2 Is it proof?
Mathematical proof is different from proof in other subjects such as science and law because every step in the argumentation has to be based on known and undisputed facts – proof cannot be based on empirical evidence. This means that in mathematics, you cannot state ‘this is always true’ from simply observing occurrences. In mathematics, alljustification of proof should be based on accepted or established statements, and rules of logic.
To learn, and to make sense of new learning, students need to be able to build on their existing knowledge and experiences, something that is hard to do when moving straight into formal mathematical proof. The remainder of this unit will suggest ways to offer students support in identifying and developing the important aspects in mathematical proving, including:
- the argument that mathematical proofs are not built on empirical evidence
- knowing and recall of mathematical properties
- developing effective argumentation techniques
- building on what is already known.
Sometimes it is hard to know when mathematical reasoning can be considered a ‘proof’ or not. An example of this is described in the following scenario, in which Mrs Kapur asked his students how they would prove that the sum of angles of a triangle is equal to 180°.
Case Study 1: Mrs Kapur asks her class to prove that the sum of the angles of a triangle equals 180°
| Teacher: | What do the interior angles of a triangle add up to? |
| Student: | 180°. |
| Teacher: | Are you sure? |
| Student: | Yes, absolutely. |
| Teacher: | How do you know this is so? |
| Student: | Because … you tell us and it says so in the book. |
| Teacher: | OK, I want you to now think carefully for a moment about how you would prove mathematically that the sum of the angles of a triangle equals 180°. This is a mathematical proof, so your reasoning has to be rigorous and convincing. Imagine you are a mathematician trying to convince the prime minister of India – how would you go about it? Think about what you know already about triangles and mathematical proof. Discuss your ideas first with your partner. I will give you five minutes to do so. |
| [Five minutes later.] | |
| So, what will you say to the prime minister to convince him that the sum of the interior angles of a triangle always equals 180°? | |
| Student 1: | We would say that when you measure the interior angles of a triangle and add these up, then you always end up with 180°. You can do this with any number, and any kind of triangle. |
| Student 2: | We would say: draw a triangle on a piece of paper. Cut it out. Tear off the corner and place these corners next to each other. They always fit on a straight line. As we know that angles on a straight line add up to 180°, we can deduce that the sum of the interior angles of a triangle also equals 180°. This will always be so for any triangle. |
| Teacher: | What do you think, class – are you convinced these ideas are mathematical proofs? Will the prime minister be convinced? Should he be? |
 Pause for thought Think about how far these ideas are indeed mathematical proofs. Are there other ways that proof might be offered? How might these proofs be challenged? |
1 Why teach mathematical proof in schools?