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Background. I teach math at a German university (both undergraduate courses and courses for Master programmes), mostly for students who major in mathematics or very similar programmes. Students typically choose their major before starting their studies. Courses aimed at students who major in mathematics are proof-based from the very beginning and proofs are an essential part of both homework and exam problems.

Context. I regularly find myself in discussions about whether every course should be designed around measurable learning objectives. When I point out that I don't know how I should do this for any but the most elemantary math courses that I teach, this usually leads to one of the following two points:

  • The very generic comment that it's of course possible, if one just commits enough to doing it.

  • Some examples to show how it should be done, but which are extremely elementary (along the lines of "How to prove a formula by induction").

Both answers don't help me to see how this could be done in my courses in practice.

Empirically, when I look at descriptions of math courses (in Germany, those are collected in so-called "module descriptions" which are supposed to have a section that lists the learning objectives) I always see one of two patterns: Either the objectives are formulated extremely vaguely (which contradicts the goal to make them measurable) or they are essentially a rewording of the table of contents (sometimes enriched by a few verbs to make it sound more like objectives rather than contents).

Question. I'm looking for concrete examples of somewhat advanced math courses that are based on explicitly stated measurable learning objectives.

Criteria for a good answer:

  • It should be a proof-based course which requires a certain level of abstraction. I'm thinking, for instance, of an introduction to point set topology or measure theory or group theory or functional analysis, or anything of a similar level.

    I am not interested in "Introduction to proofs" like courses, because for such courses I can see myself how this approach can work.

  • The description of the learning objectives should be sufficiently concrete to make it clear that and how they can be measured. Moreover, they should not merely be a slight rephrasing of the course contents (because if they were, I wouldn't see the point of it).

  • It's not so important for me to have access to all the courses materials. I'd mainly like to see the description of the learning objectives and a (maybe brief) description of the course contents.

  • Bonus points if the course is even part of an entire math programme that is designed around measurable learning objectives (but that's not a requirement).

I'd be happy with sources in English or German (or, if there's no help otherwise, in French).

Note. This question is not about the following things:

  • I'm not asking about opinions on whether basing courses on measurable learning oucomes is a good or bad idea. I'm only asking for concrete examples where it has been done.

  • I'm not looking for advice on how I could create such a list of measurable learning objectives myself for a course. This would also be an interesting question, but it is not part of this question.

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    $\begingroup$ For clarity, can you give an example of a measurable learning objective that is sufficiently concrete and acceptable for you? You state that "the description of the learning objectives should be sufficiently concrete to make it clear that and how they can be measured."And "they should not merely be a slight rephrasing of the course contents." What is an example of what you will accept? $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Dec 10, 2023 at 18:52
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    $\begingroup$ You may want to investigate courses that use a standards-based grading (SBG) system of assessment. One of the resources available from the Grading Conference is a shared folder of syllabi which includes some courses you mentioned. $\endgroup$
    – Brendan W. Sullivan
    Commented Dec 10, 2023 at 18:52
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    $\begingroup$ @MahdiMajidi-Zolbanin: I can try. Here are two examples, say for a course in functional analysis: (i) "The students are able, for examples of linear operators that they have not seen before, to prove boundedness, determine the operator norm, and derive information about the spectrum that is not completely obvious but easy to obtain by using the results proved in the course." (ii) "The students are able to use the main theorems from the course to prove easy theoretical corollaries that they have not encountered before." $\endgroup$
    – Jochen Glueck
    Commented Dec 10, 2023 at 19:13
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    $\begingroup$ @MahdiMajidi-Zolbanin: I think I could do this for the entire course, but would face the following difficulties: (i) It would give a really huge list of objectives. (ii) It would take me half an eternity. (iii) Most functional analysis instructors might consider the list more or less obvious consensus, thus raising the question what's the point (beyond listing a table of contents). (iv) Most importantly, the list would miss many objectives which I find important but am unable to phrase in a measurable way. So I chose to ask for concrete examples to see if/how they solve those issues. $\endgroup$
    – Jochen Glueck
    Commented Dec 10, 2023 at 19:23
  • $\begingroup$ @Raciquel: Thanks a lot for the links! Regarding your first link: Yes, I think this is very vague. I actually believe that the list consists of good and worthwhile goals for a math programme (and I think many other math educators would probably agree that they are worthwhile). I just don't see how these are measurable goals (or what the point of listing them is at all, since the goals are so vague that they probably apply to almost all math programmes in the world). [...] $\endgroup$
    – Jochen Glueck
    Commented Dec 10, 2023 at 22:34

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