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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 a more efficient mathematical subject in comparison to EG, 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.

Ali Taghavi
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smyrlis
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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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    Qiaochu Yuan, are you really suggesting that geometry is no longer an appropriate subject for secondary math education?? – Monroe Eskew Dec 19 '13 at 18:40
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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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    A Google Scholar search for "geometry education proofs" turns up considerable literature on this debate. For example, see http://eprints.soton.ac.uk/14691/ and the references therein. – Timothy Chow Dec 19 '13 at 21:35
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    I'm often happy to see mathematics-education questions on MO, but I must agree that this one is too opinion-based and not carefully worded. (There are also several questions posed together.) Instead of answering in depth, I will leave only a book recommendation: "Introduction to the Foundations of Mathematics" by Raymond L. Wilder. – Benjamin Dickman Dec 19 '13 at 22:26
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    Geometry is too demanding to the teachers, so the main problem might be not political. – Anton Petrunin Dec 19 '13 at 22:51
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    @smyrlis, I think you should get additional opinions from mathematics educators. Try also asking this question at The Mathematics Teaching Community. – JRN Dec 20 '13 at 01:28
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    Sounds like the real question is, "how do I best teach my students proofs?" To which I'd say, "if proofs are important to learn, they should be used throughout the math program across all math subjects." Whether discussed negatively or positively, proofs are inextricably linked to geometry around here (southeast michigan) and few appear to think of proofs as a separate skill altogether. – Adam Davis Dec 20 '13 at 02:24
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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 04:47
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    @Anton Petrunin: your quotation of Sharygin is interesting, where is it from? – Sergei Akbarov Dec 20 '13 at 07:11
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    The news of Euclid's death is greatly exaggerated. By the way, Bourbaki is not a person. Bourbaki is a committee. – Wlodek Kuperberg Dec 20 '13 at 04:21
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    I'm voting to close because the question is beginning to attract bad answers. I think it has run its course. – Benjamin Steinberg Dec 20 '13 at 15:19
  • Ah, yes! Thank you! I'll add this to my armoury. :) – Sergei Akbarov Dec 20 '13 at 15:55
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    I took a break from my break to close this a second time. The first problem is that it is apparently not even clear what teaching Euclidean geometry even means precisely. So the debate is vague from the outset. –  Dec 20 '13 at 23:51
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    Lucio Russo in The Forgotten Revolution highlights the importance of Euclid for appreciating the Hellenistic idea of the axiomatic method. He also presents evidence that the axiomatic method was used more widespread in the Hellenistic period, for example also for geometry on the surface of a sphere. Sadly, this question was asked in a quite polemic way, mixing facts and preconceptions, such that I'm not surprised it got closed. There are good reasons why Euclid should be taught in high school! I was taught it, and started to see and solve problems... – Thomas Klimpel Dec 21 '13 at 01:36
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    Related meta question: http://meta.mathoverflow.net/q/1301/12357 – JRN Dec 21 '13 at 13:26
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    Please note that I have moved some answers and comments to http://tea.mathoverflow.net/discussion/1630/copy-of-joels-answer-and-comments-to-is-euclid-dead/ – Todd Trimble Dec 21 '13 at 22:06
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    I sympathize you campaign, if you will succeed it will be very good for Cyprus. But even after the edits, the question is too provoking, this does not help to keep the discussion in the right direction. I think you need to split the question into parts. For example, you may ask what is the best textbook in EG. You may ask to share experience with geometry courses in different countries. – Anton Petrunin Dec 22 '13 at 03:27
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    To be pedantic, MO is not really well-suited as a platform for discussion, much less debate that the question calls for. I voted on closing it, and I hope this question goes away---but that some of the answers after cleaning up remain... – Suvrit Dec 22 '13 at 05:01
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    This question currently has 3 delete votes. While I am not a fan of the question for MO, it seems to me that a question with this many votes and answers should just remain closed. – Benjamin Steinberg Dec 27 '13 at 03:57
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    You might want to ask a similar question at http://matheducators.stackexchange.com/ to get answers from people closer to highschool education. – András Bátkai Apr 29 '14 at 08:00
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    The question currently has 4 votes to reopen, and 7 to delete. But how can a question have 7 votes to delete, and not be deleted? – Gerry Myerson Aug 16 '16 at 12:51
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    I was taught constructions with ruler and compass in middle school. It was very fun. The only thing - it should have been in maths class and not in the arts class. – Per Alexandersson Aug 17 '16 at 12:43
  • I learned that the US and Russia still have something in common with regards to teaching Euclidean geometry in high school. To be honest, except for the fact that tangents are perpendicular to circles which comes up over and over again in elementary physics, and maybe the fact that the sums of angles in a triangle is 180 degrees made memorizing the values of trigonometric functions easier, I really wish I had learned graph theory or something more applicable in high school instead of Euclidean geometry. Admittedly my interests are more applied than theoretical though, so I'm not unbiased. – Chill2Macht Aug 20 '16 at 01:46
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    John Stillwell, The Four Pillars of Geometry - "Many people think there is only one “right” way to teach geometry. For two millennia, the “right” way was Euclid’s way, and it is still good in many respects. But in the 1950s the cry “Down with triangles!” was heard in France and new geometry books appeared, packed with linear algebra but with no diagrams. Was this the new “right” way or was the “right” way something else again, perhaps transformation groups?". – Anthony Jan 16 '23 at 15:39
  • Mathematicians have the worst judgment in deciding what math to teach in schools. Non-mathematicians have no judgment at all in such matters. – Michael Hardy Jan 17 '23 at 00:34
  • "I am interested in hearing arguments both for and against the return of EG to high school curricula." This is very much not what Mathoverflow is for. I would vote to close (again), but there is such strong support for the question that I'm inclined to let it pass (and hope nothing else like it ever gets posted here). – Gerry Myerson Jan 18 '23 at 02:37

