What is the point of teaching calculus or pre-calculus?

[LegionOnomaMoi]

Have any members been required to take a calculus I course in college to complete there degree? Have any been required to take more than this (and if so, what courses- Calc I, Calc II, Calc. III, differential equations, or perhaps a course with "analysis" in the title)? Do any remember having to learn about matrices in pre-calculus? If so, what from required mathematical courses have such members ever used (including those who are in fields in which analysis (in the traditional real/complex case, not data analysis), calculus, and/or systems of equations are essential)? Thanks.

 

[Buttercup]

What's your major?

My son is majoring in Mechanical Engineering. He is taking tons of math, never gets a break from it, in fact.

My daughter graduated with a BA in Theater and didn't have to take math at all. Just depends on your major.

 

[LegionOnomaMoi]

What's your major?

I'm a research consultant and graduate student whose field is the physics and mathematics (modelling and statistical) of neurobiology (including computational neuroscience). I'm working on a project concerning the teaching of mathematics in college and have just finished comparing some ~100 calculus college textbooks and am now working on how and what former or past students have learned from studying what they were required to or desired to study in terms of mathematics including and beyond calculus I.

My son is majoring in Mechanical Engineering. He is taking tons of math, never gets a break from it, in fact.

I have a younger cousin who has just started a doctoral program in biological and chemical engineering. She's one of the reasons for my study, as after three semesters of calculus and a course in differential equations she couldn't even describe how she might answer non-trivial calculus I questions (which reflects the disparity between what students are taught that "calculus" and "applied math" are vs. what they really are and how disparate topics relate or do not relate to one another).

One purpose of this is to determine whether or not those like your son are paying for courses which teach them essentially nothing, as whatever is taught will end up being re-taught in a form that works or re-taught in the form already learned but expanded on.


I'm looking for counter-examples to my hypothesis.

 

[Buttercup]

One purpose of this is to determine whether or not those like your son are paying for courses which teach them essentially nothing, as whatever is taught will end up being re-taught in a form that works or re-taught in the form already learned but expanded on.


I'm looking for counter-examples to my hypothesis.

I'm not sure if this is what you're looking for, but here's the math he's required to take for a BS in ME from Oregon State University and he'll finish up with these courses in June 2015.

MTH 111 College Algebra
.....................................................
5
MTH 112 Trigonometry
.........................................................
4
MTH 251 Calculus 1
...............................................................
5
MTH 252 Calculus 2
...............................................................
5
MTH 253 Calculus 3
...............................................................
5
MTH 254 Vector Calculus 1
...................................................
4
MTH 255 Vector Calculus 2 (Electrical Engr. Only)
..............
4
MTH 256 Differential Equations
............................................
4
MTH 260 or 261 Linear Algebra
............................................
2-4
MTH 265 Statistics for Scientists and Engineers
................
4
PH 211 General Physics w/Calculus
.....................................
5
PH 212 General Physics w/Calculus
.....................................
5
PH 213 General Physics w/Calculus
.....

 

[LegionOnomaMoi]

I'm not sure if this is what you're looking for, but here's the math he's required to take for a BS in ME from Oregon State University and he'll finish up with these courses in June 2015.

MTH 111 College Algebra
.....................................................
5
MTH 112 Trigonometry
.........................................................
4
MTH 251 Calculus 1
...............................................................
5
MTH 252 Calculus 2
...............................................................
5
MTH 253 Calculus 3
...............................................................
5
MTH 254 Vector Calculus 1
...................................................
4
MTH 255 Vector Calculus 2 (Electrical Engr. Only)
..............
4
MTH 256 Differential Equations
............................................
4
MTH 260 or 261 Linear Algebra
............................................
2-4
MTH 265 Statistics for Scientists and Engineers
................
4
PH 211 General Physics w/Calculus
.....................................
5
PH 212 General Physics w/Calculus
.....................................
5
PH 213 General Physics w/Calculus
.....

Thank you! That's very helpful.

 

[LuisDantas]

Have any members been required to take a calculus I course in college to complete there degree? Have any been required to take more than this (and if so, what courses- Calc I, Calc II, Calc. III, differential equations, or perhaps a course with "analysis" in the title)? Do any remember having to learn about matrices in pre-calculus? If so, what from required mathematical courses have such members ever used (including those who are in fields in which analysis (in the traditional real/complex case, not data analysis), calculus, and/or systems of equations are essential)? Thanks.

I had some matrices teaching in high school. I wish I had been taught more about the true depth of mathematics at the time. Failure to give me such forewarning meant that I had to face it under rather fragile circunstances at college, barring me forever from an actual education in computer science.

