Saturday, June 4, 2011

The Common Core Standards: Geometry

I have to admit it. I like the new Common Core Standards (CCS) for geometry. (For those not in the know, these are the new federal standards. As of present, all but a few states have adopted them.) If you haven't read them yet, take a moment to do so.

First I'll run down what I like about them. After I'll give my only objection.

1. The CCS demand that congruence and similarity - the fundamental relations of elementary geometry - be defined in terms of the concept of transformation. (How so? Here are quick and dirty definitions. Two figures are congruent just when one can be carried onto the other by a sequence of rigid transformations, i.e. translations, rotations and reflections. Two figures are similar just when one can be carried onto the other by a sequence of dilations and rigid transformations.)

Implicit here is the concept of superposition. Two figures are congruent just when, by a sequence of rigid transformations, one can be superposed upon the other. Likewise for similarity, with the addition that we may dilate too.

Such a strategy has strong historical roots. Moreover, it conforms to current mathematical practice. Euclid employs it at times, for instance in his proof of side-angle-side triangle (SAS) congruence. Felix Klein generalized it and in the Erlangen program made that generalization the basis for the categorization of the different geometries. Mathematicians seized upon Klein's work. Today his methods are ubiquitous within mathematics, and no doubt the CCS wish to bring secondary geometry in line with this.

Of course one might question the pedagogical value of the twin concepts of superposition and transformation. Charles Dodgson certainly did. But I for one find them quite intuitive. Indeed they seem to me just the right way to motivate SAS, AAS and ASA triangle congruence. (SSS is another matter. The only route to it, I think, is through the Isosceles Triangle Theorem. But still transformation and superposition do play a role.)

2. In the CCS, we find a continual insistence upon the importance of proof. I wholly agree. Only a small part of the value of geometry lies in the particular results at which it arrives. (Students will forget most of them. Most of them they will never use.) The greater part of that value is in the way that it arrives at them. It isn't guesswork. (It sure looks like that angles opposite congruent sides are congruent.) It isn't simple induction. (Every triangle that we've looked at so far has at most one right angle.) Instead it is proof, proof from first principles. Students need to know how to put together a logically watertight argument where their basic assumptions have been made fully explicit. That's a genuinely valuable skill.

3. The CCS has students tackle problems of some difficulty. They thus seem to me like a much-needed return to rigor.

Some examples:

Proof of the triangle congruence principles, i.e. SAS, SSS, ASA and AAS. Many texts make them all postulates. (Gah! I hate that. It leads students to believe that a postulate is some damn thing that someone somewhere just made up. Why in the world wouldn't we take the opportunity to explain why they work?)

Prove angle-angle similarity. (I love this one. AA~ has never been obvious to me. I'm sure that it isn't for students either.)

The derivations of the Law of Sines and the Law of Cosines. Explain the so-called "Ambiguous Case" of the Law of Sines.

The derivations of the sphere volume and surface area formulas. Many texts today don't touch the former. What a terrible, terrible mistake! That derivation is one of the most beautiful proofs in all of elementary geometry. Archimedes was so proud of his discovery of it that he had it engraved on his tombstone! I propose a rule: if a text does not reproduce Archimedes' derivation, that text should be banned from use.

So ends the praise. Now for a bit of criticism. The CCS shifts study of conic sections from Algebra II to Geometry. The problem with this is obvious. Geometry students have had only a year of algebra and simply do not have the requisite algebraic sophistication to take on a study of the conic sections. I hope that a later version of the CCS fixes this problem.

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