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

[Sapiens]

Have any members been required to take a calculus I course in college to complete there degree?

Not required, I took it out of interest.

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)?

Ditto.

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.

Yes I remember. Seems to me that matrices were the basis of Linear Algebra and were used in a lot of other classes.

 

[freethinker44]

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

Introduction to Linear Algebra - Gilbert Strang

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

No, it introduces matrices. The entire first chapter is an introduction. It was the professor who expected a familiarity with matrices so he didn't really expand on much in the first chapter and we jumped right in with solving matrices on the first day. It took me a couple weeks before I realized that a matrix was actually a system of equations.

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.

My calc II teacher was pretty good with proofs. Maybe because he was older I guess, like late 60s - early 70s, if they changed the way calculus is taught. For every new concept introduced we would basically go over the proof, do various examples and once he thought we understood the proof well enough then he would give us the "easy way" to do it with rules or some algorithm.

 

[LegionOnomaMoi]

Ditto.

You mean that you took these out of interest, correct? Could you indicate which of them you took? If all or if analysis were one of these, could you elaborate on the nature of your analysis course(s)? Thank you.

 

[LegionOnomaMoi]

It took me a couple weeks before I realized that a matrix was actually a system of equations.

Um...you might not want to get too wrapped up in that definition. It may make some of the most important topics covered in Strang's text and in linear algebra courses in general more difficult to grasp. Or it might help you make the transition (what do I know?). However, a matrix is really just a an array of entries. In graph theory we find e.g., adjacency matrices, path matrices (actually these are a kind of incidence matrix), and more typical uses of matrices for linear transformations. In quantum mechanics you find Pauli spin matrices and far more importantly Hermitian matrices as well as other matrix operators.

Introduction to Linear Algebra - Gilbert Strang

I love Strang (as a teacher, anyway; I've never used his textbooks to learn anything)! Mostly because I love mathematics but am terrible (relatively, anyway) when it comes to mental arithmetic and am always making trivial errors such as forgetting a " - " sign or leaving out an exponent, and he does the same thing when teaching.

I have the 3rd an 4th editions to his text and somewhere I have an edition of his Linear Algebra and its Applications (Yes, I waste way too much money on books in general and in particular those that I am using for reasons other than the reasons for which they were written; unfortunately for me, one can eat up grant money like nothing else by purchasing textbooks for a study and telling a committee or PI that you can't get them from the library because you are obsessive about owning the book doesn't get much sympathy).

One difference between the 3rd & 4th editions is the addition of a section introducing matrices in chapter 1 (Sect. 1.3). In the 3rd edition matrices are introduced in chap. 2 in the context of systems of linear equations (which is the standard way that they are introduced in pre-calculus texts that cover them as well as many linear algebra textbooks). This context remains in the 4th edition, it's just that they are introduced first in direct connection with vectors and combinations of vectors. I feel this is an improvement, as a central use and reason for linear algebra is linear combinations and transformations, and I believe the standard introduction of matrices in terms of systems of equations makes it harder to connect with matrices as "operators" or "functions" that act on vectors, scalars, and/or other matrices as entities in their own right. However, I have the advantage of hindsight.

Did you find Strang's treatment beneficial or did you look to other self-study resources (e.g.,online information)? Also, were you familiar with vectors from either calculus I or II (typically, calc. I courses don't get into vectors but it isn't unusual at all for a university to offer a multivariable mathematics course combining linear algebra and more advanced calculus or for a calc. II course to include something about vectors)?

My calc II teacher was pretty good with proofs.

Excellent! Another favor: do you happen to recall the textbook he used?

For every new concept introduced we would basically go over the proof, do various examples and once he thought we understood the proof well enough then he would give us the "easy way" to do it with rules or some algorithm.

Sounds like a good math teacher. Thank you!

 

[Smart_Guy]

(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.

Thank you for sharing

As for me, it was the other way around. I loved Chemistry and lived much better with it than with math. I just moved to the English department (that happens to be part of the Social Sciences College not the Arts, here in Saudi Arabia) because, as I think you have already figured it out, I loved English more.

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.

I might have confused definitions here, sorry about that. By practical majors I means those of practical sciences like physics, chemistry, math, algebra, pharmaceutics, etc., and by theoretical I meant language, arts, poetry, psychology, etc. I don't think I, as an English guy, needed calculus. Well I utilized what I learned in chemistry department in my current job, but that's just because this job happens to be a Technical Secretary.

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.

To clarify, I meant in earlier education stages.

