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.
The question isn't whether we should do away with calculus as it is useless. It is absolutely essential. The question is how we should structure mathematics education including the ways in which calculus is taught.
For a long time, the (then-developing) calculus was cutting edge mathematics and about as applied as was possible. So it became a standard part of pre-College and College education. The problem is that the more the teaching of calculus spread, the less calculus was actually taught.
Moreover, numerous majors in universities in multiple countries require a calculus course that is virtually incapable of teaching anything which can be applied an is largely bereft of pedagogical value given that the concepts aren't taught. The people who use calculus rely multivariate calculus, differential equations, integrals more advanced than the Riemann integral, etc. The calculus knowledge required for most majors that require some calculus is essentially useless. Learning linear algebra without the faintest idea of what calculus is would make one vastly more prepared for applied mathematics across the sciences. Finally, there is much truth in the assertion that calculus courses don't actually teach calculus. They supply rules in the hopes that unnecessary time spent in calculus courses will enable students to grasp the concepts through practice. Also, just to ensure this, most of high school mathematics is designed to prepare students who won't ever take calculus for a calculus course rather than vastly more applicable and immediately useful skills/knowledge in mathematical topics like statistics, logic, probability, etc.
Generally, it is three courses followed by a differential equations course. For real mathematicians or those who desire or whose majors are sufficiently akin to mathematical majors calculus leads into real & complex analysis. Here, students are taught integration and other concepts from "calculus" that have little to do with what they were actually taught in calculus courses. Multivariable calculus tends to poorly unify linear algebra an calculus such that e.g., vectors are intensely studied, but not only restricted in general to R3 but also with little to know coverage of matrices.
It helps you realize how things work and it can help your decision making and strategic thinking.
The question isn't whether we should do away with calculus as it is useless. It is absolutely essential. The question is how we should structure mathematics education including the ways in which calculus is taught.
For a long time, the (then-developing) calculus was cutting edge mathematics and about as applied as was possible. So it became a standard part of pre-College and College education. The problem is that the more the teaching of calculus spread, the less calculus was actually taught.
Moreover, numerous majors in universities in multiple countries require a calculus course that is virtually incapable of teaching anything which can be applied an is largely bereft of pedagogical value given that the concepts aren't taught. The people who use calculus rely multivariate calculus, differential equations, integrals more advanced than the Riemann integral, etc. The calculus knowledge required for most majors that require some calculus is essentially useless. Learning linear algebra without the faintest idea of what calculus is would make one vastly more prepared for applied mathematics across the sciences. Finally, there is much truth in the assertion that calculus courses don't actually teach calculus. They supply rules in the hopes that unnecessary time spent in calculus courses will enable students to grasp the concepts through practice. Also, just to ensure this, most of high school mathematics is designed to prepare students who won't ever take calculus for a calculus course rather than vastly more applicable and immediately useful skills/knowledge in mathematical topics like statistics, logic, probability, etc.
Depending on the major, the level of calculus you need to know differs. I think that the calculus course must be taught differently based on the student's major. For many tech and engineering courses (I study CS) it's extremely essential, and as you definitely know advanced calculus is the most important course they ever take.
But there are majors who won't need advanced calculus, so I think the approach should rely on problem solving. Some simple calculus, and then ask students to use that to solve problems that have a close connection to simple real world problems.
Generally, it is three courses followed by a differential equations course. For real mathematicians or those who desire or whose majors are sufficiently akin to mathematical majors calculus leads into real & complex analysis. Here, students are taught integration and other concepts from "calculus" that have little to do with what they were actually taught in calculus courses. Multivariable calculus tends to poorly unify linear algebra an calculus such that e.g., vectors are intensely studied, but not only restricted in general to R3 but also with little to know coverage of matrices.
In my school it's actually two followed by diff. equations and engineering mathematics and/or linear algebra. I agree that when you enter diff. equations and other calculus related courses the subjects you had studied in Calculas 1 would seem to be pretty inane and insufficient.
As I said above Calculus 1 needs to be separately taught for different majors.
Could you expand on this please? Thank you very much.
