Mathematical Proof

  1. Jul 21, 2011 #1
    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.
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  3. Jul 21, 2011 #2
    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 surface-level 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) = (2x3 - 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.
  4. Jul 21, 2011 #3
    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.
  5. Jul 21, 2011 #4
    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 non-obvious 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.
  6. Jul 21, 2011 #5
    Your understanding about what Math is about would have strengthen many times more if you had learned abstract algebra.
    Last edited: Jul 21, 2011
  7. Jul 21, 2011 #6
    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.
  8. Jul 21, 2011 #7
  9. Jul 21, 2011 #8
    In high school? No, that's a bridge too far.
  10. Jul 21, 2011 #9
    Perhaps back then what you understood was satisfying enough to answer your curiosity about what maths is about. However, to believe you have the general idea of what maths is about just from your high school experience is too assuming.
  11. Jul 21, 2011 #10
    Well, opinions could differ on that point. I didn't feel that learning algebra changed my ideas of what math is really about in any essential way from the conception I got in high school geometry. Now, of course it is possible that I still don't fully understand what math is all about, and you do, and thus from that superior position you are justified in proclaiming my naivete. But I, on the other hand, could claim that you don't really understand what math is all about, and that if you did, you would see that the essence of the matter hasn't changed since Euclid, or even since some prehistoric human noticed that some little piles of stones could be divided into smaller piles of equal number, while others couldn't.

    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.
  12. Jul 21, 2011 #11
    Fair enough, but I think of knowing maths as more of a relative measure of what one knows in comparison to the current progress in mathematics. That's why I just had to disagree with you.
  13. Jul 21, 2011 #12
    Maybe. But I don't think that's relevant. I didn't claim, and never will claim, that I know all about math. THAT would be naive! Knowing what math is all about is quite a different, fundamentally simpler matter.

    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 trans-spliced leaders or gene control networks.
  14. Jul 21, 2011 #13
    Well one aspect of knowing maths is that there is this concept of generality that throughout history, we can see the trend that it keeps on growing more and more "general" or abstract. It's not really the raw knowledge of maths that I was trying to point out but there are many features of what maths is about that are just hard to capture in words.
  15. Jul 21, 2011 #14


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    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 socieo-political reasons as well. Here's my advice for what it is worth:
    • Show enthusiasm when teaching the subject,
    • Ease the students into proofs in small steps, giving them frequent chances for success,
    • I use the analogy of "learning how a tool works to better know when and how to use the tool" to explain the importance of proving calculation methods,
    • Competition and group activities help but be sure all participate,
    • There is a strong correlation of skills between mathematical proof and computer programming, if you have the facilities you can try to integrating some programming into the curriculum. OR you can use this as a motivating factor, (proofs are an exercise of one's programming ability which is growing more and more important in our increasingly automated world.)
    • And the old standby... "Because I say so, and your grade depends ond it!!!"
  16. Jul 22, 2011 #15
    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 proof-based math.
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