Yes, those plus using it in real casual life, to be at least called educated, to side-support jobs and to give more priority credentials with others. There is also the preparation to be able to join higher education and take higher priority to get accepted in it. Higher education usually give tests to the candidates.

When equations are taught then applied, they, to me at least, don't run in the mind as those shapes and signs with numbers among them, the brain learns to apply them on the run, in real time. Like for a simple example knowing that simple multiplication table of 9 means adding +1 to the left digit and -1 to the right digit; e.g. 9x2=18 & 9x3= 27, helps in getting the result of the next multiplication if it was unknown but the earlier is. To be honest, I forgot what 9x3 equals and I just figured it out now after I typed 9x2 .

Not an Eisenstein post, I know

 

[LegionOnomaMoi]

Not an Eisenstein post, I know

Far better: exactly what I asked for and more. So thanks again!

 

[Smart_Guy]

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"?

You're welcome

Oops, I forgot to cover that part in my reply. My idea in that specific part was that basic math or basic calculus is important for earlier stages of education which most of the time targets young audience with early maturity levels. Other approaches like logic of statistics could be complicated, in my humble opinion, to the majority of the young students. If it is not, I'd say it is a different worthy approach than calculus.

 

[Sapiens]

You mean that you took these out of interest, correct? Could you indicate which of them you took? If all or if analysis were one of these, could you elaborate on the nature of your analysis course(s)? Thank you.

That was forty years ago, so memory is dim and I'd need to have my transcripts in front of me to be exact. I remember taking Classical Geometries, Calculus 1A, 1B, and 1C (we were on quarter system); Multivariate Calculus, Linear Algebra, Differentials, Probability, Stochastic Process, and Mathematical Methods for the Physical Sciences.

I also took a three quarter sequence in Analysis. Looking them up today, for two semesters, covered: "The real number system. Sequences, limits, and continuous functions in R and Rn. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral, Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable." That seems about right.

 

[LegionOnomaMoi]

That was forty years ago, so memory is dim and I'd need to have my transcripts in front of me to be exact. I remember taking Classical Geometries, Calculus 1A, 1B, and 1C (we were on quarter system); Multivariate Calculus, Linear Algebra, Differentials, Probability, Stochastic Process, and Mathematical Methods for the Physical Sciences.

I also took a three quarter sequence in Analysis. Looking them up today, for two semesters, covered: "The real number system. Sequences, limits, and continuous functions in R and Rn. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral, Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable." That seems about right.

Thank you!

 

[LegionOnomaMoi]

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.

I didn't develop these questions. A study ("Can Average Calculus Students Solve Nonroutine Problems") published in Journal of Mathematical Behavior sought to determine whether they were teaching calculus, or teaching a bunch of computational algorithms bereft of the underlying calculus concepts. They offered students "average" students who had just completed the first portion of the calculus tract money to answer the above 5 problems, with liberal partial credit, and gave them an hour to do so. None of the students were able to answer any of the questions. I posed the above questions to students who had passed 1 to 3 (or 4, if you include a course on differential equations) university level calculus courses and found that the more advanced the student was, the more trouble they had with even understanding how to approach the problems in question.

 

[Yerda]

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.

What's generally in calc I?

 

[freethinker44]

What's generally in calc I?

Limits and derivatives.

 

[Yerda]

Limits and derivatives.

Thanks. I've done that.

 

[Yerda]

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.

Right I actually sat down with a pen. I got a=3, b=-1/9.

Is that right?

 

[LegionOnomaMoi]

What's generally in calc I?

It's standardized enough to allow the College Level Examination Program (CLEP) to provide students with an exam that counts as college credit and as having taken calculus I. You can view the breakdown of the material here: CLEP Calculus at a Glance.

 

[LegionOnomaMoi]

Right I actually sat down with a pen. I got a=3, b=-1/9.

Is that right?

I'm actually more interested in how people go about solving these or even just how they might go about solving them if they were sufficiently motivated. Also, I'm crossing my fingers hoping that perhaps a couple others will respond to the questions and confirming answers kind of biases any future answers. I'm not going to hold anybody in suspense indefinitely, of course. Would you be willing to describe in whatever way you wish how you derived your answer? It's fine if you don't even use complete sentences but write instead something like "rewrote equation as [blank], took derivative, used slope-intercept" (only instead of a "blank" and an incomplete method the actual approach used). If not, I'm still grateful for the response, of course.

