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Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (see King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, p. 157), often associated in the popular mind with Bourbaki's general stance on rigorous, formalized mathematics (eschewing pictorial representations, etc.). See Dieudonné's address at the Royaumont seminar for his own articulated stance.

In brief, the suggestion was to replace Euclidean Geometry (EG) in the secondary school curriculum with more modern mathematical areas, as for example Set Theory, Abstract Algebra and (soft) Analysis. These ideas were influential, and Euclidean Geometry was gradually demoted in French secondary school education. Not totally abolished though: it is still a part of the syllabus, but without the difficult and interesting proofs and the axiomatic foundation. Analogous demotion/abolition of EG took place in most European countries during the 70s and 80s, especially in the Western European ones. (An exception is Russia!) And together with EG there was a gradual disappearance of mathematical proofs from the high school syllabus, in most European countries; the trouble being (as I understand it) that most of the proofs and notions of modern mathematical areas which replaced EG either required maturity or were not sufficiently interesting to students, and gradually most of such proofs were abandoned. About ten years later, there were general calls that geometry return, as the introduction of the alternative mathematical areas did not produce the desired results. Thus EG came back, but not in its original form.

I teach in a University (not a high school), and we keep introducing new introductory courses, for math majors, as our new students do not know what a proof is. [Cf. the rise of university courses in the US that come under the heading "Introduction to Mathematical Proofs" and the like.]

I am interested in hearing arguments both FOR and AGAINST the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs? If so, is there is a more efficient mathematical subject, for high school students, in order to learn what is a theorem, an axiom and a proof?

Full disclosure: currently I am leading a campaign for the return of EG to the syllabus of the high schools of my country (Cyprus). However, I am genuinely interested in hearing arguments both pro and con.

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closed as off-topic by Steven Landsburg, Gerald Edgar, Chris Godsil, Igor Pak, Lucia Dec 22 '13 at 2:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Steven Landsburg, Gerald Edgar, Chris Godsil, Igor Pak, Lucia
If this question can be reworded to fit the rules in the help center, please edit the question.

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Most parts of discrete math (e.g. elementary number theory, elementary combinatorics, elementary graph theory) seem like a better fit. Among other things, students might actually use that material (e.g. in parts of computer science). But in any case this is probably too opinion-based for MO. –  Qiaochu Yuan Dec 19 '13 at 18:24
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This web site doesn't really cover pedagogical issues in pre-university education, and doesn't usually permit questions whose answers are opinions. I don't know of a suitable place to ask your question. Maybe start a blog. –  Ben McKay Dec 19 '13 at 18:48
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Ben McKay, the author clearly connected the issue to the effects on undergraduate math education. –  Monroe Eskew Dec 19 '13 at 18:52
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I do not understand why the question was closed. It seems to be a legitimate question for this site, and the number of answers and votes seems to show that there is substantial interest. –  Alexandre Eremenko Dec 19 '13 at 20:47
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What's this absolute nonsense about the series of articles in the 60's ? Which articles ? in which journals ? signed by whom ? What does that mean, "by the Bourbaki's" ? –  Joël Dec 20 '13 at 4:47

10 Answers 10

When I was in high school (in the early 1960's), Euclidean geometry was the only course in the standard curriculum that required us to write proofs. These proofs, however, were in a very rigid format, with statements on the left side of the page and a reason for every statement on the right side. So I fear that many students got an inaccurate idea of what proofs are really like. They also got the idea that proofs are only for geometry; subsequent courses (in the regular curriculum, not honors courses) didn't involve proofs. The textbook that we used also had some defects concerning proofs. For example, Theorem 1 was word-for-word identical with Postulate 19; Theorem 1 was given a proof that didn't involve Postulate 19, so, in effect, we were shown that Postulate 19 is redundant, but the redundancy was never mentioned, and I still don't know why a redundant postulate was included in the first place. Another defect of the standard courses in geometry was that, because of the need to gently teach how to find and write proofs (in that rigid format), very little interesting geometry was taught; the class was mostly proving trivialities. I was fortunate to be in an honors class, with an excellent instructor who showed us some really interesting things (like the theorems of Ceva and Menelaus), but most students at my school had no such advantage.