Like any other math courses, you need to develop some level of problem solving skill in order to be able to pass the course. I personally think a basic knowledge of derivatives and Integrals can help you have a better insight in many totally math-unrelated areas.
As you probably know using derivatives and integrals you can detect and analyze behavior which can be quite helpful in real life.
I'm not saying you can transform your life issues into calculus problems (even tough in some instances you could), but if you get the hang of how to solve basic calculus problems, it will train your brain to be more smart when you have to deal with an unknown situation in life.
A basic example would be finding points of inflection and extrema. by doing basic problems you learn how to find out the most important parts of a problem, and what you need to pay attention to while learning what needs to be ignored.
So far I've been thinking that as I can find almost no real-world problems that can be addressed with Calc. I and given the way the math continues to be taught mostly in terms of rules, that it should be replaced for majors that don't require more than Calc. I. However, you raise an interesting question. It is entirely possible to restructure elementary calculus courses such that those students who will be required to take subsequent calculus courses can be taught more or less what current Calc. I courses teach and others may be taught a far more concept-oriented calculus. I've tutored students in e.g., "managerial calculus" or similar calculus courses that were intended to be simpler than your standard first semester calculus course, but they simply taught less. That doesn't mean, however, that such courses could be made more valuable. I have to give this some serious thought. Thank you.
Like any other math courses, you need to develop some level of problem solving skill in order to be able to pass the course. I personally think a basic knowledge of derivatives and Integrals can help you have a better insight in many totally math-unrelated areas.
Would you argue that elementary calculus does this more so than courses in logic, probability, & statistics (all three of which are sometimes combined in discrete mathematics courses or similar "university mathematics" courses)?
As you probably know using derivatives and integrals you can detect and analyze behavior which can be quite helpful in real life.
Certainly. Almost all real-world phenomena are non-linear and that was the motivation for most of the development of calculus. One thing I'm looking into, though, is at what point your average calculus student is able to solve non-trivial problems.
Thanks so much for your detailed feedback- it's much appreciated.
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.
That's alarming. I did spend a while staring at it wondering why it wouldn't just solve itself. I imagine a lot of people would go straight for panic mode.
LegionOnomaMoi said:
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.
Harvard, unfortunately. However, I left my program to make some money so that I can continue my program, so of late my research has been contract work with companies designing research products/software, government, academic research labs, even lobbying groups (mostly to my dismay). On the plus side, despite my distaste for most of such work I have gained a considerable amount of experience in research methods and conclusions in fields I would otherwise likely never have become familiar with (at least not directly). Technically, I'm still a graduate researcher at Harvard, but I haven't been to the lab in months nor have I worked on any projects there for some time.
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.
Legion, in case you haven't seen it I recommend that you look at the Humongous Book of Calculus Problems. It is an inexpensive tomb of step by step solved example problems. There are also ones for Statistics, Algebra and Geometry. Most students can afford them, and they do a good job of explaining how to solve things.
Thanks! I haven't included in my survey of calculus texts any "teach yourself" calculus books or those meant to complement one's course for struggling students, but now that you mention it I think I should (albeit as a separate category). I have (in the course of teaching/tutoring math for high school and college students) tried to find and review many such materials, but I am not sure if I've come across that one. I'll certainly take a look though, so thanks!
Relatedly, I've come to realize that even though a surprising number of texts I pay for can be found online, it's usually not because they are free (which is why some of the links in my thread "Math Books/Resources for Free and for Learners" are dead).
However, some ebooks (including, but not limited to, those found in my thread above) are really free/open access. I have included some in my survey but only those that are intended to be used as textbooks, which differ qualitatively from the type of book you recommended (one need only browse the table of contents of e.g., Theory of the Integral or Real Analysis to see by how much)
There are also ones for Statistics, Algebra and Geometry. Most students can afford them, and they do a good job of explaining how to solve things.
When it comes to physics, I have a heard time understanding how much can really be taught without calculus (which was developed largely for physics). I never took high school physics, but from those I've talked to who have they do sometimes do things like approximate the area under a curve and of course the difference quotient formula for derivatives has even more easily relatable approximation analogues as until you include limits it's just algebra.
