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My student's family won't let her study what she wants
Yesterday I had a tearful encounter in office hours with the most avid participant in the math class I teach, in which she related the following situation.
She is an international student whose parents are well off by local standards but have made significant financial sacrifices for her to study abroad at the university where I work. Part of the plan was that she should have a scholarship, but it has recently been denied on the grounds her grades are not strong enough, and as a result she will have to switch to a nearby university in the fall. Part of why her grades aren't strong is because she is a very specifically focused student: professional mathematics is all she really wants to do with her life. She already knows. When she is able to do what she wants, she just studies math. She works extremely hard, but at end of the day she couldn't care less about her other courses and views the struggle to get good grades as a distraction from what's important to her.
Her parents disagree that there is any future in math for her. They tell her it's not worth it and that she's bad at it. They're medical doctors in a country where teachers are treated poorly, viewed as people lacking the talent to do anything better, and are worried that a career in math would be both unremunerative and undignified—which might be the case in her home country, but my student also doesn't hope to return home. Nevertheless, for because her parents are footing the bill and because she's not willing to openly defy them, my student will finish a pure CS major when she enrolls in her new university, and won't be allowed to enroll in any future math courses during undergrad.
I watched her just crumble as she told me this, weeping and apologizing for it, unable to make eye contact. She's about to embark, without any choice in the matter, on a course that's going to make her miserable for years, if not longer, and moreover, from what I know of her, she is just the kind of person who should be in math.
I've told her there's nothing anyone can do to stop her from learning on her own, and that absolutely no instructor would refuse to let her sit in on whatever math course she wants, so that at most this will be a hiccup resulting in her getting a sort of bachelor's degree she doesn't really want at the expense of what she'd rather do, and that at the time she's financially independent, she can really study what she wants, but one could tell she's not completely buying this. She's devastated, and I can hardly blame her.
What can I (or anyone) do to help in this situation?

Comments are not for extended discussion; this conversation has been moved to chat.

I went to school for math, and continue to study math at the graduate level. Myself and many of my peers have jobs in software/computer science related fields for one reason or another. I think it's just worth noting that many companies that hire CS majors will also hire math majors for the same jobs, with the same salaries, and often choosing them over those with an educational background in programming. Maybe finding some statistics on this could help convince the parents that a math degree is not a waste of time, and in fact might lead to better paying jobs in the future.

What's her nationality?

Please update this page to let us know what happened eventually (say, an update after a year and another one after 5 years if you can manage that).

"will finish a pure CS major [...] and won't be allowed to enroll in any future math courses" I don't get it. I have a BSc and MSc in CS and I studied a lot of math. Computer science is heavily based on math (plus some other things, but still). How is it possible that she won't be able to enroll in math courses? Some of them are even mandatory for graduating in CS.

Not enough rep to answer -- but I would suggest talking to the scholarship committee and seeing if they'll award to her if she gets her grades up this term (more motivation to get good grades in the other classes). Sounds like if she fixes the financial issue, she can stay at your university...

Depending on your relationship with the deciding body on her being 'forced out' of this university because of her grades, you may have sway to instead get her placed on academic probation? Having a professor 'going to bat' for her could make a difference (I've seen it happen). She could also apply for student loans at a financial institution.

I was in the same boat, and I eventually transferred out of the premed major that my parents wanted. I know this type of enormous pressure, and it is not uncommon for students to cave. I would suggest to her this: find a university where it is easy to transfer between majors. Then, she can appease her parents by giving CS a go. Perhaps she will like it, then everyone is happy; or it may strengthen her resolve to return to mathematics, then her parents are more likely to finally accept her aspirations, and ending up with a minor in CS may give her an extra edge.

@foo There's no such thing as not enough rep to answer. Asking and answering are the two activities that you can always do.

If you update the answer or tag it with something indicating the student's nationality, you might get better answers, as this seems to be very deeply rooted in the student's family culture.  I don't think you would be revealing too much, there are probably a large number of students from that country in Canada, at many different schools.

@BenI. There's an exception to that. Answering a protected question requires a minimum amount of rep earned on the same site the question is on before answers can be posted. This question is protected, so all unregistered and some registered users are not able to post answers to this question.

Any idea if the student has pointed out non-teaching high-income careers that are available to people with a strong math background? My impression was that talented math grads could often command extremely good salaries in a number of areas like finance. In professional settings math grads seem to have quite a bit of freedom with many firms keen to hire them. CS is a reliable career choice but a math degree with some programming knowledge will get your foot in the door to almost any job a CS degree will.

Sounds like someone from the Philippines.

@AndreaLazzarotto, I am sorry to tell you that math is not equal to "math".  I do not think you have studied a lot of math. Some basic analysis and algebra together with some applied stochastics and numerics. But that's roughly a first year math course. You're most probably talking about "applied math". Which is a completely different things than pure math. Did you ever calculate something in your course? Or did you just prove things (exclusively!)? If your answer is 1. yes 2. no, you did not have math classes but applied math.

@Mayou36 of course I had to study and prove a lot of theorems in calculus, discrete mathematics, error correcting codes, algorithms, formal languages, mathematical logic, artificial intelligence and information theory, plus some other minor stuff. Sorry to disappoint you. Is your comment written for the sake of arguing? Also, the OP mentioned that this person won't be able to enroll in any math related course which is clearly impossible when studying CS.

@AndreaLazzarotto, from what I read I really get the impression that she is interested in pure math. Math is often mistaken as applied math as people think very differently, which matters IF she really loves pure math. CS and physics (my field of work) are more of a natural science (we think of 10^1000 being a big number, right?), whereas pure math has a completely different way of thinking. So: math is not equal to math. That's my comment. On the courses: sure, but as they belong to the CS, I don't think this is gonna be a problem, only additional courses are.

@Mayou36 it is certainly possible that additional courses will be a problem (probably less than anticipated by her, though, given that a "math inclined" mind is a great asset for studying CS... See Alan Turing) but my first comment was only addressing the "won't be allowed" part.

@AndreaLazzarotto, I agree, it sounds contradicting, but I would assume, math classes within CS are fine. Otherwise, she would not be able to study a whole bunch of things. Sure, A.T., the example from a CS :)

@Mayou36 I graduated with a CS Major and Math minor. Took classes on number theory, graph theory, abstract algebra, etc. all for elective credit. If you want to do pure math in CS, it is highly feasible.

@mascoj, of course, but you had math in minor. And I think that is what the parents forbid her to do. Not the math courses in CS (which are "surface scratching" for mathematicians by heart) but to take any additional courses (like taking a math minor) are the problem. Wouldn't you agree that math courses from the CS Major are significantly different than courses from the Math Minor? (applied calculus vs abstract algebra for example)

@Mayou36 Absolutely not, generally CS degrees are based upon discrete math and complexity/computability theory which more closely tie to pure math. Sure, it varies from school to school, but my program was largely based around the theory and not application. (i.e. Crypto course was mostly explorations in basic number theory.)

@mascoj, good example! I remember some colleagues from algebra/analysis mentioning how the hated crypto and number theory when studying as for them, it wasn't "real" math. My point is simple: math and CS are two different studies. Yes, depending on your specialization, there can be a rather large overlap, but this is 1. not generally true and 2. most of all not true for the ones who (in my experience) "really love math". Turn around the argument: If someone loves to code, setup servers and so on and she is forced to study math, what would you say? Don't worry, it's the same?

