26

Allow me to contribute from the student perspective. I've taken College Algebra classes three times at three different levels and schools due to unusual circumstances. To make things even more interesting, I started out as an English Lit major, and later became a computer programmer. First, you have already realized that confronting the students in an ...


20

I was unofficially advising a student the other day who told me "I am struggling, but I know more than my friends in the class." This was the excuse she used for not attending meetings of a popular study group. How could she increase her grasp of the subject if her friends couldn't help her? An answer to this question (if not the answer, as I believe there ...


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 am a woman in my last year of a PhD in low-dimensional topology, I'll be a postdoc in the fall, and I have research aspirations. I'm mostly trying to articulate how I got to this stage in my life. In a real sense, my answer shouldn't matter very much since the many things that keep women away from mathematics clearly didn't work on me too well, or on the ...


16

I think there is no clear answer, although there has been some research on this topic. I remember one study which focussed on gender differences of university math students: Mischau, A., Blättel-Mink, B., Daniels, J., & Lehmann, J. (2004). Doing gender in mathematics: indications for more gender equality in German universities? Bielefeld: IFF. The ...


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 ...


13

Let me suggest two things. First: I think a lot of the misjudgments about how much students know stem from having an incomplete or wrong idea about what the learning objectives for the course are. For example, calculus students think that the sole objective of calculus sometimes is learning how to differentiate polynomials by hand. That's part of it, but ...


13

At least as a counter-point to the other good answers, I must confess that I have misgivings about the standard undergrad math curriculum in the U.S., primarily because I think it presents the subject as a plodding, exaggeratedly fastidious version of the most elementary parts of it... pointedly ignoring genuine motivations, historical motivations, ...


12

First, we need to be aware of the ways that women are discouraged, and when those are not related to what mathematics is, we need to change the culture. Does math have to be approached as a competitive sport? No. Related to this concern, please note that I am the only obvious woman in the top 40 users here. There are plenty of women blogging about math and ...


12

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge. The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it. ...


10

Two fundamental issues, in my observation: pathetically, quite a few (male) people subconsciously decide that, while they are not athletes or whatever, their machismo can be proven in ... mathematics. By math contests and being aggressive in class. Vehicle for ego. As a consequence, there's the "oop, it's not macho if chix can do it, too" unfortunate-riff... ...


10

I've mentored roughly a dozen year-long undergraduate senior research projects, and I've always used a mix of the following techiques to keep students motivated. Set clear goals, both short and long term. Students often flounder when they don't understand quite what they should be doing. Research is hard to figure out, and students often don't know how to ...


9

It seems to me that you're wondering about two different issues: are some topics or areas in mathematics more or less difficult to understand? the nature of developing an understanding. I can't speak to the first point, but the second point... there is something about arriving at an understanding such that you cannot ever imagine what it was like to NOT ...


8

I'd put a proof-based linear algebra course at the top of the list. I've found that people sometimes overlook the central role of linear algebra. Linear algebra provides not only important tools and results, but the basic language for huge swaths of both pure and applied mathematics, including geometry (Euclidean geometry obviously, but non-Euclidean ...


8

Well, there's a fairly standard answer to this question, which is used by most universities in the United States. A student who has completed algebra and trigonometry and wants to be a math major should learn the following subjects: Advanced topics in precalculus mathematics, including logarithms, functions, basic mathematical modeling, advanced algebra ...


8

How Does One Do Mathematical Research? (Or Maybe How Not To), by Lee Lady Mathematics as a creative art, by Paul Halmos I Want to Be a Mathematician: an Automathography, by Paul Halmos Also, this MSE thread ("what is mathematical research like?") might be of interest.


8

Here are two from famous mathematicians who have tried to explain how they approach mathematics: George Pólya. This book is actually targeted at introductory students. It has great examples, and an explanation of a method to approach them. Terence Tao is one of the best problem solvers of our time. He explains what I believe you're looking for in great ...


8

I post this upon request, but I immediately must excuse myself as it doesn't give a practical resolution to what the OP asks. Having said this, this does respond to an underlying component of the question, that of using grades to "assess a student's knowledge", as the title states. My original comment, [...] there's proffesional research in the way of ...


7

Some ideas that may help ~ Help them understand the point of any difference in approach If the emphasis turns to proof or derivation from more fundamental principles, you can make the analogy of "you know how to drive a car, let's learn how the engine works". They can argue that being a better driver makes you a better mechanic and visa-versa but they can'...


7

The other answers made good points... and suggest continuations and qualifications. For example, some students seemingly benefit from being informed that courses and grades are merely a stylized approximation to the genuine goals... while other students are confused or see opportunities to game the system upon hearing "an advisor" say any such thing. Gauging ...


7

It seems like some of the other answers are aiming at PhD programs. I would suggest (as your question on academia.sx suggests) that you may wish to look at a Master's program (at a non-PhD-granting institution). Obviously not all have the same quality, but a lot of them have high-quality coursework, with assistantships of various kinds. I regularly see ...


6

The Princeton Companion to Mathematics has a section Advice to a Young Mathematician, which seems available for free, where each of Atiyah, Bollobás, Connes, McDuff, and Sarnak give advice and most of it is of the form that it fits what is asked for in the question. Terence Tao's blog contains various related information under Career Advice and On Writing; ...


6

G. H. Hardy's A Mathematician's Apology is a nice read.


6

I see the original question as actually two questions asked here: How do we engage women into studying mathematics? How do we get women to become mathematicians? Studying mathematics for the goal of becoming mathematicians (pure, applied, etc.) is a subset to the first question. For the first question, I have read about some of Jo Boaler's research in ...


6

I feel your angst, word problems are a consistent weakness across most grade/subject levels. Learning to decode and solve word problems is a learned skill and so it needs to be practiced. However, just giving students 50 word problems and telling them to solve them for homework wont help anyone. I'd suggest to focus on each of the following items ...


6

The answer provided by celeriko seems like it hits all of the main points. However, I have found in my experience teaching word problems with students that a few additional strategies are most effective for struggling students. Rewriting the important mathematical "givens" in a problem. By having students rewrite these ideas, they're taking possession of ...


5

Perhaps because all math is simple once you understand it? ;-) On a bit of reflection, you'll see that each area of human endeavour advances until the going gets though for the geniuses in the area, who struggle at the forefront. They are followed by bright people who struggle to understand them, and sometimes try to make the genius' findings understandable ...


5

Class time is a designated time for students to sit and focus on the material of the course. From experience advising college students, here are the most valid rationalizations I've heard for skipping all classes: The professor just reads from the book / copies down things from the book. The class moves [too slowly / too quickly] and as a result I don't ...


5

Not to blow my own horn, but a significant portion of my doctoral dissertation was devoted to an analysis of memoirs and biographies of mathematicians at work. Chapter 2 is probably the most relevant.


5

In anything remotely computer-related a solid foundation in discrete mathematics (combinatorics, graph theory, ...) is a must. Number theory shows up in the most unexpected places, more so in computer science and related fields.


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