As for whether it is essential, yes, it most certainly is essential to learn at the very least calculus if one hopes to have any proficiency whatsoever in exact sciences. It is absolutely indispensable for any engineering, for instance, and in fact it is only the first step in a long list of higher mathematics disciplines.

Unfortunately, I failed to realize that anywhere near soon enough.

 

[LegionOnomaMoi]

I had some matrices teaching in high school. I wish I had been taught more about the true depth of mathematics at the time.

This is a central issue for me. So far as I can tell, there does not exist any linear algebra course (the essential course for understanding systems of equations and matrices) which doesn't already assume you have no idea what matrices and related concepts really are, thus rendering any such notions at best irrelevant.

As for whether it is essential, yes, it most certainly is essential to learn at the very least calculus if one hopes to have any proficiency whatsoever in exact sciences.

Certainly, "calculus" is essential for understanding everything from the central limit theorem to nonlinear dynamics (which it was really developed for). However, a critical question is whether calculus as it is taught actually teaches the essential notions that make calculus essential (by "as it is taught" I mean up to & including those who take at least a college level calculus course and including those who take 3 calculus courses as well as a course on differential equations). Similarly for most of pre-Calc.

What of what you were required to or chose to take do you use (and how?) and how much of both would you say you don't use or weren't given enough of a basis of in order to use (e.g., one can be taught matrix multiplication without ever understanding how this could be used let alone be critical for vast numbers of application that outnumber calculus)?

It is absolutely indispensable for any engineering, for instance, and in fact it is only the first step in a long list of higher mathematics disciplines.

Very true (IMO). One of the things I am looking into, however, is the ways in which the almost universal methods for teaching calculus don't actually teach "calculus" (but rather a bunch of rules for extending pre-calculus). This isn't always true and is never fully true, but one has merely to look at a "real" calculus textbook (i.e., some textbook on {elementary} real analysis) to see how far standard calculus textbooks do not cover the concepts essential to understanding calculus.

Unfortunately, I failed to realize that anywhere near soon enough.

That's another aspect I'm investigating. Can you remember a time and/or a reason you decided not to continue to study mathematics (if there were ever such a time in which you made such a decision)? Thanks.

 

[LuisDantas]

Can you remember a time and/or a reason you decided not to continue to study mathematics (if there were ever such a time in which you made such a decision)? Thanks.

Sure. I entered college under particularly fragile circunstances (way too young and with a very disfunctional "family", for two) and had less than no preparation or warning for the actual demands. I was 16 years old at the time. The first class of calculus overwhelmed me at various levels all at the same time. I simply had no idea of where to run for succor. "Family" was if anything a serious aggravant, leading to the start of a series of cycles of pushing myself into classes with the emotional state of cattle going to the abattoir until I simply could not take it anymore and dropped out.

The cycle repeated itself for far too long and led nowhere. It is a pity, among other reasons because I liked high school math. It was in fact my favorite discipline. But there was some serious neglect to give me any hint of the kind of environment one needs for college, and mathematics was where the realization was most blunt.

 

[LegionOnomaMoi]

Sure. I entered college under particularly fragile circunstances (way too young and with a very disfunctional "family", for two) and had less than no preparation or warning for the actual demands. I was 16 years old at the time. The first class of calculus overwhelmed me at various levels all at the same time. I simply had no idea of where to run for succor. "Family" was if anything a serious aggravant, leading to the start of a series of cycles of pushing myself into classes with the emotional state of cattle going to the abattoir until I simply could not take it anymore and dropped out.

The cycle repeated itself for far too long and led nowhere. It is a pity, among other reasons because I liked high school math. It was in fact my favorite discipline. But there was some serious neglect to give me any hint of the kind of environment one needs for college, and mathematics was where the realization was most blunt.

Thank you very much! Such information isn't just key in and of itself but also in terms of how I will structure my actual empirical study so as to make it sound. I very much appreciate your help.

 

[LegionOnomaMoi]

NOTE:

It just occurred to me that confidentiality and similar issues could play a role here. So to clarify: I will not use any responses here in any study, publication., or any other communication (without permission, but I don't intend to require it as this is used to structure my research). If I ever wish to, I will ask for specific and explicit permission to use any and every response from any an all members.

 

[Smart_Guy]

When I was in the chemistry department, I took one of those courses and flunked it (oops). When I moved to the English department, non was there. I guess it is because 1+1 is never 2 in English

I believe in practical majors, unlike theoretical, it is needed. Um, is gardening a practical major? Is it even a major?