I conjecture that Euclidean geometry can be used for a good introduction to mathematical proof, but, as the preceding paragraph shows, there are many things that can go wrong. (There are other things that can go wrong too. I mentioned that I had an excellent teacher. But my school also had math teachers who knew very little about proofs or about geometry beyond what was in the textbook.) So my advice is, if you want to develop a course such as you described in the question, proceed, but be very careful.

Incidentally, many years ago, I recommended to my university department that we use a course on projective geometry as an "introduction to proof" course. The idea was that there are fairly easy proofs, and the results are not as obvious, intuitively, as equally easy results of Euclidean geometry. My suggestion was not adopted.

Qiaochu Yuan's suggestion of discrete math instead of geometry might have similar advantages as my projective geometry proposal, but it will still be subject to many of the pitfalls that I indicated above, plus one more: Most high school math teachers know less about discrete math than they do about geometry.

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+1. Axiomatizing Euclidean geometry properly is tricky and full of insidious points that a high-school student cannot understand and appreciate. For instance, configuration details and intersection points. If I am not mistaken, it took Hilbert to do it rigorously, a couple of millennia after Euclid's attempts. I am no math education expert, but to me abstract algebra looks like a much better starting point: it's much clearer what needs to be proved and what the axioms are. (yes, I studied EG in high school). –  Federico Poloni Dec 19 '13 at 19:39
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This sounds as if you used bad textbooks in your school. Andreas, was that in USA? :) –  Sergei Akbarov Dec 19 '13 at 19:44
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Essentially you say that school teachers were not competent. That is the biggest problem indeed. –  Anton Petrunin Dec 19 '13 at 22:21
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My middle-school geometry class (age 13) was a proof-based geometry class. We started out with synthetic affine geometry and built up to Euclidean. It was surely not the most rigorous axiomatization of geometry, but it was an excellent class and made every "introduction to proof" class I've encountered since entirely redundant. Every assignment and every test was entirely proof-based, and some were quite challenging to me at the time. Do I remember any geometry? No, not really. Was it one of the most valuable classes I've ever taken? Absolutely. –  dfeuer Dec 20 '13 at 7:04
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@FedericoPoloni, Using abstract algebra instead of EG because it avoids tricky arcane foundational issues seems like a misplaced priority. Abstract algebra is, well, awfully abstract, and won't be as accessible or intuitive to young math students as EG. And accessibility, imo, is far more important at that stage than total rigor. –  Jonah Dec 20 '13 at 11:13

I try to keep my answer short.

Fact: Euclidean geometry is still taught in Iranian middle and high schools.

Observation (based on research): Most teachers do not like to teach geometry. They say, when you teach geometry, you are always faced with problems that you don't know how to solve them. But, it seems that they haven't got that problem with the rest of mathematics taught in school! Thinking of your campaign, ask yourself, have you got enough teachers willing to teach geometry and able to do so?

Fact : There is at least one mathematician who is in love with triangles. Here is a quote from his paper in The Mathematical Intelligencer:

No object has ever served mathematics better or longer. Compare the number of nontrivial results which are true for all topological spaces, rings, groups, etc, without putting extra assumptions on them with the number of nontrivial results which are true in any triangle...When it comes to deducing results in mathematics just from the definition of an object, nothing can hold a candle to the triangle. The triangle will serve mathematics forever.

Opinion: There is a big difference between teaching geometry as a source of fascinating problems, and as a rigid body of axiomatic knowledge. Personally, I favor the former. Go to observation above!

