Proofs

This piece was suggested by an article by Deborah Ball and Hyman Bass (http://www-personal.umich.edu/~dball/BallBassMakingMathReasonabl.pdf, Ball, D. L, & Bass, H. (in press). Making mathematics reasonable in school, in G. Martin (Ed.), Research compendium for the Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics),about mathematical reasoning in the early grades. The central question was, “Is the sum of two odd numbers always even?”

General remarks.

A proof is a logical reasoning that derives a new fact (new to students) from facts they already know.

Modern mathematical proofs that use variables are clear and concise for people who can read them. But often, when the proofs are short, they can be paraphrased and restated in everyday language. There is also a third way to present a proof, which has been used in antiquity: A proof given through a “generic” example. This is done by considering one or a few examples in a way that convinces others that no specific properties of the objects at hand were used. This method is still dominant in geometric proofs based on drawing, because every drawing is a specific figure.

All three approaches can be mixed together depending on the audience and purpose of a proof.

Example.

1. An advanced proof.

Definitions:

A whole number n is even if and only if it is the sum of two equal whole numbers.

Therefore any even whole number n = m + m, where m is another whole number.

A whole number n is odd if and only if n-1 is an even whole number.

Therefore any odd whole number n = m + m + 1, where m is another whole number.

Theorem. The sum of two odd numbers is even.

Proof.

Let n1 and n2, be two odd numbers. n1 = m1 + m1 + 1, and n2 = m2 + m2 + 1. Therefore n1 + n2 = m1 + m1 + 1 + m2 + m2 + 1 = (m1 + m2 + 1) + (m1 + m2 + 1). Thus n1 + n2 is the sum of two equal whole numbers, and therefore it is even.

2. A paraphrased proof.

When is a whole number even? It is even if it is built by adding another whole number to itself. For example, 3 + 3 = 6 is even, and 22 + 22 = 44 is even. Also 0 + 0 = 0 is even. But 5 is not even; 5 = 5 + 0 = 0 + 5 = 4 + 1 = 1 + 4 = 3 + 2 = 2 +3, and that is all. It cannot be built by adding another whole number to itself.

When we count, every second number is even and every second number is not. The numbers that are not even are called odd. So each odd number is built by adding one to an even number.

Proof.

An odd number looks like this: one number + the same number + 1. Another odd number looks like this: a second number + the same second number + 1. We can add these two odd numbers in any order, so we get: one number + a second number + 1, all added to itself. So we get an even number.

Let’s look at an example.

17 = 8 + 8 + 1 is odd, 23 = 11 + 11 + 1 is odd. When we add 17 and 23, we get 8 + 11 + 1 plus 8 + 11 + 1 = 20 + 20, which is an even number.

Do you see that this works for any two odd numbers? (If not, then we can make a few more complex examples.)

Remarks.

Proofs depend on the details of definitions. So, for example, if even numbers are defined as 2*m where m is a whole number, then this can be paraphrased by saying that an even number is a sum of twos. In this case, both an advanced proof and a paraphrased proof would be different from those given above.

The main reason that students do not understand proofs is that
they do not have the background knowledge that we assume to be “given”.
In this case: Do they know what being even and being odd mean? Do they know
that every odd number is bigger by one than some even number?