Some thoughts:

Suggest counseling.  They are much more likely to be able to help support her in expressing her goals to her parents, understanding their position, and perhaps in negotiating some sort of compromise with them.  Family relationship issues are likely to be a pretty common concern for college students, and a university counseling service will have had lots of experience with them.
Suggest talking to the financial aid office. They may be able to explore other options for funding if she decides to go on without her parents' financial support, though admittedly there may not be a lot of options for international students.
Talk to your contacts at the other university. See if you can connect her with someone who may be sympathetic to her situation, and who might be able to help her with some of the other things mentioned here.
Help her explore degree and course options. Is there some other degree offered by the institution that might represent a compromise between what she wants and what her parents are willing to pay for?  Are there ways for her to take more classes that she's interested in?
She needs to be able to succeed at things even when they don't interest her.  You write "She couldn't care less about her other courses and her grades".  Regardless of the degree she pursues, this is a recipe for disaster in US academia, where breadth is considered essential and degree requirements are designed accordingly, and overall GPA is widely used as a metric of success,  e.g. for graduate schools and jobs.
(I am guessing from your terminology that you are in the US.  If not, please correct me.)
She doesn't have to like all her courses, but she does have to be able to push through and do her best work even in those she dislikes.  That's a necessary academic and life skill, and as a faculty member you may be able to help her develop that skill.  Counseling services may also help with this, if she has some emotional resistance to get past.
She might find these courses more interesting if she gives them a chance. Even pure mathematicians can benefit greatly from knowledge about other fields, and you or other faculty members might be able to help her see connections with things that interest her more.

I've overstated "couldn't care less." She's aware that the grades are important, though she resents them, and studies hard, but her heart isn't in it. She feels, strongly, the tension between education as a series of hurdles and evaluations and education as learning, and feels like there's only value in the latter. I only know what she's reported about this.
I'm in Canada now, although my vocabulary remains American from force of habit.
These are all great suggestions. Thank you.

@jdc I’d also add that you could help her to see other career options opened by a mathematics degree, besides teaching. Even if she ultimately wants to be a professor, it’s relatively common to turn a Math PhD into a job as, e.g., a data scientist or analyst of some sorts. Having those as “backup plans” might help to assuage her parents’ fears. Depending on how she wants to play it, she could emphasize those sorts of paths over being a professor. Not pushing back too much if they latch on to those jobs as the goal might let her pursue Math without family strife.

Fellow Canadian here: University in Canada is cheap compared to many other countries and is generally very open in terms of immigration. If she's willing to commit to staying here and start the immigration process she could very easily get scholarships, especially since her parents would be abroad.

No.1 life lesson I got from college.  "She needs to be able to succeed at things even when they don't interest he"

I have had a small number of students in the same situation over the decades.
My first rule is: Don't get between the student and the parents. That's a complicated and stormy dynamic and I can't do anything but harm there.
My second rule is: Advise the student to cede to the parents for the undergrad degree. To the student, the 4 years of undergrad study seems like a lifetime, but it's really just dust in the wind.  I say, "Suck it up for a couple more years.  In grad school you'll get tuition waived and a stipend and you won't need your parents' support.  Do what they want now.  Do what you want in grad school."
But mostly, don't mess with parental authority.  It's a nuclear volatile area.

Absolutely!  From the many hints about her home culture, for a professor to contact the parents could be ineffective: “How dare he interfere with our reasoning our daughter!”  Or worse, make things really bad for her: “How could you discuss family business with a foreigner and make us lose face!”

This started like a great answer, as I absolutely agree with your "first rule". But then the second bit hinted to some outdated ideas (as much as I protest about it, all staff at my institution undergo training on how to deal with student's ... more personal troubles if the situation arises). Telling the student to "suck it up" and "deal with it" comes across as so cold and disinterested. You don't know the situation - it could be anywhere between headstrong parents and psychological abuse, and telling somebody to "suck that up" could mess them up proper in the latter case.

I guess what I'm getting at is, I understand that most of researchers and professors are not qualified nor do they feel comfortable getting into issues like that, especially including parents - and that's perfectly fine. But somebody not just tearing up but "crumpling" before their advisor suggests that they do need help - I think acknowledging that, and maybe steering the them into a direction of somebody who could help and who is qualified to listen to their troubles just sounds like a much better idea than betting on the assumption that that person can take 4 more years of the situation.

The second point of this answer is probably the worst advice I've read in many years.

I just don't think that sucking it up is ever a good advice in any situation, as it solves no problem - especially when it comes to University and career studies: those are the times you must enjoy the most, and the choices that build your personality: sucking it up won't ever do any good, neither to her, nor to her parents (who won't learn the lesson).

@gented So: 1.  you find that perseverance is "the worst advice", and 2.  the reason seems to be that if one suck's it up, it might interfere with the 4-year party one is entitled to.  "Sucking it up won't ever do any good"?  Seriously?  That's how most of us made it to retirement.

@B.Goddard 1. that's not even nearly what I said: sucking it up is not equal to perseverance (where did you get that from?). 2. You might consider 4-year of career study a party, I do not consider it so: on the opposite, I believe it is the most formative experience in your life. 3. Don't claim that most people make it to retirement sucking it up - it is just untrue (some may, some may not, it all comes down to how you want to live your life).

This is obviously a very difficult situation that raises tricky issues to do with psychology, parenting, and the sociology of the unnamed country the student is from, which are far above my StackExhange pay grade. I feel rather at a loss to suggest any concrete measures that would tip the scales in the student’s favor in a situation involving such powerful forces of adversity and delicate sociological factors.
However, one thing that I feel may be worth exploring is the student’s parents’ actual state of knowledge regarding the value of a higher education in mathematics. Are they aware that Forbes magazine declared “mathematician” to be the number 1 job in the U.S. in 2014? Or that the American Mathematical Society has a web page listing many examples of industries employing mathematicians? That page has links to many additional resources touting the virtues of mathematics as a career - for example, this web page on the US Bureau of Labor Statistics website, which taught me interesting statistics and facts, such as:

The median annual wage for mathematicians was $105,810 in May 2016.

Employment of mathematicians is projected to grow 29 percent from 2016 to 2026, much faster than the average for all occupations.

Thus, the one concrete idea I can think of is for either you, the student, or maybe a senior professor at your university (if you are not one yourself) to communicate some of this information to the student’s parents, in writing or over the phone or Skype or maybe even a face to face meeting (if that can be arranged). I would be a bit more understanding if they wanted to discourage their daughter from pursuing an educational track with poor employment prospects (though that would still be sad and tragic from the student’s point of view), but as it happens, math is actually an extremely useful and even lucrative profession these days — so I’m wondering if by giving the parents some actual data about how attractive math can be as a career it just might be possible to persuade them that a comfortable life and their daughter’s personal happiness are not mutually exclusive goals.
In any case, good luck to you and to your student.

I upvoted this answer because it's an excellent one. However, I have a concern. What if the OP's student insists on pursuing pure math all the way and stay in Academia forever? Then the future may not look too good financially.

@scaaahu there are no guarantees in life. “What if” can work both ways, and she can end up unemployed and bankrupt with a medical or engineering degree as well. Or “what if” she goes into pure math and wins a cushy professorship and a $100K prize by the age of 34? Or she can just end up as a “normal” mathematics professor - some of us are doing all right, believe it or not. Anyway, glad you liked my answer.

