Teaching Euclid's Elements

[mrg]

Good evening all, first post here, and I'm hoping I've come to the right place to find some answers.

I'm a math teacher at a small private school, and I've been debating about what to do for my Honors Geometry class next year. I have this strong desire to use Euclid's Elements as our primary textbook, and I'm eager to show work with students through a traditional, rigorous, and pure math book.

However, I'm concerned with pacing, assessment, and meeting the Common Core Standards. For example, reading through a book like this can be draining on high school students, even if they're gifted. I also am concerned with assessment. How would they be graded? How would I ensure that they are studying their book? How would I find out what they have or have not learned? Finally, in today's world of high-stakes testing, would Euclid's Elements alone be enough to prepare students for the ACTs/SATs?

I have a feeling that those that survive the class and complete the text will be much better math students, and will have no trouble adapting to anything thrown their way, much less some silly standardized tests. But in order for this to be successful, I need to make sure that I have planned their course with careful detail.

Any help or direction would be appreciated. If there is a better place to post a question like this, I'd also love to hear it.

Thanks to everyone for their time.
Joe

 

[Tobias Funke]

Does @mathwonk work on these forums? Check out the bottom of his page: http://www.math.uga.edu/~roy/

I think your questions apply to any book. No matter what, very few high school students will learn any geometry without you going through it in class anyway, so the fact that it's Euclid doesn't matter much. I don't recommend book V or anything like that, though. Maybe you can start introducing coordinate geometry about halfway through the year but still (mostly) follow Euclid's order.

You should also tell them that because of the language, to even understand the wording of a theorem they might have to look at the proof and piece it together from there (that was the case for me at least). That's the only real stumbling block I can think of.

 

[mrg]

Thanks for the reply Tobias. The link you've provided contained some very useful information and I thank you for sharing it.

While I agree with you that I will need to go through the material for most all high school students for them to truly get it, Euclid's book is vastly different from what is used today. Textbooks today contain little "reading." Mostly, they're a book full of examples, steps, and practice problems. Even geometry books are stripped of much of the beautiful pure math that they're supposed to contain. As such, many students simply are not prepared to read even a single page of the book. I'm hoping to find as much information as I can about how to help students transition from a modern math textbook to something very rigorous like Euclid's Elements.

They're also not "aligned with Common Core," which I could certainly spend some time this summer doing myself (adding certain things here and there, skipping certain sections here and there, etc.). But finding someone that has already done it would save me some time, but more importantly, it will make sure my students get the best class that they can get.

 

[AlephZero]

I'm in favor of kids learning some geometry, and getting a head start on the idea of formal proofs at the same time, but I don't think Euclid's order is the best. For example the first two books are entirely about straight lines. Do you really want to work through more than 60 theorems (including the proof of Pythagoras's theorem) before you even mention circles? Probably not.

It's fairly easy to find charts of which theorems in Euclid are prerequisites for others, so you could make your own course syllabus working through several books in parallel. But it might be easier to find a geometry textbook from 40 or 50 years ago and use that.

If you haven't found this site already, check it out: http://aleph0.clarku.edu/~djoyce/java/elements/

 

[micromass]

I agree with AlephZero, it doesn't sound like a good idea.

First of all, Euclid's Elements is a huge book consisting of many volumes. I doubt you will be able to cover them all. However, only covering the first book seems limiting. Covering the later volumes is problematic since Euclid didn't even like the idea of infinities and irrational numbers (which explains his weird formulation of some of the axioms and theorems). Many of the proofs are incomplete and some are flat out wrong. Furthermore, Euclid doesn't do any analytical geometry (because it didn't exist yet of course), which I think is quite essential today.

If you want students to work through a real math book written by a real mathematician, then I highly recommend Lang's geometry: https://www.amazon.com/dp/0387966544/?tag=pfamazon01-20 Be sure to read his preface on the choice of topics. Yes, it doesn't exactly follow Euclid, but that might not be a bad thing.

 

[homeomorphic]

Another interesting book with some material you could consider incorporating is Lines and Curves: A Practical Geometry Handbook. Originally used to teach "gifted" Russian high school students. More intuitive approach.

 

[mrg]

AlephZero, micromass, and homeomorphic: Thank you for your excellent replies. I greatly appreciate the guidance and wisdom you've given me. I'll be looking into the books that were suggested, and giving them serious consideration.

I'd love to hear from others, as well. What are your opinions on the matter?

 

[eigenperson]

I understand the desire to use the Elements - it is traditional and has been used successfully for thousands of years. There is no question that it is a good textbook.

However, I would hesitate to use it on a modern high school class. I think it must take a lot of patience to learn geometry from the Elements, and while students today have many advantages over those of the last 20 centuries, patience is not one of them. I'd worry you'd lose your audience.

