Global Math Education: Exploring Calculus & Precalculus

[matt grime]

I realize that in all likelihood the predominate user base of this site is in the US, but (just as we now (eventually) have a disclaimer pointing out courses are differnet at differnet univeristies) how about some nod to those parts of the world that do not use the distinction of calculus and precalc.

This is a particulalry confusing distinction to those of us who would, upon seeing the two choices offered, have no idea where to post a question on algebra. OK, I'd know where it went but then not everyone has been to both US and UK universities.

I know it is bad to merely criticize and not offer a solution, however I can not think of a good addition to the description of each subforum. So consider this also a call for suggestions (and probable flaming and accusations of rampant anti-americanism, which would be most unfounded).

In the UK precalc roughly corresponds to A-levels (this is a qualificiation taken between the ages of 16-18 after high school and before university) and calc to University level courses. But that doesn't help anyone not in the UK does it?

 

[Fermat]

As someone (else) from across the pond, thanks for the clarification/explanation of the difference between calc and pre-calc.

I had no idea what the diff was !

 

[Galileo]

Thanks for the nod.
I never understood what precalc contained. And what are the math courses 'before' and 'after' calculus? Is there really a general consensus about that in the US?

Now that I`m at it. Here there is no distinction between algebra and abstract algebra. There is just... algebra. Abstract algebra really sounds like a first order pleonasm. So what's the difference?
Also, what you call calculus we call analysis, so what's the difference there?

 

[motai]

Thanks for the nod.
I never understood what precalc contained. And what are the math courses 'before' and 'after' calculus? Is there really a general consensus about that in the US?

Precalculus is generally the algebraic (polynomials [factoring, division, synthetic division, rational expressions], nth roots and radical exponents, linear equations, quadratics, complex, inequalities [including rational functions], everything that has to do with graphing, logarithms).

and

Trigonometry (angles, radians, trig functions, sine/cosine functions, graphs of trigonometric functions, inverse trig functions, double/triple/half/product-to-sum/sum-to-product formulas, law of sines/cosines).

In short, precalculus is mainly just everything short of limits and derivatives.

 

[quantumdude]

This is a particulalry confusing distinction to those of us who would, upon seeing the two choices offered, have no idea where to post a question on algebra.

Is it really not evident from the subtitle? Surely any student of any nation knows whether or not he's taken calculus yet.

In the UK precalc roughly corresponds to A-levels (this is a qualificiation taken between the ages of 16-18 after high school and before university) and calc to University level courses. But that doesn't help anyone not in the UK does it?

Probably not, and "a-levels" makes much less sense than "precalculus". Even if you've never heard of a course called precalculus, a basic understanding of the English prefix "pre-X" should indicate, "everything that comes before x", no?

I never understood what precalc contained. And what are the math courses 'before' and 'after' calculus? Is there really a general consensus about that in the US?

There is a general consensus on it. There is standard background material in algebra, geometry (both classical and analytical), and trigonometry that must be learned before one is allowed to take calculus.

Now that I`m at it. Here there is no distinction between algebra and abstract algebra. There is just... algebra. Abstract algebra really sounds like a first order pleonasm. So what's the difference?

Abstract algebra refers to groups, rings, fields, etc. In the US we don't teach that to students at the precalculus level (of course they learn about fields, but they don't know that they are learning it). Typically what is known as "abstract algebra" is "that algebra that is studied after linear algebra" (vector spaces and whatnot).

Also, what you call calculus we call analysis, so what's the difference there?

In short: In calculus courses students are typically taught how to use theorems, whereas in analysis courses students are taught how to prove them. But since we group calculus and analysis together at PF, I don't see how this could cause any confusion.

Guys, if you have suggestions for the titles and subtitles to make PF more user-friendly to international students then I'm all ears. But if all you've got is, "that sounds funny to me" then I'm going to lose interest in this thread pretty quickly.

So I guess I should open with the following questions.

1.) Since calculus is the "dividing line" between the two math help sections, what does "calculus" mean in your country?

2.) Are there more or less standard prerequisites for calculus?

3.) If so, what are they?

 

[Moonbear]

We thought we were resolving the problems with non-US visitors by removing the grade levels that applied only to the US educational system.

Do you not call the subject calculus there? Pre-calculus would mean anything you need to take before you know enough to take calculus. (Some schools in the US have an actual class called pre-calculus, but I never had that and have no idea what it would include.)

It's a little tough coming up with a descriptor for math courses that makes an easy distinction because the same names are used for the introductory level classes as for more advanced classes (i.e., Algebra could mean the stuff you learn in secondary school [but that some students don't learn until they get to a university], or it could mean an advanced math major's course that would, at least in the US system, be taken after calculus and differential equations).

