This is the third in our series of guest posts for Math Week on Afterthoughts. I broke Willa’s original post up into two parts, one focusing on the more philosophical and historical aspects of teaching Euclid, the other 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} <–you are here
- Teaching Euclid in the Homeschool, Part II {by Willa}

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. |

Why would a student of today want to learn Euclid — not just about him, but from him and his ideas? And if a student did want to do this, how would he or she start? And how would a homeschool parent guide this study?At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.

Euclid’s Elements of Geometry was *the* core math text for close to 2000 years. Abraham Lincoln studied Euclid’s books, as did Albert Einstein. Bertrand Russell was dazzled by Euclid, though he became disillusioned later in life. Anne of Green Gables was glad to be “done with geometry, learning or teaching it,” as she “thumped a somewhat battered volume of Euclid into a big chest of books, banged the lid in triumph, and sat down upon it”. The men of the Scientific Revolution, though they discarded most of Aristotle and Galen and other ancient scientists, kept the format and much of the substance of Euclid for their new physical models of the universe and its workings. Even some living today remember studying Euclid. In spite of the advances of the modern world, if Euclid looked in the Plane Geometry of a modern high school geometry textbook he would find much familiar to him, albeit ordered differently and somewhat altered in format.

Geometrical proofs are according to some reports slowly dropping out of the high school curriculum, perhaps because they can’t easily be put on standardized tests. But something similar to the Euclidean method of constructing proofs is still employed by mathematicians today, so this de-emphasis may be a matter for concern. Some have proposed that Euclid, because of its emphasis on close reasoning and precise formulation, is a good bridge to college mathematics both for mathematics-oriented students and for those who major in the humanities.

Recently, the Great Books and classical education revival has led to a return to Euclid by some high school programs — Great Books Tutorial is one, Regina Coeli is another, The Lyceum is a third–and some colleges — St John’s College is one, Thomas Aquinas College is another. The University of Denver offers Greek for Euclid.

If you are reading this, you probably have some interest in the idea of studying Euclid. Perhaps you have heard that the children in Charlotte Mason’s PNEU schools studied Euclid in middle and high school, and want to follow that example; perhaps you prefer to focus on Great Books and avoid textbooks, especially modern ones; or perhaps you are a classical homeschooler who wants to prepare your child to tackle Euclid, Descartes and Newton in a liberal arts college.

My family is sort of a combination of the above. I have four grown children and three still in the homeschool. We use a classical/Charlotte Mason blend in our homeschool, and try to include a fair quantity of great books and a minimum of textbooks in our studies. But the immediate practical motivation for two of my older children to read Euclid was their admission into a liberal arts college. Both of them wanted to get some familiarity with the Euclidean method before actually studying him formally in a college environment. I would like to do some Euclid with the three remaining students at home before they graduate. That’s where I am now.

From that perspective, here are some steps I think are important in teaching Euclid in the homeschool, especially since Euclid is not a standard textbook, and most of us {I, for one} were not taught mathematics in this way during our school years.

**Know your goals.**

The traditional goal of studying Euclid was to learn to reason well. A proof-oriented geometry course is a course in basic reasoning. There is a useful {and short} booklet in public domain: On Teaching Geometry. It makes the point that proofs are extended syllogisms. Given the premises, the result must follow, if the reasoning is valid.

Abraham Lincoln apparently used Euclid’s system of logic in order to build his arguments in speeches and debates throughout his life, and recounted of his earlier days:

I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? … I consulted all the dictionaries and books of reference I could find, but with no better results. At last I said, — Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.

One of Euclid’s excellencies is that every word and detail is chosen for a reason and so those words and details matter. They are not interchangeable. My son who studied Euclid in college said that he found that if he could not demonstrate a proposition in precise words, it usually meant that there was an error in his thinking — he had taken something for granted, or made a shortcut in his reasoning. Since geometry is simpler than, say, political or philosophical argumentation, it is a suitable training ground for younger people.

One aim of education is to enable us to articulate what we see, and Euclid is superb at occasioning this activity. Euclid begins with simple geometrical objects that are presented to the mind in three ways: through canonical depictions on paper, through images readily held before the mind’s eye, and through careful language.

