Tuesday, August 1, 2017

Calculus III

In the last year or so, I decided to apply for a master's program in applied statistics, but I was missing one of the prerequisite mathematics courses: Calculus III. I had taken calculus courses in high school and college, but those courses were more focused on applications. Furthermore, I hadn't covered any of Calculus II in those courses.

Instead of taking the entire series, which would have taken quite a bit of time and money, I decided to do something rather daunting: I took Calculus III online and used Khan Academy and other sources to catch up on Calculus I and II. I read reviews that the first few weeks were tough even if you had taken Calculus I and II just before III. Undeterred, I started the course earlier this year, and the first few weeks were indeed tough.

The online program I used - NetMath - used an online math tool for running code and submitting homework. Each student is assigned to a mentor who grades assignments, answers questions, and ensures each student is on schedule. Students receive feedback on their homework and are able to re-submit corrections a couple of times. The two midterms and final must be taken in person with a proctor.

My first mentor was not very responsive. On week 2, a critical week in the program, my mentor did not respond to emails or grade my assignments in a timely fashion (within 3 days, as noted in the program handbook). I notified the program administrators and they assigned me a new mentor. She had quite a bit of catching up to do, but she did her best and eventually graded the outstanding assignments and responded to my questions. Honestly, she was amazing, and I'd write her a letter of recommendation if she asked.

Lesson 2 is quite difficult. It's really the first lesson on the topic of the course, where lesson 1 was review of parametric equations and other necessary concepts, and it's there in case anyone missed or forgot these topics. With the combination of difficult content and slow responses from my mentor, it took me 2 weeks to finish lesson 2. In addition, I got sick for a couple of days and there was a death in the family, which put me behind another 2 weeks or so. Fortunately, the program offers a two-month extension, and I planned on using it if needed.

However, there were additional, serious problems with the course. One of the most grievous was incomplete or incorrect content. There were often no terms given to ideas, preventing students who have taken this course from communicating the concepts effectively. For example, vector projection was just called "vector push on another vector". It took me quite some time to find the right term to be able to research this concept online.

The course also neglected saddle points and claimed that whenever a gradient was {0, 0} (or more zeros depending on the number of dimensions), that the point was a minimum or maximum of the function. This is blatantly not true when a saddle point is present, and it would be terrible for students to internalize this falsity since it is profoundly meaningful in calculating predictive models with machine learning, specifically neural networks. You can't assume you've optimized a function when the gradient is {0, 0} without looking around it to see if you've found a saddle point.

All told, I was quite unhappy with this course. Not only was I spending a lot of time catching up, but I was spending time trying to learn the material through other sources since the course material was incomplete or inaccurate. Nearing the end of the course, I was able to catch up to the point where it looked like I could finish the course if I just had another week or two. I emailed the administrators for a course extension, noting the reasons I had for the delay and the issues I had with the course, and they offered a shorter extension so I could finish it without rushing and without taking the full extension. (My schedule to finish without an extension would have been very demanding for my mentor to complete all the grading in time.)

Despite all the delays and issues, I finished all the material within the original time frame, and I just needed to take the final. I studied for a few extra days and took the final about 2 weeks after the course originally ended. Since the final was comprehensive and included the last three lessons, I was very nervous about it. I had aced the midterms, but there was just so much to remember (e.g., the curl of a 3D field is difficult to remember). To my great surprise and delight, I not only aced the final, I got an A+ in the course. I was relieved!

Now I just have to re-take the GRE and apply for the master's program.