Nursing is another matter. My younger sister went to Upenn's nursing school (she's now getting her master's in nursing at BU or BC- I can't recall which at the moment) and as you said was required to take calculus. She was actually the first person I asked to look at and try to answer the 5 questions from the quiz that I posted earlier, as well as an early source for other information about her calculus experience. She never uses anything she learned (and there really isn't anyway she could) and doesn't remember most of it.
Ironically, she was better able to address the 5-question quiz (which, again, is from a study on whether first semester calculus students can answer non-trivial first semester calculus questions/problems) than my cousin, who had at the time I asked her to look at it taken 3 semesters of calculus, a course in differential equations, and linear algebra (I'm not sure what other math courses she took if any). To me, this suggests that the material covered in a typical calc. I course isn't just largely useless for any real application, but also more divorced from advanced courses in calculus or analysis than they are foundations for them.
The point behind it was teaching people how to think rather than developing practical application of calculus.
You aren't the first to proffer this as a defense for teaching or requiring elementary calculus. Perhaps you're right. I have several reasons for thinking otherwise, but of course a major point of this thread is to hear views contrary to my own. I am also realizing that I don't know if I've yet summarized clearly my justification so that others can question them, so I will do so in my next post.
This post will lay-out clearly (I hope) my reasons for believing that elementary calculus (i.e., high school calculus & at least first semester college calculus) doesn't help students with abstract/analytical reasoning, quantitative reasoning, or provide them with a tool that even could be applied in any real-world application that wasn't contrived the way "real-world" problems in calculus textbooks are. I will also address why I believe that such skills are far more easily and more fully acquired in other courses (or could be).
1) I don't think that elementary calculus does much encourage critical thinking, but rather continues to offer computational techniques for solving particular problems mainly with pre-calculus and little understanding of the reasoning & concepts behind their rote application of various rules. There are several lines of evidence I think support this view:
a) For students who go on to analysis courses rather than follow the more typical Calc. I-III and one to two semesters of differential equations, the texts are structured in order to account for the fact that the students aren't just expected to know but are expected not to know rudimentary but fundamental components of analytic/abstract reasoning.
One of the best calculus textbooks around (Spivak's Calculus) is little used compared to the big, commercial calculus packages (e.g., Stewart's, Larson & Edwards', Tan's, Thomas', etc.). In part this is because it is clearly more difficult than most calculus textbooks. Yet while most of these start with a review of functions and then immediately move to limits, Spivak's Calculus starts with what is mostly arithmetic ("Basic properties of numbers"). He doesn't get to limits until "Part II" of his textbook in chapter 5 (previous chapters include e.g., "Numbers of various sorts", "Graphs", "Functions", etc.). So what makes his textbook so challenging? He uses the basics of algebra in lieu of mathematical logic (which most students wouldn't know), so that from the beginning students are required to derive the rules of algebra they've used for years. He begins by requiring students to follow and provide proofs. His textbook is highly conceptual, and thus despite the fact that he doesn't reach what is normally covered in chapter 2 in most calculus textbooks, his textbook covers more than any calculus textbook I've seen of comparable size. He removes the vast numbers of problem sets which provide students with the necessary practice with derivative & integration "rules", because he requires students to derive these and understand them. His textbook isn't designed as part of some larger package of textbooks like the "big names" (which not only create several textbooks out of one by dividing a complete textbook into smaller ones and/or rearranging material), because his presentation is linear. It is designed to teach students to understand calculus in terms of its relation to "real" mathematics complete with rigor, conceptual depth, and conceptual unity.
b) Math professors have widely recognized for years that students having taken calculus are incapable of the kind of analytical, abstract reasoning expected of "intro. to logic" students:
"Eight years ago my department instituted a course in mathematical reasoning to serve as a transition between calculus and higher-level math classes. We had found that the students entering our higher-level classes woefully unable to construct the most simple proofs or to figure out answers to easy abstract questions. The idea of the course was to give students a batter chance for success in more advanced classes...Over the next several years I had primary responsibility for developing the course. During this period I came to realize that many of my students difficulties were much more profound than I had anticipated. Quite simply, my students and I spoke different languages. I would say Of course, this follows from that or As you can see this means the same as that and my students would look at me blankly.
