still important to understand both.

An Applied math mathematician in particular would work with Vector Analysis, and Differential Geometry methods and these require both disciplines.

Another is Algebraic Geometry:

https://en.wikipedia.org/wiki/Algebraic_geometry

and Geometric Algebra:

https://en.wikipedia.org/wiki/Geometric_algebra

Here's a teacher's perspective:

https://www.nctm.org/Publications/Mathematics-Teacher/Blog/Geometry-and-Algebra/

Notice how theorems can be proven using either algebraic techniques or geometric techniques.

This is a key strategy that mathematicians use when exploring new fields of math. They look for similar systems of math and then map the new system to the known ones and then can deduce new theorems in the new system through the mapping.

Check out category theory and morphisms:

https://en.wikipedia.org/wiki/Category_theory

https://en.wikipedia.org/wiki/Morphism
 
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PeroK
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Hi PF,
What would you say is the level of importance of Geometry, as opposed to say, Algebra I or II, in a future Mathematician's mathematical foundation? Would not covering it thoroughly leave holes that would show up later in higher mathematics? Any thoughts?


Thank you,
Chandller
Geometry may shape a mathematician's career!
 
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Stephen Tashi
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What would you say is the level of importance of Geometry, as opposed to say, Algebra I or II, in a future Mathematician's mathematical foundation?
Fifty years ago the title "Geometry" might have defined a particular course - elementary geometric material in Euclid presented in a two column approach of "steps" and "reasons". When I last looked at a high school "Geometry" texts ( 30 years ago!) the course had changed. Textbooks took a more modern approach and the content varied a lot from textbook to textbook. There were editorials in educational journals debating what a Geometry course should look like - or whether there should be a full year course called "Geometry".

As to future effects of the two types of course, I don't know. It's one thing to visualize Geometry as course that is exclusively a ruler-and-compass approach. It's a different matter to visualize Geometry as a course that introduces analytic geometry and trigonometry - or even set theory.

The old style of course , ruler-and-compass, steps-and-reasons, left students with the impression that proof in mathematics is what the current teacher says it is. In those days (and perhaps today) the steps-and-reasons approach wasn't carried over into the Algebra texts. So in 3 consecutive years, a student might take Algebra I, Geometry, and Algebra II - being bounced back and forth between two different standards of mathematical proofs.
 
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This is so true of many subjects where the teachers use outdated concepts to teach math and science and these concepts must be unlearned or amended when the student goes to college and now has to relearn them.
 
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So in 3 consecutive years, a student might take Algebra I, Geometry, and Algebra II - being bounced back and forth between two different standards of mathematical proofs.
I took high school geometry in the '59-'60 school year. We didn't do anything with a compass and straight edge, but we did quite a few proofs that involved triangle congruences and such, with acronyms such as SSS (side-side-side) and SAS (side-angle-side) for proving that two triangles were congruent.

IMO, the two most important concepts taught were putting together a reasoned argument to prove some point, and a basic understanding of the areas of certain geometric figures - circles, triangles, rectangles, and a few others. The parts of the course that were useful in later math courses could probably be done in a semester, rather than in an entire school year.

Some years later, I must have taught a couple of courses in high-school geometry, but I don't have any recollection of the book that we used, other than it was probably very similar to the one I used when I was taking the class.
 
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Stephen Tashi
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I took high school geometry in the '59-'60 school year. We didn't do anything with a compass and straight edge, but we did quite a few proof that involved triangle congruences and such, with acronyms such as SSS (side-side-side) and SAS (side-angle-side) for proving that two triangles were congruent.
By "compass and straightedge", I mean the approach to geometry that does not use coordinates. The subject of congruent triangles is often taught in that fashion - as are topics such as "When parallel lines are cut by a transversal, the alternate interior angles are equal".
 
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symbolipoint
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Hi PF,
What would you say is the level of importance of Geometry, as opposed to say, Algebra I or II, in a future Mathematician's mathematical foundation? Would not covering it thoroughly leave holes that would show up later in higher mathematics? Any thoughts?


Thank you,
Chandller
YES.
 
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symbolipoint
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So in 3 consecutive years, a student might take Algebra I, Geometry, and Algebra II - being bounced back and forth between two different standards of mathematical proofs.
The colleges (community colleges) and universities (in case they offer those three courses) readily permit another more desirable sequence: Algebra 1, Algebra 2, AND THEN Geometry.
 
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symbolipoint
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This is so true of many subjects where the teachers use outdated concepts to teach math and science and these concepts must be unlearned or amended when the student goes to college and now has to relearn them.
I studied the same Mathematics courses in college as I did in high school, as far as the college prep Math courses. I did not need to unlearn anything; although everything became better the second time through.
 
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As someone who never took a high school geometry course, I think I turned out fine. Most of the concepts about chords of circles and the various ways you can draw lines through triangles turn out to just not be that interesting in most of math. Geometry in modern math is important, but it just doesn't look like geometry in high school.
 

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