12 Answers12

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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.

Andreas Blass
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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
  • @SergeiAkbarov Yes, this was in Detroit. – Andreas Blass Dec 19 '13 at 21:48
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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 high-school Math courses (mid-2000's) were exactly the same as in your first paragraph. – BlueRaja Dec 19 '13 at 22:22
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    @AntonPetrunin I say that some of the teachers were not competent. There were also some excellent teachers. Let me add that, thinking back on the situation now, I believe that some of my best math and science teachers in high school could easily have gotten jobs in industry with far higher salaries; they were teaching essentially as a public service, for which I am very grateful. – Andreas Blass Dec 20 '13 at 00:15
  • @AndreasBlass, still it is extremely difficult to teach Geometry. Your first par tells me that the teachers were not competent. For me, big question is: if you want to move EG back to school, where you will get those high professionals? You will not get them in one day just by political decision and as you say these people can get jobs in industry with far higher salaries... – Anton Petrunin Dec 20 '13 at 00:31
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    To avoid international dipute I think one should give references to good textbooks. My generation had been taught geometry (in Russia of 1970es) by Kolmogorov's books: http://www.twirpx.com/file/489992/. As far as I understand, they were not translated into English. Later a lot of reproaches to these textbooks appeared (and it was impossible to understand what the authors of these reproaches had in mind), but for me these textbooks are examples of well-done work. – Sergei Akbarov Dec 20 '13 at 06:57
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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 07:04
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    In addition: straightedge and compass constructions are wonderful ways to get hands-on with mathematical proofs. – dfeuer Dec 20 '13 at 07:08
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    Let's make an example. At some point in your course, buried deep inside a more difficult proof, you will need a property like "if a point $P$ is internal to a polygon $\Gamma$, every line through $P$ intersects $\Gamma$ in at least two points". How do you handle it? Do you add a postulate? Do you add a postulate for the triangle, and then you prove it for polygons? (and what about circles, then?) Do you silently assume it? Do you say "it's a theorem, but it's too difficult to prove for you? Do you ignore the configuration issues? How do you answer to a student asking "isn't this obvious?" – Federico Poloni Dec 20 '13 at 07:47
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    @Federico Poloni, math and science classes are full of "lies told to children" and things left unexplained. What high school number theory or discrete math course would build up from Peano arithmetic or ZF theory? What high school calc 2 class includes a rigorous proof of Stokes's Theorem? – dfeuer Dec 20 '13 at 08:47
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    @dfeuer That is true in general, but the main (or rather the only) purpose of studying EG is because it teaches proofs. If you start hand-waving in the middle of a formal proof, the whole purpose of your lecture is lost. So, when one teaches how to demonstrate things, it is of utmost importance to pick a "playground topic" in which the basic proofs can be made fully formal without traps and delicate points; and in this respect Euclidean geometry looks like an egregiously bad choice to me. – Federico Poloni Dec 20 '13 at 09:53
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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
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    It seems to me that Qiaochu's suggestion to teach, say, graph theory instead solves both problems: you can do everything "honestly", and at the same time the objects are completely concrete and you can play with examples. Having said that, I pity students who don't learn about straight edge and compass constructions, incircles, excicles, etc., because they are a wonderful source of puzzles. I think they should be taught somewhere (perhaps maths circles) for the sheer enjoyment of it, not because it's "important". – Alex B. Dec 20 '13 at 12:59
  • Do you remember which textbooks did you use? – Anton Petrunin Dec 22 '13 at 01:12
  • @AntonPetrunin No, I don't remember the name or author of the textbook. We can hope it's out of print by now. – Andreas Blass Dec 22 '13 at 21:58
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    @AntonPetrunin I should perhaps add that the "good stuff" I mentioned, like the theorems of Ceva and Menelaus, wasn't in a textbook. The teacher just added that material on his own. In contrast to the textbook, the teacher's name was worth remembering, Mr. Garritano. (I hope I spelled that right.) – Andreas Blass Dec 22 '13 at 22:00
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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. 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 The Mathematics of Mathematics Houses (The Snaky Connection) 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!