Very few of my students had an intuitive feel for the equivalence between a statement and its contrapositive or realized that a statement can be true and its converse false. Most students did not understand what it means for an if-then statement to be false, and many were also consistent about taking negations of and and or statements All aspects of the use of quantifiers were poorly understood [I am omitting an extensive survey of specific failures to understand what I learned in an Intro. to symbolic logic course]. The fact is that the state of most students conceptual knowledge of mathematics after they have taken calculus is abysmal. The most dramatic formal studies on this subject found that a large majority of calculus and post-calculus students at universities throughout the country could not set up or even correctly interpret simple proportionality equations.
c) Almost without fail, calculus textbooks rely on the rote, mechanical application of rules to remarkably similar derived problem sets. Proofs are often given for many of these rules, but not only are students frequently never asked to prove anything (nor, actually, provided with the means/language to do so), the "proofs" for central theorems rest upon unjustified axioms. Perhaps astute students are able to catch such pervasive glosses, but the textbooks themselves go to some lengths to hide or minimize the extent to which the entirety of their content is not supported by anything more than unproved assertions.
2) The way that integration is taught:
"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."
(The Calculus Integral).
3) Here is a limit definition as mathematicians would represent it, with a few bits left out:
I have found that few calculus students, including those with several semesters under their belt, can "translate" that. Their familiarity with set notation is far less important than their inability to precisely formulate and understand the mathematical/symbolic representations of derivations, proofs, or arguments that students receive in logic courses with no mathematical pre-requisites. The universal and existential quantifiers in particular are absolutely essential to precisely formulating mathematical statements that are given in English (or whatever the native language may be) in elementary calculus texts. The problem, though, isn't just an inability to construct or understand the formal representation of basic calculus concepts. It is the inability to understand their implications. Again, this is something required in elementary logic courses in college.
4) In addition to calculus textbooks, I have acquired a number of textbooks on critical reasoning, argumentation, and/or logic. Most of these require nothing other than literacy for background. All of them are superior in communicating critical, analytical, and abstract reasoning of the type much needed than any calculus textbook, as even a textbook such as Spivak's is limited by the need to impart calculus concepts. I do not see how realizing that rectangles can be used to approximate area and trapezoids even more is somehow supposed to challenge students intellectually in ways that will best (or even likely) ensure superior analytical/abstract reasoning or critical thinking. The same holds for almost all of elementary calculus, which is mostly challenging in the ways students need to apply pre-calculus skills (e.g., algebraic manipulations, trig formulae, etc.).
The following is continued from my last post, but somewhat different. Here, I will address the issue of whether or not elementary calculus teaches mathematics that can realistically be useful to anybody.
Even the rote mechanical approach to calculus covers absolutely essential material for students in many sciences. However, this is only after several semesters. There is no point to a single semester of calculus from an applied mathematics point of view.
1) First, there isn't actually much "calculus", but rather the introduction of terms from calculus which are taught by carefully formulating problem sets to minimize the "calculus" part and maximizing the "pre-calculus" steps.
2) Second, there are far too many students who are required to take calc I. (but no more), such as those in economics, nursing, psychology (this is rare and limited to B.S. degrees in psychology), managerial sciences, and often a blanket requirement for B.S. degrees across the board. First semester calculus is prima facie incapable of use in modelling as if it were even close to what is needed than there would be no need for such extensive subsequent courses. Also, it's single variable calculus which is inadequate for even extremely simple real-world problems and cannot be extended to the 3D space in which we live (presenting a fundamental obstacle not just with respect to its use, but in any readily conceptual connection between the material and the "real-world" which isn't the 1D & 2D world of elementary calculus).
