Introduction to Mathematical Thinking
Keith Devlin
Learn how to think the way mathematicians do - a powerful cognitive process developed over thousands of years.
Dear FIX,
I hope you are continuing to enjoy the course and are getting some benefit from it. Many students have been emailing me with questions, comments and requests. In a regular, physical class with maybe 25 students, I encourage this. But with almost 60,000 students (our current total is 59,738), it's clearly impossible to structure a course that involves one-on-one contact with the instructor. MOOCs are different. The basic mechanisms to resolve issues are student-student interaction and crowd sourcing. (This has what I see as a positive educational value: it de-emphasizes the instructor and puts the onus for learning on the student, where it should be.) Whereas I can't handle individual emails (even if only 0.1% of the class emailed me each day, that would be over 50 emails a day!), I do monitor the forums, and when I see a thread that has a lot of votes, I take a look, and occasionally add a comment if I think it will help. (Often I don't jump in, because I think the learning experience is better for all if the entire class arrives at a conclusion through discussion and debate.) You can be sure that if you have an issue that many others in the class have, I'll see it when it gets a lot of votes on the forum. This is pretty impersonal, which I don't like, but the team putting this course and the Coursera platform together are all working to try to improve things as we go along. This is new new territory for all of us. (And as with all new territory, I hope you find it as exciting as I do.) -- KD
Dear FIX,
Looking at the forums, it's clear that not everyone realizes what we are doing. This is not a course about mathematics (i.e. the formal study of rigorously defined abstractions), rather the development of **mathematical thinking**, which can be applied in the real world as well as within formal mathematics. Our current **goal** is to make language precise so we can use it in domains where absolute precision is critical. We do that by starting with language as used in the everyday world. (How else could we start?) But in the real world, absolute precision is rarely possible; we have to make judgments. The real world is (in many ways) sloppy and so is the language we use in everyday life. Yet people rarely have any trouble communicating, so in a very important way it's not sloppy, indeed language as used is remarkably precise. The challenge is to take that de facto precision and use it to motivate and guide precise definitions that can then be used in domains where there is no familiar context to rely upon. At this stage we are making the first step - introducing precision in a sensible way that accords with our intentions. In week 5 we'll take a look at some important examples of the second step, using language in formal mathematics, where total precision is crucial. This first step is in many ways a lot harder than solving a formal mathematics problem, since it requires often sophisticated judgment (no rules!) as well as mathematical skill. Incidentally, what we are doing now is an example of what is called mathematical modeling. Many uses of mathematics in today's world require this kind of reasoning. I hope this helps. Oh, and don't forget my exhortation to SLOW DOWN. Rushing into formal mathematical derivations and computations is very definitely not what this course is about. [I wasn't joking when I said the best students in high school math often have the hardest time with this transition material. :-) ] Hang in! -- KD
Dear FIX,
If you are one of a number of students who had trouble with the two questions about Alice in Problem Set 1, make sure to "attend" the tutorial session this Wednesday, released at 10:00 AM PDT, when I'll be explaining some of the problems in Assignments 1 and 2 and in Problem Set 1. If you are not following the forums closely, do make sure to check out this thread, which is a superb example of how MOOC learning should occur. Also, check out this blog post from one of your fellow students who contributed to that discussion. By my thinking, the role of the "instructor" in a MOOC is essentially the same as the individual described in this classic song from 1977. -- KD
Assignment 2 (from Lecture 2) can now be found on the "Course assignments" page. (See the left-hand navigation column.) Every lecture has an associated assignment, released at noon that day. Please look at the "Course assignments" page after each lecture, to find its assignment. Remember, although the assignments are not graded, working on them (coupled with the forum discussions they generate) forms the central part of this course. The lectures play a minor role. No one ever learned to think mathematically by sitting in a lecture, even less by watching one on a screen. The lectures merely set the agenda, give a bit of context and motivation, and provide some examples of mathematical thinking. Recognizing this fact is a key to success (or even survival) at university-level mathematics.
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