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When teaching an upper-level proof based course, what criteria do you use to determine which and how many problems to assign? And, as a corollary question: How do you determine the level of success on problems indicating success in the course?

In an advanced course, problems are intended to be hard. A student may struggle, even fail, to do them, yet still master the course:

A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced not so much to be solved as to be tackled (Herstein).

The homework problems are the most important part of the course. Many of the problems are extensive and difficult, and require hard work. They are not meant to be completed in one sitting. (Artin)

It's interesting to note that when teaching a course from his own textbook, Professor Artin himself only assigns about 15% of the problems in the book.

How do you determine which problems to assign? And how do you determine how many? Is it simply a matter of taste ("this one was interesting")? Do you try to ensure that each subtopic has one problem? How many is too much?

And, given the vast range in difficulty (from quick and easy to bordering on research), are there multiple dimension to the criteria? For example:

  1. Ensure that every concept, definition, theorem, or technique appears in at least one problem
  2. Have at least one problem that merely tests comprehension, one that requires insight or recognition of the true meaning of problems (in different contexts), and at least one that is too hard to be solved.

Or is it budget based: I need to cover X sections in Y weeks, and can assign Z hours of problem sets per week.

What criteria do you use to say "This course has enough problems. If you do them, you can master the material."


Update & Clarification

To refine the scope of the question and make it easier to answer:

This question does not seek a universal formula, but rather individual data points.

Obviously, what's appropriate for one setting is not appropriate for another. That doesn't imply, however, that it's impossible to learn from experience (your own or others). We can collect individual data and learn from it; learning from data is not mimicry.

Therefore, an answer should not be "Here is a universal formula for these criteria" but rather "In setting X, I used criteria Y, and got results Z". Extrapolation and generalization are the job of the reader of the answer, not the poster of the answer.

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    $\begingroup$ Since each college and university is different in its expectations, your best answer might come from another faculty at your institution. $\endgroup$
    – Sue VanHattum
    Commented Mar 14 at 19:53
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    $\begingroup$ @SueVanHattum I'm interested in learning what different faculty at different institutions do, and why. I don't intend to blindly say "Oh, someone posted they do X, I'll do the exact the same thing!" but rather to learn from others' decisions and experiences. $\endgroup$ Commented Mar 15 at 9:19
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    $\begingroup$ @MichaelHardy Yes, available time is a key constraint. But you can respond to it in two ways: 1. By stating "Since mastery requires at least these problems, we'll have to allocate more calendar time to this section, even if it means omitting a different section". Or 2. "Mastery can be achieved with fewer problems". I'm interested in learning how others determine what's required for #2. $\endgroup$ Commented Mar 15 at 9:21
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    $\begingroup$ @Pedro Good points. I've refined the question to make clear that what's being asked is not a universal criterion that is valid for all upper-level proof based courses but rather specific data points in specific settings. Data is valuable, even though (or perhaps because?) it requires its user to extrapolate and interpret, not copy and paste. $\endgroup$ Commented Mar 15 at 11:24
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    $\begingroup$ I upvoted, but I think "This course has enough problems. If you do them, you can master the material" is misleading. In my experience it's simply impossible to "master" the material taught in a course by participating in the course, even if one does a large number of exercises. In order to understand a topic really well I usually need several iterations - for instance when I attended a course as a student, was a teaching assistant for it later on and then taught the course myself. And even then I don't feeling like "mastering" the topic, unless I use it extensively in my research. $\endgroup$ Commented Mar 16 at 11:46

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The post has many very broad questions. Here are some principles I try to apply.


(1) Which problems?

  • If the subject has "classic exercises", they must be on the list.

  • The list must contain a sufficient number of "easy" exercises (that is, those that the average hard-working student can solve).

  • The list must contain exercises that really prepare the student for the tests.

  • The list can contain interesting results that were not covered in class and/or results (with accessible proofs) used in class without demonstration.

  • The list can contain some "hardy" problems or challenges (to emphasize that things may not be as simple as they seem). But in this case, I believe the student should know that they are not easy (personally, I like to include hints).

Warning: The meaning of "easy" and "hardy" depends on the audience.


(2) How many?

  • If all exercises will be graded, I choose just a few. (I believe this can only be done if the class is very small; I only did this once due to very specific circumstances).

  • If no exercises will be graded, I choose many. I seek to ensure, for example, that important techniques are applied more than once.


(3) How do you determine the level of success on problems indicating success in the course?

  • Type I problems (requires only mastery of standard demonstration methods and the main concepts/theorems of the theory). I believe that success on this type of problem indicates success in the course.

  • Type II problems (requires creativity, tricks or very long solutions). I believe that failure in this type of problem does not indicate failure in the course.

Warning: Possibly, not every problem is of type I or II.


(4) Is it simply a matter of taste ("this one was interesting")?

No. See (1).


(5) Do you try to ensure that each subtopic has one problem?

Not necessarily. But as I think that the main examples worked in class generally emphasize the main points of the theory, for each main example I always try to include at least one exercise that uses the same idea. And since the main subtopics usually have main examples, the main subtopics always have at least one exercise.


(6) Are there multiple dimension to the criteria?

Certainly yes.

Ensure that every concept, definition, theorem, or technique appears in at least one problem

Certainly not.

Have at least one problem that merely tests comprehension, one that requires insight or recognition of the true meaning of problems (in different contexts), and at least one that is too hard to be solved.

Desirable, but not mandatory.


(7) Is it budget based?

This is an important issue. I personally do not assign problems for which there is not at least a sketch of solutions available (either in the book or in my class notes). Therefore, the problems I assign are subject to my time in writing the solutions.

Warning: The exact moment of making solutions available for the students depends on many factors.


(8) What criteria do you use to say "This course has enough problems. If you do them, you can master the material."

It is enough when (at least) the three "must" criteria in (1) are fulfilled, with the majority of problems being Type I (as defined in (3)).

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  • $\begingroup$ Very insightful! One question: You write Certainly not... every concept, definition, theorem, or technique appears in at least one problem. How do students master the concepts and theorems that do not appear in any problem? Or is the idea that indeed, mastery of every concept isn't required for success? $\endgroup$ Commented Mar 15 at 16:03
  • $\begingroup$ @SRobertJames There are levels of mastery. In a standard course, at least according to my understanding, it is not necessary (or even possible) to master all aspects of a subject (and that's why we sometimes come across the warning "omit on first reading"). If students demonstrate satisfactory mastery of the assigned problems, I think it is safe to assume that they would be able to master additional exercises on related subjects not included in the list. I think this is an assumption that every instructor has to make, as it is not possible to exhaust a topic in a single course. $\endgroup$
    – Pedro
    Commented Mar 16 at 23:53
  • $\begingroup$ In practice, mastery of subjects not covered will result from future studies, probably arising from specific needs. n this sense, Jochen Glueck's comment above is very enlightening. Warning: the meaning of "mastery" depends on many factors. $\endgroup$
    – Pedro
    Commented Mar 16 at 23:53

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