LSpice
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Amir Asghari
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    Iran will stay with geometry since the ornament on your flag is made with a compass-and-straightedge construction http://www.isiri.org/portal/files/std/1.htm – Anton Petrunin Dec 20 '13 at 00:39
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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 01:27
  • Triangles are wonderful and concrete - until you enter curved space. o.O – Kupiakos Dec 20 '13 at 02:10
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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 05:15
  • Amir, would you be willing to give a specific citation for the research mentioned as supporting your observation? This would be helpful for anyone who wants to counter the suggestion that the answers are primarily opinion-based. – Todd Trimble Dec 21 '13 at 03:49
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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
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    As it can be seen, the point is not to teach or not to teach geometry. The point is to teach geomety as mathematics and to teach mathematics as geometry. That is to see mathematics as a lively connected knowlege. This is what most curriculums and accordingly most teachers fail to do so. – Amir Asghari Dec 21 '13 at 11:23
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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.

Qiaochu Yuan
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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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    So you suggest that they should teach only what is useful, and you define useful as something immediately applicable in life. Euclidean Geometry is useful because it helps you understand what is proof, what is a theorem, an axiom, a foundation. After all proof was born in the context of Euclidean Geometry. Young student get very excited when they manage to prove something they see its true, once they made a beautiful geometrical figure. – smyrlis Dec 19 '13 at 22:37
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    @AntonPetrunin, your comment looks pretty extremist to me. Personally, I believe I had a good math school, yet in retospect I wouldn't say that EG, especially stereometry, taught me any "rigor", if you don't identify rigor and bourbakist formalistics. Stereometry was especially a prolonged tedious exercise on calculations in vector calculus. Would you support your PoV by any scientific studies? – Anton Fetisov Dec 20 '13 at 00:08
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    @AntonFetisov, Be careful about "scientific studies" in this subject, most of them are tools in the political games. (I am sorry that you did not get much from EG, I've learned a lot from it.) – Anton Petrunin Dec 20 '13 at 00:19
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    I don't think the issue is that EG is a great place to start with proof as much as calculus is a terrible place. – Sam Hopkins Dec 20 '13 at 04:30
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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 06:48
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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 08:16
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    @Monroe: obviously engineering is very useful, but I don't see why we should force (I want to keep emphasizing this word) students to learn its foundations if most of them aren't going to become engineers. – Qiaochu Yuan Dec 20 '13 at 08:19
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    @smyrlis: why do most people need to understand what a proof, a theorem, or an axiom are? Students are excited about many things, many of which have nothing to do with school. In fact in my experience it's much easier to get excited about a subject when no one is forcing (there's that word again) you to learn it. – Qiaochu Yuan Dec 20 '13 at 08:20
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    @Victor: you're twisting my words. – Qiaochu Yuan Dec 20 '13 at 08:20
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    @Qiaochu Yuan, I can explain this: "why do most people need to understand what a proof, a theorem, or an axiom are..." Because in human society there must be a critical mass of those who understand what logic is, and as a corollary who can estimate the reasonableness of what different other people say among those who pretend to be leaders ("political, economical, intellectual", etc.) This is very important. I dare to say that all the main horrors of social life of 20 century were results of the underestimation of this need. :) – Sergei Akbarov Dec 20 '13 at 08:42
