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27

Hmm apparently I will be the dissenter here. I think that long proofs taught in lectures are very much a good thing. This is particularly true for hard proofs. I will try and split the reasons why I think so into a couple of points. Hard proofs are no only hard to create but also hard to learn on your own. Have you tried learning a complex proof on your ...

26

The Moore Method is alive and well, and so are a great many variants. These days the community is more likely to use the term Inquiry-Based Learning (IBL), because the Moore Method can be seen as a restrictive set of practices, and people use the same underlying ideas in a lot of different class structures. If you want to learn more and meet people who are ...

22

The Moore method is used at the University of Chicago in some sections of "Honors Calculus", which is really an introductory real analysis course for top incoming freshmen. I assisted with it a couple of times and taught it on my own once. It absolutely depends on having a well-constructed sequence of notes to use; we started with a truly excellent set of ...

21

I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes unnecessarily complicated proofs are skipped, but not often enough to my taste. But there is a point to proving these "big theorems". The proofs contain new ...

20

Isolines and isosurfaces Isolines and isosurfaces (i.e., lines and areas of equal whatever) correspond to the graphs of implicit functions and are relevant in many sciences, e.g., isopotentials (physics), isobars and isotherms (metereology). Probably the best-known example of this kind are topographical contour lines (lines of equal altitude, see image ...

20

Memorization per se should not be the primary focus. When I learn something new, I type up notes on the computer in a reverse-indented outline format. Then as time goes on and my understanding improves, I edit the notes to reflect that. When I work an exercise or read a paper, I refer back to my notes. Memorization, to the extent that it happens, is just a ...

18

I'm a female who was often 'the only one' and later became a teacher in classes with 'only one' or very few females. When I was a student in a normal (say 100th-ranked university), I just worked hard (and screwed up the grading curve for everyone hehehe) and didn't notice the gender ratio or whatever. Then I moved to grad school at a really elite school ...

17

Of course there are many differences between undergraduates and graduates, including: Mathematical independence. In a graduate class, one may generally presume a much greater level of mathematical independence and self-motivation than in an undergraduate class. Mathematical sophistication. Graduate students generally are more knowledgeable than ...

17

One of the comments above mentions "the huge increase in cost" for using color in a book. The large cost increase for using color in a book was true twenty years ago. However, now the cost differential is quite modest. My book Measure, Integration & Real Analysis, which was mentioned in the question as an example of a math textbook that uses ...

16

I took a number theory class at University of Cincinnati that was taught using a modified Moore method. The class size was pretty small, and credit-wise it was an "upper level mathematics elective" (i.e. not a core class), so our professor had some room to be experimental. Like with the Moore method, the students presented all course material. The ...

16

In searching for information about hours of work per week by teachers at the secondary vs. tertiary level (for an earlier question) I came across a nice report by Scholastic and the Bill and Melinda Gates Foundation. A summary-type page can be found here; the full report (pdf) can be found here. (From the same study) A nice summary of "teachers on ...

16

Perhaps a bit of a rant...: As long as we condone "mathematics" being treated as a competitive sport, in whatever mode, it becomes a vehicle for ego and machismo. "Math contests" stir passions, but by appealing to some unfortunate human weaknesses. Similarly, conducting mathematics courses in a fashion allowing or encouraging "competition", as opposed to ...

16

For the project, you would need to define what are "classical results in mathematics". I suspect that different people would disagree on the classicalness of various results. Furthermore, it is doubtful if everyone should learn the same classical results - mathematics is a big field. You can, of course, cut it down by a sufficiently strong definition of "...

16

For context, I have a lot of experience self-learning mathematics. I spent a summer learning additional algebra, point-set topology, linear algebra, and analysis (to extend my undergraduate degree) before entering my current graduate program. This was sufficient to skip a literal year in the program. I am now well into the program and have to self-teach ...

14

This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the ...

12

Researchers have philosophized about and demonstrated that there is a specific kind of knowledge that teachers need that is likely different from just knowing the content itself. So, there is data that demonstrates that knowledge of content alone is not what good teachers need. Rather, they need knowledge of things like how students make sense of the ...

