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[SOLVED] Amending the Pre-Calculus sub-forum description

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MarkFL

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Feb 24, 2012
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I think some students may be understandably inclined to post analytic/coordinate geometry questions in our Geometry sub-forum, so I would suggest adding:

Analytic Geometry

to the Pre-Calculus sub-forum description for clarification.
 

Klaas van Aarsen

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Mar 5, 2012
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I think some students may be understandably inclined to post analytic/coordinate geometry questions in our Geometry sub-forum, so I would suggest adding:

Analytic Geometry

to the Pre-Calculus sub-forum description for clarification.
Do you mean calculations with vectors?
If so, how about Vector Calculus?
 
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MarkFL

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Do you mean calculations with vectors?
If so, how about Vector Calculus?
No, I actually mean geometric questions that involve using a coordinate system for the solution. A good example is locus problems, such as:

Find the locus of points equidistant from the line $y=k$ and the point $(a,b)$.

While it is technically a problem in plane geometry, it involves using coordinates and the distance formula, rather than Euclidean theorems/postulates/axioms.
 

Klaas van Aarsen

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Mar 5, 2012
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No, I actually mean geometric questions that involve using a coordinate system for the solution. A good example is locus problems, such as:

Find the locus of points equidistant from the line $y=k$ and the point $(a,b)$.

While it is technically a problem in plane geometry, it involves using coordinates and the distance formula, rather than Euclidean theorems/postulates/axioms.
I just checked and currently our smallest category is actually Geometry (58 threads), except for Other Topics (34).

Rather than introducing yet another category, I propose to broaden the current Geometry category.

Perhaps it can also be the home to analytic geometry, and while we're at it, perhaps also vector geometry.
 
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MarkFL

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I'm not suggesting that a new category be created, only that the description of the Pre-Calculus sub-forum be amended to include Analytic Geometry. :D

I would tend to want to put Vector Geometry problems (not involving the calculus) in the Pre-Calculus sub-forum as well, as we have done in the past, but this may be to a bias on my part towards U.S. mathematics education conventions.
 

Klaas van Aarsen

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I'm not suggesting that a new category be created, only that the description of the Pre-Calculus sub-forum be amended to include Analytic Geometry. :D

I would tend to want to put Vector Geometry problems (not involving the calculus) in the Pre-Calculus sub-forum as well, as we have done in the past, but this may be to a bias on my part towards U.S. mathematics education conventions.
Ah, my bad. I didn't read carefully enough.

Either way, when I see questions where lines and planes are represented by vectors, and one has to calculate the distance between points and planes and such, I don't really consider that Pre-Calculus. Actually, it really sounds closer to what you name Analytic Geometry.

But to be sure, in the Netherlands we did not have these distinct categories like Pre-Calculus, so I'm going with the description given: "Simple Limits, Functions".
Note that regardless this narrow description, it is still our 2nd biggest category (156 threads) after Pre-Algebra (242 threads).
 

Deveno

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Feb 15, 2012
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Pure Geometry is a subject that has fallen on hard times in the United States. The fact that the subject matter is Geometry is almost immaterial, a better name would be:

A first look at logical thinking.

I feel that logical thinking (rather ironically) is more of an art-form than a science...the idiots (and I would rather use a more actionable term) that oversee our educational system apparently feel that pumping out engineers and accountants is MUCH more useful than producing philosophers and artists (I mean, my God, who needs those anyway, really?). Euclid is going the way of Beethoven..."optional" figures relegated to (sniff) "culture".

If anything, we here at MHB should be fighting this trend, by encouraging posters to see Geometry as an integral part of all mathematics, rather than playing along with the linear "fast-track" educational system currently in place. Obviously, this is a biased opinion on my part :)
 

Jameson

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Jan 26, 2012
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Let me try to make sure I understand what is being discussed by summarizing the thread.

1) MarkFL is suggesting adding "Analytic Geometry" to the forum description under "Pre-Calculus".

2) I like Serena pointed out that our Geometry section is underperforming so it would be wise to try to get more posts in that category by broadening its description as well.

3) MarkFL explains (1) another way to clarify and adds the suggestions that "Vector Geometry" to the "Pre-Calculus" section.

4) I like Serena and Deveno point out some issues with this.
[HR][/HR]
I don't see a problem with adding "Analytic Geometry" to the "Geometry" forum description but maybe that could cause some confusion? This seems tricky when considering international educational systems that differ from the US so we want to be as universal as possible.
 

Klaas van Aarsen

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As I see it, there are 3 categories involved here that don't really have a proper place yet:
  1. questions about distance to a locus or a line using coordinates (analytic geometry),
  2. questions about solving geometric problems using vectors (vector geometry),
  3. questions about a basis of vectors, orthogonality, normality, and so on (pre-linear algebra)

Mark wants to put all of them into pre-calculus, since apparently in the States they are taught in that class.

Deveno points out that "pure" geometry (axioms, congruences, and such) is hardly practiced any more, although he does not seem to show a preference where the 3 categories should go.

