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 ...
  • 968
28 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

I think a more correct view is that proof is the LAST of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case. Before ...
22 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 ...
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 ...
  • 196
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 ...
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 ...
  • 4,746
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) ...
  • 3,694
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 ...
  • 13.6k
14 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

I have taught Discrete & Computational Geometry to US undergraduates project-based, as opposed to assignment- and test-based. Some of the projects do involve proofs, but others are more ...
14 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

With the technological advances of the past couple decades, computational mathematics is now somewhat accessible to undergraduates. The wikipedia entry for computational mathematics lists out the ...
  • 6,514
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 ...
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, ...
  • 19.1k
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! ...
  • 19.1k
11 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

In other words, "knowing" math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew. You appear to believe that somebody ...
  • 211
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 ...
  • 13.6k
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: ...
10 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

When I describe undergraduate research to students majoring in mathematics, I ask them to browse the abstracts of the most recent MAA undergraduate poster session. Here is a link: Abstracts for the ...
  • 8,093
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 ...
  • 13.6k
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 ...
  • 4,746
8 votes
Accepted

How can I discourage proof by patchwork?

Instead of providing your student a counter-example, you could try to ask her or him about its faulty proof. Justifying every step until the faulty one is identified by the student herself might ...
8 votes

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

This is a great question. Here are some thoughts on it. A theorem statement is a sign of an idea that tends to be useful in the pattern of mathematical inquiry in a given subdomain. A good theorem ...
  • 5,857
8 votes
Accepted

How much prior math should I review in teaching a graduate-level course?

Even for math grad students, I'd forcefully review much more than many traditions seem to indicate. That is, I would not presume perfect recall of the standard curriculum, especially either in detail ...
  • 13.6k
8 votes
Accepted

What does one full year of calculus mean?

In the US, the stereotypical "one year of calculus", means "calculus one" and "calculus two" (semester system). It would rather approximate what is in AP Calculus BC, except perhaps without graphing ...
  • 96
8 votes
Accepted

Is the Wronskian still assumed for graduate education?

I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true. Certainly, anyone who has studied a semester of ...
8 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

Not quite an answer to your question, but if you have students who are interested in mathematics but not interested in (generating their own) proofs, encourage them to go into mathematics ...
  • 2,224
7 votes

How can I discourage proof by patchwork?

I think the situation is tricky. Without knowing more about the actual problem it may well be that this proof by patchwork will be finished after a few iterations. This happens when every ...
  • 2,942
7 votes

What is gratifying in being a mathematics teacher?

With respect to Benjamin's answer, the top reason in his graphic "to make a difference." I received the note above at the end of the last school year. The first few reasons listed are slight ...
7 votes

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

I view avid19's frustration as an argument for presenting proofs within some historical context. Few major theorems have been achieved without a struggle, often involving several mathematicians over ...
7 votes

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

When teaching calculus I like to include L'Hospital's Rule because it exemplifies how the subject is an impressive arsenal of calculational methods, and because it is easy to explain why this rule is ...
  • 8,093

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