12

(Perhaps this should be a comment, but it's pretty much an answer.) Mathematical Knowledge for Teaching (MKT) is based on the more general term Pedagogical Content Knowledge (PCK) due to Lee Shulman. For MKT, see the work of DL Ball. Link. Deborah Ball is currently at the University of Michigan; you can find courses and related research projects on MKT ...

12

Graphic novels are an underappreciated means of pedagogy. Please look at: Galois' Dream by Michio Kuga It teaches: Group Theory Differential Equations To first-year undergraduates from a course at University of Tokyo. Graphics should certainly make the material more engaging, but I suspect difficulty is getting an artist and mathematician to ...

11

Based on extensive (if anecdotal) experience, undergrads really cannot cope with more than a single reference, which must be traversed in order, possibly omitting some sections. Having any other source is somehow beyond imagination. Subject=course=textbook. Grad students can do somewhat better, but are not happy about having to operate at a higher level. ...

11

The university I worked as a tutor at, involved an interview with a brief teaching situation. This was done for all students who applied to a job as a tutor at the math department, regardless of the intended course (e.g. linear algebra for engineers, analysis for physiscists, numerics for mathematicians). You had to prepare a short presentation (5 -10 ...

11

I believe allowing students to prepare notes for use on an exam is a valuable way to help them focus their exam studying. I do not see the creation of the sheet as a waste of time. To make the notes, they need to reflect on the course and think about what was important, then summarize the information. However, I allow a handwritten notecard on exams rather ...

10

One difference that is often overlooked is the availability (or non-availability) of mathematical vocabulary. When teaching an advanced course, I can talk essentially as I would talk to a colleague (or to myself). In a lower-level course, I have to watch my language more carefully to avoid using terms that are second nature to me (and to other ...

10

Since one point was not made very forcefully in the other answers: I like teaching mathematics because, even on the worst days, I get to talk/think/engage about mathematics... which I somehow find endlessly entertaining.

10

Given the student's computer science background, I'd draw a programming analogy. Consider the example here: You start with this: <p> <label for="field">My field</label> <input type="text" id="field"> </p> then you get rid of all that annoying boilerplate and put it in a function: createFieldHtml( id, label ) ...

10

Asking random math educators about the policies of a specific graduate school makes absolutely no sense at all. The page you were looking at has this contact information at the bottom. Use it! Adolescence Mathematics Contacts 45 Columbus Avenue 2nd Floor New York, NY 10023 Tel: 212-636-6400 Fax: 212-636-7106 Email: gse_admiss@fordham.edu Dr. Alesia ...

9

The difference between undergraduate and graduate is a matter of degree, not a clear cut. The difference between first and third year students are as large (or larger) than undergraduates and graduates. If anything, graduate classes are easier, the students take the courses because they are interested, not because they have to take it as part of the ...

9

What I find gratifying about teaching, math specifically, is learning from my students. I cannot count how many times I have had students come to me and ask me to check their work because they were not sure they were doing it correctly, and they end up teaching me a method of solving the problems that I have never thought of before. I love showing students ...

9

Anecdotally, based on self-observation and observation of many faculty and grad students: "if it's not in your head in some form, you can't think about it". A funny point here is that it seems not strictly necessary to "completely understand" something, if one can keep it in one's mind. Indeed, I don't see how to make the transition from not-understanding ...

9

This is an answer to the title. Defining APOS & RME framework would make answering the question easier. As Massimo Ortolano mentioned in a comment, l'Hôpital's rule is one tool in a box. Maybe you use it often, maybe rarely, but it is very nice when you can use it. Just the other day, when trying to understand how ultrasound mediated electrical ...

8

Some ideas: Literature (sorry, don't have a reference handy) says that when you ask one student, other students stop thinking. So, if you are going to do that, pose the question to the whole group, give a bit of wait time (6-10 seconds)so everyone is thinking about the problem. Spending some time at the beginning of the course to establish communication and ...

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