What makes a good teacher? What is a formula?

[Kirby]

Personally, I think the definition of "function" is more strict than the definition of "formula".

But anyway, yeah... The horse is dead. RIP

[Bill Spight]

In what way?

In the sum up to 2ⁿ.

A sum is a sum. We know how to sum.

Knowing nothing about primes, you can calculate one from the expression, no?

But then why not allow the Sieve of Eratosthenes? See above.

If you can express it with math symbols like that, why not allow it? I don't see a problem. It's a formula.

The problem arises when the unknown, which you are trying to find, is part of the formula. Yes. there is such a thing as recursion, but then you can use subscripts.

As for the Sieve of Eratosthenes, presumably the teacher had shown the elementary students how to tell whether a natural number is a prime or not, which in turn implies the Sieve of Eratosthenes, even if she did not mention it. Whatever formula she had in mind, if it included the Sieve, it could not have been part of the formula.

[Bill Spight]

Personally, I think the definition of "function" is more strict than the definition of "formula".

But anyway, yeah... The horse is dead. RIP

Indeed. Formula is an informal term.

[Kirby]

The problem arises when the unknown, which you are trying to find, is part of the formula. Yes. there is such a thing as recursion, but then you can use subscripts.

Indeed, there are recursive formulas

But this is not recursive. It's iterative. You can often convert recursive formulas to iterative ones, which is what is done here.

Indeed. Formula is an informal term.

If so, it cannot be used to make strong mathematical claims.

[AloneAgainstAll]

Ah so all that fuss was beacuse you assumed that formula is informal term?? Why the hell you didnt said it when you said teacher was correct?

You indeed cant give formula for literally ANTYHING if term formula is not formally well defined!

Isnt that called strawman in english?

[Bill Spight]

If she assumed such a restriction she should express it beforehand, and thats true for any restrictions.

Consider your audience. Why complicate matters for a group of elementary students? She used formula in an informal, non-precise way, as it is used in ordinary language.

There is a similar problem, BTW, in teaching go to adults. For instance, some people are told that you need two eyes for life. Later, these people see a seki and are confused. And the question of what is an eye is a mess. See discussions on Sensei's Library. It is probably not a good idea to tell raw beginners about Two Headed Dragons, which they may never see in real life. OTOH, why lie to them and say that a false eye cannot help you make life? My solution has been to define a false eye, not by its local shape, but by the fact that eventually the opponent can put at least one stone that forms it into atari. What to tell beginners, or elementary school students, is not easy to say.

[Bill Spight]

Kirby he means that we can sum only up to given number by his assumption that only first n-th prime numbers ar given, and only those we can use in our formula. Its sad that if we assume that we cant give formula for volume of ball (and other standard formulas), even if we have radius, but thats his definition.

It is not my definition, thank you very much. My definition has nothing to do with balls. I have reasons for thinking that the teacher had a restricted sense of formula in mind.

[Bill Spight]

The problem arises when the unknown, which you are trying to find, is part of the formula. Yes. there is such a thing as recursion, but then you can use subscripts.

Indeed, there are recursive formulas

But this is not recursive. It's iterative. You can often convert recursive formulas to iterative ones, which is what is done here.

Indeed. Formula is an informal term.

If so, it cannot be used to make strong mathematical claims.

You think that the teacher was making a strong mathematical claim to a bunch of elementary school students? I think she was talking to them using normal speech.

[Kirby]

If she assumed such a restriction she should express it beforehand, and thats true for any restrictions.

Consider your audience. Why complicate matters for a group of elementary students? She used formula in an informal, non-precise way, as it is used in ordinary language.

There is a similar problem, BTW, in teaching go to adults. For instance, some people are told that you need two eyes for life. Later, these people see a seki and are confused. And the question of what is an eye is a mess. See discussions on Sensei's Library. It is probably not a good idea to tell raw beginners about Two Headed Dragons, which they may never see in real life. OTOH, why lie to them and say that a false eye cannot help you make life? My solution has been to define a false eye, not by its local shape, but by the fact that eventually the opponent can put at least one stone that forms it into atari. What to tell beginners, or elementary school students, is not easy to say.

If she assumed such a restriction she should express it beforehand, and thats true for any restrictions.

Consider your audience. Why complicate matters for a group of elementary students? She used formula in an informal, non-precise way, as it is used in ordinary language.

But she (is it a she?) used that imprecise term to say to make a bold statement about non-existence.

Sometimes when people are referring to numbers, they are talking strictly about the natural numbers. Using casual, ordinary language, someone might claim that there aren't numbers less than zero. Obviously, folks who are considering negative numbers might take issue with that.

Then you can get into a 100+ post argument about how a number is defined. When really, there are different ways of specifying numbers (natural numbers, reals, extended reals, etc.).

It'd be much safer for the teacher to say that there are no *natural numbers* less than zero. Because that's the class of things she's talking about.

For instance, some people are told that you need two eyes for life.

So the teacher had better not say that!

[Kirby]

You think that the teacher was making a strong mathematical claim to a bunch of elementary school students? I think she was talking to them using normal speech.

Normal speech that was wrong.

Anyway, I already paid my respects to the dead horses. I've expressed my thoughts on this enough. I'm out.

[hyperpape]

I believe the formula in question is not a closed form formula: https://en.wikipedia.org/wiki/Closed-form_expression.

Summation notation is problematic, because given an initial set of values given by elementary (and some non-elementary functions), the number of such function values you must compute grows arbitrarily large as n goes to infinity.

The distinction between a closed-form-expression and others is quite important. I'm not sure how formula is used, but I think the worst you could accuse your teacher of doing is being slightly less precise.

[Bill Spight]

It'd be much safer for the teacher to say that there are no *natural numbers* less than zero. Because that's the class of things she's talking about.

Consider her audience. Math was not taught as well when I was in grade school as it is now, but nearly all of our ten years olds understood number to mean a non-negative integer. Why confuse them with the term, natural number? Maybe counting number would have been OK, though. But then the teacher would open the door to questions about non-counting numbers before she was ready to teach about them.

[AloneAgainstAll]

Kirby he means that we can sum only up to given number by his assumption that only first n-th prime numbers ar given, and only those we can use in our formula. Its sad that if we assume that we cant give formula for volume of ball (and other standard formulas), even if we have radius, but thats his definition.

It is not my definition, thank you very much. My definition has nothing to do with balls. I have reasons for thinking that the teacher had a restricted sense of formula in mind.

I didnt wrote that you definition has sth to do with balls, i said that using this definition we cant give formula for ball volume. Btw, who "invented" this definition of formula?

But really whatever, as i said, if formula is not defined then we cant give formula for anything. Case closed.

[Bill Spight]

I didnt wrote that you definition has sth to do with balls, i said that using this definition we cant give formula for ball volume.

Well, then, your complaint was irrelevant.

[AloneAgainstAll]

What complain? I didnt made any complain.

Irrevelant is discussion about what definition of formula assumed teacher.