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Iran will stay with geometry since the ornament on your flag is made with a compass-and-straightedge construction isiri.org/portal/files/std/1.htm –  Anton Petrunin Dec 20 '13 at 0:39
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An excellent observation! It is especially relevant in the situation OP finds himself in, namely, when geometry has been absent from the school curriculum for a while. As an example, the state of New York dropped EG from its school curriculum for a number of years and now faces a paradoxical situation of teachers instructing in a subject that they themselves have never learned in school! This effect is felt for several generations, e.g. we have pre-service teachers who took geometry in school from someone who had never learned it himself/herself. –  Victor Protsak Dec 20 '13 at 1:27
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@Amir Asghari: Apparently, you have addressed a very serious issue: If EG returns, then who is going to teach it? Even now high school teachers avoid to teach several difficult things. –  smyrlis Dec 20 '13 at 5:15
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@ToddTrimble Dear Todd. The research I mentioned is an unpublished master thesis I supervised in 2009: (Leila Mansouri), The differences between the teaching of geometry and the teaching of mathematics in highschool! Unfortunatly, the result is not available in English. Thus, let me summerize the results here, hoping that it comes handy. –  Amir Asghari Dec 21 '13 at 10:54
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First of all, consider that the title speaks itself. Separating the teaching of geometry from the teaching of mathematics (including calulus) reflects the belief of most teachers. Indeed, whatever a mathematician may say in favour of teaching geometry falls into disfavour from teachers' point of view: Geometry is problem based, each problem could have several solutions, solving most problems needs creativity (and you cannot teach creativity), geometry has a unity (that is to say its different parts are closely related to each other) and so on. –  Amir Asghari Dec 21 '13 at 11:14

I strongly recommend to read this paper of Sharygin. (It is in Russian, but it worth to translate.) You will see the reasons to return EG in school, you will also the reasons why it disappears.

Sharygin is my hero, he is the author of many very good math books for school students, he also wrote the best (the opinion is mine) text book in Euclidean geometry for school.

P.S. Let me share what I know about the history of geometry curriculum in Russian school. We had textbook of Kiselev, which served for more than half century. It was changing slowly, at the beginning it was quite close to Euclid's Elements. (If you ask about geometry someone from the generation of my parents, their eyes start to radiate with positive energy and they start to explain how wonderful was the experience.)

After that (60-s) changes start. First Nikitin's book --- a big step back. After that, instead of coming back to Kiselev, many books were written by very prominent mathematicians (including Alexandrov and Pogorelov) these books were yet worse than Nikitin's book. Later Sharygin's book appears; it is a very good book but extremely demanding from the teacher (say absolute geometry was not discussed, but if the teacher is not familiar with absolute geometry then he can not teach properly).

Now we get so called "Unified state examination" (the worst reform ever made in Russia) it is either too expansive or impossible to check proofs on this exam; the later wipes geometry from the school curriculum; formally it is still there but since it is not needed to pass the exam, no one needs to learn it.

Conclusion: It seems that every big reform makes education worse. The right direction would be to change things gradually, and it has to be done by teachers with help of academia, not other way around.

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An interesting read! Sharygin was also my hero because of his problem books in plane and space geometry. His observations about geometrical skills of strong math olympians are right on target. However, overall this is a rambling article with strong conspirological overtones and, frankly, it should be classified as a panglossian manifesto extolling the virtues of geometry rather than an objective study relying on rational analysis. It is also surprisingly devoid of concrete examples of "good geometry" that he promotes. I would have expected more from such a great mathematician and educator. –  Victor Protsak Dec 20 '13 at 6:28
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@VictorProtsak, Yes the article is written in a very emotional way, but Sharygin was actually teaching geometry, so things he says relying on experience (which is not comparable with mine or yours). OP asks for arguments for the return of EG in high schools. I think it answers his question. –  Anton Petrunin Dec 20 '13 at 15:52
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P.S. when I hear "objective study relying on rational analysis" I think "now it is time for the lie". –  Anton Petrunin Dec 20 '13 at 15:53
    
Anton, are you familiar with "Higher geometry" by N.V. Efimov? Do you have an opinion on the text? (Probably it is not a text suitable for school teaching, but I would still like to know what you think of it on its own terms.) –  Andres Caicedo Dec 21 '13 at 17:10
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Very interesting answer! –  Gil Kalai Dec 21 '13 at 18:41

With the caveats mentioned by Andreas, I think Euclidean Geometry makes excellent sense as a high-school course. (My high school experience was not dissimilar to Andreas's -- still the two-column format, but I also had a teacher who understood mathematics beyond what was in the textbook.)

