Beals’ Lament

So much of math instruction is learning and mastering vocabulary. Parallel lines, isosceles triangle, rotations, 180 degrees: Students can learn all of these terms and more through movement. Here’s an example warm-up sequence we do to get ready for math class.

First, my students start with what we call “isosceles triangle,” or “mountain” in yoga terminology: We spread our feet about two feet apart, firmly planted on the floor. Then, we take a moment to trace the three sides of the triangle with our hands. We start at the top point (our belly button) and move down one side, across the base, and up the third side.

The latest nonsense from Edutopia has me thinking of the Lament for Geometry Proofs I once posted on Out In Left Field:

One of many casualties of Reform Math–along with simultaneous equations involving more than two variables, rational expressions involving polynomial denominators with more than two terms and of degree > 3–are traditional geometry proofs like this one:

Proofs are gone for many reasons. Some people find them soul-killing. Here, for example, is what Paul Lockhart’s (in)famous Lament has to say about this particular proof:

In other words, the angles on both sides are the same. Well, duh! The configuration of two crossed lines is symmetrical for crissake. And as if this wasn’t bad enough, this patently obvious statement about lines and angles must then be “proved.”

Instead of a witty and enjoyable argument written by an actual human being, and conducted in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic formletter of a proof. And what a mountain being made of a molehill! Do we really want to suggest that a straightforward observation like this requires such an extensive preamble? Be honest: did you actually even read it? Of course not. Who would want to?

(Everyone I know loved to construct these proofs–precisely because they forced you to abandon intuition and work things out logically, building up from Euclid’s postulates, in the wonderfully exotic, unnatural language of math.)

Others, I suspect, find these proofs too difficult to teach–especially in the era of Reform Math, with incoming geometry students having spent so little time with problems that demand any kind of rigorous, multi-step logic in the unnatural language of math.

Finally, there are the Common Core Math Standards. The only mention of geometry proofs anywhere in the math standards in the Introduction to the High School Geometry section:

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later incollege some students develop Euclidean and other geometries carefully from a small set of axioms.

Within this introduction, the only other mention of proof is in a subsection entitled Connections to Equations, where the type of proof under discussion appears to be algebraic rather than geometric:

Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.

Nowhere in any of the specific goals for geometry do we find the word “proof.”

And yet, surely the Common Core authors, with their obsession with higher-level thinking and deep understanding, would use the powers vested in them by Bill Gates, The National Governor’s Association, and Education Industrial Complex, to revive this dying art?

Or perhaps we can come up with a proof as to why this isn’t happening:

Geometry proofs don’t lend themselves to standardized tests.
Proofs can’t be captured in multiple choice format; only expensive experts could grade them.

The Common Core authors don’t like goals that aren’t easily measured by standardized tests. Reason:
Most of them are affiliated with the educational testing industry (see here).

Or, as Lockhart would put it, in terms that are so much less bureaucratic, and in one of the world’s many natural language: “Well, duh!”

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