Saturday, July 16, 2016

Geometric Logic in the Summer...and Classical Education


One of the *other* tasks that kept me enormously busy last summer was that I was busy teaching geometry to my daughter. The university required her to take a "how to teach math with manipulatives" class, which I thought would be incredible. Once upon a time I had taken one of those classes to get my education degree. Yet that was only the name of the class, not the intent (post modernism strikes again). Instead the book looked crazily and disappointingly Common Core-ish. In other words it looked more difficult than it needed to be. In class the professor would teach a lengthy method for doing elementary math to which my daughter would shoot her hand in the air to comment that she had an easier method. The professor was always flabbergasted by my daughter's method and the classmates often commented that they had never been taught that simple method before. When asked where she learned such simple yet highly effective methods my daughter replied, "My mom taught me."

Well, turns out the intent of this class was not to teach education majors how to teach math to elementary level students. Instead the intent was to prepare them for a state test for education majors. Because geometry was lacking in class, but on the test, my daughter asked me to teach her.

Well, I don't teach Common Core. I raise the bar. So after she finished her summer college "Common Core" math class we crammed in Teaching Textbook's Geometry curriculum in every spare moment. We had used this in high school but my daughter struggled with it at the time. It turned out that she needed Vision Therapy which revealed that she had spatial reasoning deficits. One way to build spatial reasoning skills is to study geometry. This time, I sat through it step-by-step, helping her through each step, redirecting her as needed for each step of the process. We had a few weeks to get ready for the big state test. 


The reason why I purchased Teaching Textbooks for our junior high and senior high years, is that the CD's have lecture and solutions for every single lesson and problem in the book. I needed that. Even though I took honors classes in high school and college, math was never my forte. Thankfully, though, geometry was easier for me, thanks to a terrific high school math teacher, Mr. Broyles. He insisted we memorize every definition and be capable of detailing exactly what they meant within our own environment. He also had us do formal proofs.Because of him I nailed geometry. I brought his teaching methods alongside the Teaching Textbooks curriculum.


This meant that we did every single proof. This is where the logic is built. Shortening the task short circuits the objective and the results. We also talked through every single step. In college I specialized in reading, which focuses on decoding skills for the younger students and comprehension skills for all. One of the key phrases one of my professors always used was, "make the communication process public." That's exactly what I did. Forcing ourselves to think out loud pulls in the audio, and allows me to know what was going on with my daughter's thinking so that I could realign her thinking as needed.


To streamline the lessons I used a pile of old printouts from homeschool that were no longer needed and simply used the backs. (An old map is on the other side of this page.) I wrote out each proof to be done at the top, then drew any required elements for my daughter, since she struggles with a lot of that with her spatial reasoning limitations. Precision is of the utmost importance in geometry. Also I had a short time to conquer so I trusted that the process of doing the proofs would build some of the other skills (and they did) while achieving an immediate objective (the test). She wrote out all the proofs after I approved each step of what she verbalized to me.


One thing lacking in public school and public college education today is the building of logic skills. Sadly there is a push for informal proofs in geometry. Not in my class. We did the full, formal proofs. No short cuts here.

Besides, I was a Classical homeschooler. Classical study entails, in part, going to the primary source.

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.… Just as the eye was made to see color and the ear to hear sounds, so the human mind was made to understand quantity. -Johannes Kepler, 17th Century German physicist
Around 300BC, the Greek, "Euclid brought together much of what was known in mathematics up to that point and organized it in such a way that, beginning with short list of abstract statements assumed to be true and armed with reasoning, he pieced this body of knowledge together as an extended chain. He did so in such a way that the Elements became the standard textbook in geometry for the next twenty-two hundred years." (quote from this website article, The Purpose of Mathematics in Classical  Education)

As a side note, my son's classical education liberal arts college uses Euclid's Elements for their sole geometry text.

What did Plato have to say about geometry?
The knowledge at which geometry aims is knowledge of the eternal, and not of anything perishing and transient. Geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Therefore, nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry. -Plato's Republic (Book VII)

 And in regard to statesmen...

We must endeavor that those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; … arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible and tangible objects into the argument. -Plato's Republic (Book VII)
To expound upon this idea, even our Founding Fathers had a Classical Education. They studied geometry.

Furthermore a modern book says this...

The abstractions of mathematics possessed a special importance for the Greeks. The philosophers pointed out that, to pass from a knowledge of the world of matter to the world of ideas, man must train his mind to grasp the ideas. These highest realities blind the person who is not prepared to contemplate them. He is, to use Plato’s famous simile, like one who lives continuously in the deep shadows of a cave and is suddenly brought out into the sunlight. The study of mathematics helps make the transition from darkness to light. Mathematics is in fact ideally suited to prepare the mind for higher forms of thought because on one hand it pertains to the world of visible things and on the other hand it deals with abstract concepts. Hence through the study of mathematics man learns to pass from concrete figures to abstract forms; moreover, this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. -Mathematics for Liberal Arts by Morris Kline 

For more on the importance of geometry, read this article: The Purpose of Mathematics in a Classical Education.

Oh, and how did my daughter do? Although she was told that the state test is enormously difficult, she passed! But more importantly she grew in her ability to use logic skills. That will carry her further in life than passing a state test.


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