I would prefer to put them all into geometry since as a category that seems a better fit to me, and also because geometry is pretty small right now.

It would be nice if some others would put in a comment.
 

Opalg

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Feb 7, 2012
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As I see it, there are 3 categories involved here that don't really have a proper place yet:
  1. questions about distance to a locus or a line using coordinates (analytic geometry),
  2. questions about solving geometric problems using vectors (vector geometry),
  3. questions about a basis of vectors, orthogonality, normality, and so on (pre-linear algebra)

Mark wants to put all of them into pre-calculus, since apparently in the States they are taught in that class.

Deveno points out that "pure" geometry (axioms, congruences, and such) is hardly practiced any more, although he does not seem to show a preference where the 3 categories should go.

I would prefer to put them all into geometry since as a category that seems a better fit to me, and also because geometry is pretty small right now.

It would be nice if some others would put in a comment.
I would be happy to see categories 1. and 2. put into the geometry subforum. There are plenty of problems in both analytic and vector geometry that can be solved either by analytic or synthetic methods, and it can often be instructive to see both methods of solution.

I have no opinion on category 3. It is somewhat more tenuously related to geometry than the other two categories, but it could comfortably go either there or in pre-calculus (whatever that may mean: it's not a category that is used in most places outside the US).
 

Deveno

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Feb 15, 2012
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In a an oblique way (your score increases by 3 points if you chuckled) I was suggesting an opinion along the lines of what Opalg has posted...it is my firm belief that symbolic manipulation should work together with visual intepretation in mathematics; as I am fond of saying:

One should not accept something as true unless you can prove it in at least two ways, and one of them ought to be a picture (do equations count as "pictures"? Hmm...I'll get back to you on that...).

As far as the red-headed stepchild "category 3" is concerned, I'm fine with questions in it going in Linear Algebra, Pre-calculus OR Geometry. I also think that the staff here is adroit enough to use their "thread moving powers" with discretion, and I believe this forum has the capability of leaving a thread originally posted in some sub-forum and moved to another with a pointer redirecting it to its current location.

The abstractionist in me feels some sympathy for MarkFL's position, but abstraction without realization is ultimately sterile...while mathematics is certainly beautiful in its own right, its applications are also beautiful, and "Earth-measuring" deserves some props...and by this I mean: we have a chance in the Geometry section to illustrate unity AND diversity in mathematics, which is perhaps a greater challenge to the staff...but you guys can do it, I have faith in ya.
 

MarkFL

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Feb 24, 2012
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First, I want to express my gratitude to everyone who has expressed their thoughts on this rather sticky topic. It serves well to highlight the plight of the student trying to decide which sub-forum is the best fit for a "geometry" problem. (Thinking)

From a distal viewpoint, I see two types of geometry, those done in flat spaces (Euclidean) and those done in curved spaces (non-Euclidean). This is perhaps the broadest of simplifications. It is of course the former which we are addressing here.

Now, when I think simply of "Geometry," I think of those things ingeniously captured within the 14 volumes of Euclid's The Elements, gathering dust in my library as I type. Five axioms, from which the many postulates and theorems are logically deduced, devoid of non-positive numbers and the concept of coordinates, but which trains the student to think in a logical and consistent manner. It still stands as a beautiful monument to the power of the human mind.

When we introduce a coordinate system and the concept of the function to deal with geometry, as fathered by René Descartes, I think of this as analytic or coordinate geometry. I see this as a significant enough mathematical advance for me to want to distinguish the two approaches to geometry.

However, in my mind, and as a general rule here as well, we make no such distinction with regards to trigonometry, which is really just a subset of geometry. So, to be consistent, we should therefore not treat geometry as a subject any differently. I believe this is what the consensus here is pointing towards as well. So I happily concede that geometry, whether coordinate based or not, fits well in the Geometry sub-forum, and let our Topology and Advanced Geometry sub-forum catch those topics dealing with curved spaces.

As far as vector geometry goes, oftentimes this is an application of trigonometry, at least in the simpler cases. In many problems, these can be approached in either a purely trigonometric fashion, or using a coordinate based approach. So, I agree that there is a lot left to interpretation when it comes to vector geometry. These can arise in courses on Trigonometry, Pre-Calculus, Calculus and even Physics, to name a few.

Most outside of the U.S. probably wonder what this Pre-Calculus thing is all about. When I was a student, I witnessed the birth of a course called Pre-Calculus at my school. When I first enrolled at the local community college, a student took College Algebra, Calculus I, Analytic Trigonometry, Calculus II, Calculus III, and then Differential Equations to finish. As I was finishing College Algebra, my professor approached me telling me that he was going to be teaching Modern Analytic Trigonometry during a summer term, and that the following fall term, Pre-Calculus would be taught and the course in trigonometry would no longer be offered. He said he could get the Calculus I prerequisite waived in my case. He said the treatment of trigonometry in the new Precalculus course was much more cursory, and he wanted to give me the opportunity to get the "full treatment" before it disappeared. So I wound up taking both courses, and was truly glad I got to take the course in trigonometry.