This answer has a pleasant optimistic outlook to it, though I'm not sure if an objective observer, especially one who was already critical of this argument, would be swayed.  For example, the cited statistics for mathematicians are for 3,100 jobs across the US, and that 29% growth projection is due to an estimate of 4,000 jobs in 2026; it seems unlikely that a critical observer would find those figures to be reassuring.

@scaaahu then they are, by definition, not in it for the money and will be happy doing what they love at a lower salary. However, that's not something  the student would need to mention to her parents, and dangling this sort of information before them might be enough to convince them that math isn't some sort of dead end career.

@terdon  That's right, the student may not care about money at all. But, what about her parents? We don't know the student's cultural background. From most westerner's poin of viewt, the student is already an adult. The student doesn't need to care about her parents' thoughts. In some culture, they do.

@scaaahu that's precisely my point. This is an argument to convince the parents. This shows that, contrary to what they believe, math can offer remuneration.

It's probably worth finding out if the parents expect the student to return to her country. If so, they won't be swayed by US wages.

@scaaahu "The student is an adult and doesn't need to care about parents thoughts" Doesn't matter if she's 10 or 60... if her Parents are paying, they get a say in it. Getting a good "return on investment" is a good desire for parents to have. I don't want to spend $200k on an education for my kid to work at Starbucks... this answer (good jobs exist for Mathmaticians) is the answer to that part of the question (Math isn't a good investment).

You seem to believe that someone would easily renounce their perceptions of reality the moment they see a couple of estimated figures from a website of some obscure agency in another country. To me, this is very naive.

@DmitrySavostyanov I don’t “believe” any such thing - where in my answer do I give hints about such a belief? My suggestion may or may not make a difference, and quite likely it won’t. However, it’s the only idea I could come up with, and it seems worth trying. If you have a better suggestion, please post it.

Depending on her mathematical interests, it might be worth looking for career options at the intersection of mathematics and computer science - there's a lot happening there, and some of it even pays quite well ;-)
My father has been a "Professor of Computer Science" for most of his career, but to a large extent he's a mathematician who uses computers as a tool and who applies mathematics to getting more out of computers.
Areas like operations research combine computing and mathematical concepts, and there's high demand for people who can master both.

Hey, I've implemented a number of algorithms you dad came up with! Please tell your dad a rando on the internet appreciates his work!

Computer Science and Math are close already, especially if she chooses  specializations like theoretical computer science. Tell her that many of the subjects can be seen as applied mathematics. And later in work she will probably need at least programming anyways. Other subjects like set theory, logics, signal processing, cryptography, automata and formal languages, complexity theory either contain large amounts of Math or can be useful for a mathematician. I think Math is even one of the biggest reason for CS students to drop out at my University. If she for example takes a Bachelor Thesis topic like cryptography she can do Math for half a year.
She can still do a PhD in Math with a Master in CS.
She can still study Math after studying CS.
She can still study Math in parallel to CS and also online during semester break.
In Germany we have a main subject and a "subsubject" ("Nebenfach") where you can take whatever you want with a much smaller amount of hours, is that possible there as well?
If all that doesn't help and she really wants to change the subject and it is to expensive to pay for studying on her own she can try to come to Germany where studying is nearly free (around 200 € / semester). There is a "Studienkolleg" where foreigners learn German for a while and then they can attend University.

So I would not try to convice the parents, it sounds as if they have their fixed opinion, but instead try to sidestep the issue and give CS a chance while choosing as many Math or Math-like courses as possible. If you want to support her more, can you give her a job as a teaching assistant or something similar? Also for the relation with their parents I would refer her to student counceling, they will be better trained to help with this sort of problem.

The asker clearly said that the student will not be allowed to take any more math classes for the remainder of her undergrad studies.

Seems clear that the relationship between parents and daughter is a major factor, and some answers are missing the fact that this is a bigger thing with her and with her home culture than it would be in USA or Canada.
Her parents may be rightly, or erroneously, trying to do what’s best for her.  Or they may be trying to protect themselves from the embarrassment of having an unsuccessful daughter.
However, IF she is willing to risk a break from her parents, might there be someone in Canada who would vouch for her to obtain immigrant status?  My parents are far from wealthy, but their “sponsorship” was enough for a young lady from an Asian country to be allowed to stay.  That’s USA, though, and her parents were not opposed.

Well, life is hard.
I think unless you're able and willing to directly fund her, you can't help her and she'll have to help herself. The main ways to do this would be:

Persuade her parents. It sounds like she's given up, in which case you'd probably have even less chance of being successful.
Get a loan of some kind, or another scholarship. Your institution (or the new one) might have some scheme to help students facing financial difficulties. Alternatively, there might be a scholarship out there for mathematics students. I did a quick Google search and found this; something similar might be available for her.

Having said that there're some caveats. If she breaks with her parents to fund her own education, she'll likely be facing lifelong consequences. This might be culturally acceptable in some countries, but in others, their relationship could be permanently damaged.
The other issue is that if she's completely convinced she wants to be a mathematics researcher, her career path won't be lucrative for quite a while - perhaps until she gets a permanent position. She might have difficulty paying off loans. This is also risky in the sense that if she later realizes how hard it is to get a permanent position in mathematics and / or how different graduate studies is from undergraduate, she might get depressed looking at the student debt she's accrued - all for naught.
On a personal level, I'd suggest she follow her parents' wishes. The risk of permanent damage to their relationship is too high, and besides they have her best interests at heart. She can still have a fulfilling and possibly better-paying career outside of mathematics. You could even show her the compound interest chart and how fast exponential growth is, and suggest that she start saving money after graduation. Eventually, she can fund mathematics research directly. She might not be able to do it herself, but she can still have a direct positive impact on the field.

I strongly disagree with the last paragraph. The damage her parents are doing now may be far greater than the damage of fighting this battle.

@AustinHenley I suspect we're from different cultural backgrounds then =)

Not doing mathematics isn’t the end of the world either. I find this a very presumptuous and offensive statement. It’s not your place to tell someone else what is or isn’t the end of the world from their perspective, especially when the thing in question is a choice of vocation, which is something that some people feel extremely strongly about. The rest of your answer is sensible and well-written, but I am downvoting because of the last paragraph, and especially the “not the end of the world” statement. I’ll reverse my vote if you make an edit to fix this.

@DanRomik that could be another cultural thing - I've had people tell me variants of that in genuine earnestness, and reacting in an offended manner would undoubtedly have surprised them - but sure, I removed it.

Thanks. I removed my downvote, but I still disagree with your last paragraph.

Very likely to be cultural differences ... for illustration, I know a woman who left home in her 20s to work as a maid in a foreign country. She sent a large chunk of her salary home, which was used to set up her siblings. Her family would constantly ask for more and she would oblige. She effectively gave up her own life and dreams (too old to have children now) for the sake of the family. Against this kind of background, breaking with the family is almost incomprehensible. The consolation for giving up her own life is the ones she helped enable, which might sustain the student here as well.

You could even show her the compound interest chart and how fast exponential growth is — Seriously? Do you really think explaining exponential growth to a mathematician would do anything other than insult her intelligence?

@JeffE it's one thing to see exponential functions on paper and another to see it applied in real life. Don't tell me you're not able to explain this to her in a way she'll appreciate and not find insulting, because I'm confident I can. Start by saying "have you seen the compound interest calculations" and if she says yes jump directly to the conclusion; if not show it to her, point out it's an exponential, and then go to the conclusion. How hard can it be?

