Is Euclid dead? or Should Euclidean geometry be taught to high school students?

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.

Note. This question was also asked in mathoverflow.

• Speaking from South Africa, I can testify that Euclid was recently resurrected. That is, Euclidean geometry was taken out of the syllabus about 5ish years ago, then reinstated 2 years ago. – David Ebert Apr 30 '14 at 14:12
• you bring up a great point and i think both curricula (EG and set theory/soft analysis) have amazing implications and applications to learning. personally, i would love to see both available for secondary students to take! – celeriko Nov 6 '14 at 3:24
• Euclid est mort. Dieudonné aussi. – Selene Routley Jul 21 '16 at 0:10
• You might distinguish between Euclidean geometry and Euclid's program. By the former, I mean the geometry of Euclidean spaces, usually $\mathbb{R}^2$ and $\mathbb{R}^3$ equipped with the Euclidean norm. By Euclidean program, I mean the process of proving his own theorems from his postulates. My opinion is that students should master Euclidean geometry but Euclid's program should be reserved for discussions of historical context. Nearly all of his first principles can be proved with algebra, and algebraic arguments should be heavily incorporated into curricula that explore Euclidean geometry. – Andrew Mar 29 '17 at 15:52

I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, than Euclidean Geometry is a great way to implement this:

Visuality Euclidean Geometry deals with objects that can be easily visualized. It can be properly served on all three represantative layers: enactive, iconic, symbolic.

Scaling of Argumentation level Theorems in Euclidean Geometry can be proven or argumented for on different argumentation levels: intuitively formal-rigorous, abstractedly formal-rigorous (Euclid's way), with generalizable examples, using intuitive knowledge (symmetry, movement invariance, …).

Loss of calculations Many proofs in Euclidean Geometry include no calculations at all, others only as small substeps. Consequently, students can learn that way, that maths is not just calculations. This sounds trivial, but it is often a problem: They hesitate in Euclidean Geometry, because they cannot simply calculate a result. The result isn't even a number or a value, but the theorem itself.

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?

I am going to address part of this question, specifically the changing role of the notion of proof as it appears in standards (influenced by research).

My brief response borrows heavily from the document I am referencing (Yackel & Hanna, 2003); by responding I intend to connect your question to something underlying the inclusion of mathematical practices (and/or mathematical processes) in recent standards documents here in the USA. This inclusion is based on mathematics education theory. I do not address arguments for including Euclidean geometry and geometric proofs as curriculum content. As proof and proving are not my particular area, I am hoping that what I can provide here points you towards research and arguments that help you address the part of your question that is tied to "proofs."

Many math ed researchers tightly associate learning and reasoning (see a quote from Thompson in Yackel & Hanna, 2003, p. 227). Von Glasersfeld (the prominent theorist) "posited that knowledge is built up by the cognizing individual (2003, p. 227)."

The importance placed on mathematical reasoning in the learning process has been embraced by some recent research-based approaches to math education, Being able to communicate how you know something now becomes an inherent part of mathematics instruction (and classroom activity for both teachers and students). Therefore, at all levels, argumentation and justification become a practice that must be suffused throughout the curriculum rather than a specific content target for a particular grade band. This is not only true for the NCTM Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), but also for the recent Common Core State Standards with their section on mathematical practices.

You will find some research-based justifications for this approach to standards-based education in the Yackel and Hanna (2003) chapter. A more recent book (Stylianou, Blanton, & Knuth, 2009) also provides insights into how research informs the inclusion of proof throughout the grades, and as early as kindergarten.

I hope this acts as a decent starting point to sources that argue strongly for proof as a mathematical practice, which would provide a foundation for secondary-level proof and proving that might be more recognizable in traditional curricula. Yackel and Hanna (2003) note the difficulty some students face in secondary school when they have not had an underlying mathematical education that includes justification and argumentation (p. 234).

In short, we can see a research-supported change in the way proof and proving is approached in K-12. We see the idea that argumentation and justification are practices students need throughout their mathematics education rather than being introduced to "proofs" as an activity late in their public school education. Additionally, that these practices support students in learning "proof" later on.