The basic point of agreement (between those Bourbakistes and those who would uphold EG) seems to be that there is need for a course that expounds mathematics as an axiomatic discipline, and the careful modes of reasoning that go into that. In some sense just about any system based on axioms (be it EG, set theory, "discrete mathematics", or something else) would serve that purpose, except that Euclidean Geometry has the big advantage of being visual and readily accessible to intuition. (The downside to that might be Isaac Newton's criticism [see Arnold's Huygens and Barrow, Newton and Hooke, pp. 49-50] that most of the theorems are intuitively quite obvious, so that the typical course can seem a painful exercise in pedantry.)

I like Andreas's projective geometry proposal. Among other things, this would help promote the idea of the power of unification in mathematics: that things that might look very different, such as ellipses and hyperbolas, are often the same thing in disguise.

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Euclidean geometry is still taught in American high schools, but I am strongly against it. I think it should be replaced with linear algebra.

Arguments against Euclidean geometry:

  • Most of what you prove in a high school Euclidean geometry class seems pretty obvious until you learn about non-Euclidean geometry. It makes students think that proofs are pedantry for its own sake.

  • Euclidean geometry is basically useless. There was undoubtedly a time when people used ruler and compass constructions in architecture or design, but that time is long gone.

  • Euclidean geometry is obsolete. Even those students who go into mathematics will probably never use it again.

Arguments for linear algebra:

  • $\mathbb{R}^2$ with the standard inner product is a model for the Euclidean axioms, so in particular you can still prove the same theorems if you really want to.

  • Linear algebra generalizes easily to dimensions larger than 3 where most students' geometric intuition breaks down, so it is easier for them to appreciate the need for axioms and theorems.

  • Linear algebra - particularly eigenvalues and eigenvectors - is ubiquitous in modern science and engineering. I would argue that the average person is much more likely to encounter an eigenvalue problem than a calculus problem.

  • Linear algebra is, of course, still the basic language in which most of mathematics is expressed and thus a linear algebra class is a more honest taste of what math is all about.

  • Providing students with an early foundation in linear algebra would make later education run more smoothly. Even many non-scientists use software that is based on solving linear systems or computing matrix decompositions, and it might help for such people to have a little more context. And those who go on to take further science classes - particularly physics - would more obviously benefit. If nothing else, we might finally be able to teach our students the correct second derivative test in multivariable calculus classes...

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Things like the Pythagorean theorem, the theorem about inscribed angles in a circle, the volume relation between a pyramid and a parallelepiped, are these all so intuitively obvious? –  Monroe Eskew Dec 19 '13 at 21:30
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@MonroeEskew, sadly, I doubt you will see a proof of these things in a high school geometry course (in the US, at least; I think the Russian curriculum is another story). –  Adeel Dec 19 '13 at 21:47
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"Euclidean geometry is still taught in American high schools" --- your first sentence is wrong, I do not see the point to read further. –  Anton Petrunin Dec 19 '13 at 22:02
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Something that goes by the name Geometry is still taught in American high schools, no question about it, and Euclidean is as accurate as any other single adjective (what else would one call it?). If Anton's point is that the course is some denatured form or deformation of what he understands by Euclidean Geometry, then that of course is a separate point. Otherwise, Paul is correct. –  Todd Trimble Dec 19 '13 at 22:41
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Paul: a personal story. When I was in the 9th or 10th grade (in the USSR), I bought an experimental geometry textbook which exposed Euclidean geometry from the point of view of vectors, using what the book called "H.Weyl's approach". Although I was an accomplished math olympian and had spent quite a bit of time improving my understanding of elementary geometry using problem-based approach, I remember how comprehending this fairly elementary linear algebra based geometry was HARD. –  Victor Protsak Dec 21 '13 at 3:04

As long as this question is open I might as well throw in my two cents. I think it is not useful to teach Euclidean geometry to high school students. Here are some reasons I can think of for people to teach Euclidean geometry to high school students and why I think they are bad reasons:

  • As an introduction to the notion of a proof. As I said in the comments, I think there are better options here, such as areas of discrete math like elementary number theory, elementary combinatorics, or elementary graph theory. Unlike Euclidean geometry, at least some of this material has nontrivial applications: for example, the application of elementary number theory to cryptography or the application of combinatorics to analyzing algorithms. Also unlike Euclidean geometry, this material offers a lot of opportunity for computer-based exploration: for example, Project Euler. But it's not even clear to me that high school students really need an introduction to proof.

