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  • Instructor: In fact this expression is equal to that expression. Let us see how we can convince ourselves that that is true.

  • Student: I'm more than willing to take your word for it! You're the expert!

This student feels that he is reassuring the instructor that his word will not be doubted, so he is sparing the instructor some work, reassuring the instructor that the instructor's goal is already accomplished, and he is sparing himself what he expects would be the agony of learning whatever it was the instructor was leading into.

How do we know that $\pi$ is irrational?

Well, duh!!! That's really basic. We learned that in 7th grade!! Everybody knows that!! Every textbook and teacher says so.

In fact, typical students don't know that the subject matter is precisely: how can we learn how to tell, independently of professors and authorities who tell us, what is true?

In order to explain that point to them, I would like some good examples, perhaps in elementary arithmetic, of situations where students as naive as the one quoted above (who is a fictitious composite of a number very real students) of easily explained problems in which even students like the one above would understand that the way they know the answer is correct is that they can see why it must be correct. That would illustrate the point that learning how to do that in more advanced subjects is precisely the subject matter of courses in those more advanced subjects.

Are there any sexy, or at least good, examples of that sort of problem, that could serve that purpose well?

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Well, there is the dialogue Meno, though perhaps Plato's sexiness score is on the low side. –  André Nicolas Aug 30 at 22:28
    
Things like the infinitude of primes are totally inappropriate for this sort of thing, for a reason I would have thought (until it was proposed in an answer below) to be glaringly obvious. –  Michael Hardy Aug 30 at 23:25
    
The irrationality of $\pi$ is clearly NOT appropriate for this occasion, for reasons that will become clear if you read the question. Does what I wrote really make it appear as if I thought the irrationality of $\pi$ could be an example of the kind I was asking for? –  Michael Hardy Aug 30 at 23:30
    
I think the Pythagorean Theorem is well-suited here. Every student has heard of it, few know why it's true. The usual geometric-minded proofs are easily understood and convincing. (It also brings up a reason to discuss what a "theorem" is in mathematics.) –  David Mitra Aug 31 at 10:14
    
Also, at least to my mind, that the Pythagorean Theorem holds is not obvious. –  David Mitra Aug 31 at 10:22

4 Answers 4

I would try throwing some false, but not necessarily too obviously false, statements at them. Here are some that I might try.

For every positive integer $n$, at least one of $6n-1$ and $6n+1$ is prime.

$(x+y)^2=x^2+y^2$.

$2=1$ One 'proof' of this goes: Let $x=y=1$. Then $$x=y$$ $$x^2=xy$$ $$x^2-y^2=xy-y^2$$ $$(x-y)(x+y)=y(x-y)$$ $$x+y=y$$ $$1+1=1$$

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How do we know that π is irrational?

There's no “we” here, Michael. There's just you, and me.

I know it by trusting the experts; a.k.a. reality check. Though proofs abound, I've never been able to understand a single one of them, nor will I ever be able to, in this life time. Furthermore, I also know that $\gamma$ and all odd $\zeta$ constants are transcendental, despite the complete lack of any proof in this direction, apart from the mere irrationality of Apery's constant, which I also don't understand, by the way. Even if by absurd all these were proven to be at least irrational, it would still not mean anything to me personally, since, again, I would not be able to understand these proofs. To make matters even worse, even the proofs that I do understand, I've always hated, because most of them never made any sense to me, since they convey no actual understanding or intuition as to why it is so.

how can we learn how to tell, independently of professors and authorities who tell us, what is true?

We can't “learn”. Either you already understand something for yourself, or you rely on trustworthy sources. Sometimes, for no reason, while pursuing something completely different than that which we set out to, we discover something which, to our utter shock and surprise, suddenly illuminates the matter which escaped our understanding. But such moments of insight are rare in the life of an average person.

the way they know the answer is correct is that they can see why it must be correct

But that insight or understanding can't be artificially created if it's not there already. It cannot be “taught” or “transmitted” to those that do not possess it.


I know that this may sound a bit strange, but no one can actually “teach” anyone anything. All one can do is “present” things to people, and then try to guide their understanding through them, to the extent to which it exists. If someone lacks the power to understand something, then no amount of “explanations” will change that, ever – which is not to say that if someone does not understand one particular explanation presented to him under some particular circumstances, then he is hopelessly lost for all eternity: maybe he just needs some time to think more about it, or hear a better one, etc.

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No matter how right your main point is, you're missing something. There may be examples simple enough that the students do understand them. OBVIOUSLY proofs of things like the irrationality of $\pi$ are wildly inappropriate for this sort of student. Furthermore, even if they fail to understand everything even at the level of elementary arithmetic, nonetheless one may be able to communicate to them that there is such a thing as understanding the reason why something must be true. –  Michael Hardy Aug 30 at 23:29

The infinitude of primes as explained in Euclid's version seems to me to exactly the kind of proof that can show a student that proofs are not arguments by authority.

It has enough surprise to it ("why shouldn't the primes just stop appearing after some very large number?") but is also simple enough for an inexperienced student to get the feel, perhaps his first ever feel, of how numbers work independently of us.

And therefore how proofs work independently of teachers.

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It seems to me that almost anything in mathematics could be an answer to your question. It is probably best to confront such a student with something which they have not already received on authority. For example:

$1=1$

$1+3=4$

$1+3+5=9$

$1+3+5+7=16$

Present this to the student and ask if they notice some pattern. Ask them if they can figure out why this pattern must hold. Let them think about it. Perhaps illumination will come from within.

Another possibility is to have them draw an arbitrary quadrilateral and connect the midpoints. Do they notice that a parallelogram alwats results? Why could this be?

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Such small examples can hardly help, though. What is really required is to change the entire culture of the "math class": away from producing answers, and towards producing explanations. A lot of lip service is payed to "showing your work", but we ask for essentially no original thought from students until (perhaps) late undergraduate math classes. This can and should change. –  Steven Gubkin Aug 31 at 3:19

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