The chapters in Stylianou, Blanton and Knuth (2009) make the argument for the broadening of the notion of proof, and are a much more detailed look at how and why researchers are seeing the need for this perspective change. This doesn't argue against Euclidean proof, but I think it does argue against that being the first introduction to the idea of proving, and of the practice of justification as an inherent part of knowing in mathematics.

I checked, and there is proof used to "refute or support conjectures" in the 9-12 Geometry Standard of the NCTM Principles and Standards for School Mathematics. (Link is behind a paywall)

Cited

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.

Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge. Retrieved from http://books.google.com/books?hl=en&lr=&id=8RCRAgAAQBAJ&oi=fnd&pg=PP1&dq=proof+blanton&ots=c5wYvMHgOT&sig=-5-wM5dmwLG_ZWCJeLfIpyEUDpA

Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.

• +1 Many sources which support, what is already theoretically clear: a spiral approach to mathematical argumentation which results in formal, rigorous proofs. It's clear, that the abstract proofs in Euclidean Geometry, which often use non-direct arguments, are a mere but good intermediate step in this spiral. – Toscho Apr 30 '14 at 19:56
• If whomever is downvoting me on this answer reads this, I'd like to know how they recommend I improve the answer. I think I accurately connected one point of view on proof with the research justifications it rests on. But I am willing to listen to constructive criticism. – JPBurke May 3 '14 at 1:26

I teach at the college level in the US, where geometry is a standard part of the high school curriculum, so my students are all supposed to have had previous exposure to it. However, probably only a certain percentage (50%?) of students here take a geometry course that requires them to write proofs. I assume that in many cases in the US, the high school geometry teacher simply isn't willing to grade a stack of proofs from five classes, each with an enrollment of 40.

What frightens me is the results I get when I ask students to write even a trivial proof. For example, I introduce the vector cross product and give both its characterization in terms of components and its geometrical characterization. I then assign a homework problem in which the student is asked to prove that $\mathbf{A}\times(k\mathbf{B})=k(\mathbf{A}\times\mathbf{B})$. The most common response (among students who write an answer at all) is to make up an example and demonstrate that the property holds for that example.

If students think this is a valid mode of reasoning, then I suppose they believe that all presidents of the United States are African-American, since we have an example that proves it. This is not a defect in their mathematical preparation, it's a defect in their critical thinking skills that could perhaps have been addressed through their mathematical training.

These are the results in an educational system that does, at least in theory, require proof-writing in high school. I shudder to imagine how much worse it would be in a system where there had not even been any attempt to teach proof-writing. When Abraham Lincoln was a country lawyer in Illinois, traveling on horseback to county seats and sleeping in boarding houses, he always carried a copy of Euclid with him, which he would study at night by lamplight as a model of logical argumentation. I wonder how Lincoln would have turned out if he'd never been exposed to Euclid.

• @MichaelE2: Point taken, but I think what's relevant is that in the 19th century, Euclid was an expected topic of study for young men who were going to become educated. This is a centuries-old tradition dating back to the medieval quadrivium. If you look at the contents of the trivium and quadrivium, it's pretty clear that it's not meant to be utilitarian; rather, it's meant to teach you how to think and argue. – Ben Crowell May 6 '14 at 15:43
• Great point about mathematics addressing critical thought. But I think the adult makeup of Lincoln, determined to learn, in flagrant opposition to all those around him as he grew up, would have been pretty robust in the face of hypothetical perturbation. – Selene Routley Jul 21 '16 at 0:32
• I also teach at the college level, and I don't expect students to be able to write trivial proofs unless they've been shown how. Even with a grounding in Euclidean proof, I found proving basic algebraic properties from axioms to be a little strange, until I was shown some basic examples. These claims can seem so obvious that the student wonders what there even is to prove. This specific problem must be addressed head-on before such a student is comfortable doing such exercises. – G Tony Jacobs Jun 14 '17 at 15:29
• I'd like to echo G Tony Jacobs comment. I have noticed students trying to prove basic facts with examples in different countries; I think it is a natural tendency. Van Hiele claimed it was an early developmental stage in learning to prove. I don't think it means students can't reason; they're just showing the answer is intuitively reasonable. But give students a surprising, non-obvious theorem, such as all angles inscribed in a semi-circle are right, and students tend to approach such proofs more thoughtfully. – Scott Eberle Mar 8 at 8:00

Since no one here seems to be speaking against the teaching of Euclidean Geometry in highschool I guess I'll explain a few structural problems I have with EG.