One more reason for studying Euclid’s ideas is that they are part of our intellectual heritage. The school math curriculum of today is often not only disassociated from its logical underpinnings, let alone its real-life applications, but from its place in the history of thought.

**Start reasoning before Euclid.**

This article makes the point that informal “proving” or justification of arguments should start before high school, and this sequel gives some ideas of how to do so.

This doesn’t have to be complicated. If you are educating your child in a Charlotte Mason or classical Great Books style, hopefully you are already hearing narrations, discussing books and ideas with your child, observing the world around you and talking about it. This is a precursor for the kind of thinking done in mathematical reasoning, because you are already modeling and allowing time to observe, distinguish, trace cause and effect, etc.

Also, I think it’s important that the children get real-life {or at least visual and interactive} geometrical experience. Montessorian manipulatives can be helpful. There are various geometry games and visuals online. Some practice in drawing shapes also seems like a good precursor to formal demonstrating of propositions. Playing with blocks certainly can’t hurt, either.

**Understand the Euclidean method and structure of argumentation.**

When I first looked at an edition of Euclid, I had trouble seeing it as a math book. After all, there were no “problems,” no arithmetic or even algebraic equations. Perhaps that is an advantage for modern students, who become accustomed to using their brains as calculators. This kind of mathematics uses the part of the brain that thinks and wonders and puzzles. But reading the Elements is not the same as reading a work of literature or history. You have to read carefully, and understand the structure of the arguments, and keep your mind active as you read. Perhaps you might even want a notebook or slate next to you as you read.

A book called Class Lessons on Euclid, in public domain, has given me some of the background I felt I was missing. The first 3 chapters give some reasons for studying Euclid {along with some information about his life}, discuss the definitions and axioms that precede the propositions, and tell about the structure of a Euclidean proposition. Even if a student or parent got no further than this, it would not be time wasted, because the logical paradigm of defining terms, listing presuppositions, and building syllogistic arguments is used in many Great Books. These are useful habits or skills for deciding any kind of serious issue in life, as well.

Lue-Yee Tsang says

How do you identify the right age at which a child may begin to study Euclid?

Brandy Vencel says

I can’t speak for Willa, who originally wrote this, but I know Charlotte Mason began a more formal study of geometry at about junior high age… Before this they were doing practical geometry to prepare for understanding proofs, etc.

Anonymous says

I read Euclid’s Elements the summer before my daughter’s freshman year to re-learn geometry. While working through the proofs, I realized it was a much better textbook than the geometry textbook I had in public school. After finishing her freshman year, the experience of using his Elements led me to look for algebra books by Descartes, and calculus books by Newton or Leibniz (or any other primary sources by ancient/modern mathematicians). Unfortunately I couldn’t find anything that had close to the same structure/organization as Euclid’s Elements when it came to using a work as math textbook.

Brandy Vencel says

I would love to hear what you ended up using for your other math courses!

Sarah says

Thank you for those links above, Brandy. I downloaded them to my Kindle.

This is very timely information as I have a daughter who will finish Algebra I soon and a son who is finishing pre-algebra. I haven’t been sure what to do next. I think Hubby and I will discuss Willa’s post over dinner tomorrow night. 🙂 How’s that for a date night topic?

Jen says

Thank you! I was able to download the Kindle versions from those links. I tried googling and didn’t turn anything up…thanks for taking the time to look!

Jen

Jen says

Very interesting…certainly not something I had ever thought about before (Proofs in HS Geometry were a nightmare for me, so I have avoided). You have be intrigued, though. Do you happen to know if the public domain resources you mentioned (On Teaching Geometry and Class Lessons on Euclid) are available elsewhere than from Google Books? I am outside the US and so can’t download directly from Google Books. I’m curious to read more, though.

I am thoroughly enjoying this series – thanks for hosting Brandy, and to all of the great contributors too. =)

Jen

Brandy @ Afterthoughts says

I did a little hunting and found this, but I do not know if they will work for you or not. Let me know…

Class Lessons on Euclid by Marianne Nops

On Teaching Geometry by Florence Milner