@JeffE Knowing mathematics is one thing; applying it to someone else's problem is another thing; applying it to your own situation is yet another thing. The academia.SE is a perfect example of this: even mathematicians sometimes need an advice.

Hopefully I can provide a different outlook here as I’m a CS major trying to get into software development, not an academic. But this question touched me so much I had to answer.
It’s less common, but certainly not impossible to get a job as a software developer as a math major. So she may be able to sell that to her parents. She should tell them that employers will like to see that she did so well in such a hard subject and it will give her an advantage over CS majors (this is probably stretching the truth a bit, but not a complete lie).
I’ve also heard that medical schools and law schools like math majors. But I don’t know enough about either of those to verify if it’s true.
So I would say plan A is for her to try and convince them that there are other things she can do with a math major besides academia.
Plan B would be to find a compromise. Maybe applied math or statistics. From what I understand those are heavy on the math, but a little more practical than pure math.
And if all else fails, CS involves a lot of math too. Especially when you get to the more theoretical side, such as algorithms and language theory. If she’s going to get a masters or phd, that’s even better as she can focus more on the mathematical side and less on the hardware side. Find out if she has any interest in machine learning, data science or artificial intelligence. She may enjoy it more than she thinks.

The main problem here is what is going to happen in the new school.  If she doesn't study and do well she might as well be defying her parents.

She could talk to her parents about taking say 1 extra course in math at the same time as she does her main course.  Offer to pay them back after she graduates. However, this will likely fail as she blew off her other courses and her parents have learned she can't be trusted.  If she had established good faith with her parents by completing the existing courses with good grades this would be an easy sell.

Maybe if she succeeds in the new school this can be revisited.

Get a part time job somewhere and pay for her own math classes.  Danger: She has to do well in all her classes or this will back fire.  Her parents will see she failed again, and probably cut her off and bring her home.  Where she will have to take up a terrible job in her own country until she is old enough to strike out on her own.

Your only hope of changing things is to prove math is a viable career field.  Using actual job listing and a significant body of evidence.
Another issue is "doesn't hope to return home".  The big question here, is does her parents know,understand, and accept this fact, because it is critical.  Their main argument is that in their own country her math skills will be wasted.  If her parents were to accept this, then she could produce articles from countries she is thinking about moving to on how math and/or teaching is highly valued in that country.
However, "doesn't hope to return home" I am assuming she is there on a student visa or some such permit.  After that expires then what?  Will she have already applied for citizenship in the new country?  Otherwise she will be forced to return to her country.  The citizenship process can be multi-year process, hope doing her homework on this process now.

Maybe its time to impress on her how badly thing can get if she doesn't temporarily do what her parents want.  Then if she turns her grades around maybe her parents will be receptive to what she wants.
We aren't give hardly any cultural information, so I am just making up a worst cast ending.
She fails, or does not succeed to her parents expectations, they bring her home.  Then what probably no math jobs, and she will be forced to take another job, maybe even a house maid.  Maybe even forced into an arranged marriage, have children and never do math more advance than the math needed for cooking dinner.
I am not a fan of scare tactics, but if she does badly another semester her parent are going to be mad.  They may take action that make her life 10x more miserable than they are now.

I just read the question and realized that it could be quite a specific case if your student is from China. If she is from China, do NOT try to convince her to go against her parents will. I've been teaching thousands of Chinese students and majority of them study what their parents chose for them, not what they want. Their family culture is very complicated. Parent-child bond is pretty much sacred and maintaining it is essential for the child happiness in life.
What I do is usually find a way they can use their passion in the profession that was chosen for them. Math is easy actually, it is used everywhere. She can apply math to any profession. Give her some ideas, examples, advice on math applications in her field. Chinese parents choose professions for their children mostly using two criteria: expected income and lifestyle suitable for their culture. It's not that bad if you think about it.

FWIW, I noticed that the asker said that the student's parents are doctors and that teachers are looked down upon in her country.  I recently read an unrelated article that stated that Chinese typically see teachers as being on par with doctors, so that would suggest the student's parents are likely from somewhere else, but it's all speculation.  Google Fu: http://www.bbc.com/news/education-24381946

I'd council not getting too involved in family conflicts.
However it sort of sounds like they're of the belief that math can only lead to a job as a math teacher and are trying to push her into a "safe" major.
Perhaps giving the student some tables of income projections by major along with the kinds of jobs they open up might give her something to reassure them with.
Math does pretty well on that score and in her shoes I'd probably be asking "why are you asking me to step down into a (likely) slightly worse-paid, slightly lower status career path? If I was training to become a doctor would you demand I step down to nurse?"
http://www.naceweb.org/job-market/compensation/physics-projected-as-top-paid-class-of-2016-math-and-sciences-major/

With all due respect to her parents and difficult situation, it is she that must do the academic work, and later, practice in a field she finds rewarding. I'd hope her parents wish her to find happiness long term, yet they believe she 'doesn't know what's best'.
There's no easy way to combat this, but here is one potential route:
1) Student states unequivocally her desire for a different path
2) Student proves sincerity by applying for (but not necessarily accepting)  financial aid to render her independent of the parents support.
The parents can then either double down and allow her to incur debt, or back down and let her follow her interests.


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When is the right time to tell my advisor that I plan on leaving my PhD program?
I've previously asked a question on this forum about leaving grad school and moving to a new field. I'm a chemistry PhD student in my (now) second year. I've decided to leave my PhD program because my heart is not in my work, nor am I very passionate about my field (word to the wise - don't go to graduate school because you don't know what else to do). The toll of grad school has resulted in some serious damage to my mental health to the tune of depression and anxiety.
I don't have a fantastic relationship with my advisor, but it's not terrible either. There's a significant language and cultural barrier that creates a sort of disconnect between us. As far as I know, my advisor does not know that I plan on leaving, although if my lack of motivation and focus hasn't tipped him off that there's something wrong, I don't know what will. The emotional and physical toll of lying and not telling my advisor the truth is becoming debilitating. I can't focus on work, I spend most of my time feeling hopeless and scheming on how to get away as quickly as possible and I suffer from constant headaches. I never was so apathetic until about 3 months into grad school.
My main concern is that I still have two classes I need to complete in the upcoming Fall semester in order to leave with my terminal master's degree, and I'm worried that if I tell my professor now he will become extremely angry, cut my funding (even thought I'll be teaching in the fall) and make me leave the lab. I do not want to leave without having something to show for my work here. However, I think I would feel better if I just told him now and put myself out of my misery. That way he would understand where I'm coming from and why I'm acting in this way. He's paying me for the summer and the guilt I have for wanting to leave is getting out of hand.
I'm also in sort of an awkward situation where the project I'm assisting in will eventually be mine when my co-worker graduates in May and there would not be another student to take over when I leave as well. I feel like I'm just leaving my advisor out to dry.
So my primary question is, when is the best time to tell an advisor about leaving? As soon as possible? Or continue to let it sit until the fall semester starts? Is there even a "good" time to do this?

I haven't been in a situation like this, but I would wager as soon as possible so that the advisor can plan contingencies. No point in a "Surprise, 2 weeks notice!" thing where you basically admit to having strung him along.

Based on your description, I think waiting until near the very end of the fall term would be best. Telling your advisor now may lead to you being kicked out of the group, and you may not be able to complete your master's degree requirements. Similar advice goes for folks in faculty jobs: you don't ever tell someone in your department that you are going somewhere else until you have something else officially lined up. For you, you are "lining up" your degree. Hang in there.

Tell him today. In person.