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    @Sergei: I think it is important for people to learn what good and bad arguments look like but I strongly disagree that teaching people Euclidean geometry is a good way to do this. For example, you could teach them about cognitive biases instead; this would at least help them detect bad arguments. – Qiaochu Yuan Dec 20 '13 at 08:47
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    Offering people the opportunity to study something is not the same as forcing them to do so. I had to fight tooth and nail to get into appropriate math classes in elementary and middle school—no one forced me to do that! – dfeuer Dec 20 '13 at 09:00
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    @Qiaochu Yuan: my thesis is opposite. I disagreee that it is possible to explain which arguments are good and which are bad, if the listener has no idea of what logic is. There are many examples when people followed false arguments of intellectual pilferers (with horrible results) just because they looked "attractive". Hegel's filosophy is one of these examples. if you want. But I am afraid, this will be a long dispute, we can chat if you want. – Sergei Akbarov Dec 20 '13 at 09:02
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    I forgot to add that in my opinion so far there are no other effective tools to teach people logic at school but giving them Euclidean Geometry. You (and other people) mentioned combinatrics, projective geometry, etc., but first, there must be good textbooks for children (I do not know them), and second I don't believe that one can think up something that can compete with the visuality of EG. – Sergei Akbarov Dec 20 '13 at 09:17
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    There are no theoretical reasons preventing people from exploring elementary geometry computer-based. We've got lots of dynamic geometry software to discover results, and I don't understand why noone has made a user-friendly proof checker for the limited kind of reasoning commonly present in geometric proofs. Probably that will come when generic proof checkers become more usable and develop a more substantial ecosystem around them, but I think it could just as well be done by now if math educators would be up to date on modern CS. – darij grinberg Dec 20 '13 at 15:18
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    What is the use of "visuality" when the student does not have the spatial intuition to take advantage of it? Indeed, is it not a disadvantage if we wish to teach students formal reasoning? – Zhen Lin Dec 20 '13 at 16:03
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    @Zhen Lin: I don't understand your point. It is desirable to have illustrations when you explain something, that's why visualization is important. If there is a possibility to explain logic WITH illustrations, it is better than explaining it WITHOUT them. Or what are you talking about? Again, I feel that this will be a long discussion, will it be better to chat? – Sergei Akbarov Dec 20 '13 at 16:44
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    It is clear for me that the attitude of different people to this subject depends exactly on the culture of teaching geometry in their countries. As far as I see, Russians are more satisfied with the way of how EG is taught at school, than Americans and Chinese. The solution I believe is sharing the experience: translating textbooks, teaching school teachers, and so on. – Sergei Akbarov Dec 20 '13 at 17:52
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    @QiaochuYuan To continue Akbarov's explanation of "why do most people need to understand what a proof, a theorem, or an axiom are..." Proofs are not just the result of EG, they began with the arguments at Agora, the discussions of the philosophers. All people, as opposed to most people, should know what a proof is. The rules of logic are not innate (studies in psychology have shown this). To avoid people passing form "an Arab terrorist" to "all Arabs are terrorist" you need, at least, people educated in what a sound reasoning is. Whether EG is the way to do that is a different story. – O.R. Dec 21 '13 at 00:51
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I strongly recommend to read the paper Нужна ли школе 21-го века Геометрия? 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.

