What is proof?
Another obstacle in high school math are the proofs in the geometry course.
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 (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book
, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..."
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.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"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."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
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 (well their proofs usually take many many paragraphs worth of writing).
Keith Devlin says in his book
, "... being a proof means having the capacity to completely convince any sufficiently educated, intelligent, rational person..." 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.
Is this a proof?
PROBLEM: If E is the midpoint of BD, and AE is as long as EC, prove that the two triangles are congruent.
"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."
What do you think? Is that a proof?
If it isn't, tell me why. What is missing?
And while thinking, you can also check three other ways to write the proof of the same.
Tags: philosophy, geometry
Comments
Did anyone else have an 8th-grade geometry teacher who cared more about the WAY you wrote proofs (i.e., put this here and that theorem there) than about what a proof actually was?
Teaching in that way certainly highlights one example of the madness of emphasizing method over understanding.
On a side note, there is almost certainly a way to show this proof in another way--using parallel lines cut by a transversal. No?