This is the last in our series of guest posts for Math Week on Afterthoughts. I broke Willa’s original post up into two parts, the first focusing on the more philosophical and historical aspects of teaching Euclid, today’s focusing on more practical matters. You may want to read the preceding posts first. Here is the series Table of Contents:

- Series Introduction
- Teaching Maths the CM Way {by Jeanne}
- Five Strategies for Teaching Mathphobics {by Tammy Glaser}
- Teaching Euclid in the Homeschool, Part I {by Willa}
- Teaching Euclid in the Homeschool, Part II {by Willa} <–you are here

Willa is the wife of a computer game designer and mother to six sons and one daughter, currently ages 9 to 26. Her family has been homeschooling for 18 years. She currently blogs at Take Up and Read and has contributed chapters for Literature Alive! and A Little Way of Homeschooling. |

[dropcap]T[/dropcap]here are many resources available for studying Euclid. The Google Books repository has the classic Heath-Heiberg annotated version of Euclid in three volumes {Books 1-2, Books 3-9, Books 10-13}. An interactive {Java-enabled} online version of this definitive work is available. Personally, I find Heath confusing because the commentary is so extensive, though the Dover facsimile edition is the one both my high school son and daughter used. The Green Lion edition in one volume is recommended at the college where they studied.

Another route might be to use one of the several Euclid textbook editions in public domain. This one arranged by HS Hall covers books 1-6 and 11-12 of Euclid and is targeted towards secondary schools.

Though it only covers the first two books of the Elements, I like the Class Lessons on Euclid which I mentioned in my previous post, because it explains the significance of some of the Euclidean ideas and words. For someone like me whose degree is in literature and who was not taught math using Euclidean methods {to say the least}, this provides extremely helpful context.

There are video demonstrations of the first 26 Euclid props on Youtube. These were not available when my older children were in the homeschool, but I plan to use these or similar ones with my current high schooler.

In the late 1800’s, when Charlotte Mason was still alive,the PNEU schools founded by her used Euclid in the curriculum, but by the 1920’s A School Geometry by HS Hall was being used in Form III {grades 7-8}. It is very Euclidean in format, with some adaptations in arrangement and method.

For younger students in Form II {grades 4-6} a book called Practical Exercises in Geometry was in use in the PNEU schools. Also, for younger students, Sam Blumenfeld has recommended a book called First Lessons in Geometry, which is from the 1850’s and has a simple, conversational format.

### Put Euclid in the weekly/daily schedule

There are close to 500 propositions in the 13 books of the Elements, plus sundry definitions. However, not many courses I have seen cover the whole 13 books. More commonly, high schools in the past couple of centuries focused on the first six books, which cover Plane Geometry and have not been outdated by advances in mathematics, and sometimes on only the first two. The first six books have only 171 propositions and 72 definitions. If you tackled only the first two books, that would be 61 propositions and 25 definitions. That seems doable for 1-2 lessons a week.

From looking at the PNEU Timetables, which used the methods of Charlotte Mason, I see that in 1908 there was a Euclid lesson 2-3 times a week, while Arithmetic or Algebra was studied 4-6 times a week.

My two highschoolers, who were seniors when they read Euclid, simply spent a few weeks on it alongside their ordinary work in a math textbook. My son carried the book around with him and did the demonstrations in his head {being that type of thinker}. My daughter used a whiteboard and colored pens and sometimes corralled a younger brother to teach the prop to in order to help her retention and memory.

Alternately, a student could work on Euclid while studying Algebra 1 as a sort of introduction to geometry before taking geometry as a course. That might be valuable in getting the student introduced to proofs and logical reasoning before embarking on the infamous double columns of the standard US proof.

I plan to work with Euclid once or twice a week with my present high schooler. We plan to review the proposition from the preceding week or two, then read the new one, discuss, and work through on the whiteboard. Since he is in the middle of Jacob’s Geometry right now, we can use Euclid just to deepen understanding and contrast the older method with the newer one.

### “Deliberate Practice” takes time

…They say that Ptolemy once asked {Euclid} if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry.

“some one who had begun to read geometry with Euclid, when he had learned the first theorem, asked Euclid, ‘But what shall I get by learning these things?’ Euclid called his slave and said, ‘Give him threepence, since he must make gain out of what he learns.’ ”

The importance of Euclid’s elements was recognised by the Greek philosophers, who posted on the doors of their schools: “Let no one enter here who is unacquainted with Euclid.” {Science-History of the Universe}

Looking through the Google Book repository and through the Parents’ Review magazine archives at AmblesideOnline, I found that around the turn of the 20th century, there was quite a bit of criticism leveled at the use of Euclid in middle and high schools. If you are interested, you can look at these articles from Parents’ Review. The criticism is targeted as much towards poor, mechanical teaching as towards the propositions themselves, and brings to my mind Charlotte Mason’s quote of Ruskin in regard to students:

‘they cram to pass and not to know; they do pass and they don’t know,’

We are fortunate not to have our kids tested on Euclid, but exams and hurry are characteristics of modern education, and if we are going to study a work like Euclid’s, we should keep in mind that it should be something we do liberally, for the sake of becoming better, not just to get through it. It would probably be better to cover less, and do it thoughtfully and conversationally, perhaps with lots of drawing of lines and circles, even including a kindergarten sibling in the lessons as my daughter did, rather than memorize huge sections in a rush.

Euclid’s Elements are part of a liberal, philosophical education. Such an education cannot take place in haste, though timeliness certainly has a place.

Recently I read a book that discussed the concept of Deliberate Practice.

When you practice deliberately, you identify the tasks or knowledge that are just out of your reach, strive to upgrade your performance, monitor your progress, and revise accordingly. Deliberate Practice … takes intense concentration … requires deep motivation, often self-generated … involves working on the task that’s most challenging to you personally.

The book mentioned that this best takes place with some time and space to think and work. I think that freedom from too many media distractions and activities on the schedule helps with the kind of thinking required for Euclid and other “great books”.

## 5 Comments

Thanks for these great resources !!!

Check out this resource, it is also real good !!

http://anthemprep.greatheartsacademies.org/wp-content/uploads/sites/34/2018/04/RESOURCE-A-Teacher-s-Guide-to-Euclids-Elements.pdf

How do you balance doing Euclid as a Geometry course with students having to take standardized tests which require a more algebraic form of Geometry?

For the sake of my sanity, I usually like to only commit to one year of homeschooling at a time, but I’m now feeling like I must teach through the upper grades just so I can learn this along with the kids. I loved the smattering of logic I had in high school, and I bet this would be right up my alley. In fact, I don’t think I’m going to wait for the kids. Thanks, Willa!

It is too funny to me how different types of personalities have to make such different decisions in order to survive homeschooling! I am exactly the opposite–I need to know that this is what we’re doing and that it won’t change on me. I mean, of *course* if the Lord has other plans, I have to be open to them…but I am comforted by that feeling of constancy. Making a decision every year would stress me out. But I find a *lot* of people are like you, Sara. 🙂

Personally, it is the upper school years that excite me. I think that is why I like things like Plutarch. They are Big Kid studies. I do the little kid stuff–the reading, writing, addition, subtraction, etc.–but I wouldn’t teach that for anyone other than my own children if I could help it. 🙂 Teaching something big like Euclid gets me excited.

I want to start now, too! 🙂

By the way–I am jealous that you received some logic in high school! I didn’t get that until college…

Thanks so much for hosting this series, Brandy. I so appreciate the work you have done here and elsewhere to make things doable and understandable for CM homeschoolers. I totally enjoyed my part in it!