But in school, (or pre-college if I get the word right) I believe calculus is so very important. It kinda opens the mind and makes it more active.

Note: I'm still good at basic math

 

[Revoltingest]

Some meandering thoughts.
Math was always fun.
Discovering negative numbers in elementary school was an eye opener...
...things were more complicated than we were being taught.
Beginning algebra was fascinating because it was powerful.
Geometry was fun. Proofs were a game.
Trig was useful.
Analysis was less interesting.
Calculus was the reward for suffering thru analysis.
It was easier, more fun, & powerful....dividing by zero!
Diffy Qs ramped up the power to analyze systems.
I majored in architecture, math, & finally settled in mechanical engineering.
Had me some: statistics, probability, linear algebra, number theory
After this, there were more useful math tools things to learn, but most of'm were acquired outside of math courses: Lagrange stuff in kinematics, tensors in rheology, convolution in probabilistic systems analysis, numerical methods in heat transfer
Working IRL, I used calculus on the job several times for structural analysis problems which weren't addressed in handbooks.
I've found that the basics of calculus are pretty easy to teach to people who disliked math.
The tricks:
- Make it seem relevant to their lives.
- Make it as easy as it really is. (Schools so often make things harder than they should be.)
Calculus, trig, algebra, &

General observation:
Math should be better understood in general.
Lack of facility (especially algebra & trig) hurts many on the job.
Schools need to make it easier & more relevant to avoid turning off students.

 

[suncowiam]

Have any members been required to take a calculus I course in college to complete there degree? Have any been required to take more than this (and if so, what courses- Calc I, Calc II, Calc. III, differential equations, or perhaps a course with "analysis" in the title)? Do any remember having to learn about matrices in pre-calculus? If so, what from required mathematical courses have such members ever used (including those who are in fields in which analysis (in the traditional real/complex case, not data analysis), calculus, and/or systems of equations are essential)? Thanks.

I took all those courses you listed for my Computer Engineering degree which is a mix of EE and CS. This was way back around 1997. I now specialize in SW engineering and frankly don't use much of that math. The closest I came was developing a control algorithm for power, temperature, clocks, and time for our products which touched on some simpler notions of calculus I.

 

[Wirey]

Have any members been required to take a calculus I course in college to complete there degree? Have any been required to take more than this (and if so, what courses- Calc I, Calc II, Calc. III, differential equations, or perhaps a course with "analysis" in the title)? Do any remember having to learn about matrices in pre-calculus? If so, what from required mathematical courses have such members ever used (including those who are in fields in which analysis (in the traditional real/complex case, not data analysis), calculus, and/or systems of equations are essential)? Thanks.

I'm in electrical, and I use the crap I learned in calculus all the time.

 

[freethinker44]

This is a central issue for me. So far as I can tell, there does not exist any linear algebra course (the essential course for understanding systems of equations and matrices) which doesn't already assume you have no idea what matrices and related concepts really are, thus rendering any such notions at best irrelevant.

I've taken math up through calculus 2 and I'm in a linear algebra course now. They definitely expected at least a basic knowledge of matrices in linear algebra, which I had none from pre-algebra through calculus 2, like it didn't even come up as a side topic as something we might want to research a little on the side.

So the first month has been a little tough as the "refresher" period of the course was all new material for me. But I'm a good self learner so I wouldn't say it has been irrelevant, just wish I was better prepared before I took the course. The pre-requisite is calculus 1 so I assumed I would be fine with having calculus 2. Nope.

 

[Wirey]

You're just jealous because we math types get the girls with the Hoiven and the glaiven and the boobs and the hair so soft it smells like happiness.... sorry, where was I? Oh, yes, pancakes please.

 

[LegionOnomaMoi]

When I was in the chemistry department, I took one of those courses and flunked it (oops).

(semi-)Interestng aside: One of the things I do (and have for a while) is review sources, tools, & material to aid those either studying on their own or having trouble with some course(s) (mostly because I tutor & teach a lot so I am always looking for good resources to help those whom I am not able to help directly). For undergrads taking the standards three calculus semesters for some science major, I have recommended The Teaching Company's "The Great Courses", which includes three different calculus "lecture series" corresponding (more or less) to Calc I, II, & III. The lecturer for all three is a distinguished mathematician and noted math professor who started out in chemistry and was flunking but had taken calculus so he switched his major to mathematics. For him, I guess the chemistry part of chemistry proved much more difficult than the calculus courses he was required to take.