3) Calculus was motivated by many things (e.g., the typical two beginning explanations to differential & integral calculus: "instantaneous rate of change" and "area under a curve", respectively). However, it was primarily developed by natural philosophers (the pre-cursors to scientists) like Newton who sought to create models for how things worked, particularly motion (i.e., the field of mechanics). Because real-world phenomena are almost always non-linear, the mathematical tools before Newton & Leibniz couldn't account for e.g., celestial motion or model basic things like "acceleration" or "work". At the heart of elementary calculus (and all analysis really) is- simplistically- that when you "zoom in" or "magnify" a curve enough, it looks like a line and can be approximated by one. However:
a) Students coming out of an elementary calculus course couldn't solve problems that those using the insufficiently rigorous & woefully incomplete "calculus" that Newton & Leibniz provided. Mostly this is because elementary calculus doesn't provide the tools for dealing with functions of more than one variable, although there is also the problems with how much students rely on rote application of rules and mechanical computation to solve problems they don't really understand. After all, fundamental to calculus and all analysis is the notion of "limits", which receives a single chapter in most textbooks because it is perhaps the most conceptually challenging component. Alas, by limiting "limits' this way, we get an elementary calculus which can't go far beyond trivial, rote application of specified rules. The logical foundations for these rules involve the kind of nuanced treatment of limits that are deliberately avoided.
b) The most widely used mathematical tool across all sciences is without question the "Generalized linear model". It is encapsulates virtually all of multivariable statistics in research ranging from sociology & psychology to climate science & statistical mechanics. It is called "linear" because despite the fact that the kinds of curvature in problems researchers address can't even be defined (at least without recourse to definitions of "curves" and "spaces" that are far beyond virtually all undergraduate level mathematics), it is vastly superior than anything elementary calculus could provide when it comes to approximating "curves". More importantly, the statistical tests/methods that can be learned and understood with a pretty elementary grasp of algebra so utterly outstrip any "real-world" applications of calculus that, while we see undergraduate-level statistics used across all the sciences & beyond, most of the calculus taught in calc. I isn't used anywhere.
c) The GLM merely a kind of special "package" of techniques that are built upon linear algebra. Most linear algebra courses do not involve any calculus (although it is a fairly typical prerequisite; the reason for this has to do with an attempt to ensure that students have some facility & familiarity with higher level quantitative reasoning). Linear algebra has become more important than calculus in all the sciences and so much so in physics that it basically makes up quantum mechanics. True, most of the more advanced uses of linear algebra involve multivariable mathematics (and therefore calculus), but like elementary statistics a first course in linear algebra allows for vastly greater applications in any science. Put another way, calculus is largely useless without (concepts from) linear algebra, but the reverse isn't true.
In short, there are far superior ways to deal with curvature than single variable calculus, including elementary statistics, basic probability, and the far more challenging courses like linear algebra. There are really no applications that elementary calculus could offer that are exceeded in spades by courses that better prepare students for real-world problems and even every day problems. So what is the point of requiring students to take a single calculus course?
I'll stop here & summarize what was covered. I want to emphasize that I am not saying that we shouldn't require students in fields like physics, engineering, etc., to take calculus or that we shouldn't encourage those in other fields to do so as well. Much of probability theory (and therefore statistics) depends upon calculus. The world is nonlinear, and the limitations of multivariable mathematics without calculus (e.g., multivariate statistics or linear algebra) are too great (in fact, graduate level statistics courses are a far more insidious problem than the question of elementary calculus.
However, a student who has taken one or two semesters of calculus (and often more) is less capable of dealing with our nonlinear world, large datasets, etc., than someone with a course in elementary statistics and one in linear algebra. All one does by requiring B.S. students to take single variable calculus is ensure that they will spend at least one semester on a topic virtually useless without several more semesters that are not required.
Finally, elementary calculus is designed such that the challenges are almost all matters of pre-calculus. It is not designed and is ill-equipped to provide students with a grasp of mathematics, analytical/abstract reasoning, or critical thinking. There are other courses which are fundamentally concerned with critical thinking and analytical reasoning and do not require knowing trig and other pre-calculus material (most of which is never used except by those who end up in fields that require far more than a single calculus course and much of which is structured to solve problems in elementary calculus which is itself designed to be the bridge from pre-calculus to mathematics). And despite the more challenging mathematical computations required in pre-calculus, students leave such a course with almost no greater ability to grasp higher-level mathematics and less knowledge of useful mathematics than other courses like statistics provide.