On the other hand, I'm not sure this problem is remedied by choosing a different text...

 

[mrg]

You're correct Eigenperson. I want to share with my honors students one of the most read and studied works of art of all time. Plus, there is a great deal to learn from the structure of it, the amount of proofs, etc. As you say though, there are some initial concerns with how the modern-day student will handle such a text. Which is why I'm here - to figure out how I can alleviate those concerns, or if there is a useful alternative.

Keep in mind, this would only be for honors students. The only kids that would be learning from the Elements (assuming that's what I go ahead with; I need to buy a few books and read through them first) would be honors kids - kids who have proven to me from previous years that they are ready for a rigorous course, and hungry for challenging, pure math.

 

[Robert J]

Just wondering how your class did with Euclid's Elements. Any updates or feedback? How were you able to align it with common core curriculum?

I recently finished working through the entirety of Euclid's Elements, and found it insightful. The only book that seemed less relevant, however, was book 10. I'm probably preaching to the choir here, but the ancient Greeks didn't have as firm of a grasp on irrational numbers as we do with modern mathematics, and a lot of the language seemed antiquated, compounded by the fact that Euclid used multiple meanings for the term "rational". For example, we know today that the square root of 2 is 1.4142135623..., an irrational number. However, the ancient Greeks might have considered this a "binomial" or an "apotome" with both a rational and an irrational part, the 1 being the rational part, added to the irrational part 0.4142135. They also didn't use the decimal system and would have represented such numbers as lines or rectangles. Furthermore, Euclid might have called the square root of 2 "rational" simply because it can be constructed in two dimensions with ruler and compass, as in the hypotenuse of a right triangle with remaining sides each equaled to 1, or a perpendicular line drawn to the diameter of a semicircle from the point where the diameter is divided into a ratio of 2 : 1 units to a point on the circumference. Euclid might have called this a "rational straight line commensurable in square only with the remaining parts". Despite the shortcomings of book 10, it was ground breaking for it's time, still very insightful today, and provided a greater appreciation of mathematics as a whole from a historical perspective. The most fascinating books of all were books 11 through 13, where it all seems to come together, and Euclid accurately describes the construction of figures in 3 dimensions. It's daunting to imagine he was contemplating such things 1800 years before artists in the renaissance began incorporating perspective geometry, and 2300 years before the dawn of computer-assisted graphics!

It would be great if there were a more modern and updated version of Euclid's Elements for the young reader. The antiquated language is probably a barrier for people that would otherwise enjoy and benefit from it.

I'm sure your students must have had a real eye-opening experience, and I'm eager to hear how it went.

 

[mrg]

Just wondering how your class did with Euclid's Elements. Any updates or feedback? How were you able to align it with common core curriculum?

I actually didn't use it; I thought about some of the drawbacks and decided it wasn't in my students' best interests.

As you've said, it is a bit antiquated in some respects. But my biggest gripe was that it was missing some skills that students need (the complete lack of any coordinate plane, for example, is a huge drawback. Not only in the context of geometry, but in the math continuum. Most of these students will go on to Algebra 2 and beyond, and missing an entire year's worth of coordinate plane concepts can't be a good thing).

There are things that I like about the book though, and I have gone through a proof with my students from the Elements (I am forcing them to do paragraph proofs for the rest of the year, so I wanted to expose them to Euclid's proposition style). I just don't see how it can be adapted for use.

One of the other posters on this thread suggested an excellent text which I will use for my Honors Geometry class (probably two years from now) which is Serge Lang's Geometry. Lang is an excellent author (and his Basic Mathematics will be my Honors Pre-Calculus text).

Of course, I stand ready to be corrected. I'd love to use a book that served to teach Geometry to the greatest mathematicians throughout history.

 

[td21]

I think the axiomatic and logical aspect of Euclid’s Elements are more important than the geometry knowledge in it.
One of the best way to learn Euclid's Elements is to work out the logical dependence of the propositions. This should be a good exercise for students.
For example, the logical dependence of in the first 5 propositions is:
1----->2---->3---->5<------4

 

[mrg]

Agreed completely, td21. It's a wonderful, elegant, and beautiful way to approach mathematics that has been mostly lost, I'm afraid. Although many contemporary math books contain formal definitions and proofs, the emphasis is squarely placed on the skills and procedures by which to calculate answers. This, in my opinion, leads to the arduous, painful subject we now call "math."

 

[Robert J]

mrg,

Following our chat here I've been putting together some examples of select propositions from Euclid's Elements adapted for the coordinate plane. Would you like to take a look at a sample and let me know your thoughts? I'm hoping that perhaps you might even find some of it useful.