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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 used in high school geometry textbooks. Mathematicians, on the other hand, typically write out their proofs in sentences, in so-called "paragraph proofs."

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 that: for something to be a proof, you need to COMMUNICATE (not just write a jumbled mess of symbols and numbers) it 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 that is the 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?

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"!

A proof needs to COMMUNICATE your thoughts clearly. That is why we use "line segment 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, such as your parents, your math teacher, and a mathematics professor.

But, the form of the proof or numbering your arguments is NOT the most important thing! You don't necessarily have to write proofs in 2-column format if you like writing them as plain text (prose) better.

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"

Children can – and should – experience proofs before high school also. You as the teacher will want to convince your students that what YOU ARE telling them is indeed true. That is proving in a general sense. But proofs in elementary and middle school don't have to be on the same level of rigor as later on.

For example, when you show students how and WHY multi-digit multiplication works (2 × 371 is the same as 2 × 300 + 2 × 70 + 2 × 1), you are proving, or "justifying" something.

When you use fraction manipulatives to demonstrate why 5 × 2/3 equals 3 1/3, you are "proving" or demonstrating. (Take 5 copies of two thirds. Combine the thirds until you get 3 wholes and one third.)

Often, diagrams or visual illustrations work as proofs in grade school. When students "see it", they become convinced.

Those proofs are not as rigorous as proofs that mathematicians do, 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 justification.

Showing students the "why" of math keeps alive that small voice inside them that asks and wonders, "Why is that?"

Then, after you have proven something as the teacher, change places. It's time for YOU to ask the students WHY something works. A good place for this is after the student has solved a problem (and even if the answer is incorrect): ask why he or she did it this particular way.

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

It doesn't have to be anymore complicated than that. Simply take time to reflect and consider the "why" amidst all the drills and worksheets. It will pay off!

See also

Proof that square root of 2 is an irrational number

An example proof of a property of logarithms

An example of a two-column proof versus a paragraph proof

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