Proof Progressions

Two-column proofs are not used in this course, and this is the lesson where that decision is explained directly to students. The format they find easier is easier because it does less of the mathematical work. Two-column proofs allow individually justified statements that do not chain together into a genuine deductive argument. Flow chart proofs and paragraph proofs both require students to show the shape of the argument. In a flow chart, boxes and arrows make the law of syllogism visible: you can literally see how the conclusion of one step feeds into the next. When students ask why they cannot use two-column proofs, this explanation is worth sharing directly.

 

I share a story about Ray Allen before any proof work begins. Students are about to attempt proof writing seriously for the first time. The story does something essential: it gives them permission to struggle without feeling that struggle means failure. What looks effortless from the front of the room — a proof flowing cleanly from given to conclusion — is the result of working it out in advance. The teacher already knows which dead ends to avoid. Students are seeing the finished version. Their first attempt will probably be messy. That is what writing a proof actually looks like.

 

The proof progression is the central activity. The idea largely comes from “A Proof Progression for Geometry” by Wayne Nirode (The Mathematics Teacher, Vol. 111, No. 7 (May 2018), pp. 512-519). The parallelogram problem is revealed in three stages, each one pulling back one layer of scaffolding. Progression 1 gives the alternate interior angles as given information — students only need to apply ASA then CPCTC. Progression 2 removes the angle marks; students must justify the alternate interior angles from the parallel sides. Progression 3 gives only the definition of parallelogram; students must establish parallel sides from the definition before proceeding.

 

After all three progressions, name the structure explicitly: the three problems were one proof at three levels of scaffolding. The intermediate steps were always there — the scaffolding hid them. This is what complex proof writing always looks like: the path to the final conclusion runs through intermediate results that are not announced. Part of developing as a proof writer is learning to see those intermediate steps.

 

Name three planning strategies students now have: (1) start from a familiar intermediate step and build in both directions; (2) start from the conclusion and walk backward asking what must have come before; (3) write everything you know from the given and diagram and see where it leads. All three are valid. With practice, all three combine. The card sort task (Task 4 in the full rewrite) offers a fourth approach — arranging statements physically before committing to paper — which is particularly useful for students who find the blank page intimidating.