 

[Yerda]

I'm actually more interested in how people go about solving these or even just how they might go about solving them if they were sufficiently motivated. Also, I'm crossing my fingers hoping that perhaps a couple others will respond to the questions and confirming answers kind of biases any future answers. I'm not going to hold anybody in suspense indefinitely, of course. Would you be willing to describe in whatever way you wish how you derived your answer? It's fine if you don't even use complete sentences but write instead something like "rewrote equation as [blank], took derivative, used slope-intercept" (only instead of a "blank" and an incomplete method the actual approach used). If not, I'm still grateful for the response, of course.

Eh, so I took the derivative of the curve at x=3 and got 6b. I set that equal to the slope of the line and got b=-1/9.

I then rewrote the straight line as y=6bx -9b and found that when I plugged in the b value I had y=1-2x/3. It dawned on me that the 1 was my a/3. So a=3.

I'll be honest, I'm not the most confident or competent mathematician. I don't think I've ever attempted a problem quite like that (with generalised constants) but the approach was the only one I could think of. Is it clear what I was trying?

 

[icehorse]

Perhaps from a different angle... part of my job is to spot popular trends in the computer software development industry. For the last several years "big data" and relatedly "machine learning" have been hot trends and they both use a lot of matrices and some basic calculus.

 

[LegionOnomaMoi]

Eh, so I took the derivative of the curve at x=3 and got 6b. I set that equal to the slope of the line and got b=-1/9.

I then rewrote the straight line as y=6bx -9b and found that when I plugged in the b value I had y=1-2x/3. It dawned on me that the 1 was my a/3. So a=3.

Interesting. I can't be sure how I first did it (because I did these myself a long time ago; the study didn't include the answers), but when I posted it here and did the problems again I re-wrote the equation in terms of y first to get

Then I did what you did and took the derivative of f(3)=6b, set

, and solved for b (-1/9). Then I plugged in the points (x, f( x ) = (3, 9b) = (3, -1).

Then I solved for a:

I just found it easier to get the slope from the equation for the line first (actually, I didn't really think about using the derivative as the slope rather than find the slope and set the derivative equal to it).

I'll be honest, I'm not the most confident or competent mathematician.

Well, you solved a problem that none of those studied (fresh from calculus) did despite the fact that
1) The first thing they're taught is finding tangent lines
&
2) They had an hour for five problems and this was the first.

Also, one of the reasons for my study is that there was a time when I thought I hated math (except proofs in geometry) and that I wasn't very good at it because most of high school math is a lot of algebraic manipulations and I was always making minor computational errors. For example, after posting the problem set I did them all again. When I solved for b by setting 6b=-2/3 I initially wrote down -18/2 instead of -2/18, but caught my error as I'm used to making that kind of mistake.

Unfortunately, almost all math up through calculus consists of such calculations. It wasn't until I took statistics and got into linear algebra, graph theory, and the higher level mathematics that I realized a few things:
1) My conceptions of what mathematics consists of was very wrong. Nothing brought this home like linear algebra, which was one of the first subjects I learned after getting into mathematics. Most of the computations are simple arithmetic, but the concepts were beautiful, elegant, and useful for all manner of applications.
2) I had been taught to solve problems as if I were a calculator. I'm not good at being a calculator, I'm good with mathematical reasoning & concepts. Now I let the computer do the calculating, because like most who work with datasets, do modelling, etc., a lot of the computations aren't really possible for humans to do.
3) I still hate algebra. I love algebras. The elegance of abstract algebras and algebraic structures I find wonderful.
4) There's no such thing as being "good at math". The studies I've read on mathematics education bear out my own impressions. A lot of students who excel in math in high school and get A's in Calc. I (and often later calculus courses) are generally very good at taking a set of rules and mechanically churning out answers (they are good with computations and algebraic manipulations). Often, these same students will suddenly find themselves in a math course that is practically unrecognizable to them and in which they flounder. Some people are good at logic and algorithms, some at mental arithmetic, some at problems involving highly abstract structures and spaces, and so on. Yes, some people are brilliant at all the different kinds of mathematical reasoning/math skills and some struggle with practically all of them, but there is no singular "math" for one to be good at.

I don't think I've ever attempted a problem quite like that (with generalised constants) but the approach was the only one I could think of.

The questions were designed to be "non-trivial" (i.e., require more thought and greater understanding of the concepts behind the procedures/rules).

Is it clear what I was trying?

Yes. And I very much appreciate your taking the time to both do the problem and explain. It was a lot to ask and I am extremely grateful.

 

[Maldini]

Without Calculus we wouldnگt have any of the modern tech we have invented in the past years. The invention of Calculus helped human beings to reach heights that they never even dare to think about.

And It's a beautiful course if you look at it the right way. It helps you realize how things work and it can help your decision making and strategic thinking.