The Pre-Calculus course, in addition to exploring concepts of trigonometry, fortified much of what was taught in College Algebra, and went again although more deeply into the basics of polynomials, functions, matrices, arithmetic and geometric series, and then proofs by induction, which was not taught in College Algebra. We were not introduced formally to simple limits until the first Chapter of Calculus I, however, we did deal with asymptotes of which the horizontal variety is in fact at least a disguising of the notion of limits at infinity. I will admit though, that these simple limits, which I tend to think of as limits by substitution, including those with removable singularities, really is a Pre-Calculus topic as we were taught this before actually beginning differential calculus.

Thus, there is a great deal of overlap between Pre-Calculus and Algebra/Geometry/Trigonometry and we need to keep this in mind when a student may post inappropriately. Although it does state in the rules, specifically rule #5:

"The key to posting a question in the correct subforum is to consider the content of the question, not where the question has come from."

A new user most likely has not thoroughly read all of the rules, and so may be inclined to to go straightaway to our Pre-Calculus sub-forum to post a problem from his/her Pre-Calculus course, even though it should really go into one of our other Pre-University sub-forums. While we seasoned users of math forums see this as a logical policy, a user new to math forums in general may not see this at all.

Most (the vast majority, if not all) questions that arise in a Precalculus course would in fact not go into the Pre-Calculus sub-forum. I must admit, at the moment, I would be hard pressed to say what kinds of questions definitely do belong in the Pre-Calculus sub-forum and nowhere else. One of the best candidates for this sub-forum in my opinion is the type of problems where the student is asked to express one quantity as the function of another. To me this is definitely a skill that need to be honed before studying the calculus. This was also taught in the first chapter of my Calculus I course along with simple limits, and so I see why the description of our Pre-Calculus sub-forum contains these two topics. I now see the light! (Sun) So I do think our forum description for the Pre-Calculus sub-forum is fine just the way it is now.

So, in the end, I agree with Deveno when he lays the ultimate responsibility on the staff to decide on a case by case basis, which sub-forum is best for a particular topic, and to recognize the difference between a topic carelessly posted in the wrong sub-forum, such as a differential calculus question posted in the Pre-Calculus sub-forum, and a topic on logarithms posted there, which most would agree technically should be in the Pre-Algebra and Algebra sub-forum.
 

Ackbach

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Jan 26, 2012
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So we can mark as solved, then?
 

MarkFL

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Feb 24, 2012
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Klaas van Aarsen

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So we can mark as solved, then?
I would like to see the description of Geometry expanded to include coordinates and vectors, so that members can tell where to put such questions.
 

Ackbach

Indicium Physicus
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Jan 26, 2012
4,191
I would like to see the description of Geometry expanded to include coordinates and vectors, so that members can tell where to put such questions.
Added "Analytic or Coordinate Geometry; Basic Vector Geometry" to the Geometry description. It seems to me that more advanced vector operations, such as the cross product, belong in a more advanced forum than geometry. Pre-Calc would be fine, or even Linear Algebra. Thoughts?
 

Klaas van Aarsen

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Mar 5, 2012
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Added "Analytic or Coordinate Geometry; Basic Vector Geometry" to the Geometry description. It seems to me that more advanced vector operations, such as the cross product, belong in a more advanced forum than geometry. Pre-Calc would be fine, or even Linear Algebra. Thoughts?
Nice!

I'd say that the basic application of the cross product to find a normal vector can also go into Geometry.
More advanced theorems and outer product generalizations should go into Linear Algebra.
I think that curl and Stokes fit in Analysis.
Advanced applications, such as in 3D mechanics, should fit in Advanced Applied Mathematics.
 

Ackbach

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Jan 26, 2012
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Nice!

I'd say that the basic application of the cross product to find a normal vector can also go into Geometry.
Or they could go into Other Topics, as I think many geometry courses do not cover the cross product, but any physics course worth its salt will.

More advanced theorems and outer product generalizations should go into Linear Algebra.
I think that curl and Stokes fit in Analysis.
Both of those are also taught in Multi-variable Calculus, so it could go into Calculus as well. If it's more theoretical, then Analysis would be better.

Advanced applications, such as in 3D mechanics, should fit in Advanced Applied Mathematics.
Sounds good. I'm not sure any of these thoughts would need to go into the forum descriptions, though. I think these nice judgements would be better done simply by moderator.
 

Klaas van Aarsen

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Sounds good. I'm not sure any of these thoughts would need to go into the forum descriptions, though. I think these nice judgements would be better done simply by moderator.
Agreed.
 

Deveno

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Feb 15, 2012
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I'd like to add here, that I think this thread is a beautiful example of what discussion can DO:

A problem is posed, and the discussion not only illuminates the original problem, but also helps all concerned gain some clarity into PURPOSE.

I only wish the leaders of this world could all be so clear-minded. Good work, people!