I think your advisor would appreciate you letting him know your plans as soon as possible.

In these sorts of situations, follow the golden rule (https://en.wikipedia.org/wiki/Golden_Rule) and behave the way in which you would want the other party to behave if your roles were reversed.  Also, unless you have strong evidence to the contrary, expect your supervisor to react appropriately to your request.  I have been in the supervisors position myself, and the only time I have been disappointed was when the students were not straightforward with me.  BTW I suspect if you were punished, there would be an appeals procedure - not in the supervisors best interest to treat you unfairly!

If you do choose to string your supervisor along, I would suggest perhaps not asking them for a reference afterwards! ;o)

A couple things: 1) A 2-week notice is pretty standard and generous on all professional contexts. I don't see why academia is any different. It's pretty hard and expensive to replace a good employee in every context.

In regards to the reference. I have worried about this for similar and different reasons. Don't forget your lab-mates can vouch for your work, the same as using a co-worker as a reference. Great if your boss (in this case advisor) can speak positively of your skills, but I've seen an increasing trend of HR departments calling previous employers to confirm employment, and that's about it. Some employers make it a policy to not give out detailed references out of legal worries, as has been the case with my significant other, in the last 3 jobs no one has actually talked to a previous boss.

I was in this exact situation. Since I knew I had funding via TA-ing I talked to my advisor at the beginning of the academic year (had two semesters to go), and offered to tie up loose ends on my project. I asked him to be a reference for a laboratory position I applied to and presented my project at, but otherwise I just told him when I'd be starting my new job. It was pretty awkward still going to research group meetings that year, true. But I think we both knew before it wasn't working out, and it would have been /more/ difficult to keep pretending I was still trying.

It seems to me that @Mad Jack's comment, though perhaps practical, is unethical.  Funding in a PhD program is conditional on one's intention to complete the program. If you accept funding under the expectation that you will take a certain long term action, then if you clearly realize that you no longer have the intention to complete that action then you are behaving deceptively, hence unethically, by hiding that information.  In an academic job by contrast, you sign a yearly contract.  Unless/until you have tenure, you could be dismissed at the end of any academic year.  Even when you have tenure you are rarely receiving university resources which would be wasted if you did not have a multi-year commitment, and if you do find yourself in a position of receiving those resources with the intention of leaving, then you are behaving unethically.
(Example: most universities which offer sabbaticals view them as investments for the future of the faculty member's academic career and their career at that university.  Many universities have a policy that one has to return to "active duty" for at least one year after a sabbatical.  Violating this policy would be unethical in a similar way, although it is also more explicitly against the rules.  A yet closer unethical faculty behavior would be to receive a multi-year grant to do X, to realize after the first year that you have no intention of continuing to do research on anything at all related to X but continuing to take salary from the grant in subsequent years.)
Moreover the suggested unethical behavior is in response to hypothetical undesirable behavior by someone else.  The OP does not know that behaving unethically is necessary in order to receive the outcome which is in her personal best interest.  In fact she does not definitely know that she wants to leave the program: rather, she has some good reasons to leave (lack of interest and enthusiasm for a future academic career) and other good reasons to stay for a while (a desire to get a degree).  The right thing to do is to talk to her advisor, be candid about her doubts, and get on a track that would allow her to gracefully complete a master's degree if she so chooses.
Someone who cuts your funding for expressing doubts about the future is not behaving ethically.  In all programs that I'm familiar with, funding is guaranteed conditional on satisfactory progress, and having a frank conversation does not make satisfactory progress unsatisfactory.  If the OP's advisor behaves so badly as to cut her funding following that conversation, then the OP can seek recourse from the director of graduate studies or the department head and has an excellent case.  The fact that she's scheduled to teach in the fall strengthens the case: if the funding is cut and she leaves because of it, who teaches the class?
The fact that not informing the advisor is setting him up for a crisis later on is also a significant consideration.  To accept mentorship and funding from someone while planning all the while to leave them in the lurch later is quite indefensible.  I'm not surprised that the OP is having feelings of guilt about it.  As she is learning, we know deep down at a visceral level when we are doing something that we think is wrong.  Doing the right thing has intrinsic rewards -- call it karma or neuropsychology, it's true nevertheless.  I think she will sleep far better at night knowing that she did the right thing and suffered financial hardship for no fault of her own than if she did what she knew was wrong for purely selfish reasons.  I know I would.
Finally, having a master's degree from a program and advisor who will only say negative things about you when asked is far from ideal. Being able to play well with others is valued even more in "the real world" than in academia, so burning bridges in this way is not a smart move.

The best time to tell your advisor is when you are comfortable, and when you have an exit plan that works for you.
There's no ethical conundrum involved with waiting until it's convenient for you.  PhD programs have attrition rates approaching 50% on average, and any advisor or graduate director who has been doing this for a length of time knows that some students are going to leave. They prepare for these contingencies.
However, first I think you need to assess how likely it is your advisor will react the way you predict. I told my advisor in graduate school that I was quitting and although he was disappointed, he didn't threaten to cut my funding before I found something. (I changed my mind and stayed.) I recently told my postdoctoral advisor I was leaving academia for a job in the private sector, and although I thought she'd be upset, she was very supportive and happy for me. Are you sure that your advisor will threaten to cut your funding? Is there a chance that he'll be supportive of your decision to leave and will help you finish the terminal master's? Only you know your advisor, so only you can decide.
If you think he would, then you have to decide whether your mental health and your professional goals would be better served by telling your advisor now/soon, or by waiting until closer to the end of the semester. I understand your feelings of guilt and stress about not liking academia and wanting to leave, and feeling like you are living a lie. But the truth is people leave jobs and careers all the time. They job search while they are working in a specific job; they tell their employers they have a dentist appointment when really it's an interview; they stay employed when they really want to go to graduate school in 6 months because they want to eat food and live inside. People want to act like academia and PhD programs are different, but at their core, they're not - you're leaving a job/career. Wanting to leave does not mean that you need to sacrifice eating and sleeping inside for the sake of your advisor.
So if you think he'll cut you off, then wait until towards the end of the semester. Give him about a month's notice, maybe a little more, so you can finish up handing off projects. If you get so bogged down you simply can't wait anymore, then tell him earlier.
You have guilt about wanting to leave because that's normal, to because there's anything wrong with you. Everyone I've talked to who has ever left academia (including myself) has felt some amount of guilt about it. The reasons are too complex to get into here, but if you do an Internet search about it you'll find lots of stories from former academics who feel super guilty about leaving. It eventually goes away. In fact, reading about other people's experiences may help it go away. Your advisor will eventually take on another student to finish that project. They may get a couple undergrads to help in the mean time. There may be a student from another department that switches over. They may drop it for a couple months. PIs adjust. He'll be fine. More importantly, you don't want to soak in misery for years on end because you want to help him finish.

Most of this advice is good, but a few comments: (i) cutting someone's funding because they verbally express a desire to leave is extreme, borderline sociopathic behavior.  In a well run academic department or university it should not be permitted.  If the OP has any real expectation that this might happen she should approach it as a systemic problem: e.g. an ombudsperson can go the department chair saying "A student in the program is considering leaving with a master's degree but is concerned about punitive retribution for saying so.  Please tell us how s/he can safely proceed."