  • As preparation for other topics that high school students ought to know. Euclidean geometry might not be a bad way to prepare students for trigonometry and eventually calculus, but I don't think high school students ought to learn these things either. The same goes for physics.

  • As preparation for using mathematics in daily life. Here I think topics like Fermi estimation and some basic probability and statistics would be more useful (e.g. for helping people make better political and medical decisions). As far as I can tell most people have no use for Euclidean geometry in their daily lives.

  • As preparation for jobs involving mathematics. If students want to take such jobs, the relevant mathematics can be taught to them as part of their job training, or they can pick it up themselves. Note that there are many people with programming jobs despite the general lack of programming in most high school curricula.

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Your opinion is quite extreme. Perhaps you think mechanical and electrical engineering are becoming obsolete? –  Monroe Eskew Dec 19 '13 at 22:26
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There is nothing better than EG as "introduction to the notion of a proof"; there is nothing on the second place and nothing on the third --- you examples say way below. –  Anton Petrunin Dec 19 '13 at 22:28
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It's not just the notion of proof; it's the nature of mathematics seen as an axiomatic deductive discipline that should be part of one's broad cultural education of where mathematics fits into general human knowledge. This aspect is badly underappreciated. (It might be tempting to overplay the applications aspect, but that would be missing the real point of such a course.) Of all the traditional high school curricula, EG comes closest to capturing that essential aspect of mathematics as it is understood by mathematicians. –  Todd Trimble Dec 19 '13 at 22:35
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Granted, majority of people get by with very little mathematics and none of it too deep. Still, I am SHOCKED that "daily life" of "most people" is reduced to "political and medical decisions". How about making and fixing things with your own hands? And by the way, for 99.99% people spatial imagination is way more important in their daily life than any kind of mathematical proof. –  Victor Protsak Dec 20 '13 at 6:48
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@Todd: it's not clear to me that teaching high school students Euclidean geometry does anything to address this. I think as mathematicians we should be careful to separate our experience of mathematics from the experience of the masses and appreciate that not everyone finds it as engrossing as we do. To be clear, what I am mostly against is forcing children to learn things that many of them will neither enjoy nor use. As long as we're going to force children to learn anything we might as well think carefully about what we're forcing them to learn and whether there are better options. –  Qiaochu Yuan Dec 20 '13 at 8:16

Also in Israel, Euclidean geometry is taught in schools for quite some time (judging from my parents, me, and my children). I personally like the idea of it being taught and being the first encounter with mathematical axioms, definitions and proofs, as well as an encounter with geometrical thinking. For learning what a mathematical proof is, I doubt if any of the suggested substitutes will even come close.

But it is not clear to me how crucial it is to teach (everybody) the notion of a mathematical proof in high school at all.

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Gil, when children at school are taught religion or military training, is it doubtful for you as well? Wouldn't it be better to explain them that when a "great intellectual leader" tells something strange there is a possibility to verify whether what he says is indeed wise, or on the contrary stupid? :) –  Sergei Akbarov Dec 21 '13 at 19:04
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Unfortunately, I was born in USSR, where those stupidities were presented without hesitation in school education as "great truth of modern science", and "modern logic" (since the author, G.W.F.Hegel suggested his own understanding of logic in his great masterpieces marxists.org/reference/archive/hegel/works/hl/hlconten.htm). Fortunately, we had been also taught geometry, where we could understand what actually logic is, and this saved us from total intellectual degradation. –  Sergei Akbarov Dec 21 '13 at 19:22
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@SergeiAkbarov, I agree, if one wants to remove the proofs from high school, one has to think what will come instead. –  Anton Petrunin Dec 21 '13 at 23:18
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@SergeiAkbarov I think your first comment has little to do with math proof, especially for a typical student. Actually for this I think training in language, logic, philosophy could be more useful. I think we discussed this already once but for my taste you blow out of proportion the relevance of math proof education on such matters. In my opinion it is neither sufficient nor necessary for critical and independent thinking in the context you bring up. –  quid Dec 22 '13 at 1:38
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"quid and quim are different people!" -- Ah! Excuse me, I did not notice! Yes briefly that was my point: "if you take proofs out of the curriculum, bullshit like this will come in instead." Anton understood me correctly. –  Sergei Akbarov Dec 22 '13 at 10:15

