28 votes

What am I supposed to be learning with long proofs of the main theorems in class?

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 ...
user avatar
  • 896
26 votes
Accepted

What happened to the Moore method?

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 ...
user avatar
25 votes
Accepted

What are some good examples to motivate the implicit function theorem?

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 (...
user avatar
  • 2,366
23 votes
Accepted

What am I supposed to be learning with long proofs of the main theorems in class?

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 ...
user avatar
22 votes

What happened to the Moore method?

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 ...
user avatar
  • 6,210
18 votes
Accepted

How to improve atmosphere in male-dominated courses

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 ...
user avatar
  • 196
17 votes
Accepted

What are the differences between graduate and undergraduate classes, relevant to course design and teaching?

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 ...
user avatar
  • 3,986
17 votes
Accepted

What is gratifying in being a mathematics teacher?

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 ...
user avatar
17 votes
Accepted

What famous graduate math textbooks use color?

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 ...
user avatar
16 votes

What happened to the Moore method?

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" (...
user avatar
16 votes

Teaching and "The Two Cultures"

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 ...
user avatar
  • 13.4k
16 votes

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

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 ...
user avatar
  • 4,544
16 votes

How do I learn advanced mathematics without forgetting?

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) ...
user avatar
  • 3,477
14 votes

Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation?

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 ...
user avatar
  • 13.4k
12 votes
Accepted

Is there any research on the correlation between a professor's research ability and their ability to teach graduate level courses?

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 ...
user avatar
  • 1,239
12 votes

Mathematical Knowledge for Teaching

(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. ...
user avatar
12 votes
Accepted

Teaching advanced math using books with cartoons

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 ...
user avatar
12 votes
Accepted

A4 paper of notes in an exam

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, ...
user avatar
  • 19.1k
11 votes

Using Several Textbooks in a Course

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 ...
user avatar
  • 13.4k
11 votes
Accepted

How to choose/test future tutors?

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 ...
user avatar
  • 1,791
11 votes

What does one full year of calculus mean?

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! ...
user avatar
  • 19.1k
10 votes

What are the differences between graduate and undergraduate classes, relevant to course design and teaching?

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 (...
user avatar
10 votes

What is gratifying in being a mathematics teacher?

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 ...
user avatar
  • 13.4k
10 votes

How can I discourage proof by patchwork?

Given the student's computer science background, I'd draw a programming analogy. Consider the example here: You start with this: ...
user avatar
9 votes

What are the differences between graduate and undergraduate classes, relevant to course design and teaching?

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 ...
user avatar
  • 12k
9 votes

What is gratifying in being a mathematics teacher?

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 ...
user avatar
  • 334
9 votes

The interplay of memory and mathematical performance

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 ...
user avatar
  • 13.4k
9 votes

L'Hopital's Rule: Why do we need it?

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 ...
user avatar
  • 4,544
8 votes
Accepted

How to design exercise groups in very advanced course?

The exact situation you are in is not completely clear to me, I thus give a two-part answer assuming two different scenarios. Scenario 1. The exercises are given by somebody else (the instructor of ...
user avatar
  • 7,572

Only top scored, non community-wiki answers of a minimum length are eligible