What makes a good teacher? What is a formula?

[Bill Spight]

You mean to explain what is under attachment? You ask me to explain this formula? How this formula works? Or what?I still dont get it, really.

If you ask how formula works, it works (by works i mean "produce n-th prime number) beacuse:

The sum contains three factors which we multiply k, and 2 complicated fractions. We sum from k=2 to k=2^n, but
k is always different from 0, first fraction is different from 0 only when k is prime, and 3rd factor is equal to 1 only when there is exactly n prime numbers below k(otherwise is equal to 0). Thanks to this, this big sum, has terms always equals 0, except 1 time, when its n-th prime number.

Why we summing from k to 2^n? beacuse we need to have n-th prime number between, thanks to Chebyshew rule, which says that between x and 2x there is prime number, which follow that n-th prime number is lower than 2^n (and thats what we needed).


You need to try to examinate first this (its element of this function) to get how this works.For given a, this function is equal to 1 only when x=a, otherwise is equal to 0 - look at F1 attachment.

Then you need to examinate next funtion, which is equal to 1 if k is prime number, otherwise is equal to 0 - look at F2 attachment.

If you get this 2 steps, and examine my formula, you will see how it works. I hope that this close case. I am exhausted really.


I am not the best explaining mathemathical things in english, i hope i didnt make any mistakes.

Thank you.

[AloneAgainstAll]

Even having this restriction, my formula is still a formula, right? I didnt used any "infinite things".

[Kirby]

I’m reminded of an incident with my cousin’s wife, who is a second grade teacher. For a math test, she had a problem saying something to the effect of “color the circle” or something like that. Based on definitions of circles that I’d read, the circle is basically the border definition. So I would have prefer she wrote “color the area surrounded by the circle”. Because someone who colored only the border would be correct, in my opinion.

But there may be definitions of “circle”, which include the area and not only the border. But no definition of “circle” was given.

Gee, Kirby, haven't you heard of the area of a circle? (That is, instead of "the area bounded by a circle".)

I have. But I don’t like that phrasing. I think the circle definition here is somewhat accepted:

http://mathworld.wolfram.com/Circle.html

[jlt]

Even having this restriction, my formula is still a formula, right? I didnt used any "infinite things".

Yes, what you wrote is indeed a finite sequence of mathematical symbols, but so is my "formula" f(n)=f(n)... You say that "f(n)=f(n)" is not a formula but you never gave a precise definition of (what you mean by) a formula.

[Kirby]

• In my last few messages, I showed that according to Kirby's definition of a "formula", I could produce dumb "formulas" that "compute" the n-th prime number.

It's not "Kirby's definition" of a formula. I took that from a reputable math site, which I linked. This is the only thing close to a real definition that I've seen presented in this thread. What is your definition of a formula? Or what other definition are you referencing? I've given an example of a site giving a definition - do you have something with a more constrained definition of formula for which AloneAgainstAll's professor would have been correct?

If we say "formula" is just a wishywashy word without a definition, then the truth of AloneAgainstAll's teacher's statement is undefined - and a useless statement at that. You might as well say that prime numbers can't be expressed as a diddlywomp, without explaining what a diddlywomp means.

[Kirby]

Here's a link to the site, again: http://mathworld.wolfram.com/Formula.html

In mathematics, a formula is a fact, rule, or principle that is expressed in terms of mathematical symbols.

So I suppose the claim is that "f(n) = f(n)" is a fact that is expressed in terms of mathematical symbols. Typically, the "f" mathematical notation here is used to represent a function (http://mathworld.wolfram.com/Function.html). What is a function?

A function is a relation that uniquely associates members of one set with members of another set.

So I suppose the fact being represented mathematically in "f(n) = f(n)" would be that "a function on n is a function on n". I suppose that's OK. Just not useful. Like saying that "5 = 5".

[jlt]

As I said, I have no definition of a formula. If I want to be precise, I say "a mathematical expression involving symbols among some list". Now, in that list, do you allow summation symbols? Integration symbols? Infinite series? Limits of sequences? Quantifiers? Sequences defined by induction?

[Knotwilg]

Given the enthusiasm with which you have been in search for the formula for the n-th prime, it seems that your teacher has managed to inspire you with the thirst for knowledge. At the least he didn't obstruct it. His claim that something didn't exist made you wonder. That seems more like a success.

A teacher doesn't have to be 100% correct, not even about what is true or known. In fact, you'll be hard pressed to find even among the brightest scientists people who never made a wrong claim and passed it on.

Here's an interesting anecdote (I hope).

In my math curriculum, I had two opposite teachers. One was didactically great. She had a very well structured syllabus on topology. The other was a mumbling piece of chaos, with no syllabus, whose classes on Lie Algebra I came home from with incomprehensible notes. So I went to the library and set up a work group to find order in the chaos. Through self study and hard work, and long evenings with one fellow student, we finally found that order and because I had done such hard work, I mastered the subject far better than the well devised, easy to digest topology course. It may also be that Lie Algebra was just more my piece of cake than topology, but it makes you wonder ...

