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Mathematical Proof 
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#1
Jul2111, 04:59 AM

P: 153

I have found that for a number of students doing proofs is an unnecessary evil. For some of them it is like a painful visit to the local dentist. I thought I had explained the need for proves but it still puts a number of students off.
What motivating reasons can I give students for them to at least understand the logical deduction of proofs and why they need them? I have informed them of things like absolute truth, permanent, make sure our statements are correct, more powerful than a scientific proof etc. 


#2
Jul2111, 05:06 AM

P: 120

I think it's an issue of interests. I mean literally being interested. Proofs lead to a deeper understanding of concepts, but a lot of students only need a surfacelevel grasp of mathematics to get a decent grade. I find that someone who is disinterested in math will not be interested in knowing the "why" and "how" part of anything.
I tutor high school students. They are always more interested in how to use the chain rule than how to prove it. Of course, high school exam problems in elementary calculus are things like "Differentiate f(x) = (2x^{3}  2)^{2}" so of course the more interesting results of the chain rule aren't important to someone who just wants to pass their course. 


#3
Jul2111, 07:03 AM

P: 20

From the perspective of a student, being thrown in a bunch of new terms and ideas is like trying to absorb gibberish (no offense).
I would suggest to start with something really simple yet interesting in that it has some intriguing logical implications. Perhaps, it would be best to start the session with a talk about proofs to prevent killing whatever little interest left in the students from doing other activities first such as math problems. I first found my interest in math while studying linear algebra. In the process, by wanting to answer all of the questions that I raised, I pushed my brain to think logically and after a while, I discovered some "entertaining" facts, tricks and math. I think self study is the best way to build interest and I personally blame bad presentation to account for 80% of the reasons why students generally do not have interest in math. 


#4
Jul2111, 07:23 AM

P: 455

Mathematical Proof
I thought I hated math until I took High School geometry. Previously every math class had been taught as a way of doing computations. Algebra and word problems were a little more interesting, but still, it all seemed to be just utilitarian grind. But in geometry, we did proofs, and I came to understand, what I had never realized, that this was what math was really all about. It was fascinating to me to first discover nonobvious things, like that the triangles formed by two lines crossing in a circle were similar, and then to be able to show by argument why that had to be so.



#5
Jul2111, 07:30 AM

P: 20

Your understanding about what Math is about would have strengthen many times more if you had learned abstract algebra.



#6
Jul2111, 07:30 AM

P: 925

I'm assuming this is high school students rather than college level math majors. During my time in the M.Ed. program, I had come up with seevral ideas for teaching various topics. One (regarding proofs, although I would not have been teaching math) involved using a courtroom as a sort of "proof" lesson, where students were given several pieces of evidence (the axioms, theorems, etc.) and teams had to prove something resulting from those axioms. By making it within the context of a courtroom, the lesson also showed how using evidence to support conclusions can be used just about anywhere.



#7
Jul2111, 07:39 AM

P: 20

A method purely developed for the purpose of training deductive skills for maths:
http://en.wikipedia.org/wiki/Moore_method 


#8
Jul2111, 07:51 AM

P: 455




#9
Jul2111, 08:01 AM

P: 20




#10
Jul2111, 08:16 AM

P: 455

There is no way we will settle this, of course. But I don't feel that abstract algebra, beautiful though it is, is uniquely the key to mathematics. 


#11
Jul2111, 08:21 AM

P: 20




#12
Jul2111, 08:33 AM

P: 455

Look, I'm a geneticist. I'm willing to bet I know far more about genetics than you. But here is what genetics is all about: it's answering the question, why do children resemble their parents? Why do dogs have puppies and cats have kittens? Why, when you're exposed to someone with measles, do you catch measles and not smallpox? If you get this, you know just as well as I do what genetics is all about, whether or not you've ever heard of transspliced leaders or gene control networks. 


#13
Jul2111, 08:43 AM

P: 20




#14
Jul2111, 08:52 AM

Sci Advisor
PF Gold
P: 1,781

Two observations:
I find many of my college freshmen students, when applying rusty algebra to course problems tend to "invent" bazaar new (and quite incorrect) simplification steps. They misapply rote learning skills out of context. Example: Canceling through sums [itex] \frac{x+2}{4y} \to \frac{x+\not{2}}{\not{4}y}\to\frac{x+1}{2y}[/itex] I do think, by taking them through the logic of the rules they use via proofs better enables them, when rusty, to correctly recall calculation steps, and more importantly the habit of verifying a weakly remembered rule. Secondly, IF the student is going to move on to more general topics e.g. linear algebra, differential equations, higher calculus, by seeing the proofs at the base level and thereby seeing the dependencies of a theorem on the properties of the mathematical objects to which it applies, they are better able to generalize and see the reason behind various conditions of validity. As far as motivation goes, well that is the $64,000 question. I'm seeing a decline in the teaching of rigorous logic in the college prep. for my students and it worries me not only for the sake of good future mathematicians and scientists but for socieopolitical reasons as well. Here's my advice for what it is worth:



#15
Jul2211, 09:15 AM

P: 474

To be honest, I think what turns a lot of people off from proofs is that they're hard to understand and they're REALLY hard to come up with on your own if you've had very little exposure to them. If we actually eased people in to it, starting with high school, I think people would be a lot more comfortable with the rigorous method, and have a much easier/more enjoyable time learning it later.
Right now, it seems like you go through high school and the first couple university math courses and there's next to no emphasis on proofs. Very rarely are you asked to really justify what you do beyond "it says to do that in the text book." Then you take your first linear/abstract algebra, and proofs hit you like a sack of bricks, and the prof expects everyone in the class to be an expert proof writer after 2 weeks of class. At that point a lot of people are convinced that proofs are the worst thing ever, and/or they're convinced that they suck at proofbased math. 


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