Kindred spirits, those two. Too bad they were born about a hundred years apart; I think they would have enjoyed knowing each other.
So. When The Liberal Arts Tradition says “geometry,” it pretty much means Euclid.
Euclid was the first fixed point of the discipline.
[H]e is considered as the consummation of Plato’s vision of geometry.
How long did this last? I’m glad you asked!
For millennia the liberal art of geometry was Euclid’s Elements, and this persisted until the end of the nineteenth century.
So, if you want to teach geometry as a liberal art, Euclid’s your guy.
[F]or those searching for a classical liberal arts paradigm for the study and teaching of geometry, the answer is found in a return to Euclid.
In 1908, the PNEU (Charlotte Mason’s organization), published their weekly time table. This is, as far as I know, the only time they made that sort of proprietary information public. You might find the junior high schedule especially pertinent to this discussion:
Or, check out high school:
As you can see, Miss Mason doesn’t even list geometry. She lists Euclid.
Now, was she actually using a translation of The Elements? Well, sort of. From her old programmes, I see that she often used A School Geometry by H. Hall and F. Stevens. I searched around online because I feared this meant she was using a boring geometry textbook, and then where would we be? Thankfully, this is not the case. I learned that the full title is A School Geometry: containing the substance of Euclid books II and III and part of book IV. In her third volume, she lists (for a 12-year-old) two Euclid-based books: A First Step in Euclid, by J. G. Bradshaw (which covers the first twelve propositions of Euclid) and Inductive Geometry, by H. A. Nesbitt, which she notes is for beginners.
The Liberal Arts Tradition concurs with this sort of selection:
While Euclid’s Elements has a total of thirteen books, a normal high school geometry class will usually cover only material from books I-IV and VI.
Now, as you can see, Miss Mason’s approach is what we sometimes call threaded. This means that she wasn’t focusing on one type of math at a time. Her junior high and high school students were doing mental arithmetic, geometry, and algebra all at the same time. (Of course, this was before calculus would be expected in high school.) This is where Miss Mason seems to part company a bit with Clark and Jain, for they say:
Because historically geometry provided the foundation for the very concept of proof in mathematics, and because its constructions make it a more concrete subject, it should be placed as the next subject in mathematics after elementary arithmetic.
I don’t know about you, but I took Geometry between Algebra I and Algebra II. Clark and Jain tell us this is not traditional, nor is there a good argument for doing so. Instead, The Liberal Arts Tradition brings up the history of math — the fact that algebra was an outgrowth of geometry — as compelling evidence for having geometry precede algebra in the curriculum. The question as to whether calculus and algebra even have the same standing as arithmetic and geometry when it comes to liberal arts education is also raised. I won’t be going into the discussion about where algebra and calculus might fit, but it’s very interesting, and I encourage you to read it and think about it.
For the purposes of our conversation, what is more pertinent is this thought:
The emphasis on proofs in geometry forms wisdom in students to such a degree that it has for thousands of years captured the imagination of a multitude of philosophers from Plato to Descartes.
Well, now, isn’t that interesting? Miss Mason, too, seems to see this, though she doesn’t state it as directly. In her chapter on the use of reason in her final volume, Miss Mason says:
We know how Florence Nightingale received, welcomed, reasoned out the notion of pity which obsessed her, and how through many difficulties her great project for the saving of the sick and suffering of her country’s army worked itself out; how she was able to convey to those in power the same convincing arguments which moved herself. That was a happy thought of the medieval church which represented the leading idea of each of the seven Liberal Arts by a chosen exponent able to convince others by the arguments which his own reason brought forward. So Priscian taught the world Grammar; Pythagoras, Arithmetic; and the name of Euclid still stands for the science which appealed to his reason.
This might seem a bit nebulous out of context, so let’s think about it for a moment. The chapter in which this quote appears is discussing what Miss Mason called “the way of the reason” — we are to teach children the proper use of reason, and to know that reason is a good servant but a bad master. If you recall, Miss Mason’s 18th principle:
We teach children, too, not to ‘lean (too confidently) to their own understanding’; because the function of reason is to give logical demonstration (a) of mathematical truth, (b) of an initial idea, accepted by the will. In the former case, reason is, practically, an infallible guide, but in the latter, it is not always a safe one; for, whether that idea be right or wrong, reason will confirm it by irrefragable proofs.
The proper use of reason — and the way to gain an understanding of reason — is through Euclid. I think it’s possible that there was no formal study of Logic (I’ve never seen it listed as a separate subject) in Charlotte Mason’s schools because she believed that certain things — and perhaps Euclid foremost — would suffice.
This chapter really got me thinking about what I’m doing right now with my oldest. If you recall, I hosted a Math Week a few years ago, and one of my big interests was the teaching of Euclid. Willa blessed us with two wonderful posts on teaching Euclid in the home schoolroom (you can read them here and here). And from that point on, I was all gung-ho about teaching Euclid in eighth grade.
And then I got cold feet.
I started thinking about how I learned geometry after algebra, so maybe that was how it was supposed to be done. Also, I wondered how this would impact our ability to finish our year of math curriculum “on time.” And also, it sounded like more work, and with four students, I already have a lot on my plate.
But then this. Oh, how I love this book! And this sort of thing is why. I’m remembering all over again of why we do what we do, of how we’re searching for wisdom, not running a race. And also I remember that Euclid was sufficient for thousands of years.
So, I ordered The Elements, and it’ll be arriving soon. I’m going to take some time to go over it (and also to reread Willa’s posts), and then I think we’ll add in an hour or so per week, and see how that goes.
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