Robert Ghrist produced the most beautiful course I've seen on Coursera. It is a single variable calculus course designed for those who have already seen calculus at some level. I'll talk a little more about why these kinds of courses are necessary in a bit. I'm starting off with this course because there is very little that I can say about it that isn't good.

The production quality of the video lectures is second to none. They were designed specifically for the Coursera course and they display how effective a mooc can be in the hands of a great teacher like Professor Ghrist. The course is split into five separate chapters: function, differentiation, integrations, applications, and discretization with a grand total of 53 lectures (54 if you count the introduction). The videos average about 15 minutes of length and EVERY video is followed by a problem set of roughly 10 problems. In other words, by the time you complete this course you will have done over 500 problems, not including the chapter tests and final exam. This is a far cry from many other Coursera courses and is an example that should be emulated especially in other math and science courses.

When I took the course in its first offering I actually thought that even more homework problems could be offered. Some topics are more difficult than others and in math it is really only after doing problem sets that you can begin to have an understanding of the material. For more difficult topics a second problem set would be very useful. In the second offering I know that the problem sets were broken up into a basic problem set and a more challenging one. I'm not sure, however, if they added more problems or just broke up the problem sets from the previous course into smaller ones.

This is not a course designed to teach beginning calculus, nor is it at the level of a real analysis class. This begs the question, what goal does it serve? If you have already taken calculus at say the AP level (they recommend having experience at the AB level as a prerequisite) then what is the goal of this class? The AP Calculus AB exam is supposed to be at the level of a first semester calculus course. The exam is graded up to 5 points, with 5 being the max and a 3 being the minimum passing score for many schools. Here is the problem: to get a 3 you only need to get about 40% of the available points and a 5 is generally about 60%. In other words you can get AP credit and not have a strong grasp of the material. (As a complete aside I am teaching AP Calculus AB for the first time this year and I went to a workshop over the summer for beginning calculus teachers. Most of these future teachers were incredibly bad at calculus and so the workshop became mostly a review of the material. I believe that if you only understand the material at the level that you expect your students to understand you are almost certain to fail as a teacher and that might be a large part of why students score so low. Who knows though, ask me again in a year after I have taught my first AP class!) That is where this kind of a course comes in. It teaches all of the materials that you would see in a second semester calculus course but also goes over much of what you should have learned in AB. To keep students from getting bored Professor Ghrist teaches this material from a much different viewpoint than students would have learned, introducing the basic ideas of Taylor Series right from the beginning. Once he gets to techniques of integration the material is covered slightly more traditionally as this would generally be the start of the newer material.

The applications section is phenomenal and Professor Ghrist teaches it in such a simple way that I am surprised it isn't taught this way normally. I have a degree in mathematics and when I took calculus 2 the integration applications were taught in a high-schoolish way, i.e. here is the formula for arclength, here is the formula for the volume of a solid of revolution using shells, etc. After finishing up the calculus sequence most of that material was never used again in upper level math courses so I forgot many of the specific formulas. I could probably derive them with some work but the way that Prof Ghrist teaches it makes it impossible to forget and easy to derive. That section of the course should be studied by calculus teachers everywhere!

As I said in the title I believe that this is the epitome of what a mooc can be. Professor Ghrist shows a clear understanding of how his students learn and how to translate that to an entirely-online medium. The discussion forums were incredibly helpful with discussions on every homework question so that if you were stuck you could quickly and easily find help. There are quite a few upcoming Coursera courses that require knowledge of single-variable calculus so even if you took calculus before and even if you have a strong grasp of the material I highly recommend taking this course the next time it is offered. I would also strongly suggest that anyone thinking of offering a mooc in the future look at how Prof. Ghrist developed this course. While I don't think that you have to develop the lectures as artistically as he did the overall structure, grading and assignments should inspire you.

I enjoyed this course so much I took it twice.

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