Course MAT198 in Spring 2014

Useful Fictions: How and why mathematics is developed and then changes the world

This course is geared toward students with no prior university math experience.  It aims to provide a view of mathematics as a living, growing, creative human endeavor that classifies as both a science and an art, to give a feeling for, and some mastery of, the mathematical way of thinking (including "doing mathematics") as well as an awareness of some of the many applications of mathematics in today's world.  Active class participation is an essential component of the course, required along with participation in Professor Keith Devlin's (free) Stanford MOOC "Introduction to Mathematical Thinking."  (The course was very popular at Stanford among non-science majors looking to satisfy their Quantitative Reasoning requirement.)

We start out with a general discussion of the nature of mathematics, adopting a broadly historical approach.  After that, the main aim will be to investigate several important areas of contemporary mathematics.  The exact choice will depend in part on the interests of the class, as determined by a student questionnaire completed during the first class meeting.   Students will gain extensive experience at "doing mathematics" in Professor Devlin's online course "Introduction to Mathematical Thinking", which will run at the same time, and students should come to precepts prepared to discuss the issues introduced in the online lectures and present solutions to problems assigned in the online class.

Expect to read, to write a paper, to discuss ideas, and to be creative.  Expect to examine our world and ourselves in a novel way, a way developed by thousands of years of combined human intellect.  Do not expect to spend a lot of time learning how to 'solve problems' (in the sense of a typical high school math course).  This is definitely not a course to improve your math skills (though that may be a by-product for some).  The MOOC component focuses on developing mathematical thinking ability, enabling you to learn how to set about solving novel problems.

Topics

After a broadly historical introduction to mathematics we will investigate selected areas of contemporary mathematics.  Professor Devlin will illuminate the abstractions of mathematical thought via everyday examples from human behavior and social trends, political science, and linguistics, as he did when he was a consultant for the CBS television series “NUMB3RS” . He will also introduce some of the ideas behind the video games his new startup company InnerTube Games is developing, including the first one, Wuzzit Trouble, released a few months ago for iPad, iPhone and Android devices.

Unlike traditional courses on mathematical proofs, grading of your MOOC work will emphasize evaluation of mathematical proofs (or purported proofs) rather than their construction.

Description of Classes

The class meets Mondays and Wednesdays from 1:30 to 2:50PM.  There will also be a weekly precept, with times to be determined.   The instructor's office hours are tentatively scheduled for Monday 3:30-4:30, Wednesday 10:00-noon and by appointment (arrange by e-mail).

In addition to your active participation in the class, you should expect to put in several hours a week in the form of further study (mastering any mathematical idea takes time and effort), reading, doing homework assignments, and discussing ideas with each other.  The course will be assessed on the basis of:

  • Final paper in lieu of Final Exam (35%): The final 8-10 page paper must be treated seriously.  It will involve a lot of time and effort.  Be prepared to go through several draft versions.  So start early to ensure that you will be able to consult your colleagues, a TA, or the instructor when you meet the inevitable difficulties.
  • Class/precept participation (30%): Student presentations and group discussions primarily based on the material in the MOOC.
  • Weekly quizzes on the readings (10%): Note these are short, closed-book challenges to let you know if you are mastering the weekly readings sufficiently well.  They do not require execution of mathematical procedures.  If you do the reading with enough reflection, you should score full marks on each quiz with ease.
  • Problem Sets (25%):  Students are required to complete the weekly Problem Sets in the MOOC, including the final (anonymous) peer evaluation process, and to discuss them in the precepts.  Discussion prior to MOOC submission should not involve providing others with the answers, since performance on those Problem Sets is your main indicator of progress in understanding the nature of mathematical concepts, argumentation and proof.  Joint work on the Problem Sets is permitted, though not encouraged.  If you are handing in joint work, then you must indicate this clearly on the top of the first page, listing your collaborators, in keeping with the usual standards of academic honesty.  If is preferable that you work out your own final version of any joint work and write it up alone.
Notes

The course is based loosely on The Language of Mathematics by Keith Devlin. You will be required to read chapters and then discuss them in the precepts.  The book is 13 years old now (as of 2013), and some things have changed, but we can talk about updates in class.  Because it is old, you can probably get a used copy pretty cheaply.  An excellent supplementary (free) online source is Mathigon.  The textbook for the MOOC part is Introduction to Mathematical Thinking and can be obtained for $9.99 (new) from Amazon (print or Kindle).  This text is optional, but many students of the MOOC find it helpful.

Each MOOC lecture has an associated Assignment sheet of problems (not assessed).  These will form the core of the precept discussions.  You should complete as many of them as possible, ideally in collaboration with other students, before attempting the (machine graded) Problem Set, for which they are preparatory.

You are strongly encouraged to participate in the MOOC discussion forums --- especially if you are unsure of your mathematical ability.  The student body will be global, comprising people with very different ages and backgrounds, some 70% or more of whom are likely to have at least one degree, but all of whom are eager to improve their mathematical thinking skills.  You will encounter 14-year olds whose mathematical ability will astound you, and fifty-year-old college professors who are finally trying to fill in a mathematical hole in their education.  The instructor will occasionally ask TAs to initiate discussions of particular forum threads.

Who Takes This Course

This course is aimed at students who would like to fulfill the QR requirement with an interesting and challenging course.