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25

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 ...


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 ...


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 ...


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

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

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 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

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.


7

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 ...


7

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 ...


7

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 ...


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 ...


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

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


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

It helps to place such assessments in light of the goal that inspired the question. I think this goal is most often "How do I manage my time to learn topics from area X?", although it could also be from a desire to master a subject independent of time constraints. Assuming time management is the goal, it helps to run through some scenarios: a) the student ...


5

I think the question is one of perspective, and that really the answer should be about changing perspective and not about trying to answer the question from the perspective suggested by the question. To illustrate, supposed that some system is put in place to study and answer the question as asked. They might decide that students who routinely score in the ...


5

Ok so I suppose if your College was at all typical then after your exams they suggested to you that you might want to transfer to a different university/subject. And as you are still doing maths at Cambridge you told them you would like to continue and presumably you have them some kind of indication that you would be changing your ways. As lent term is ...


4

You mentioned that I've tried to directly address the misconception on the first day You should try to start with a complex and practical example, not just in order to see the reaction of your students but also to create gossip like "that professor was doing some weird stuff so I was wondering if I was in the wrong place" Regarding the next one I'...


4

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.


4

A course that require proofs will be vital, as their high school career probably did not ask them to do enough serious proving. Other than that, there are many directions they can take. Rather than ask for an ideal for all such students, I'd ask for a list of good starting points. It would be great if every such student had a mentor who could help them ...


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