What is proof?

Writing proofs is often considered an obstacle in high school geometry. But, what is proof? Certainly, two-column proofs are not the only kind. In fact, they are mostly popular in high school geometry textbooks. Mathematicians, most often, just write their proofs out in sentences, and that's called "paragraph" proof .

Keith Devlin says in his book:

"... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..."

So proof is about communicating in a convincing way. Remember those things: you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) in a CONVINCING way.

I'd like to show you an example. Is the following a valid proof?

PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.

PROOF: "Look at this picture that I drew. It's not drawn to scale or to be accurate. See, this line is as long as this line. And, since this is a midpoint, then this line is as long as this line. And now look at these two angles, here and here. They are the same, I mean have the same measure, obviously, because they are formed in the two corners when these two lines cross each other, or you can also say they are 'vertical angles'. We've studied that.

So, looking at these triangles, there's a side that's as long as this side, there's an angle taht is teh same as this angle, and there's a side that is as long as this side. Well that's SAS, or side-angle-side. I mean, by SAS congruence theorem we know these two triangles are congruent."

Do you think that is a proof? Is anything missing from it or not? While thinking, check also three other proofs of the same. Are they 'better' proofs than mine?

Well, the answer is, my proof above IS a fine proof - and yet it isn't. It depends!

It IS a fine proof if it was SPOKEN to someone while pointing to the various parts of the image. But it isn't the best proof if it was written in a book. You probably had to spend some time figuring what I meant by "this line" and "that angle"!

Proof needs to COMMUNICATE clearly your thoughts. That's why we use "line segment AB" or AB in text, instead of "this line".

Then also you need to CONVINCE - to be logical in your reasoning. And it is not enough to convince a fellow student, but any sufficiently educated rational person - like your parents, your math teacher, and a mathematics professor.

But, the form of the proof is NOT the most important thing! Or, numbering your arguments is NOT the most important thing.

See also an example comparing a proof written in two-column form or written as text.

Proving before high school: justify and consider the 'why'

You might think, "Proof? You need it before high school geometry?" Sure! But, we are talking about a different form of 'proving' here.

What IS proof, first of all? It is something that the person hearing or reading the 'proof' will become CONVINCED that whatever you're proving is, indeed, true.

You will want to convince your youngsters or students that what YOU ARE telling them, is indeed true. That is 'proving', in a general sense. But it doesn't have to be on the same level as later.

For example, when you are showing them how the multi-digit multiplication works and WHY it works (2 × 371 is the same as 2 × 300 + 2 × 70 + 2 × 1), you are proving - or maybe we should say "justifying".

When you take fraction manipulatives and demonstrate why 5 × 2/3 is 3 1/3, you are 'proving' - or demonstrating. (You take 5 times 2/3. You combine the thirds until you get 3 wholes and one third.)

Oftentimes diagrams or picturial illustrations work as 'proofs' in grade school. When students "see it", they become convinced. For example, using algebra tiles to show factorings in pre-algebra.

Those proofs are not as "rigorous" as proofs that mathematicians do, or they are not on the same level. But they are important, nonetheless. That way students get used to demonstrations as to WHY something works, and DON'T get used to "announced math" where rules are just given as step-by-step instructions without much explanations.

They can keep that small voice inside alive that says, "Why is that?"

Then, after you as the teacher have done it, change turns. It's time for YOU to ask WHY. A good place for this is after the student has done some problems: ask why he/she did it this way.

"You added right. Now tell me WHY did you put that 2 up there above the other numbers?"

And, of course you will take special notice of the problems asking the student to justify his/her reasoning.

It doesn't have to be anymore complicated than that. Just don't forget it amidst all the drill and "you need to complete the worksheet" stuff. Take time to reflect, sometimes, and consider the "why".

See also:

Proving triangles congruent - my blogpost about this topic, with lots of links to other resources.