I completely agree with you. It is important for everyone to be exposed to proofs, because it shows them what math is really about--reasoning, not computation. The mathematical way of thinking is very valuable for developing general critical thinking skills and the most careful and precise reasoning. I believe there is no substitute. You also hit the nail on the head when you point out that Euclidian geometry is a great medium for learning what proofs are all about. The subject matter connects with intuition, and the propositions and arguments are easily seen to be well-motivated and accessible to the novice. As you mention, it is empirically found to be hard to do proofs for the beginner with other topics.

How can we expect our undergraduates to do well when the secondary education is lacking in the prerequisite training? If mathematics education is an appropriate topic of discussion here, then certainly the relation of high school curriculum to the preparedness of undergraduates is relevant.

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Huzzah! I hate it when people try waste the young mind's time with paper computation. Use a computer. What kids should be doing is learning reasoning and honing their creative ability (with tools which they will have to learn, yes, but not those darn pieces of paper!). –  bjb568 Dec 20 '13 at 2:57

There is something important besides rigor introduced in Euclidean Geometry classes: a connection between visual perception and sequential reasoning.

In "Mathematics in the 20th Century" Atiyah likened Geometry with space-bound visual perception and Algebra with sequential time-bound reasoning. If we continue that simile a course that naturally combines both would be a movie, something much more than the ingredients. And every time we encounter one of those movies it usually generates quite a bit of excitement.

Algebra, however, is not the only sequential process in Mathematics; the other one is the sequential reasoning of a proof.

What I find important EG is that it's the first course in High School that connects visual perception and sequential reasoning, making it the first "movie" the kids ever see, and for many of them the only one. Replacing EG with Number Theory or Combinatorics as other suggested would replace the marriage of visual to sequential with a marriage of sequential to sequential.

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You need to establish a goal, and the reason for the goal. An example is "Have my high school require everyone to take a course covering this syllabus in Euclidean geometry" for the goal, and "because it is intellectually enriching and potentially useful" as the reason.

I don't think the above is a good example. Here is a different example: "Require knowledge of Euclidean geometry and its applications to graduate from high school" as a goal, with the reason being "our society needs engineers, technicians, and other workers who will use the knowledge and applications to improve our community." I like this example a little better because the reason feels more concrete; sadly, I do not know if the reason is valid.

As your present question stands, I do not see a good combination of goal and reason. When you have that, you will have a foundation for arguing for your goal.

If the goal is to help students learn proofs, I might suggest looking at Common Core education standards happening in the United States. Good communication and expression in a broad range of areas of study is emphasized, and I would couple this with the ability to produce arguments in a variety of styles: logical, emotional, inspirational, to start. I would suggest a course or two which presents arguments in geometry, algebra, analysis, discrete mathematics, and logic, so that one can taste the different flavors of proof that occur in the fields.

Gerhard "Also Gives Fresh, Minty Breath" Paseman, 2013.12.19

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Christopher Moore has a character named Minty Fresh in at least one of his books. There is a bit of explanation as to the reason for the name, I don't immediately recall. I really like his books, though. en.wikipedia.org/wiki/A_Dirty_Job –  Will Jagy Dec 19 '13 at 22:39
    
introduced in a different book: "A few characters from Moore's earlier novels participate in this story: Minty Fresh from Coyote Blue" –  Will Jagy Dec 19 '13 at 22:55

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