Mathematical Proof: What’s the Point?
The sum of two odd numbers is even. Two pictured triangles are similar. There are infinitely many prime numbers. Tell these facts to a student and the overwhelming response is “ok”; ask them to prove it, and the overwhelming response is “What’s the point?” In History class, my teacher doesn’t ask me to prove that Hannibal crossed the Alps. My literature teacher doesn’t ask me to prove that Herman Melville wrote Moby Dick. Even in Science, the subject most closely aligned with, and dependent upon Math, I’m not asked to prove Charles’ and Boyle’s laws. Why do I have to prove things in Math? Tell me what I need to learn, how to use it, and I’ll do it.
In a way, Mathematics has over-succeeded. It has been so successful in explaining, and, horror of horror, quantifying, so many natural, physical, economic, and yes, human relationships, that our modern Math and Science curriculum has become obsessed with conveying the many successes of Mathematics without conveying how those successes were achieved. Most students are well aware of the story of Isaac Newton observing an apple falling and being inspired to quantify gravity. But little or no time is devoted to how such a common observation could lead to the Law of Universal Gravitation, and its proof. Educators will lament, there is simply not enough time. We have to move on to the application. This argument is probably correct, reflecting the fact that we are trying to teach too little about too much. We should not be teaching, or trying to teach, our students every application of Mathematics. Rather, our Mathematics curriculum should emphasize the beauty of Mathematics with the emphasis on thinking, that is, proof.
Where in the busy Secondary Mathematical curriculum should proof be emphasized? The answer is in the first form of Mathematics in which rigorous proof was emphasized, Geometry. But the modern curricular movement is to teach practical application rather than proof. While Heron might be proud to know that modern High School students were being compelled to memorize his formula for the area of a triangle, in reality, few, if any, of the students are ever going to be required to calculate the area of a triangle using only the lengths of the sides. If they ever are confronted with such a problem, they will Google it, and get the procedure and formula from the internet. Heron would be far more proud if students were being instructed on the thought processes he employed to derive his formula and prove that it worked, every time.
Rare is the gainfully employed adult who is compelled to employ any of the Mathematical maneuvers, practical or otherwise, learned in High School. However, all productive citizens will, multiple times in their lives, find themselves having to make a point, or refute a point, using critical thinking and logic. In other words they will have to use proof!