(ii) Pointing out that behavior is commonly done counts zero as an ethical argument.  50% of marriages end in divorce, and most of the causes of divorce (adultery, mental cruelty...) are most certainly unethical.  If you want to say "Ignore the ethics and do what's best for you.  That's what most people do in most situations anyway, so anyone who tells you otherwise is not being practical" then okay, that's a reasonable thing to say.  But denying the existence of ethical issues here is simply wrong.

People want to act like academia and PhD programs are different, but at their core, they're not - you're leaving a job/career. Wanting to leave does not mean that you need to sacrifice eating and sleeping inside for the sake of your advisor. — Bingo. (+1, by the way.)

Pete, you're right in that the frequency of a behavior doesn't make it ethical. I still contend that there's nothing unethical about leaving an advisor to figure out how to finish their own projects, just like there's nothing unethical about leaving a job before they've been able to replace you. Just because it makes life a little more difficult for them doesn't mean it's wrong.

Leaving right away is not what the OP is asking about.  The question is whether to continue to accept funding for some time in the future knowing that one is not going to do what the funding is being given for.

I tend to agree with Mad Jack and roseofjuly. In a perfect world devoid of complexity we all have perfect moral compasses, and when we decide we are going to look for new work we tell our coworkers and employers as soon as we start that process. I'm not going to take food out of my mouth or my families mouth to live by that ideal, and as such ascribe to the saying: "never leave a job until you have something else lined up." Working in a lab is a job.

The OP will be teaching in the fall, taking classes and presumably doing research. He will absolutely be doing what the funding is being given for.

RE the funding issue:
(I'm presuming by "support" you mean paid lab work)
In regards to feeling bad about the funding. This is  something that drives me a bit crazy about academia. In my lab I'm paid 19K for 9 months of work to work on all of my advisor's projects first. That's fine. But can we all be honest about the system and just be real about what PIs get out of having RAs work on their projects? Cheap and highly qualified labor to move the grant along. In the event where the student wants to go into academia we can argue about the merits of all the experience this low-paying work adds to the CV in terms of conferences and paper submissions; but in the end we are cheap labor.
I've also seen other labs where the PI financially supports a student in the student's own research. In quite a few instances the PI has not been that involved in the student's project(s) outside of the financial support, but attaches his/her name to everything the student does. This arrangement seems common if the primary data is the PI's, or if as part of of PI's support the student collects new data. However, I've also seen the students in this kind of arrangement use publicly available data for projects. This is all fine too. In many ways this is similar to working at a high-level in R&D in industry. Your name might go on a patent, but the company owns all intellectual rights. In the case of academia, where publications are everything, the PI benefits from this arrangement greatly.
If your situation is similar to one of the previous two examples then you don't owe it to anyone to tell them before you've got your Master's and a job offer in hand. The PI will still benefit from your work. Would they benefit more knowing a student was committed to project over the course of the next several years? Sure, but the PI still makes out pretty good. Now, you say you've been pretty bummed by your situation too. If you are being paid and just at home watching Netflix all day then that's a big problem.
Now in a third situation the advisor is paying you to work on what you want to pursue, totally orthogonal to their projects, and giving you full credit of everything (i.e., not attaching their name to everything you do as a point of lab policy). If this is the case then yes... let your advisor know now and let him/her do something different with his/her funding.
I do believe your hunch is correct: Leave without a terminal Master's and you've wasted 2-3 years of your life for nothing. You've added no income potential to your future without that degree on your resume, and (depending on your field) if you're going into industry you'll likely find that selling 2-3 years of work in an academic lab is a bit more difficult than you think, so best to get yourself in the best position possible for your next phase.

You don't seem to address the part about being a funded graduate student where they study: i.e., take courses, learn from faculty, and are guided to do their own work.  This is something that an unfunded student would pay about $30K a year for.  So saying that being a graduate student is just like a worker is disingenuous in that regard.  Also, the question is not whether the OP should leave right away.  If the OP has severe health problems then extricating herself from the situation giving a week or two of notice sounds very reasonable.

The question is rather whether the OP should continue accepting funding and instruction for another six months that has been given to her under the expectation that she will perform certain actions which she no longer intends to perform.

I should also disclose though that I come from a part of academia where they idea that my graduate students could do my work for me is so perfectly backwards that it is a thing of wonderment.  If there are really areas of academia in which PhD students are lab workers first and students second, then I don't have any experience there, and I have a strong suspicion that I would not be able to stick around as that kind of student for nearly long enough to earn a master's degree.

Even if we want to play the game of factoring in tuition as part of pay, graduate students are still under paid. Also, so what if you learn from your advisor? Great. You can learn from a boss in industry too, and gain alot of skills you didn't have prior to the job. It doesn't entitle you to more than two weeks notice.

Fair Pete! Thanks for giving more of your context. I'm in the social sciences. In my department we work for our advisors in their labs to move their grants. "Support" simply means you have an RAship to work on their stuff. If it works out well you might be able to carve out your own piece.  That's my context.

Given that tuition is what pays my salary, it feels pretty real to me.  Most PhD students I know are being given about $40K to $50K a year of funding.  In order to get this, the amount of work (in the W2 sense) that they do is no more than 20 hours per week in any given semester, and there are many opportunities to get entire semesters off.  (I did zero work for more than half of my semesters as a PhD student.)

I have to say: if you feel underpaid (and you're not wrong; the amount of money you're putting in your pocket is very small) and especially if you feel like "You can learn from a boss in industry too" then...maybe do consider transitioning to industry.  Your academic experience doesn't sound very pleasant or rewarding.

Hmm, it turns out that I have said similar things to you before: http://academia.stackexchange.com/questions/41016/navigating-the-thorny-issue-of-receiving-job-offers-while-in-a-phd-program.  To be clear: yes, you can leave academia when you want, and many people stay longer than they want to for no good reason.  However, you should not accept someone's money while plotting against them for half a year at a time...especially when there is no reason to think it's necessary to do so.  This ought not to be that contentious.

You're getting ad hominem and giving unsolicited advice in this thread. Glad your grad school experience was 100% different than everyone I know at an R1. I hope you're trying to replicate that with your students? It sounds like an awesome gig. Also, you are the only one in this thread continually defending the view that graduate students should be treated as "other." Just was disagreeing with you man. You're taking things a little too far.

Also the original poster is not "plotting against" his advisor. Where are you getting that that was implied in any comment he/she made?

The OP wrote "I'm also in sort of an awkward situation where the project I'm assisting in will eventually be mine when my co-worker graduates in May and there would not be another student to take over when I leave as well. I feel like I'm just leaving my advisor out to dry." Plotting against is indeed a bit strong, and to be clear I have nothing but sympathy for the OP, but the question is precisely whether or not the OP should deceive her advisor in a way that she feels will have negative consequences for him.  (My comments reiterate what I said in an answer to your question posted here.)

In terms of comparing information: my profile is public, and you can see from my CV that I'm speaking about the math departments at Harvard, McGill and UGA.  Things are not as cushy at UGA as they were for me at Harvard, but yes I am a co-PI on a $2.5M NSF grant which allows us to buyout some graduate student teaching.  Yesterday I had a conversation with a grad student in which I said that the more students things are asked to do aside from their own research, the thinner they get spread, and we discussed ways to generate additional funding.  This is one of my top priorities.

Hey Pete: I'm gonna unplug from this convo. Hopefully we got to a place where this is still constructive for the OP. Even though I'm just a grad student I stand by what I have said. I'm glad it seems like you keep a dialogue going with your students. Peace.