LSpice
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Anton Petrunin
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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 06:28
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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
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    Very interesting answer! – Gil Kalai Dec 21 '13 at 18:41
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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...

Paul Siegel
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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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    "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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    @ToddTrimble "What man among you, if his son asks him for bread, will give him a stone?" – Anton Petrunin Dec 20 '13 at 01:53
  • @AntonPetrunin Yeah, that's what I figured you meant. – Todd Trimble Dec 20 '13 at 01:59
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    I suppose I should have at least put "Euclidean geometry" in quotes. – Paul Siegel Dec 20 '13 at 03:44
  • Also, I should add that when I wrote my third bullet point about EG being obsolete I very nearly included the caveat, "though I did have know a professor in graduate school who could prove just about anything using only triangles". – Paul Siegel Dec 20 '13 at 03:50
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    Linear algebra is useful, but that doesn't make it the right introduction to proof for students. Why do you say linear algebra is "the basic language in which most of mathematics is expressed"? I see neither hide nor hair of it in set theory, formal logic, general topology, etc. – dfeuer Dec 20 '13 at 08:54
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    Well, I'm not sure what your "etc." represents, but it's hard to extend your list any further. Perhaps it is just my personal biases, but I think most mathematics falls under PDE theory, algebraic geometry, number theory, differential geometry, functional analysis, and representation theory. – Paul Siegel Dec 20 '13 at 13:21
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    @PaulSiegel, essentially the same reform you suggest fails badly. You have to have experience teaching in school to suggest such things; from your answer it is clear that you have no such experience. – Anton Petrunin Dec 20 '13 at 16:44
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    @AntonPetrunin I indeed have not tried this in the classroom, and I believe that it has failed, but I nevertheless have hope that it can succeed. Do you by any chance have references to past experiments? – Paul Siegel Dec 21 '13 at 00:51
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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 03:04
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    Independently and around the same time, I tried to teach myself linear algebra out of Kostrikin and Manin's book. An utter fiasco! Once again, even for someone with extremely strong algebra background, this required an extreme effort, primarily due to the need to internalize the new mathematical language stripped of all intuition. – Victor Protsak Dec 21 '13 at 03:08
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    A couple of decades later, I taught a very down-to-earth linear algebra course (the title was "Matrix Algebra") which had a large population of pre-service teachers. As it happened, many of them had not yet taken the third semester of Calculus, where vectors in 2 and 3 dimensions, equations of lines and planes and other geometric underpinnings of linear algebra are first taught in the US math curriculum. This was a bitter experience: whether due to their poor algebra background, lack of geometric intuition, or some combination of both, the students could not connect with the material. – Victor Protsak Dec 21 '13 at 03:13
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    Based on all these experiences, I can confidently predict that implementing your suggestion to replace elementary geometry with linear algebra will fare even worse than the hallmark of the NewMath movement, teaching abstract algebra to the feeble young minds. This could only possibly work in the environment where top students are paired with really competent teachers. Even then, the lack of geometric intuition will remain a serious impediment to those bright students in the future. As Anton told you, without actual teaching experience to back them up, such ideas are bound to fail. – Victor Protsak Dec 21 '13 at 03:21
  • @PaulSiegel, I am sorry, I did not find a proof. My comment was based on something I heard long time ago. Maybe such a reform dies on the very early stage and that is the reason I could not find trace of it now. – Anton Petrunin Dec 21 '13 at 05:16
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    EG is not taught in American high-schools. At least not the kind I am campaining for! – smyrlis Dec 21 '13 at 22:20
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    @smyrlis Your last comment is not at all helpful, and exemplifies one of the big problems people are having with this thread: that the question is not precise and focused enough, as put well by quid here: http://meta.mathoverflow.net/a/1303/2926, and is primarily opinion-based, which MO frowns upon. I have by now put some time and effort in cleaning up messes, but now I think I'm done. – Todd Trimble Dec 22 '13 at 01:59
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    One of the biggest problems one encounters teaching linear algebra in the university is that because students have seen little or no geometry in high school they are unable to relate linear equations to geometric objects such as lines and planes, and this makes it very difficult for them to develop any intuition about linear algebra. – Dan Fox Feb 20 '19 at 07:49
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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.