You aren't the first to proffer this as a defense for teaching or requiring elementary calculus. Perhaps you're right. I have several reasons for thinking otherwise, but of course a major point of this thread is to hear views contrary to my own. I am also realizing that I don't know if I've yet summarized clearly my justification so that others can question them, so I will do so in my next post.
You know the basis behind classical education correct? For those that don't its the idea that by developing your skills and ability to think from learning from "established" great thinkers you were able to raise your overall intellectual level. This doesn't equate well to the modern demand for skilled work. There is some measurable basis to it however.
You will have to excuse the rudimentary style in which I bring these cases about. This is from a layman perspective.
My computer is running badly right now but there have been a few studies in the areas of training children's minds with certain things that will enable them for greater mental fortitude down the road. A few examples would be learning a second language, learning a classical instrument and taking critical thinking or logic (preferably both). These courses give them a skill but develops other areas of the brain that can help overall. A second language helps the connections between both the left and right portions of the brain, the instrumental training assists with pattern recognition and abstract conceptualization of ideas. And I think critical thinking and logic stands on its own.
Teaching calculus is (at least to my understanding) along the same vein. But I don't know if it would be that effective since all of the studies about the aforementioned benefits are done as children during the childhood developmental stages of the brain. I don't know if making a twenty something year old learn calculus would actually better them for a job as an RN.
I was required to take Calc 1 and 2 for my bachelor's but have not taken any since. I also have never had to use Calculus.
I like to think that at least it helped me 'learn to learn,' but I can't say it's been useful to me in my profession. Probably another course or two of statistics would have been more beneficial, if less challenging.
Legion I agree with you about 1st semester Calculus, linear algebra and statistics.
One comment I float towards you is a related item that Calculus lectures range in quality. There are some very, very bad lecturers and some very, very good ones. Does it seem to you, like it does to me, that universities could get together and record the best lecturers and use those instead of whatever researchers and grad students they have on hand? It seems to me that signing up for a Math class is like shooting in the dark. You never know what quality of lecture you're going to get.
I was required to take Calc 1 and 2 for my bachelor's but have not taken any since. I also have never had to use Calculus.
I like to think that at least it helped me 'learn to learn,' but I can't say it's been useful to me in my profession. Probably another course or two of statistics would have been more beneficial, if less challenging.
This is very true. In fact, it's a fundamental problem for me, as while I can examine 100 or 1,000 calculus textbooks these are not designed for self-study but for an instructor to use both to structure a course and as the tool for teaching calculus. However, it is still true that it is the instructor that teaches calculus, and the student uses the text to do problems or to help recall (or follow along with) in-class instruction.
My aunt (my father's brother's wife) was the son of a mathematics professor. Apparently (I never met the man) he was a very unpleasant character. Generally speaking, professors in a math department rotate who teaches which elementary math courses (including calculus). He didn't like teaching, still less teaching elementary courses, so when called to teach calc. I he essentially taught real analysis and taught it badly (expecting students to know things they couldn't in order to learn about topics far more complicated and nuanced than any they had ever encountered). It worked: he got out of having to teach by being a terrible instructor.
That's an extreme example, but it is always true that the instructor is key. The problem then becomes, key for what? Granted a great instructor, how much is possible for such a person to teach students taking elementary calculus?
Does it seem to you, like it does to me, that universities could get together and record the best lecturers and use those instead of whatever researchers and grad students they have on hand?
Many have. You can, for example, access semesters of actual lectures from both single and multivariable calculus recorded at MIT using ITunesU or YouTube, complete with the TAs/grad students who don't lecture but rather give practice problems for each lecture (although it is rather confusing how these are broken down; without watching them, I can't tell exactly how they ought to be ordered relative to the lecture clips, which I have seen little of other than that required to get a sense of them).