You say students are "cheap labor" and "underpaid", as if students don't get their fair share of the profits. Usually the only "profit" is in the form of grants, which are usually used mainly for... student salaries! It is nothing like the sort of exploitation you seem to imply, where the more you work (and the less you are paid), the more money your exploiting boss gets. Which is actually normal in a small business, and not considered exploitation unless it reaches extremes. If you feel it's a bad deal, you're free to do something else. There's no rule that says you have to be in academia.

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text:
Another instructor is pushing me out of the classroom right after my class ends
I am a graduate student in math in my final year, and for several years have been teaching at my department as a lecturer. This semester, in the same lecture hall there is another lecture that starts 20 minutes after my class ends. It's taught by another instructor from my department. I usually have many students coming to office hours and there are also students asking questions immediately after lecture. Due to other activities, I cannot have office hours right after the lecture this semester and can only stay for about 15 minutes to answer questions.
Many times in the past I had a similar situation and never had any issues with it. This semester the instructor who is teaching right after often arrives 20-15 minutes before her class starts and tells me immediately that I have to go with my students somewhere else.
I make sure to leave the blackboard clean and take all my stuff away from the instructor's desk before she arrives, but I do believe that I have a right to stay in the classroom after my lecture for at least 5-10 minutes. There is no vacant classroom around, and I don't have time to go with students to my office, which is in a different building.
Last time the instructor told me in front of my students that I don't understand "simple things" and that I am "playing games". When I was talking to one of my students, she stood very close to us and clearly demonstrated that she wanted us out. I tried to explain her that I couldn't go anywhere else due to my time constraints, but she didn't want to listen to me. I really don't understand what "simple" things she meant and what "games" I am playing.
We leave the board clean. She doesn't need to set up a projector. She can still talk to her students before her class starts, if she wants to (even though it seems like her students don't ask her any questions before their class). So, I don't see how I cause any disruption.
I had met this woman many times before this semester, but we never talked. I didn't see her talking to other instructors/students much, and she seems to be quite reserved and a bit neurotic. She doesn't want to have any conversation with me regarding the issue.
I felt really offended after last class when she said those things to me in front of my students. What would you do in my case?
Added later: There are no official rules regarding classroom occupancy between classes. Instructors are supposed to use common sense and be reasonable. For me using 50% of the break time seems reasonable to answer questions after lecture seems reasonable. I agree that for some people it may not.
I don't block the entrance to the classroom. A few students from the next class who come earlier always go ahead and take their seats as soon as my students start leaving the room. I also had one of the students from the next class listening to my explanation to one of those after-class questions and asking me further questions before their class (which is the same class as I am teaching, just a different section). Maybe the instructor got jealous, I don't know.
The entrance to the classroom is from its front (not back), so I do stay in the front. But it is a big lecture hall, and there is a plenty of space in front of the room (the board itself consists of 8 huge panels).
Also, during my career as a grad.student who is also teaching for the department, I have had several observations from experienced professors who are considered to be great teachers at the department and are in charge of undergraduate teaching policy. In my evaluations the fact that there are always several students approaching me with questions after class considered as very positive, meaning that students find me approachable.
Thank you everyone for answers.

What about simply talking to your students in the hallway?

You should first find out what is the standard convention for when you have to vacate the room.  Since the other instructor is in your department, you can take this up at the department level: talk to your advisor and/or a trusted faculty mentor about it.
In my experience though this is often left a little fungible and people need to be reasonable about it.  In my opinion it is reasonable to expect to have access to the room five minutes before your class begins.  Ideally the previous instructor will have vacated the room at that point; if not, s/he should be occupying the room in a way which doesn't interfere with your own use of it.  (For instance, a lot of times students will not enter a room if an instructor is still writing on the blackboard.  It can be annoying to come to your class a few minutes before you want to start and find everyone waiting out in the hall.)
In the absence of a clear directive about how to split the time, it seems reasonable for the departing instructor to take half of the time, as long as they leave at least five minutes for the other instructor.  In your case 20 minutes is a very nice cushion (in my university it's 15 minutes), to the extent that I find it a little strange that the other instructor is arriving 15-20 minutes early.  What is she doing with that time?  Do more than a few students arrive that early?  If you are using the time to talk to your students and she's using the time for nothing, then I think you are morally in the right and should push back on it a bit.  You should be able to stay for the first half of the 20 minute break.
How should you do that?  As above, I would find out what other people do.  If the standard convention is that you can take those 10 minutes, then talk to the department head (or other departmental authority figure; e.g. whoever schedules the teaching) about it and get their support.  (You don't need to phrase it as a grievance or even identify the other instructor by name.)  Assuming you get backed, then you should pay a visit to the instructor in her office and explain that you looked into the matter, that you have a clear instructional purpose for how to use the time, and that the head (or whoever) has confirmed that this is a standard practice.  Definitely do this outside of the classroom so that you two can have the conversation you need to have not in front of the students and so that there is not a direct confrontation to be resolved.
In my opinion it is probably not worth trying to seek redress for your poor treatment by this instructor.  We are not obligated to be nice to each other; it's just better if we are.  If you respond to the treatment in a professional way and show that you are neither going to get pushed around nor retaliate in any personal or emotional way, then you're basically just rising above.  What some random person who you're only going to see for one more semester thinks of you isn't really so important, is it?  Especially if you're confident that you're in the right?

I agree with all of this, with one caveat: since the instructor is accusing the OP of "playing games" and "not understanding simple things" in front of the OP's students, it is worthwhile trying to put an actual end to it.

+1, there's nothing wrong with you speaking with your students in the classroom as long as it doesn't interfere with her setting up for her class. For her to suggest otherwise doesn't speak much to her character. I'm sure there's plenty of room for you to speak to one or two students in one area and she setup at the blackboard or desk or wherever, afterall, it is a classroom :)

@Lightness: My thought was that by speaking to the instructor outside of class, they can work out the issue once and for all. With this particular point of contention aside, they need not and probably will not have much interaction in the future.  I agree that if the instructor continues to make disparaging remarks in front of the students, that becomes a separate problem to solve.

I would be quite firm about this. The woman is a bully. She comes in demanding you clear out the room, you say "I believe the next lecture is not for another 20 minutes; I need ten minutes here to wrap up. Thank you" with a curt smile. She will have no choice but to comply unless she feels like committing physical assault.
If she kicks up a stink, she can take it to a higher authority (and ultimately, from the sounds of it, get shouted down by them). When they ask you about the situation, then you have an opportunity to also tell them about the inappropriate things she's said about you in front of your students, which is a bonus because then there is no impression that you went complaining about her: you were asked about it.
If you have good reason to believe that she would actually cause a big scene there and then, frankly I'd let it happen — but only after quietly consulting the relevant authorities on the rules of room use, so that you may be absolutely certain as to your position before she makes a total fool out of herself.

If I were in your situation, I would ask the other professor for a reason and explain that I wanted to make an announcement to my students at the next lecture. If the response is "because I want you out" then I would politely explain that I would announce to the students that we could only remain for 10 minutes after class after which time we would all need to be out of the room.
No need to make it confrontational.  I find that if you offer to help people solve a problem that helps reduce tension.  If this professor doesn't actually have a problem then offering to help solve a problem will be a non-confrontational way of pointing that out.
This worked for me last semester.  We only had 15 minutes between classes and my course was taught in a computer lab. Quite often students had questions about projects they were working on and already had their code pulled up on their screen.  Unfortunately the students from the following course would be filing in and because their course required them to have access to code on a particular machine (shared drives wouldn't work) they needed particular seats which my students were still occupying.
In my case, the professor after me simply explained why he needed the room and at the next lecture I explained to my students why they needed to exit within 5 minutes of the end of class. Everyone understood, no tension, no confrontation.