Todd Trimble
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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.

Michael
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    Exactly. Secondary school mathematics education already selects way too heavily for people who like combinatorics and algorithmic computation, which results in an extremely homogeneous group of people who think of themselves as good at math. The important part of "diversity" is encouraging a diverse range of talents. – Elizabeth Henning Apr 26 '21 at 02:43
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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.

Gil Kalai
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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
  • For example http://www.marxists.org/reference/archive/hegel/works/na/nature1.htm: "Negativity, which as point relates itself to space and in space develops its determinations as line and plane, is, however, in the sphere of self-externality equally for itself and appearing indifferent to the motionless coexistence of space. Negativity, thus posited for itself is time." – Sergei Akbarov Dec 21 '13 at 19:05
  • Or: "That the line does not consist of points, nor the plane of lines, follows from their concepts, for the line is the point existing outside of itself relating itself to space, and suspending itself and the plane is just as much the suspended line existing outside of itself.-Here the point is represented as the first and positive entity, and taken as the starting point. The converse, though, is also true: in as far as space is positive, the plane is the first negation and the line is the second, which, however, is in its truth the negation relating self to self the point." – Sergei Akbarov Dec 21 '13 at 19:14
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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 http://www.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. –  Dec 22 '13 at 01:38
  • @quid: Blow out... I must say, I don't see how training in langauge and philosophy could be useful here. As to logic do you know a way to train logic outside of mathematics? – Sergei Akbarov Dec 22 '13 at 02:43
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    @SergeiAkbarov When learning Latin, eg, often one translates political speeches and other things analysing rhetorical techniques and tricks and so on. Seems obviously useful in the given context. Of course you can train logic outside of mathematics. It starts with using language in a precise way, quantifiers, conditionals and so on everythings is there in everyday situations. –  Dec 22 '13 at 02:55
  • Maybe this explains everything... In Russia nothing of this exists at school. I even hardly imagine this. In particular, Latin and Greek authors were almost inavailable in USSR, they were published rarely and with little circulation. I read Herodotus only in 1990ies... – Sergei Akbarov Dec 22 '13 at 03:10
  • @quid, this is indeed intriguing. Are you saying that this is taught at schools in the West? Latin, Philosophy, Logic? – Sergei Akbarov Dec 22 '13 at 03:55
  • @SergeiAkbarov, I think Latin fell out of the western curricula more or less at the same time as Euclidean Geometry. Some little Philosophy and Logic do remain (at least in some countries). Anyway, how much faith can one retain that teaching proofs, in itself, will be determinant for building critical thought? Isn't the USSR system that you explain a counterexample? – quim Dec 22 '13 at 09:31
  • @quid: The critical thought existed in USSR, otherwise the Gorbachov reforms wouldn't have support at their beginning. And I would say, it was based on technical education (rested upon logic, without Latin, Philosophy and Linguistics). Quid, do you have time for a little chat? I forsee accusations in off-topic. – Sergei Akbarov Dec 22 '13 at 10:01
  • Also, it seems perfectly possible that critical thought existed, but that for a sufficiently large mass of young people the bullshit had stronger impact than the proofs. I simply don't know. – quim Dec 22 '13 at 10:13
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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
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    @quim: Yes, but there is a qualitative difference -- if you are familiar with logic, then you can say: "Excuse me, what you call "modern logic" is not logic at all! Compare this with the EG!" You have no this possibility if you know only special tricks like "proofs by induction" (which quid mentioned in my answer, that I deleted recently). – Sergei Akbarov Dec 22 '13 at 10:38
  • @SergeiAkbarov sorry I missed your last comment initially. To answer your question regarding Latin. I would say it is a lot less popular than it used to be, but if you want you can learn it (but 'the West' is cetainly not uniform here). See here for an overview http://en.wikipedia.org/wiki/Instruction_in_Latin Teaching in Philosophy also exists but not expansive (AFAIK, did not find a list and there is a lot of variation). But also I didn't want to say 'this is done and therefore critical thought exists,' only I wanted to challenge claims of math being the unique possible source. –  Dec 25 '13 at 11:03
  • @quid: It's a great surprise for me that Latin or Philosophy can be seriously considered as alternatives for studying Logic. In Russia only physicians study Latin. Philosophy exists in universities (and partly was presented in ideological disciplines like "history" or "state and law" at school), but it was considered that its aim was to provide ideological fundament for the communist party's politics. As a corollary, they did their best to make an impression that the statements like what I quoted from Hegel "follow directly from rules of logic". – Sergei Akbarov Dec 25 '13 at 13:20
