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If a maths lecturer is lecturing a higher-level undergraduate course outside their area (i.e. has not taken a course in themselves or learned themselves over the course of their research), how could you justify to the students that they are suitably qualified to teach and assess them?

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    $\begingroup$ I'm not sure I understand why you should be justifying the lecturer's qualifications to the students. Presumably your department has ascertained that the individual is indeed knowledgeable enough in the area to provide high-level instruction (that is very important). That should be suitable justification in and of itself. To maintain that trust, it is essential to never permit an unqualified individual to instruct a course. $\endgroup$ – Michael Joyce May 16 '16 at 18:13
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    $\begingroup$ I'd go with "you cannot". $\endgroup$ – Daniel R. Collins May 16 '16 at 19:00
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    $\begingroup$ It depends on the background of the Lecturer. After some point of mathematical maturity it becomes comparatively simple to assimilate new material. Also, if this person is a good teacher and cares about the students then they'll take the time to see how other folks are teaching similar courses at other good institutions. But, the students, well, as long as he doesn't grade too hard they'll be fine. $\endgroup$ – James S. Cook May 16 '16 at 19:23
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    $\begingroup$ Could you contextualize this question a bit more? Right now, you could cross out the word maths from the question and it'd read essentially the same; is there some way in which this being math-specific should/would/could affect the answer? Also confusing: Is the person a non-expert (as in the title) or generally unfamiliar with the material (as in the body)? There's a difference between a non-expert and a total naif. Finally: Is the person "suitably qualified" for this position? If yes: Why? If no: This question looks to me like it is asking about tricking students. $\endgroup$ – Benjamin Dickman May 17 '16 at 3:46
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    $\begingroup$ "I think being in maths rather than another subject could affect the answer." - Based on my discussions with colleagues in other disciplines, I do think it is somewhat peculiar to mathematics that people often are called upon to teach completely out of their 'specialty' (and not just generalist courses). Plenty of math PhDs with no statistics background whatsoever are asked to teach both intro and mathematical statistics every year! $\endgroup$ – kcrisman May 17 '16 at 14:44
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The comments are very interesting, but you are talking about how to justify to students. That may not always be necessary, but here is a stab at it if someone really asked (which I find somewhat unlikely).

The instructor has a PhD in mathematics, which means that s/he has learned not just how to learn independently, but how to become an expert in mathematical subject matter. Given the interconnectedness of mathematics, I am confident that Dr. X will do an excellent job not only with the subject at hand, but how it fits into your studies in general, both before and after this class.

Some may disagree with this characterization, but anyway I think it's more true than not.

Obviously the first time won't be the best, but that is something that can be said for any course when taught the first time, including freshman calculus/college algebra/finite mathematics or whatever your "first" college course happens to be. How many people really "get" the Mean Value Theorem until they have to teach it? Proving random stuff with it in some undergraduate real analysis course also likely lies years behind instructor Y or Z, not just instructor X.

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    $\begingroup$ (Something you might not want to say is that the instructor might actually be more motivated - and hence better - by getting to learn something they didn't learn before but always wanted to, as opposed to something they now find sort of uninspiring or boring. It's likely to be true reasonably often, but probably students won't like that rationale as much.) $\endgroup$ – kcrisman May 17 '16 at 14:57
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    $\begingroup$ Not only a non-specialist can be more motivated, but he or she can see more easily what is difficult to understand, not having years of habits to hide difficulties from her or him. $\endgroup$ – Benoît Kloeckner May 23 '16 at 8:38
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Usually undergraduate level courses are not very difficult for experienced researchers/lecturers. Even without previous research experience in the specific topic, lecturers can learn the material very fast and be able to teach the topic to students well enough afterwards.

However, it may be the case that the lecturer appears to be unqualified or unfit to teach the material, but based on my personal experience I think this often reflects on the lecturer's ability to teach rather than their knowledge on the subject.

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In France, my answer would be that we only hire faculty who can teach every mathematical subject at every undergraduate level (possibly with quite some work, of course). This is most usually included in the job description, and is pretty accurate in practice.

One thing that makes it relatively easy to ensure that a good majority of faculty have this broadness is that a good majority of us have taken Agrégation, a national competition intended to hire high school teachers of higher mathematical level, where one has to demonstrate a good mastery of a big chunk of undergraduate mathematics.

Another point is that the higher education in Écoles Normales Supérieures is very broad: one does not specialize too much too early; and a good majority of faculty in France went to one of these ENS. Of course, we also hire people whose higher education was at a university, and foreign mathematicians, but many of them are equally broad, and all in all I think the vast majority of faculty is able to teach any undergraduate course, if needed.

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The one thing I would add--hopefully not considered a segue--is to avoid justifying or opening up a discussion of justifying. The natural impetus of the students is to assume competence of the instructor. Don't weaken that.

You may feel an urge to confession ("first time teaching this" or "never took this"), but it is likely you are trying to gain sympathy or to set expectations lower. This is a natural human thing to do, especially for bright people with inverse Dunning-Kreuger. But don't do it. Stay positive, stay in control, and just do your best to teach a great course...with a stiff upper lip about your lack of experience. Nobody wants to hear your issues...they want to learn the topic.

Source: instructor training for first aid and skiing. Absolutely do not tell your classes, you are a first timer. And I believe the commonality here is psychology and pedagogy. Not course content. IOW, the "educators" part is dominant, not the "mathematics" part.

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  • $\begingroup$ I have come across situations where the students do not automatically accept the lecturer. Besides, I don't think it's really appropriate to be doing something like that if it can't be justified to the students. $\endgroup$ – Jessica B Jul 1 at 18:07
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Assessment is the easy part, anyone who is well versed in the logic of mathematics should be able to vet the student's proofs and computations.

Teaching is less clear. You want the lecturer to know what the important parts are. If they aren't familiar with the subject, how are they going to know where the most fertile ideas can be found? (This can be remedied if there is a good textbook available.) Still, you can always run the class --- not as a lecture --- but as a seminar where everyone is upfront about the fact that everyone, including the professor, is learning the subject together. I've certainly had classes like that in graduate school, for emerging topics. They were fun!

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    $\begingroup$ I think there are problems with assessment. While in theory you should be able to verify correctness, I find that even in familiar topics it can be hard to spot all the tricks that sound reasonably but are not right for some fairly subtle reason. Judging the value of an incorrect answer seems potentially problematic too. Everyone learning together sounds fine for graduate level, but I find it hard to picture that working so well in an undergraduate context where a mark has to given at the end. $\endgroup$ – Jessica B May 17 '16 at 20:19

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