What makes a good teacher? What is a formula?

[Gomoto]

primes = #@!?b + primes - #@!?b

the interesting idendities are the ones with different left and right sides.

[AloneAgainstAll]

What is the simplest formula that works?

This is argument which is backing up your claim? I must admit, i expected sth much better.

[jlt]

AloneAgainstAll: your formula is correct, but doesn't do the job that one would expect from a formula. This conversation is a bit like:

- Go Master: assuming komi is (fair komi + 0.5), there is no known method to win a go game as white 100% of the time.

- StudentAloneAgainstAll: I disagree. You can explore the entire tree and search for an optimal path.

[AloneAgainstAll]

You really think this is good paraphrase? I dont think so. There are things we cant "enclose" in mathemathical formulas (like results of integral of sin(x^2) cant be described by elementary functions, only infinite sum ) etc. So the fact that we can produce such a formula (even if its completely useless for computing) is mathemathically significant.

[jlt]

What is significant and what is not is a matter of personal taste. So let's agree that we disagree here...

[AloneAgainstAll]

But you agree that there is significant difference between:

1.There is no formula
2.There is no formula usefull for computing.

We werent talking about 2nd at all! If you want, we can talk about usefull formulas for computing prime numbers in completely different thread, but lets keep this talk about what it was.


To not derail original thread completely, i want to say that i feel jealous for ppl who believe that bad teachers does not exist - they must have met only good teachers in their whole like, which is very good for them.

[jlt]

The notion of "formula" is not well-defined, unless you specify which symbols you are allowed to use. People may object to the use of summation symbols.

I agree that bad teachers exist, however it is not easy to determine if someone is a good or a bad teacher. Certainly someone who teaches wrong ideas/mistakes, or who breaks student's motivation, is a bad teacher, but outside extreme cases like that, I think that the student's progress has much more to do with his passion and with a stimulating environment (competition against other students) than with the teaching abilities of the master.

[AloneAgainstAll]

Summation symbol is just a short form of writing many sums. I dont see any problem with it.

[jlt]

Suppose someone wants a formula that computes sum(k ; k=1...n). According to your criteria, "sum(k ; k=1...n)" would be a perfectly valid answer. However, other people may object that this answer is tautological, hence completely useless, and would like a formula without summation symbols, like n(n+1)/2. Both are right in some sense, since you didn't define precisely what you mean by "formula", so both of you implicitly started with different definitions.

[AloneAgainstAll]

Yes this is perfectly valid useless answer.

Think about this way:

1.F(x)=1
2.F(x)=1+x
3.F(x)=1+x+x^2 etc

Are all of this formulas? You all the time say about usefullness which make me think you are practicing-informatician/programmer. If so, then your point is understandable, practical usefulness is most important to you, but that is still another subject.

If you disagree with summation symbol which is just short form of writing some sums, then you also must object for integral symbol. Really think that?

[jlt]

I am not objecting to the use of any particular symbol, nor saying that a formula should have some practical use, I just said that you and your teacher had different implicit definitions for the word "formula".

[AloneAgainstAll]

How you know we had different definitions of formula?

I think no, i even didnt talked with this teacher at all - i think she would be deeply shocked if anybody would present her one of such a formulas. I was about 10 years old then, so i just remembered and thought she is right, unless was proven wrong years after that.

[jlt]

Of course I don't know for sure, I am not in her head, it's just my guess. And my guess is that, for most people, a "formula that computes f(n)" is "a formula that computes f(n) more efficiently than the naive algorithm", or at least a non-tautological formula.

You may have another opinion, but you are not in the majority, so my guess is that your teacher's statement makes sense for most people, and in any case is not a proof that she was a bad teacher.

[zermelo]

You really think this is good paraphrase? I dont think so. There are things we cant "enclose" in mathemathical formulas (like results of integral of sin(x^2) cant be described by elementary functions, only infinite sum ) etc. So the fact that we can produce such a formula (even if its completely useless for computing) is mathemathically significant.

Ok, I define S(t) as the integral of sin(x^2) from 0 to t. Now I can write you a nice short correct formula for it, i.e. S(t). Yes, it does not use elementary functions only, but your prime formula with floor functions does not use only elementary functions either. The whole issue is really about what kind of formulas we allow.

I don't dispute that maybe you had a bad teachers, and maybe he/she could have discussed the nuances of formulas and algorithms better.

[AloneAgainstAll]

Well, if for you this wrong statment makes still sense, then i give up.