  • @SergeiAkbarov especially when we are talking about education in schools I think a lot depends on how the subject is approached in general and perhaps even more on the individual teacher. If you were to ask me which subjects in school contributed most to develop my critical thinking I'd answer religion and latin; by contrast math not at all, the instruction was rather 'ideological' in the sense that due to lack of competence of the teacher one could not get an answer to various things, they needed to be accpeted as 'great truth.' –  Dec 25 '13 at 15:20
  • @quid: this is amazing: " religion and latin". Will it be difficult to you to give an example? – Sergei Akbarov Dec 25 '13 at 15:31
  • @SergeiAkbarov regarding religion, it was a well-educated teacher that besides presenting the religion he did teach presented all kinds of world views, so different religions but also atheistic philosphers and we discussed a lot about such things, and various other things. For latin I menioned rhetoric and political speeches already and then one did make some conections to more current events. –  Dec 25 '13 at 23:34
  • @quid, something similar was in the soviet education: in the course of history (and philosophy in universities) there were lessons where teachers gave some imagination of different religions. But the accent was that "the true scientific approach suggests that..." - and after that the quotations of Lenin, Marx, Hegel - and deep in chronology. So for forming critical view this seemed to be almost useless in that situation. – Sergei Akbarov Dec 26 '13 at 07:45
  • For example, I don't believe that it is possible to contest Hegel's "reasonings" which I cited, if you are not familiar with formal logic and its substantial applications like EG: all your objections will be broken on the thesis that "this is the old-fashioned understanding of the subject, but the modern science and modern logic suggest that..." - and so on. – Sergei Akbarov Dec 26 '13 at 07:59
  • @SergeiAkbarov I still do not understand how knowledge of EG could be relevant here. Math, especially using the axiomatic approach, and to teach this is the main argument in favor of EG here it seems, is very 'relativistic' You can change your axioms at will, and proceed from there to arrive in most any world you like; like just drop the parallel postulate already things are different :-) –  Dec 26 '13 at 10:09
  • Then, also in math some study/use other types of logic. Say, do you allow double negative elimination or not. Most mathematicians see no problem with it but then some do. Yet, sure, if you know some math you could do like Euler when debating with Diderot, but perhaps this type of 'logic' is not what you have in mind :-) But also I do not want to deny that it is a good thing to know what it means that A follows logically from B and things like this and that for some an appreciation of this could come from studying math. –  Dec 26 '13 at 10:23
  • @quid, I think, the problem is that you had no deal with marxist propaganda: their main thesis in this area was that there is "formal logic" (in usual mathematical sense), and this is "old-fashioned", "mathephysical", "wrong" logic. And on the other hand, there is "dialectic", "the modern logic", "true logic", "the last word of the science". And of course, a thinking person must choose dialectic, since this is "modern science". To understand what lies behind this (and to be able to find counter-arguments) you had to know what formal logic actually was. That was the question. – Sergei Akbarov Dec 26 '13 at 11:23
  • I am not against any type of formal logic, including intuitionist or any other. But in any way when you accept formal rules, they don't allow you to "deduce" everything what you want, like this was in marxist propaganda. They needed dialectic (see vordenker.de/ggphilosophy/popper_what-is-dialectic.pdf) to have this indulgence, this possibility to deduce what they needed in every concrete moment. Sometimes, when the situation changed, they changed their conclusions to the opposite ones. But every time this was under the slogans that "these are the last words of modern science!" – Sergei Akbarov Dec 26 '13 at 12:18
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    When I was arguing with people at that time the only possible argument for me was appealing to the example of Euclidean Geometry. "Look what logic actually is!" - that was my argument, and only this worked. Even the logical method of analysis could be interpreted as "just several tricks among others, includong tricks of dialectic". The qualitative difference lies in axiomatic systems (and EG is the only one that was sucessfully taught at school). – Sergei Akbarov Dec 26 '13 at 12:18
  • @SergeiAkbarov I agree I have some difficulty understanding/envisioning the situation. Thank you for the attempts at explaining. –  Dec 26 '13 at 15:31
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It might be interesting to point out that the support for removing axiomatically taught Euclidean geometry (not all geometry) from the school education predates Bourbaki. Oliver Heaviside (1850-1925), a British mathematician who also made important contributions to physics wrote in his "Electro-Magnetic Theory", vol. 1 (1893):

