"The abstraction, which makes the course attractive for us as instructors, hits many of the students like a ton of bricks." -- Professor and Chair Selim Tuncel
Students usually don't have too much trouble with the largely computational first chapter, Matrices and Systems of Linear Equations. (See, however, the last paragraph below for increasing conceptual content in this section of the course.) Although this material is an essential tool in the rest of the course, be sure you finish it in no more than the nine lectures specified in the Math Department's Math 308 syllabus.
For the rest of the quarter, David Rosoff's
advice to students also provides insight for a new instructor in the course:
"A word of caution: do not be led into complacency by the relative ease
of the first few weeks. Many students experience a 'jump' in the difficulty
of this course after the first midterm. To reduce this jump you must be sure
that you understand not only how to solve the homework problems, but also
the concepts (i.e., definitions, theorems, and, to the extent you're able,
their proofs) discussed in lecture and in the book."
"Another important thing to understand about this course, especially if you
plan to go into an engineering or other applied science field, is that
everything we'll do this quarter can be done very quickly by computers,
much more quickly (and reliably!) than a human can do by hand.
Hence the point of the course is not only for you to memorize the algorithms.
Rather, in order to program the computer properly you have to know what you
want to do and why it is possible. Knowledge of the underlying theory will
enable you to understand how and why the software doesn't give
expected results (as will inevitably happen at some point).
This underscores my remarks above concerning the due diligence you should
afford the conceptual material."
Keep in touch with where your students are with the material. Most of the students have little experience with really using mathematical definitions, and will nod happily through a lecture then be unable to give you even the most basic definitions or facts that were discussed. It's probably a good idea to break up your lecturing by asking the students to do something simple with the definitions or ideas you have just presented, either in small groups, or alone then check with a neighbor, while you walk around and get a feel for how well they are absorbing the material. Some instructors have students fill out an online survey every week, or every other week, that asks what part of the week's material was most confusing, and and for any general comments about the course: organization, lectures, homework, etc. You can give some homework credit for the survey to encourage participation.
Teach proofs. That is, give students some explicit gudiance about what is expected in answers to questions asking for explanations or proofs. Most will not be able to just learn this by experience, or realize how to extract the skills they need from the text. Examples for them to use as models can be very helpful to them, and make your job grading tests easier.
Whenever possible, draw pictures. Be especially conscious of this if you are more algebraically than geometrically inclined, because the reverse is likely true for many of your students. Even if you have to choose absurdly simple examples to make it possible to draw, do try to do at least a couple of drawings for each new concept.
Here are a couple of common sources of confusion. Early in the book, all matrices represent systems of equations, so students may think a matrix is the same thing as a system of equations. At the start of chapter 3, you may want to introduce briefly the idea of a matrix as a function from Rm to Rn, and give some simple examples \ such as rotations and projections. The book uses set notation ({x : ... }, intersections and unions, etc.) with little or no explanation, and some students don't know it already. Please note other common confusions you notice on the Math 308 Instructor Materials page (link below).
The Math 308 Instructor Materials page in the Math Department Wiki gived links to sample course web sites, and other ideas from previous instructors. In particular there are links to some problems Ralph Greenberg developed to shift some of the conceptual challenges earlier in the course.
Return to the
Index for the 3xx Instructor Guide.
Most recently updated on September 23, 2011