If anyone can offer a name or short description that would be more universally understandable to convey the distinction in content between the two forums, please suggest it.

 

[Fermat]

...
If anyone can offer a name or short description that would be more universally understandable to convey the distinction in content between the two forums, please suggest it.

Why not just school and college (or something similar) ?
We all (roughly) do the same subjects at the same age. So although comparison of differnt descriptors for different course might be confusing, we can all be familiar with what subjects we studied at school age and college age.

 

[Fermat]

About calculs. I always think of that as referring to either Integral or Differential calculus. (Integration or Differentiation)

 

[quantumdude]

Why not just school and college (or something similar) ?

We just switched from that!

If you recall, our old HW sections were "K-12" (meaning kindergarten through 12th grade, the final year of high school) and "College Level" (meaning the 4 years after high school).

This did not work for 2 reasons:

1.) In the US, "College"="University". You go to college at age 17 or 18 and you leave at age 21 or 22 (depending on when your birthday is). It is my understanding that in the UK, something completely different is meant by "college".

2.) There are college students who take very low level math, and there are high school students who study math at a higher level that many college students. Some high school students take calculus, while some college students take pre-algebra. We found empirically that it makes more sense to gather questions by subject, rather than by grade level. Under the old system (which you are proposing we go back to), we had calculus homework questions appearing in 2 different forums.

 

[Fermat]

We just switched from that!

That was the one I understood. Because I was familiar, to some extent, of the level of work covered in those age ranges.
This was my suggestion. That an indication of age be used in the description of the relevant forum(s).

1.) In the US, "College"="University". You go to college at age 17 or 18 and you leave at age 21 or 22 (depending on when your birthday is).

If age had been included in the forum header. e.g. something like, "K-12 -- 12 to 17 yrs" and "College -- 17 to 21 yrs", this might have avoided some confusion over what went where.

It is my understanding that in the UK, something completely different is meant by "college".

In the UK, College, or High School, or Grammer School would mean the last several years of K-12, i.e. 12 to 17/18 yrs (roughly).

We found empirically that it makes more sense to gather questions by subject, rather than by grade level. Under the old system (which you are proposing we go back to), we had calculus homework questions appearing in 2 different forums.

I wasn't really proposing to go back to the old system, but rather explaining why that worked for me. Because I had an idea of what level of work would be covered at at each age range.
I was completely misled by the use of Calculus and Pre-Calc forums. In the UK, calculus (integration and differentiation) would have been well covered in high School (K-12), so pre-calc seemed like something more basic. And I searched all over the place for the new forum that would deal with college-level (US) questions.
Calculus and Pre-calc may clarify where to go, for US students, but would mislead UK students.
Final suggestion then: "Pre-calculus -- 12 to 17 yrs" and "Calculus -- 17 to 21 yrs".

 

[Moonbear]

In the UK, calculus (integration and differentiation) would have been well covered in high School (K-12), so pre-calc seemed like something more basic.

That was why grouping by age or school years didn't work, because it's not consistent. Many students in the US also learn calculus while still in high school, but then there are others who never learn it until university.

And I searched all over the place for the new forum that would deal with college-level (US) questions.

That's the "Calculus and Beyond" forum.

Calculus and Pre-calc may clarify where to go, for US students, but would mislead UK students.
Final suggestion then: "Pre-calculus -- 12 to 17 yrs" and "Calculus -- 17 to 21 yrs".

But we didn't call it pre-calculus and calculus, the two forums are "Pre-Calculus" with subtitles indicating everything leading up to calculus, and "Calculus and Beyond." That means all higher level math, inclusive of calculus.

 

[Integral]

That was the one I understood. Because I was familiar, to some extent, of the level of work covered in those age ranges.
This was my suggestion. That an indication of age be used in the description of the relevant forum(s).
If age had been included in the forum header. e.g. something like, "K-12 -- 12 to 17 yrs" and "College -- 17 to 21 yrs", this might have avoided some confusion over what went where.
In the UK, College, or High School, or Grammer School would mean the last several years of K-12, i.e. 12 to 17/18 yrs (roughly).

But it is not covered till university by most US students. So your interpretations would be incorrect for the old system.

I wasn't really proposing to go back to the old system, but rather explaining why that worked for me. Because I had an idea of what level of work would be covered at at each age range.
I was completely misled by the use of Calculus and Pre-Calc forums. In the UK, calculus (integration and differentiation) would have been well covered in high School (K-12), so pre-calc seemed like something more basic.

you have absolutely correct, It is not clear what you are confused about??

