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. 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.
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. This question was also asked in mathoverflow. 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, 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 going to address part of this question, specifically the changing role of the notion of proof as it appears in standards influenced by research. 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.
Von Glasersfeld the prominent theorist "posited that knowledge is built up by the cognizing individual , p. 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. You will find some research-based justifications for this approach to standards-based education in the Yackel and Hanna chapter. 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 note the difficulty some students face in secondary school when they have not had an underlying mathematical education that includes justification and argumentation p.
In short, we can see a research-supported change in the way proof and proving is approached in K 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 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.
Link is behind a paywall. National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston, VA. Stylianou, D. Teaching and learning proof across the grades: A K perspective. Yackel, E. Reasoning and proof. Kilpatrick, W. Schifter Eds. 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.
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 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. 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.
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. 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. 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:. Ok, so, I can't trisect an angle.
Really , because if I use a ruler and a compass we can trisect an angle. He is also famous for his theories on other parts of life: in Optiks , he discusses perspective and gives insight to how we view the world through our eyes. With geometric principles, other mathematicians in later centuries were able to develop upon his work.
In this regard, he is understood to be the Father of Geometry since he paved the way for so many future thinkers to expand upon his organized ideas. Other thinkers have even used his geometric method as a foil, in which they have expanded thought in a completely different direction. Today, we understand Ancient Greek culture as classical, in which thought, discussion, mathematics, sciences, and the arts developed and flourished as never before within Greece.
Euclid was a part of that culture. Euclid existed around BC, and he was a prominent figure in Greek culture at the time as a thinker and scholar. He was part of a new tradition of questioning thought, understanding the changing world, and developing ideas so that we could better understand the patterns in the world around us. He, among other Ancient Greek scholars, has left a legacy of thought that many scholars and academics today continue to follow.
The situation is best summed up by Itard [ 11 ] who gives three possible hypotheses. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death.
The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about years earlier.
References show. Biography in Encyclopaedia Britannica. H L L Busard ed. P M Fraser, Ptolemaic Alexandria 3 vols. Oxford, Mathesis 3 4 , - G R Morrow ed. Monthly 57 , - XL Mem. MGU Istor. Storia Sci. Exact Sci. E Filloy, Geometry and the axiomatic method. IV : Euclid Spanish , Mat.
Ense nanza 9 , 14 - II, Sides and diameters, Arch. Histoire Sci. I Grattan-Guinness, Numbers, magnitudes, ratios, and proportions in Euclid's 'Elements' : how did he handle them? H Guggenheimer, The axioms of betweenness in Euclid, Dialectica 31 1 - 2 , - W Knorr, Problems in the interpretation of Greek number theory : Euclid and the 'fundamental theorem of arithmetic', Studies in Hist.
W R Knorr, What Euclid meant : on the use of evidence in studying ancient mathematics, in Science and philosophy in classical Greece New York, , -
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