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How can you design an adequate course for a subject you never formally had classes?

This is similar to teaching somebody else's class.

As an example, suppose you are an algebraic geometer and you are asked to prepare an introductory algebraic geometry course but you are entirely self-taught in the area (perhaps you only had advanced classes).

What are good methods for planning

  • Learning outcomes;
  • Homework sets;
  • Bibliography used;
  • What breadth/depth to strive for.
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    $\begingroup$ Great source of inspiration for homework problems: math.se. $\endgroup$ – dtldarek Mar 23 '14 at 8:10
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    $\begingroup$ A tangential comment. A graduate student complains: "Why do I have to take complex analysis? I'm going to be a group theorist, and I won't need complex analysis!" ... An answer might be: "You could end up in a small math department, and be asked from time to time to teach all undergraduate math courses, including complex analysis." $\endgroup$ – Gerald Edgar Mar 23 '14 at 13:40
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Most of the courses I teach I didn't take as an undergraduate, including:

  • Calculus I and II

  • Abstract Algebra

  • Dynamical Systems

  • Differential Geometry

  • Introduction to Proofs

  • Point-Set Topology

In all cases, I use the following algorithm to figure out what to teach:

  1. Choose a textbook. You can usually figure out what the choices are by asking around or searching for the course title on Google and Amazon.com. (The sales rankings on Amazon.com can be particularly helpful for figuring out which books are popular.) After you've assembled a list of standard choices, order some desk copies so that you can take a look at them, and read reviews to figure out how they differ. You can choose a book based on (1) what topics are covered (2) whether reviews are generally positive or negative and (3) your perception of the difficulty of the text relative to the level of the students. Make sure not to choose a graduate text for an undergraduate course unless you're sure it's a good idea.

  2. Find some syllabi. You can usually find some syllabi online for courses that other professors have taught using the textbook. Just Google the name (or author) of the textbook together with the word "syllabus" or "course description" or "homework". You can usually find several week-by-week descriptions of what various other courses covered, and you can use these to help plan your own course.

    By the way, especially the first time you teach a course and especially if you didn't take the course as a student, you may want to err on the side of following the "standard" curriculum. It's fine if you have your own vision for how material should be covered, but it may be a good idea to try the "standard" approach once before you try to invent your own approach.

  3. Choose some course goals. This part of course construction is often overlooked. It is vital to figure out your pedagogical goals for a course, and how those goals are related to the material that you plan to cover. For example, my goals for the students in abstract algebra are:

    • Teach the basic material of abstract algebra that they will need for future math courses.
    • Reinforce the proof-writing skills that the students learned in Introduction to Proofs.
    • Introduce them to the idea that you must sometimes play with a math problem (especially specific examples) before you are ready to write a proof.

    My goals in dynamical systems are:

    • Teach the students how to use Mathematica to model complicated mathematical objects.
    • Help the students learn to make conjectures and draw conclusions based on experimental evidence.
    • Reinforce the students' understanding of calculus and linear algebra

    Note that none of the goals for dynamical systems are related to the actual content. I don't really care if the students learn certain specific topics dynamical systems -- the purpose of the course is for the students to improve their overall mathematical sophistication in the ways I have described. (Learning something about dynamical systems is a happy side effect.)

    The way to choose goals is a combination of thinking about (1) how the course you're teaching fits into the overall curriculum (2) what course goals will mesh well with the material you plan to cover, and (3) what kind of students you will have, and what they need to work on to become better at math.

  4. Make a tentative plan. Make a tentative schedule of which sections of the book to cover on which days. You can base this plan off of syllabi you find, as well as your own sense of what you plan to cover. Make a note of some topics that are optional and could conceivably be skipped if you start running behind. You should also make the other usual course decisions, e.g. what kinds of exams there will be, whether there will be weekly quizzes, etc.

  5. Play it by ear. The first time I teach a course, I don't give the students the course schedule beforehand, and I don't announce homework assignments until about a week before they are due. When you're teaching a course for the first time, you have to allow for the possibility that your original plan may have been overly ambitious, or that you didn't realize which aspects of the material the students would need to spend more time on.

    It's also very important the first time you teach a course to have a lot of interaction with the students, to make sure that you know how difficult they find the course so far, which aspects of the course they are struggling with, and so forth. Don't be afraid to explicitly ask students how the course is going, and whether they have any suggestions. You need to make sure to find out about any problems before they become serious issues.

I hope all of this helps!

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As long as the course is somehow standard, which is the case in the example, one can ask colleagues what they teach in such a course and/or one can try to find similar courses on the internet; many individuals and institutions have a lot of information on their courses available on the internet.

Some care is needed however to judge if the context is really similar, not to have found oranges while actually needing apples. This is especially true regarding prerequisites (and in particular for a course on algebraic geometry; it should make a difference if the students know some commutative algebra or not). In this vein, it also might only help that much having taken such a class oneself (except if the context was also very similar).

It is also not unlikely there is some specification of the content in the curriculum; though sometimes what is written in such documents is more a description of an 'ideal world' so it is not clear what is written there is really achievable. And, if there is not you are more flexible.

If the course is not standard it typically will also not be a very rigid (and chances are one is an expert in the field). In this case one might only start with a very rough plan, and just make the thing up as one moves along (from one week to the next). There is a risk in this that one gets of track, but then as discussed in How do you fix a broken course plan? this risk also exists if one has a clear plan.

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Hopefully you are given enough time beforehand... apply what is said here (and at math.SE) about self-learning. Check out a few potential textbooks, grab a selection of lecture notes.

Analyze the courses that come before, talk to who teaches them (anything can be said in course decriptions); also what comes after (if any). See what the students are going to need on the subject for their future professional life.

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  • $\begingroup$ I feel like this answer could be more in depth, with real-life examples or research. I would upvote it if it were edited with such things. $\endgroup$ – Brian Rushton Mar 24 '14 at 3:19
  • $\begingroup$ I was in that position a lot of times (I was part of the group starting undergraduate informatics here, having graduated as a chemical engineer, albeit with strong interest in the primitive computers available and a short stint as a systems analyst/programmer). No research to fall back on, sorry. Plus having to teach several courses taking over from colleagues who left suddenly being hired away by industry. $\endgroup$ – vonbrand Mar 24 '14 at 3:26

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