And I searched all over the place for the new forum that would deal with college-level (US) questions.

What difference does age or location make? If you have completed calculus then your questions would go in the "Beyond" category.

Calculus and Pre-calc may clarify where to go, for US students, but would mislead UK students.
Final suggestion then: "Pre-calculus -- 12 to 17 yrs" and "Calculus -- 17 to 21 yrs".

According to what you just wrote you understood the classifications but for some reason seem to want to make it difficult. It is not. If you have not taken calculus you are pre-calc. If you are taking calculus (yes either differential or integral) then post in the calculus area, if you have completed calculus ie Differential equations, abstract algebra, linear algebra, complex analysis then the "beyond" forum is the place. I do not think that an age break down is what we want or need. In the US you have some High school students taking calculus and some University students taking Basic algebra and trig. We want the different levels of math separated, age or where the student is studying does not matter.

 

[vanesch]

I was completely misled by the use of Calculus and Pre-Calc forums. In the UK, calculus (integration and differentiation) would have been well covered in high School (K-12), so pre-calc seemed like something more basic. And I searched all over the place for the new forum that would deal with college-level (US) questions.
Calculus and Pre-calc may clarify where to go, for US students, but would mislead UK students.

I think that high school courses can follow very different patterns. For instance, my math teacher taught us abstract topology when we were 16-17 years old before introducing limits in the real number system! For instance, I was very proud that I had found all topologies on a set of 4 elements (there are 355 of them if I remember well). He used this abstract topology to have as a special application the topology introduced by a metric, and as a special application of this, the topology on R. Of course, once this machinery gets set up (which took a big part of the year), the application to limits and derivatives in R are quickly treated. The guy really had a strange sense of abstraction, going from the highly abstract to the more practical instead of the other way around. In the last year (17-18) he started by projective geometry and ended with 2-d and 3-d cartesian geometry with linear algebra.
So "pre-calculus" would include abstract topology and 'calculus and beyond" would include 2-d cartesian geometry :-)
But I think this was a very peculiar pathway that the guy chose.

 

[quantumdude]

For instance, my math teacher taught us abstract topology when we were 16-17 years old before introducing limits in the real number system!

Sweet Jesus

 

[Doc Al]

For instance, my math teacher taught us abstract topology when we were 16-17 years old before introducing limits in the real number system!

No wonder I'm always several steps behind everyone else! I remember taking a class, just for fun, in differential topology in grad school--my head almost exploded. (Unfortunately, all I can remember about the class was the fact that I took it.)

 

[Lisa!]

I think that high school courses can follow very different patterns. For instance, my math teacher taught us abstract topology when we were 16-17 years old before introducing limits in the real number system! For instance, I was very proud that I had found all topologies on a set of 4 elements (there are 355 of them if I remember well). He used this abstract topology to have as a special application the topology introduced by a metric, and as a special application of this, the topology on R. Of course, once this machinery gets set up (which took a big part of the year), the application to limits and derivatives in R are quickly treated. The guy really had a strange sense of abstraction, going from the highly abstract to the more practical instead of the other way around. In the last year (17-18) he started by projective geometry and ended with 2-d and 3-d cartesian geometry with linear algebra.
So "pre-calculus" would include abstract topology and 'calculus and beyond" would include 2-d cartesian geometry :-)
But I think this was a very peculiar pathway that the guy chose.

Could you please talk about your other teachersand professors esp. your history teacher?(not here of course) I was always surprised how one could be so knowledgeable. Although I think it's not because of your teachers.

 

[vanesch]

Could you please talk about your other teachersand professors

My biology teacher only showed us stuff about ants... He said: ants are the future!

I was always surprised how one could be so knowledgeable.

Age !

 

[Lisa!]

My biology teacher only showed us stuff about ants... He said: ants are the future!

I though it was your history teacher unless your biology teacher taught you very developed evoluton theory which said humans are ants' ancestors!

Age !

You mean 65 milion years? But you know I think even I could live as long as that, it wouldn't help at all!

 

[Diane_]

If anyone cares: the class currently called "precalculus" used to be called "trigonometry". I was in college when the name got changed, so i don't really know why it was changed, but I have my suspicions: 1) It covers a good deal more than trigonometry, and someone thought the other subjects were getting shortchanged. 2) "Precalculus" looks cooler. 3) As frequently happens with education in the United States, someone who didn't know what they were talking about got ahold of it and made a few random changes. My pet phrase for that is "randomly-firing neurons", which pretty much seems to cover it.

Choose any or all from the list above. This will be 50% of your grade.