My main complaint is what Toscho sees as an good feature. In particular:

Loss of Calculations

Ok, fair enough, standard EG following in the steps of the long dead Euclid proceeds from Axioms to Theorem with out so much as a number in sight. I studied it in high school, admittedly the construction of perpendicular bisectors etc. by compass and straight-edge alone is fascinating. But, I ask, is this loss of calculation natural in our modern age ? Have students already been exposed to math which is over a millenia beyond Euclid before the take EG ? The answer to my question is quite obvious since about the 17th century. From a student's perspective, what I think is a more accurate portrait of EG is this:

Artificial loss of calculations

Ok, so, I can't trisect an angle. Really, because if I use a ruler and a compass we can trisect an angle. Or, to be more direct, if I have a protractor and the idea of dividing real numbers by $3$ then trisecting an angle is not a big deal. In short, students already know the logical successor of EG, in particular analytic geometry is woven throughout our modern discussion of Mathematics and its application to geometry.

I would argue, students would be better served by studying proofs which involve calculation. One of the great absurdities, created in part by the existence of the "transition to proofs" class, is this idea that there is some conceptual divide between math involving equations and math involving proofs. Certainly not. In fact, what we have is merely a dichotomy in standards. On the one hand, the student has an algebra course where to find the vertex of a parabola they simply use an equation, often with no expectation of supporting argumentation or explanation, just the plug-and-chug method so to speak. Then that same student in their EG course is asked to prove something which is frankly obvious with a 5 line column proof where they have to refer to each Axiom and step with more precision than we require in some proof classes at the undergraduate level.

I can't say what is best to replace EG with, but, I would tend to agree with Burke that is not actually about curriculum. The real struggle here is about standards. Why do we insist on having standards in EG, but, not so much in other areas of math. Students should be expected to explain why they are calculating in a particular way. The over-emphasis of problem-solving verses the coherence of theory is the true root of the problem.

I can't blame teachers most places, there is simply no intellectual will-power or structural room to build good mathematics. We're too busy meeting SOL's and worrying about inane federal guidelines and such. The idea that students should be held to a higher standard is at odds with every other societal direction, so, I don't think there is a global fix. The solution is at most local, with particular educators investing their vision and coherent understanding of Math where proof and critical analytical thinking are woven into every math class, not just the EG course. This is why I am a strong advocate for every math teacher having at least an undergraduate degree in math.

• To me, the artificiality of Euclidean geometry is actually a plus, not a minus. It introduces the point that math isn't about some absolute notion of truth, but only about what is logically entailed by various reasonable sets of assumptions - and the assumptions can, up to a point, be adjusted for various reasons, including simply that of making the game more interesting. – Alexander Woo Feb 15 '17 at 21:29
• I can't tell you how much I agree with the point on separating solving from proof. Right up through grad school I encountered students who thought the idea of proving a solution to an equation is, in fact, a solution, was a strange concept. The way we do things now - putting proofs in an entirely new topic that superficially looks unrelated to anything else taught - is, well ... I don't know if you could come up with a better way to reinforce this fallacious division. – user37344 Mar 5 at 3:39

I am generally favorable to Euclidean geometry and proofs, within limits, but we do need to understand why learning proofs is so difficult that many countries have reduced such content considerably. Only by appreciating why learning proofs in high school is problematic can we hope to overcome the difficulties.

A considerable amount of research, especially in the 1980s, has shown that students are not as successful with geometric proofs as they seem. For example, in the US, where proofs are still usually taught, teachers developed a system of "two-column" proofs to make the process easier to grade and more "rigorous". Unfortunately two-column proofs often very predictably go through similar processes which teachers manage to teach in such a way that many students learn to successfully imitate them without any deep understanding of what they are actually doing. For example, a great many proofs students learn to do can be solved with a two-column proof by identifying the given information in a triangle, identifying the appropriate theorem (ASA, SAS, SSS, etc.) and continuing to the conclusion. The whole question of why we are doing this in the first place, or what it all means, is given little emphasis.