[John Fairbairn]

Because someone who colored only the border would be correct, in my opinion.

I note and welcome your own hesitation about that in your subsequent paragraph, but I still think it needs to be stressed that that's the sort of over-defining thinking that gets maths a bad name among most people, in my experience. Ordinary people want to say things like "the searchlight cast a circle of light" and expect you to know they don't mean a rim, but if they ask a group of you to form a circle, they don't expect you to cluster as a blob in the centre of the room. Common sense rules, OK?

In any case, although I'm way out of my depth, to talk about a border strikes me as incorrect, and not in line with the website pointed to - to use in a common way just two of the relevant words. The boundary in the definition is a set of points. As far as I know - very basic Euclid - a point is 0-dimensional. And if a line is given as a set points it only becomes 1-dimensional. Which can't be coloured in, no? Unless you invent 1-dimensional colours, or the like, and if you do the eyes of ordinary people just roll up heavenwards.

Ultimately - again in my experience - the real problem is that too many definition mavens - a tiny minority - can't accept that ordinary people - a huge majority - even exist. Haven't they got eyes? (Just to keep it on the topic of go )

[bogiesan]

I thought this was a discussion about characteristics that make a good go teacher.
I've been teaching people to play go forty years. It's only recently, say, ten years ago, I realized I was doing it all wrong. So I get points for being objective enough to realize I scared more people away from go than I recruited.

[Bill Spight]

Well, I was not particularly reluctant to call that expression a formula. I attempted to guess a definition of formula that was close to the one that the elementary school teacher had in mind, and which is also close to my sense of it.

By the way, what *is* this definition? I can understand the one I posted from Wolfram, which seems consistent with AloneAgainstAll's interpretation. Given that we have an alternate definition in play here, and that we can't ask AloneAgainstAll's teacher what their definition, would it be possible for you to express your intuitive definition in a precise way?

It's true, I did not state it in the form of a definition. To be clear, let me take some time to build up to it.

We start with language. There is a common pattern in statements, a formula, if you will, where you state what is known, understood, or given by your hearer, and then say something new about it or previously unknown to them.

Suppose that we have a right triangle with hypotenuse C of length c, and two sides, A of length a and B of length b, and we know a and b. Then we have the formula for the length of the hypotenuse, c = √(a²+b²), the formula being the expression on the right side. Even though √(a²+b²) = c, c is not a formula. If we knew a and c we could write a formula for b. Note that the formula is in terms of what we already know, what is given. It tells us something that we don't know, that is not given.

Suppose that we know the first n-1 primes. Perforce, we also know whether a natural number less than the n-1th prime (P_n-1) is composite or prime. That we can take as given, if we want to know the nth prime. What about numbers greater than P_n-1? Can we take them as given? That's a question. Obviously we cannot take P_n as given, it is what we are trying to find. Note that AloneAgainstAll's expression includes P_n in the sum up to 2ⁿ, so it includes more than is given, and it is not a formula, in the intended sense. I would go further and say that we cannot include any number greater than P_n-1 in the formula for P_n.

Back to language. If I tell you I have three children, but I actually have four, am I lying? Logically, no. If I have four children then I also have three. But I am not supposed to hide the fact that I have the other child. So if I am given the nth prime, I supposedly know about the numbers less than P_n, but not necessarily about numbers greater than P_n, and if I do, I have to say so. Usually when we talk about formulas, we do not explicitly say what is given, but because of the ambiguity, I think we have to in this case. Anyway, here is my stab at a definition for a mathematical formula.

A mathematical formula for an unknown is an expression in terms of what is given which when evaluated yields the value of the unknown.

[AloneAgainstAll]

Even having this restriction, my formula is still a formula, right? I didnt used any "infinite things".

Yes, what you wrote is indeed a finite sequence of mathematical symbols, but so is my "formula" f(n)=f(n)... You say that "f(n)=f(n)" is not a formula but you never gave a precise definition of (what you mean by) a formula.

Your formula is indeed good soulution to our problem (to give formula for n-th prime) - you just need to add definition of f function, and then it works (you can use function i provided).

I didnt gave definition of formula cuz for me its synonim of function. But thats aside.

And btw - by your definition of formula, some functions are not formulas. If i would make my own definition of formula, it would be wider than function - i could come with for example recurence definition of n! - its not a function (but define a function) but for me it should be counted as formula - haha, thats more and more offtopic .

[f(x)]^2=x is not a function but would be formula for me. Another example.

@Knotwilg

That was not my teacher - she taught in my elementary school, but not me. I came across this "formula" completely by accident. But i admit that if teacher inspires and motivate, its great.

I heard a story about being too precise in math:

Elementary school.
Pupil: I have 2 sandwiches for breakfast.
Teacher : You should explain precisely. You should say "i have set of sandwiches containing 2 elements".

That joke was invented to counter exactly what was pointed above - common sens rules should apply, teaching chilrens at basic level in kind of super precise manner is wrong . After i laughed hardly i realised that there is a catch - what if sandwiches were identical?

