Tuesday, August 16, 2016

expectations in analysis

I’m working on the syllabus for my (junior and senior level) analysis class this fall, and I’d like to share some parts of it, hopefully thereby eliciting feedback. The main thing I’m concerned about is the type of specifications grading I’m adopting for the class—I’ll share that later this week. This post is about establishing the expectations of the course, on which the specifications will be based. None of these are particular to analysis; they establish what I believe any student in an upper-level mathematics course should achieve.

The rest of the post is taken verbatim from (the current draft of) my syllabus.

To learn mathematics, it is essential to engage actively with the material. This is especially true at this stage in your mathematical careers, as the focus of study shifts from developing computational tools to examining underlying concepts and practicing abstract reasoning. This shift may be described as a move from pre-rigorous thinking, which is informal and intuitive, to rigorous thinking, which is formal and precise. (This terminology has been suggested by mathematician Terence Tao; he also includes a post-rigorous stage, in which professional mathematicians work, where one is able to make intuitive arguments that are grounded by formal training.)

The content of this course resides in definitions, theorems, and proofs. You will be expected to state both definitions and theorems accurately and to illustrate them through examples. Mathematics is not merely a collection of disconnected facts, however, and so you will also develop your logical skills by proving mathematical truths, linking definitions to their profound consequences captured by theorems. All of this will happen in the context of a community—two really, our class and the larger mathematical community.

Definitions. In mathematics, as in other sciences, it is necessary to quantify what is being studied and to be able to identify what is of interest at each moment. This is done by carefully establishing and internalizing definitions. This is not to say that definitions do not involve creativity; as a subject develops, often definitions evolve to encompass more or fewer cases, to be more precise, or to reorganize ideas.

By the end of the course, you should be able to:

  • state definitions accurately and explain any notation or previously-defined terms they contain;
  • identify whether or not an object meets the conditions of a given definition;
  • give examples that satisfy a given definition as well as examples that do not satisfy it;
  • test an unfamiliar definition using examples;
  • create new definitions when needed.

Theorems. A theorem has two parts: the antecedent (its assumptions) and the consequent (its conclusions). To take a familiar example, the equation \(a^2 + b^2 = c^2\) by itself is not a theorem; rather, the Pythagorean Theorem states that “If \(c\) is the length of the hypotenuse of a right triangle, and \(a\) and \(b\) are the lengths of its other two sides, then \(a^2 + b^2 = c^2\).” A theorem may not always include the words “if” and “then,” but you should always be able to determine what are the antecedent and the consequent. Sometimes rephrasing the theorem’s statement can help. For example, “Every differentiable function is continuous” can be rephrased as “If a function is differentiable, then it is continuous.” In most cases, the consequent does not imply the antecedent (e.g., not every continuous function is differentiable). A theorem that says one set of conditions holds “if and only if” another set of conditions holds is logically making two statements (the antecedent and consequent can be reversed), and both must be proved.

By the end of the course, you should be able to:

  • state theorems accurately and identify what are their assumptions and their conclusions;
  • determine whether the conditions of a theorem do or do not hold in a given situation, explain why, and determine what the theorem does or does not imply in that situation;
  • recognize logically equivalent forms of a theorem;
  • formulate and test conjectures.

Proofs. Proofs are how we as individuals and as a community determine the truth of mathematical statements, i.e., theorems. Here is one definition of a proof, due to David Henderson: A proof is “a convincing communication that answers -- Why?” The extent to which a proof succeeds, therefore, depends on how well it embodies these three properties: it should be logical (does it convince?), it should be comprehensible (does it communicate?), and it should be intentional (does it answer why?). Evidently, each of these properties depends somewhat on the others. It is thus reasonable to classify proofs into an S/Q/U system:

  • (S) A successful proof makes an argument for the truth of a mathematical statement that is fully convincing to an informed reader or listener. It employs appropriate vocabulary and carefully chosen notation. It avoids sloppy reasoning. It makes clear use of the theorem’s assumptions and, when necessary, previously known results. The best examples provide motivation for the methods chosen. Minor revisions may be advisable, but they do not hinder the overall effectiveness.
  • (Q) A quasi-successful proof contains most of the ideas necessary to make a complete argument. It may have slips in logic or notation, or it may neglect a special case, or it may be hard to read. It contains sufficient evidence, however, that the argument can be “salvaged” by filling in gaps or clarifying language. Serious revision is necessary. [Not in syllabus: thanks to Dan for suggesting “quasi-”.]
  • (U) An unsuccessful proof does not convince an informed person of the truth of the purported theorem, for one or more of the following reasons: – It makes logical leaps or omits key ideas. – It demonstrates incomplete understanding of definitions or notation. – It fails to reference previous results when appropriate. Complete revision is generally necessary.
In other words, a successful proof is of sufficient quality that it could reasonably be accepted as part of a paper in a professional journal. A quasi-successful proof has some merit, but it requires revision, after which it might or might not be acceptable at a professional level. An unsuccessful proof is sufficiently flawed that it would not be acceptable as part of a professional publication.

By the end of the course, you should be able to:

  • evaluate, on the basis of professional standards, whether a given proof is successful or not;
  • write original, successful proofs.

Community. Our class time will be structured primarily around discussion rather than lecture. The idea is to have a space that promotes sharing ideas, making guesses, taking risks, and sharpening our reasoning abilities. I will guide and facilitate these conversations, but everyone is responsible for contributing to discussions, both in small groups and with the entire class. That is, in this course mathematical authority resides not just with me as the instructor, but with every class member. I will give short lectures (20 minutes) when the entire class agrees it would be beneficial, but not more often than once a week.

By the end of the course, you should be able to:

  • engage in discussions about mathematics by sharing questions, proposals, and insights;
  • evaluate others' contributions critically and respond constructively;
  • present your own work in front of an audience and address their comments and questions.


Bret Benesh said...

You have put a lot of thought into what a student should be getting out of a senior-level math course. I am teaching abstract algebra this semester, and I might be borrowing from this language liberally.

Joshua Bowman said...

Bret: Thanks! Please do!

jsa said...

Dear Josh, this is great! I am teaching an elementary number theory class this fall, with an IBL style and a focus on proofs, may I borrow some of this language?

Joshua Bowman said...

JSA: By all means!

Theron Hitchman said...

I am late to the party, but I really like what you have done here. Good work.

Joshua Bowman said...

TJ: Not late at all! Thanks for the response!

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