I believe in practical majors, unlike theoretical, it is needed. Um, is gardening a practical major? Is it even a major?

Could you say why? And could you specify what you mean by "theoretical"? For example, one can obtain a doctorate in "theoretical physics", but this requires an enormous amount of mathematical knowledge. Thanks.

But in school, (or pre-college if I get the word right) I believe calculus is so very important. It kinda opens the mind and makes it more active.

Just to make sure I understand you correctly: calculus is important for the ways in which it can challenge students mentally and stimulate (at least some form of) "higher level" analytical thinking? If so, how would you compare it to other formal or quantitative reasoning courses(such as logic or statistics) regarding its ability to "open the mind"?

Finally, would you say that taking a course in calculus also has practical applications and if so what? Thanks again.

 

[LegionOnomaMoi]

Analysis was less interesting.
Calculus was the reward for suffering thru analysis.

I wrote an entire response and then accidently hit "delete" when I wasn't actually "in" the textbox so it functioned as a "back" button and lost it all. I'll rewrite some but in the meantime I am especially interested in two things:

1) You're the first person I've encountered whose taken an analysis course prior to calculus. Usually, if a university offers a course in elementary analysis, real analysis, and/or complex analysis they require at east calc. I for the first two and more for the second. Could you explain/describe the kind of material covered and problems/questions answered in the "analysis" vs. "calculus" you took?

2) I absolutely agree that there are serious deficiencies in general math knowledge. However, I think that the way most high schools teach math (providing rules to be applied rather than context, orienting so much under the assumption that the student will take a calculus course that includes a great deal of "fluff" and not much in the way of "calculus", etc.) discourages many students unnecessarily. If you have time, I'd be interested to your response to pages v & vi of The Calculus Integral and in particular what I've quoted below within the context of those pages:

"For these reasons we have called it the calculus integral. But none of us teach the calculus integral. Instead we teach the Riemann integral. Then, when the necessity of integrating unbounded functions arise, we teach the improper Riemann integral. When the student is more advanced we sheepishly let them know that the integration theory that they have learned is just a moldy 19th century concept that was replaced in all serious studies a full century ago.

We do not apologize for the fact that we have misled them; indeed we likely will not even mention the fact that the improper Riemann integral and the Lebesgue integral are quite distinct; most students accept the mantra that the Lebesgue integral is better and they take it for granted that it includes what they learned. We also do not point out just how awkward and misleading the Riemann theory is: we just drop the subject entirely."

Thank you!

 

[LegionOnomaMoi]

I've taken math up through calculus 2 and I'm in a linear algebra course now.

Would you mind sharing the name of the assigned textbook? Thanks.

They definitely expected at least a basic knowledge of matrices in linear algebra, which I had none from pre-algebra through calculus 2, like it didn't even come up as a side topic as something we might want to research a little on the side.

That is very interesting! I've never even come across a linear algebra textbook that didn't introduce matrices without expecting any prior knowledge.

The pre-requisite is calculus 1 so I assumed I would be fine with having calculus 2. Nope.

You mean for the linear algebra course? That's fairly standard, despite the fact that few courses or textbooks in linear algebra ever use calculus. The idea is that, given the conceptual difficulties of linear algebra, they want students to have demonstrated a certain level of skill at higher level mathematics/abstract reasoning. However, at the same time they've changed what calculus is by taking out most of the fundamental concepts and replacing them with rules that can be used and they've minimized the number of proofs students are required to perform as well as those provided.

 

[LegionOnomaMoi]

I'm in electrical, and I use the crap I learned in calculus all the time.

I have 5 questions which require only Calc. I (or the high school equivalent) knowledge for you and others in the sciences or mathematics who use calculus (whether in e.g., data analysis/statistics, modelling, signal processing, etc., and regardless of whether your use of calculus is almost entirely restricted to differential equations or far more broad). If you can't or don't want to bother answering the questions (or some of them, even one), just a description of how you might go about getting the answers would still be great. Thanks.

1) Find values of a and b so that the line 2x + 3y = a is tangent to the graph of

at the point where x = 3.

2) Does

have any roots between -1 and 0? Why or why not?

3) Let

. Find a & b so that f is differentiable at 1.

4) Find at least one solution to the equation

or explain why no such solution exists.

5) Is there an a so that

exists? Explain your answer.

If you don't wish to either attempt an answer or explain how you might answer any of the above (and using MATLAB, Mathematica, SAS, R, or similar mathematical/statistical software packages doesn't count), but you do have some comments on one or more (or any thoughts regarding them) that would be great too.