|
Why? Classical Euclidean Geometry What is the difference? In 1980's the National Association of Math Teachers decided to de-emphasize formal axiomatic proof in high school geometry standards. Thus nearly all, if not all Geometry books currently in print, (even popular home-school titles) are woefully deficient in teaching the classic axiomatic proof. Some home school texts based on incremental-review pedagogy merely prepare students for passage of SAT/ACT geometry problems and completely ignore the geometry's traditional role in introducing students to deductive logic and critical thinking skills.
Traditionally the geometric proof and understanding the accompanying axioms transcends all mathematics skills. The ability to write and understand a formal proof enhances the study of all other disciplines.
Logical arguments/proofs are at the heart of classical philosophy, rhetoric and debate. At the center of the scientific method is the very idea of proof.
The value of teaching students the fundamentals of mathematical proof cannot be denied. This is especially true for students who will study proofs in advanced algebra and calculus, physics and chemistry.
Anything less is mere imitation or memorization of problem solving. This course will aim to resurrect classical training while simultaneously developing skills in applied or analytical and algebra based geometrical problem solving as found on SAT/ACT tests.
Methodology: With Classical Euclidean Geometry, one teaches the student about “first principles” or fundamental truths, or axioms which are accepted as universally true without further explanation. In Greek, the word axio [axio] means ‘authority.” So the axioms have the authority to convince and prove. One example of such an axiom is “a whole is greater than any of its parts”. The notion, or fact, or idea is obvious and accepted without proof or question. Aristotle used the term common notions, and Euclid referred to these common notions in his work: Elements.
Then we learn that geometry is a deductive system that takes the common, accepted notions, or axioms, and we deduce all other truths from those axioms. A truth that is built up from using these axiom is named a theorem. Once we have theorems, we can use them to prove even more deduced truths or theorems. To proof something we must support the proof at each step with either an axiom, or a previously proven theorem.
Now apply it!
I have noticed once you get the students to prove a theorem, they seldom need to memorize it. When they use it to prove things, they will remember it. Now they can use it in applied or analytical problems easily, and those are the easiest of problems. Those are the types of problems on the SAT and ACT.
Because Euclidean Geometry builds from axioms to theorems, it is naturally a cumulative body of knowledge. We constantly draw from chapter 1 while doing chapter 2, 3 and so on. Thus we do not need arbitrary sets of busy drill work to review. |
