# 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.

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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 at 14:12

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.

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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.

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+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 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 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.

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@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 at 15:43
I'm in sympathy with your point, but I don't think that history so clearly supports it. According to Rouse Ball, the standard in the middle ages of the quadrivium, which were the subjects for the M.A. degree, was not all that high in England. A student's understanding of mathematics was not put to the test; the student merely had to swear he had done all the statutable exercises and was otherwise free to continue to concentrate on "logic, philosophy or theology." –  Michael E2 May 6 at 19:28