One large study (Usiskin, Z. [1982]. Van Hiele levels and achievement in secondary school geometry. Chicago, IL: University of Chicago) found that for many, perhaps most, students who had completed a full year of EG, not only did they not understand what they were doing, but they did not yet even have the prerequisites to begin a study of geometric proofs. There has been a considerable attempt to fill in the gaps found in these studies (both in geometric content and logical reasoning) in earlier grades, but it is not still clear if today's students are ready to fully appreciate EG, especially when taught all at once in a one-year course, as is done in the US.

There have been similar findings in Europe (e.g. Gutiérrez, Á., & Jaime, A. [1998]. On the assessment of the Van Hiele levels of reasoning. Focus on Learning Problems in Mathematics, 20(2/3), 27-46.)

• Agreed. I think simply deriving identities is a much better step towards proof. My feeling is that students are much more likely to internalize derivation conceptually and it's a pretty good middle ground between calculation and proof. – user37344 Mar 5 at 3:41

When I went to high school in the early 80s, I studied geometry and did formal mathematical proofs. I then studied economics undergrad and business grad school. When I decided to return in my 30s to school to pursue a master's degree in mathematics, I did not need an undergraduate math degree to apply to the graduate programs that interested me, but I did need to have completed certain courses (like Elementary Number Theory). When I started taking these upper division math classes, some required that I write proofs. Without my high school experience, I would have been lost. Further, whether proofs are introduced in geometry or in some other way in hgih school, learning and practicing logical thinking is foundational to a number of undergraduate disciplines. Thus proofs seem to be an essential college preparatory skill.

One difficult issue with Euclidean geometry is that Euclid's approach, although it was an important step historically, is highly non-rigorous by modern standards. This is especially clear in the definitions and basic notions (what does "a line is breadthless length" mean?). But there's a much bigger problem that already comes up in Prop 1, which is that there's no way to see from the axioms that those two circles intersect at all! Indeed $\mathbb{Q}^2$ satisfies all of Euclid's axioms, but the circles won't intersect since they share no rational points. As Hilbert and others explained, there are ways to fix this issue by adding a lot more axioms. But at that point you're almost doing analysis anyway.

I think number theory, group theory, analysis and graph theory are all better ways to introduce proofs than Euclidean geometry.

I am a high school math teacher and whenever I teach the geometry class, I always do full proofs. I use the geometry class to show kids how we construct all the ideas of geometry from a single point and I use proofs to explain to them the idea of mathematical truths. I teach at an IB school so this preps them for the TOK course they'll take in a few years. But proofs in the Geometry unit really gives me the opportunity to wax philosophical and some the kids from countries where they are just expected to regurgitate facts really appreciate learning that there is an answer "why is this true?" beyond "because I said so.", behind everything they have learned.

"Should Euclidean geometry be taught to highschool students?" Certainly, yes. The whole idea that it shouldn't is a misguided mistake with identifiable historical sources and based on identifiable philosophical errors.

Searching through both the question and the existing answers I notice that there has been no mention made until now of the so-called New Math. I think this should be mentioned is because the basic impetus behind Bourbaki, Dieudonne "death to triangles" call, and New Math is the same, the philosophical errors are the same, and the reasons for failure are the same. For a detailed analysis of the history of the New Math educational debacle see this publication.

More specifically, the philosophical errors involved are traceable to Piaget. Piaget was interested in analyzing the structures of the learner's mind. Piaget claimed an affinity between such structures and the structures of "modern" mathematics. Such an identification was a non-sequitur but it influenced people like Begle who were in key positions in the US to affect pre-university education.

• Why does philosophy matter here? – beroal Jun 19 '17 at 19:26
• @beroal, I will try to elaborate in my answer. – Mikhail Katz Jun 20 '17 at 6:44

While talking about proofs, I notice that nobody is mentioning the conditions: while performing mathematics you can't just do what you want: all conditions need to be fulfilled before you can actually do something.

When explaining a proof to a class, it's very important that the teacher mentions the conditions, and while writing the proof, the teacher should clearly indicate at what point the mentioned conditions are used (e.g. while proving Pythagoras' theorem, the teacher should clearly indicate why this only applies to right-angle triangles).

This way of working learns the students why the conditions are so important and automatically they will watch their steps while setting up complicated reasonings (not only in math, by the way).