Additionally, MIT also put out some instructional videos on multivariable calculus, complex analysis, differential equations, and perhaps other topics which are directed at the viewer (I haven't really looked into this other than to note that they aren't as old as they seem despite being black & white and are all "taught" by Herbert Gross who was highly invested in general math education). FIU also recorded a number of courses taught by Dr. Rosenthal which pretty much span the calculus curriculum. Other colleges have put out such material too, but I am still sorting through data. For example, thanks to my sister when I happened upon a coursera.org calculus I course offered b UPenn, I was able to match what she was taught there with the material covered by the coursera.org. However, there wasn't any actual lectures or instructional videos (just instructional clips, and while I only saw a few they didn't seem that great).
These are all free (unlike The Learning Company's The Great Courses, which also cover Calc. I-III but I can't afford to buy these just to see how good they are, although I have been able to see several examples thanks to some students).
So there are lecture series on calculus by professors at both leading universities and elsewhere. I've even borrowed examples and techniques for teaching from a semester of real analysis recorded at "Harvey Mudd College" (wherever that is) and I have used talks recorded and freely available from Cambridge University's INI to learn/study myself (e.g., the videos from the seminar Algebraic Lie Theory, Discrete Integrable Systems, & The Mathematics of Liquid Crystals).
However, we're still missing the level of collaboration and dissemination that you refer to, and I agree absolutely that it would be wonderful for universities to do as you suggest.
It seems to me that signing up for a Math class is like shooting in the dark.
True enough. And not just because the question of whether the lecturer will teach in a manner that you can learn from, but also because many people sign up for courses without knowing what they entail except that they fulfill a math requirement.
This is very true. In fact, it's a fundamental problem for me, as while I can examine 100 or 1,000 calculus textbooks these are not designed for self-study but for an instructor to use both to structure a course and as the tool for teaching calculus. However, it is still true that it is the instructor that teaches calculus, and the student uses the text to do problems or to help recall (or follow along with) in-class instruction.
...
That's an extreme example, but it is always true that the instructor is key. The problem then becomes, key for what? Granted a great instructor, how much is possible for such a person to teach students taking elementary calculus?
My comment about the videos was inspired by videos that I have seen online, including MIT's lectures. (Their Math lectures are nice.) You are speaking to a terrible student who has struggled to eventually get through one semester of differential equations, which was my paramount accomplishment in Math. I will not detail what makes me a bad student, but believe me I am. That doesn't qualify me to talk much about the quality of lectures, though I agree with you from the bleachers.
From the complete opposite end of the spectrum I tell you I will always be bad at Math, no matter who my instructor is. For me the concepts in Math are enjoyable, but the tedium of working through problems overcomes me. When I took Discrete Mathematics I wasn't good at it, but I enjoyed learning it. Logic, set, graphs and all of those things are interesting. Calculus is beautiful. Learning still requires learning, and there will always be some tedium and resistance to it.
True enough. And not just because the question of whether the lecturer will teach in a manner that you can learn from, but also because many people sign up for courses without knowing what they entail except that they fulfill a math requirement.
We are guilty of signing up for many courses that way! On the other hand the universities encourage us to do so, so that probably isn't going to change. Universities make courses obscure until the day of, and students just aren't savvy.
True enough. And not just because the question of whether the lecturer will teach in a manner that you can learn from, but also because many people sign up for courses without knowing what they entail except that they fulfill a math requirement.
We are guilty of signing up for many courses that way! On the other hand the universities encourage us to do so, so that probably isn't going to change. Universities make courses obscure until the day of, and students just aren't savvy.
That is so true. I wish they would publish the syllabus for a course so we can know what to expect. Maybe some schools/courses do, I don't know, but none of the courses I've taken have.
That is so true. I wish they would publish the syllabus for a course so we can know what to expect. Maybe some schools/courses do, I don't know, but none of the courses I've taken have.
It's worse than that. I've reviewed syllabi that are given but don't give any real information. Take linear algebra: I've read descriptions of syllabi that reference systems of equations, "algebra", vector spaces, and other terms which are either understood differently because e.g., "systems of equations" means something to those having taken pre-calc vs. those taking a linear algebra course or e.g., are misunderstood because "vectors" have as of yet consisted solely of three-component entities within classical physics. Thus even were one lucky enough to be given a description of one's course, frequently it is couched in terms that can't easily be accurately understood.