"As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous".

Margaret Friedland
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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.

Monroe Eskew
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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 02:57
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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

Gerhard Paseman
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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. http://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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I'm suggesting teaching the foundations of mathematics. I would pick a system like:

  • ETCS (Elementary Theory of the Category of Sets),
  • HoTT (Homotopy Type Theory),
  • Dependent Type Theory,
  • ZFC?

I suggest this can conclude with one of:

  • The theory of cardinal numbers.
  • Defining the arithmetic operations on $\mathbb N, \mathbb Q, \mathbb R, \mathbb C$ and proving their identities.
  • I'm perhaps overly idealistic, but perhaps some funky topos theory: Nonstandard Analysis? Synthetic Differential Geometry? Synthetic Computability?

Advantages:

  • They'll learn mathematical notation and terminology. People who don't learn this notation and terminology might otherwise struggle with more advanced maths. Examples of terms they'll learn: Functions, tuples, Cartesian product, sets, subsets, natural number, etc.

  • They'll learn a proof calculus like Natural Deduction, along with proof by induction. These are idealised, general and rigorous models (imitations?) of the proofs constructed by actual human beings.

  • Has some link to topics like Functional Programming.

Disadvantages:

  • Not sure how this can help most engineers.

  • No overlap with geometry.

  • No help with calculus, except clarifying some basic terms.

  • Could be accused of being pedantic.

Questions:

  • Will students or teachers find this easy to learn or teach?

  • Where are the problems to solve? ---- Here I suggest cardinal numbers might provide a small problem list.

wlad
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    I think “Not sure how this can help most engineers” is a key point. Realistically, a math department that only serves math majors can't thrive in most (US) institutions—so you either have to develop this curriculum while maintaining a parallel ‘traditional’ track to serve applied needs, or give up on applications and risk math being seen as even more irrelevant than (US-cultural) trends would already paint it. (I also think an absence or paucity of exercises is a huge drawback in early education especially, but that's probably a function of the novelty of this approach rather than intrinsic.) – LSpice Jan 16 '23 at 16:15
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    You would teach homotopy type theory and topos theory to all high school students, before they get to university? Those topics sound difficult. Has these topics been taught to typical secondary school students? – Ben McKay Jan 16 '23 at 16:15