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I believe that good math courses are structured around measurable learning goals. For example, "can correctly replace a line integral with an equal double integral using Green's Theorem" or "can use the Sylow Subgroup Theorems to prove the smallest non-abelian simple group has order 60."

I also believe that good math courses should give students a deeper understanding than "follow these rules". In particular, at least the major level courses should certainly include proofs of main theorems, such as the Sylow Subgroup Theorems.

But I cannot come up with a measurable learning goal (one that doesn't start with "understands") that would motivate a class activity that includes proving important theorems.

My example of using the Sylow Theorems is a good learning goal, but if that is the only goal, there is no reason to explain the proof of the Sylow Theorems. But I also feel that "Can reproduce a proof of the Sylow Theorems" is going too far.

I hope you can understand the tension I'm getting at. Have you found any way to justify teaching well-known (beautiful!) proofs in a learning-goal motivated course design?

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    $\begingroup$ @StevenGubkin: Well, my point (i) is that an undergraduate math course can mean pretty different things, in particular in different countries; see e.g. the first three answers to this question. Somebody teaching at a US college, where undergraduate education is very broad and first and second year math courses such as Calculus might not focus on proofs too much, will probably give a very different anwer to the question "what is the goal of teaching the proofs of the Sylow Subgroup Theorems" [...] $\endgroup$
    – Jochen Glueck
    Jan 3, 2022 at 0:01
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    $\begingroup$ [...] than somebody teaching at, say, a German university where first year students are bombarded with rigorous proofs of almost every single theorem that they are thaught in linear algebra and analysis (whether or not this might be a good idea). So I think OP should at least specify the country. [...] $\endgroup$
    – Jochen Glueck
    Jan 3, 2022 at 0:02
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    $\begingroup$ [...] As for the second point, I agree that courses are structured around learning goals - but it is much less clear that all or most of these goals are (or should be) measurable. The fact that we have to set exams that are supposed to measure the achievement of certain goals by the students does not imply that, conversely, every goal that should be achieved can be reasonably measured (even less so on short-term). Of course, it has certain advantages of a learning goal is measurable - but it is not at all evident to me that we should structure our courses around this type of goals only. $\endgroup$
    – Jochen Glueck
    Jan 3, 2022 at 0:02
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    $\begingroup$ Interesting discussion. @JochenGlueck, I don't feel this question is country-specific. If there is a measurable learning goal in Germany, then there is a measurable learning goal in any country. Your second point, "why does everything have to relate to a measurable learning goal?" does seem up for debate, but elsewhere. I believe the question as posed is well-defined, even if someone thinks measurable learning goals are not necessary. $\endgroup$
    – Duncan
    Jan 3, 2022 at 17:08
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    $\begingroup$ Some wise professor once said that the quality of a Syllabus is inversely proportional to its length. Those words are more relevant today then ever before. $\endgroup$
    – James S. Cook
    Jan 4, 2022 at 19:36

3 Answers 3

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I assign this reading early on in the kind of undergraduate classes where I expect students to understand proofs:

http://www.ma.rhul.ac.uk/~uvah099/Maths/HoddsAlcockInglisSelfExplanation.pdf

This handout is based on the following research:

https://www.jstor.org/stable/10.5951/jresematheduc.45.1.0062#metadata_info_tab_contents

Once students understand the difference between "self-explanation" vs. "paraphrasing" or "monitoring", I can then make a measurable standard like:

"I can write self-explanations for various proofs of the Sylow Theorems".

This does not require a student to reproduce a proof (not one of your goals), but it does require that the student can comprehend a proof that someone else has written and provide evidence. I probably wouldn't put such an assessment as part of a timed examination, but I might assign it as part of a homework assignment, or the take home portion of an examination.

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  • $\begingroup$ I'd be interesting in reading that handout, but the file is flagged as a security risk by my university, presumably because of the http instead of https. Is there another version available? $\endgroup$
    – Stef
    Jan 3, 2022 at 13:45
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    $\begingroup$ drive.google.com/file/d/1AAx3sbyx-CQnjZqiHEKbNO_mIkBPqKo0/… $\endgroup$
    – Steven Gubkin
    Jan 4, 2022 at 15:08
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    $\begingroup$ Thanks! In "practice proof 1", it should be "such that for every positive number s, 0 < r <= s", not "0 < r < s". $\endgroup$
    – Stef
    Jan 4, 2022 at 15:13
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Several professors I've had over the years, when asked why they were showing us a proof well beyond our fabrication, justified it as follows.

Mathematics essentially contains a finite number of "tricks". As you develop your skill in mathematics, you accrue more of these tricks that you have seen employed in various contexts. It is rare that a proof requires an ingenious trick -- it is more common that a proof is a series of these tricks strung together with some mathematical thread. There is ingenuity in this stringing together, of course, but the tricks are not new. The justification given is then two-fold:

  1. To show us more of these tricks so that we may go on to use them (or modifications) in subsequent proofs. Famous proofs are often famous because they used a new, ingenious trick.
  2. To show us how one goes about stringing together these tricks into a coherent proof.

I might add a third reason as well. Mathematics is beautiful, and I think it is important that students gain an appreciation for this. Showing them a proof you find beautiful, and explaining why it is so, is definitely working towards that goal.

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Your quest for measurable learning outcomes suggests a framework rooted in Bloom’s Taxonomy and the higher-education assessment industry that has been built around it. As I think about your question, I think about inquiry-based learning [IBL], especially when implemented in upper-division proof-based mathematics classes. In this Notice of the American Mathematical Society Doceamus article, we find this description:

Students in IBL proof-based courses are asked to develop the fundamental concepts and to produce the proofs of the important theorems. This may require abandoning a traditional textbook in favor of a customized sequence of tasks that meets the students where they are mathematically and is designed to guide them on a journey of mathematical discovery. As opposed to completing exercises after an instructor has covered the relevant material, students decipher definitions, explore examples, make conjectures, and prove theorems.

These videos from the Academy of Inquiry Based Learning show how IBL can look in a classroom.

Evidently, the “learning outcomes” of such an IBL approach are not neatly measured by the so called Bloom taxonomy action verbs of the assessment industry. Of course it is not realistic to use IBL in every undergraduate mathematics course, and I am not suggesting that you need to adapt this approach. Rather, I am pointing out that it is OK to give yourself license and not be a slave to measurable learning outcomes. It’s OK to design your course so that you can include proofs of important theorems in your class. Not everything you do in class as the instructor needs to be associated to an “action verb” for assessment.

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    $\begingroup$ “Action verbs of the assessment industry”: nice phrase! $\endgroup$
    – KCd
    Jan 2, 2022 at 16:33
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    $\begingroup$ Is there somewhere I can read more about counter-arguments to measurable learning goals? Perhaps I am naive on the subject, but I believe that every worthwhile classroom activity should have a learning goal associated with it. In response to your answer, my first reaction is to ask what is the learning objective that IBL is supposed to be accomplishing. $\endgroup$
    – Duncan
    Jan 3, 2022 at 14:13

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