Teaching Proof

by Henri Picciotto

Proof is, of course, a core issue in math education.

In the traditional model, it is taught as part of a tenth grade all-geometry class. This has in fact not worked so well with most students. As more and more students have been taking that class, at a younger and younger age, the intellectual challenge has been reduced by turning the writing of proof into a more mechanical exercise (two-column proofs with strict guidelines), while at the same time tracking students who are perceived to be less likely to succeed in this enterprise into no-proof classes (sometimes called "inductive" or "informal" geometry).

Much of the reform movement, reacting to this problematic situation, has worked on injecting interesting mathematics into lower-track geometry classes, with some measure of success. Several arguments are raised in favor of this approach: the old method did not work for most students; students need some grounding in actual geometry before proof is meaningful; our country does not need more mathematicians as much as more engineers, scientists, and statisticians -- in other words people who use mathematics, not people who do mathematics.

There is much merit to those arguments, but in my view some of the reform movement has gone too far away from teaching proof, true to American education's tradition of extreme pendulum swings, which go along with a loss of perspective and set us up for backlash. For one thing, this approach has reinforced the existing paradigm: upper tracks are condemned to a mind-numbing forced march through a traditional text, memorizing theorems and combining them in two-column proofs. Lower tracks do some interesting work, but are shielded from proof.

I have attempted to navigate a middle path. Along with my colleagues at the Urban School of San Francisco, I have developed a geometry course which tries to incorporate both geometric meaning and formal deduction. We introduce formal proof in the second half of the course, once students have developed some visual and logical sense while working on key geometric concepts, such as angles, circles, and the Pythagorean theorem. The heart of this process is a unit on quadrilaterals, during which students make conjectures, and then use counter-examples and proofs to sort out which of those conjectures are true, and which false. (Not unlike what mathematicians do.)

Unfortunately, even this more sensible approach is too difficult for some and insufficient for others. I try to offer more to my stronger students in 11th-12th grade electives, where I introduce proof by mathematical induction and proof by contradiction (Infinity), and some geometric proofs about transformational geometry and about polyhedra (Space) at an age and to a group where there is more chance of success.

I do all this not because I believe that writing mathematical proofs is an important job skill for most people, but because it is a foundational component of intellectual culture the world over, and what many consider to be the essence of mathematics. All our students deserve an introduction to this way of thinking.

Links

Proof in High School

How we approach it at the Urban School of San Francisco:

Math 2 (our geometry course)

Post-Algebra 2 electives:

Space (some advanced geometry proofs)

Infinity (mathematical induction, proof by contradiction)

Student-created calculus proof:

Integrating y=x², by Jacob Regenstein.

Sum of the angles in a triangle

Pythagorean Theorem

Completing the square

Parallelogram area

Interesting Proofs for Math Teachers

Why y=mx+b works: Geometry of Linear Graphs (PDF)

Alternate proofs of the quadratic formula, and proofs about the geometry of the parabola in two and three dimensions: Parabolas and Quadratics