Friday, May 1, 2015

Progress report on learning math online [updated and moved to top]

3/25/15: I've been studying math lately, as noted in the post below. I'm now in the 6th week of Prof. Keith Devlin's "Introduction to Mathematical Thinking," which runs either 8 or 10 weeks depending on whether you stop after the Basic Course or continue through the Extended Course. The latter is described in the course description as being particularly difficult:
Students who struggle with the Basic Course are likely to find the additional two weeks of the Extended Course extremely difficult, if not impossible. Note also that the final two weeks of the Extended Course are more intense than the Basic Course, being in part designed to give students a sense of the pace of a university-level course in pure mathematics. Moreover, in Week 10 there is a series of fairly tight deadlines you must meet, with 48 hour turnaround times. [Emphasis in the original.]
Me: We'll see how that goes, as I've been hoping to make it through the Extended Course and get the Statement of Accomplishment With Distinction that comes from passing it. I've found the course highly interesting and will have learned a great deal regardless of how far I get. The overall course emphasizes logic and proofs, and is designed to give a sense of how mathematicians think (in contradistinction to the emphasis on following instructions that characterizes much K-12 math). At times, I have struggled with the concepts, though my weekly test results have been generally decent (with one exception), with scores equivalent to about 87, 87, 41 (oops), 96 (comeback) and 86.

The course is not particularly oriented toward visual thinking, but I have found some visualization helps in grasping the concepts. Here is what I drew and wrote for a homework problem. [Added: SPOILER: Don't read the image first if you want to answer the question yourself.] The question was: "Prove or disprove the claim that there are integers m, n such that m2+mn+n2 is a perfect square." I drew and wrote the below, and posted it to a class discussion board (along with a question as to whether and to what extent "visual proofs" are acceptable):

A fellow student pointed out that my diagram doesn't match the algebraic expression m2+mn+n2, which is true. If I were to do it over, I'd leave one of the mn boxes out or cross-hatch it or something. My basic idea that the claim is verified by making n zero seems to have some merit.

Here is a report from someone who took the same course a couple of years ago. I agree with many of the sentiments expressed, including about finding the course very interesting and enjoyable, and also about this:
Assignments are not submitted for marking (but a helpful feedback video is made available in the next week in which Devlin explains how to answer a selection of the questions). In the first few weeks of the course Devlin puts a very prominent amount of emphasis on the need for all students to discuss the course with others in an informally established study group. In my case I chanced on and joined a Google Group called “Mathematical Thinking UK Discussion Group”. This initially had about 40 members. About seven were helpfully active in the first three weeks, but the study group has seemingly since ceased to function. So I am on my own: it feels a bit late in the day to try to find another study group, nor to attempt to breath life into this one.
Me: That mirrors my experience very closely, as I was part of a weekly Google Hangout study group, which started off very promising and then progressively wound down into nothingness. I gather the dissolution of such groups has a lot to do with people quitting the course (considerably more that, I suspect, than with people finding the course easy and deciding they don't need a study group). The statistics posted by the professor weekly show that, while a large number of people are enrolled (over 38,000, from all over the world), they vary in activity or lack thereof; and those who actually hand in the weekly test, or Problem Set, are a small and declining subgroup (under 1,500 at last count).

In any case, I've become a big fan of online courses, a remarkable and unprecedented resource (and one that for now at least is often free). Edward Frenkel, who did much to inspire my newly heightened interest in math, has an upcoming (and seemingly not too time-consuming) class, which I've signed up for as well. Whatever limits I encounter in my online education, it's clear to me that I've learned much already, and that this has been time well-spent.

The above was originally posted 3/25/15.

UPDATE 4/28/15: I've completed the course, including the extended course, and am now awaiting my Statement of Accomplishment With Distinction, which I believe I earned. Someone pathetically tried to disrupt the peer-review portion of the extended course by placing numerous bogus "zero" reviews, but this was detected and deleted. I am now on to "Introduction to Mathematical Philosophy," which looks to have many fascinating concepts but also be less time-consuming (a good thing, given my schedule, though it also means it will involve less hands-on learning, as this course, unlike the previous one, does not involve handing in weekly problem sets for grading).

UPDATE 5/1/15: My SOA With Distinction. I was one of 275 students to get one of these; another 991 earned an SOA for completion of the Basic Course.

A few additional thoughts before closing this post:

1. "The only way to really learn math is by doing it." This is something one hears now and then, and I can affirm it based on my experience in this course. If I had only listened to the professor's lectures or read his book Introduction to Mathematical Thinkingwithout actually trying to solve the (ungraded) assignments and (graded) problem sets myself, I would not have gotten nearly as much out of this course.
2. "You don't really understand something until you've taught it." Another bit of repeated wisdom that I find has validity. In evaluating other students' work and trying to explain things to them, I found I had to learn the material better than I would have just from handing in my own work.

3. My final grade was 98.5%, which is of course good but needs some context. The numerical grades are described by the professor as "akin to the points awarded in a video game: significant within the game, but only within the game." Under the scoring system, it is possible for students to get more than 100%, which then gets normalized to 100%. My grade was boosted by certain features of the scoring, such as that it includes only your highest score out of three "evaluation exercises" (evaluating proofs and seeing how close your score is to the professor's; I did great on the third one); in general, the grades are a marker of progress and persistence more than performance.

4. A substantial portion of this course is about language and communication; such as in converting between natural language and symbolic logic, and in assessing the clarity of proofs. When I signed up for "Introduction to Mathematical Thinking," I wondered if I might be going too far afield in spending valuable time on something unrelated to my financial journalism day job, Erie Canal book project and overall writing career. I'm pleased to find the course was plenty relevant to writing.

UPDATE: More here.