[Bill Spight]

I’m reminded of an incident with my cousin’s wife, who is a second grade teacher. For a math test, she had a problem saying something to the effect of “color the circle” or something like that. Based on definitions of circles that I’d read, the circle is basically the border definition. So I would have prefer she wrote “color the area surrounded by the circle”. Because someone who colored only the border would be correct, in my opinion.

But there may be definitions of “circle”, which include the area and not only the border. But no definition of “circle” was given.

Gee, Kirby, haven't you heard of the area of a circle? (That is, instead of "the area bounded by a circle".)

I have. But I don’t like that phrasing. I think the circle definition here is somewhat accepted:

http://mathworld.wolfram.com/Circle.html

But Wolfram turns around and uses a different definition in this sentence without providing it.

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

The area of a set of points equidistant from a given point is 0, thank you very much.

[AloneAgainstAll]

Well, I was not particularly reluctant to call that expression a formula. I attempted to guess a definition of formula that was close to the one that the elementary school teacher had in mind, and which is also close to my sense of it.

By the way, what *is* this definition? I can understand the one I posted from Wolfram, which seems consistent with AloneAgainstAll's interpretation. Given that we have an alternate definition in play here, and that we can't ask AloneAgainstAll's teacher what their definition, would it be possible for you to express your intuitive definition in a precise way?

It's true, I did not state it in the form of a definition. To be clear, let me take some time to build up to it.

We start with language. There is a common pattern in statements, a formula, if you will, where you state what is known, understood, or given by your hearer, and then say something new about it or previously unknown to them.

Suppose that we have a right triangle with hypotenuse C of length c, and two sides, A of length a and B of length b, and we know a and b. Then we have the formula for the length of the hypotenuse, c = √(a²+b²), the formula being the expression on the right side. Even though √(a²+b²) = c, c is not a formula. If we knew a and c we could write a formula for b. Note that the formula is in terms of what we already know, what is given. It tells us something that we don't know, that is not given.

Suppose that we know the first n-1 primes. Perforce, we also know whether a natural number less than the n-1th prime (P_n-1) is composite or prime. That we can take as given, if we want to know the nth prime. What about numbers greater than P_n-1? Can we take them as given? That's a question. Obviously we cannot take P_n as given, it is what we are trying to find. Note that AloneAgainstAll's expression includes P_n in the sum up to 2ⁿ, so it includes more than is given, and it is not a formula, in the intended sense. I would go further and say that we cannot include any number greater than P_n-1 in the formula for P_n.

Back to language. If I tell you I have three children, but I actually have four, am I lying? Logically, no. If I have four children then I also have three. But I am not supposed to hide the fact that I have the other child. So if I am given the nth prime, I supposedly know about the numbers less than Pn, but not necessarily about numbers greater than Pn. Usually when we talk about formulas, we do not explicitly say what is given, but because of the ambiguity, I think we have to in this case. Anyway, here is my stab at a definition for a mathematical formula.

A mathematical formula for an unknown is an expression in terms of what is given which when evaluated yields the value of the unknown.

If you define formula that way you cant provide formula for sum of 1+2+3...+n because it contains n+1, and n+1 is not given.

Also sometimes you have problem (thats example) - you have 3 constans a,b,c (which descibe sth, whatever). Find sth else. And "formula" for it is x=1+abc. According to you, 1 is not given so you cant use it.

Another example:

You have ball with raduis R. Find volume. You cant give formula for volume cuz it contains both 4/3 and Pi which are not given before (only radius is given).


Well, ofc someone can say 1 is not given but when a is given i can make a/a and voila. If i have numbers 1,2,3 up to n i can do same thing and get 2^n.

[Kirby]

In any case, although I'm way out of my depth, to talk about a border strikes me as incorrect, and not in line with the website pointed to - to use in a common way just two of the relevant words. The boundary in the definition is a set of points. As far as I know - very basic Euclid - a point is 0-dimensional. And if a line is given as a set points it only becomes 1-dimensional. Which can't be coloured in, no? Unless you invent 1-dimensional colours, or the like, and if you do the eyes of ordinary people just roll up heavenwards.

The first line of the circle definition is here:

A circle is the set of points in a plane that are equidistant from a given point O.

So, the "circle" with radius R does not include points on the plane that have distance less than R from O, i.e., it doesn't include the area surrounded by the circle. In the real world, if you were to try to draw the border of a circle, the width of the points you are drawing would be positive. So there would be a colored area that is not exactly R distance from O. But that's a different problem - it's not possible to be infinitely precise in the real world. If you have a problem that says to draw a line segment that's exactly 5cm, it'd be impossible to verify that it was correct to infinite precision. But anyway, having the person fill in the entire area surrounded by the circle is clearly wrong.

But that's exactly the reason I don't like the wording of the question. It could be worded in a less ambiguous way: Fill in the area surrounded by the circle with radius R. Or maybe better yet, fill in the area surrounded by the circle with radius R with X precision (to account for the real world problem that, in practice, you might have a non-zero amount of fill exiting the boundary of the circle, if you were to try to color it.