![]() ![]() If you’ve ever been a 9th-grader-or even met one-you know that such a distinction isn’t the most inviting welcome mat to geometry. But the argument hinges on the fussy technical distinction between the angle (a geometric object) and the measure of the angle (a number describing the size of that object). There are important lessons here, surely-for example, that even seemingly obvious truths demand justification. When I run such arguments by math PhD friends, they look at me dumbfounded. ![]() For example, take the “Right Angle Congruence Theorem,” which states that all right angles are congruent to one another: My beef here is that a geometry course often begins with totally mystifying two-column “proofs” of elementary facts. Logic ought to be learned through two-column proofs. Definition of Students Not Being IdiotsĤ. Experience with such proofs will help students see logic as an alien enterprise, foreign to common sense and deaf to the simplest realities.ģ. In many two-column proofs-especially those taught earliest in a geometry course-each individual step is mystifying, while the conclusion is obvious.ģ. In a good proof, each individual step is obvious, but the conclusion is surprising.Ģ. Theorem #2 : A proof is just an incomprehensible demonstration of a fact you already knew.ġ. In a two-column proof, the organic matter that holds the argument together is flushed away, and replaced with a right-hand column full of terse bullet points that students may use without understanding at all. A good proof contains not only bare statements of fact, but connective tissue of explanation. Properly understood, they function almost like diagrams of arguments, and can serve as useful tools.īut in practice, they often obfuscate more than illuminate. They offer a scaffold, a structure, a formal framework for students to lean on. Logic ought to be learned through two-column proofs.īefore sarcasm carries me too far down the rhetorical river, let me plant an oar and explain my stance. Instead, high schoolers ought to justify their arguments by reciting the names of theorems and axioms, invoked as if they were not logical statements but magical spells.Ħ. Besides, it would take too long for instructors to grade such arguments.ĥ. Fundamental Axiom of Condescension Towards Young PeopleĤ. High schoolers are too simpleminded for such techniques.ģ. In real, adult arguments, such justifications often take the form of cogent explanations, appeals to agreed-upon facts, and clear, explicit reasoning.ģ. In an argument, all steps must be justified.Ģ. Theorem #1 : “Justifying steps” ought to be an opaque, frustrating process.ġ. ![]()
0 Comments
Leave a Reply. |