Monday, December 25, 2006

Reply to Euclidean geometry/Proof Question on Math Talk

The following is a reply to a question posed on the excellent math discussion group Math Talk on Yahoo groups.

Here’s my view. Pls don’t be offended by its didactic nature, but I need to use formal logic to defend the need to teach formal logic! I know you and Mike already know all of this but I need to state the obvious anyway.
Mathematical reasoning and the structure of mathematics depends to a large degree on DEDUCTIVE REASONING. As mathematicians, we either accept a statement as true (definition or postulate) or we PROVE it is true, most often by deduction. The ‘statement-reason’ proof format was designed, IMO, to help students develop a structured approach to logic and deductive reasoning. The key to this structure is what it means for a conditional ‘if-then’ statement to be true, since every theorem can be stated as a conditional or biconditional. This is profound and requires development ONE STEP AT A TIME.

Consider the conditional
If P, then Q. Every application of this conditional MUST formally have a structure similar to:

1. P is TRUE. Given
2. Therefore Q is TRUE. Reason: If P is TRUE, then Q is (also) TRUE.

This is a variation on the 3-part law of logic entitled ‘modus ponens’ or the Law of Detachment which used to be taught in a geometry course. No, Mike, teaching syllogisms didn’t cause brain damage! Complex multi-step proofs are just compound applications of this basic argument form. This can also be taught using symbolic logic and I’ve seen those curricula too, replete with truth tables! Ok, I’m not proposing revisiting the 60’s!

For example, suppose we want to prove that, in triangle CAB, Angle A is congr to Angle B and we’re given that side CA is congr to side CB. The proof then has the basic structure outlined above (the spacing may be off here):

1. Side CA congr to side CB. Reason: Given (assertion)
2. Therefore, Angle A congr to Angle B. Reason: If two sides of a triangle are congruent, then the angles opposite these sides are congruent.

A ten-step proof is of course nothing more than a chain of structures like this.

The fact that so many students struggle with PROOFS (therefore we should avoid teaching them!) and can’t seem to supply much more than the Given and the Conclusion is that most humans cannot build a FIVE-step structure before developing MANY MANY two, three and four statement arguments! Call this the Piagetian or the van Hiele Model or just plain commonsense! There are no shortcuts to this development. Either we teach this structure or we don’t. I propose that it is still important for all students to be able to make a valid argument using sound reasoning. The 2-column method is not the only way to do this but it wasn’t arrived at whimsically or to abuse young minds despite contrary views! Certainly, any college prep curriculum should continue to expect this, but I would argue that simple proofs are accessible to most students. I completely agree with you, Anita, that the message has gone out to geometry teachers that they can deemphasize formalism. I strongly disagree with this message. As far as the argument that proof is not inherent to geometry courses, of course that is true. But Euclidean geometry was built on deductive reasoning and I just don’t buy that most algebra (or anywhere else for that matter) teachers will infuse this kind of reasoning into the curriculum!

For more comments on this and other related matters of math curriculum, I invite you to read my blog at I will probably post this entire reply there. I hope this isn’t poor protocol, Mike. Feel free to delete it if it is.
Dave Marain

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