I would like to provide two suggestions. The first one directly refers to your discussions with students:

There is no vacant classroom around, and I don't have time to go with students to my office, which is in a different building.

If the questions really need to be answered immediately, you could simply take the discussion outside into the hallway. Surely, most of the students have to or want to leave, anyway, I presume, as they might have to move to different buildings themselves (?)
For anything else, you do not have to "have office hours right after the lecture", and you do not have to "go with students to [your] office". They are adults. Surely, your students can find out where your office is on their own and come there when you are free.
My second suggestion is related to whether you may stay around in the lecture room after your class:
It generally seems like the most corteous and natural thing to me that once the class has ended, one should try and leave the room as quickly as possible. Unless otherwise regulated by your organisation, I see no basis for the fundamental belief

I have a right to stay in the classroom after my lecture for at least 5-10 minutes

... unless your departure actually takes that long (e.g. because you have to pack some elaborate experimental setups for a while). At the same time, however, I wouldn't concede the other lecturer any basic right to arrive in the classroom at least 5 - 10 minutes before the next lecture starts, unless some elaborate setup needs to be prepared for that lecture.
In other words: If you are entitled to stay for at least 5 to 10 minutes after your lecture has ended, I would argue that the next lecturer is just as entitled to claim the room for themselves at least 5 to 10 minutes before the next lecture starts.
With that said, it is usually well possible to stay around to answer such questions, though that does neither mean that the room will be reserved for you (the next lecture may already be under preparation), nor are you automatically the "main party" (in that you may block the "lecturer's desk" or any such central device, rather than cluster your discussion in a corner of the room to give the next people space).

Note that the OP did say that they have 20 minutes scheduled between classes, so staying 10 minutes after the lecture still gives the next lecturer 10 minutes before their scheduled start time.

What are your university's rules about lecture room bookings? Find them, stick to them, make sure the other instructor does too. Typically, they'd indicate a 5- or 10-minute grace period between bookings which is explicitly for the following lecturer to set up. With a 5-minute grace period (which is what my place has) a room booking for 10.00-11.00 means that we're cleaned up and out by 10.55.
Make sure you know the rules; then let her know what they are, and explain to her why they are what they are.

You don't mention where you and the students stand when having these conversations. Are you up front near the podium, or in a back corner of the room? If it's the latter, then it seems she's perhaps a bit rude and territorial; but if it's the former, she might have a fair point.
She may not look like she's doing anything before class, but you don't know what kinds of thoughts are racing through her mind – perhaps she's mentally prepping, and she's unable to do that effectively with a lot of commotion in the room.
Either way, I agree with the gist of what others have said here. The best course of action is to solve such disputes privately, locally, and cordially. When that seems unlikely, find out what the official rules are and, if you must, (reluctantly) get the department head involved.
You also might quietly mention this conflict with your room scheduler. You could make a request that, if possible, you not be assigned a room where this instructor has a follow-on class. (Sometimes you can't resolve a conflict in the present, but you can avoid it in the future.)
One last recommendation: try to keep a sense of humor about it. Even when you're in the right, being uptight and brewing animosity is rarely a better solution than an amused shrug. The worst case scenario for you here is that you have to move your after-class discussions into the hallway – something that may not be as ideal for you, but something that most instructors have to deal with eventually.

So it seems strange that you can't have office hours right after this lecture, but you can, in fact, spend 15 minutes after the lecture time helping students.  It appears you could have 15 minutes of office hours right after this lecture.  Perhaps the travel time is excessive, or you have an aversion to setting up office hours of such short periods of time?
An option not already presented in other answers is to end your lecture 5-10 minutes early and encourage students to spend the remaining time working with each other, or asking you questions individually.
This would alleviate the issues you have with limited office time. I don't think it's a great choice, though, as lecture time is also valuable.
I think to root problem is convenience.  You find you can support your students better by providing some time at the end of class, which they already attend, to support them rather than expecting them to schedule time out of their day to visit you during normal office hours. You don't seem to suggest you don't have enough office hours, so it appears you are doing this purely as a convenience for your students.
You don't have a right to the classroom outside your lecture hours, save, perhaps, for several minutes before and after to set up the classroom if necessary.  Your use of the classroom as an informal office is a convenience.
If your university or college doesn't have rules regarding classroom use between lectures, or preparation/cleanup time, then I'd suggest it's fair for you and your students to vacate the room by the halfway point between your two lectures.  This doesn't mean ending your conversations ten minutes after your lecture, but being out of the room.
I suspect that she's only becoming more and more insistent as you are consistently going well over 10 minutes, and perhaps even infringing on her 5 minute preparation time.
Regardless, I believe your best bet in this case is to simply stand firm.  If she asks, tell her you and the students will be out ten minutes prior to her lecture start, and then stick to it aggressively so it's something she can count on.  She should, then, stop interrupting your activities and passive aggressively intruding on your after-lecture time.

This was closest to my own thoughts, which were to (very slightly) speed up lecture delivery so as to end 5-7 minutes early every class, then explicitly state something like "I see I'm ended 5 minutes early, let's wrap up and be out of here no later than (whatever 5 minutes after class is)." And let the other professor know that is your plan; then stick with it.

Note that the OP explicitly said that moving the conversations to his/her office is not practical because it's in another building.

@CGCampbell: Bluntly, that’s ridiculous. You’re suggesting that the OP should give up class time so that another instructor can have extra time. Moreover, it’s a math class, so the ‘lecture delivery’ may be nothing of the kind: much of the class may be highly interactive and of unpredictable duration.

@BrianM.Scott The OP wants office hours in a room that they don't have scheduled for their use inbetween classes.  If they want the office hours badly enough, trading a little lecture time for those hours seems reasonable. They may want permission to claim the "unused" classroom for themselves, but they have no clear right to it, so no - it's not ridiculous to suggest that they use their reserved classroom time however they like - up to and including "office hours" style help.

@Adam: No, it does not seem reasonable. What the OP wants is in my experience perfectly normal, and the other instructor obviously has no more right to the time between classes than the OP, who in any case does not appear to be doing anything that actually interferes with her setting up for her class.

@BrianM.Scott I'm sorry, I thought you were saying that using lecture time for anything other than lecture time is unreasonable.  I agree that the other professor is being unreasonable.  I made a suggestion which isn't ideal, but might solve the problem if the other professor is too intrusive. It's a workaround, but it's worth trying if nothing else is working.

You don't have a right to the classroom outside your lecture hours, save, perhaps, for several minutes before and after to set up the classroom if necessary — Neither does the other instructor.

You should clean up after yourself and get out of the classroom so that the instructor and students who will use the classroom next can get into the room and prepare for their class.  Classrooms aren't the place to have these kinds of discussions with students- that's what you've got an office for.
At the very least, you should give the next instructor half of the time between the scheduled end of your class and the start of their class.  It seems that you're taking 15 minutes of the 20 minutes between classes, which is excessive.

I find this answer strange. The OP specifically mentions cleaning up the class room, and that he feels 5-10 minutes if the 20-minute gap would be fair. I am not sure where you get the idea that he stays for 15-20 minutes.

"[I] can only stay for about 15 minutes" I also understood it like he takes 15 of the 20 minutes.

In the absence of conflicts (such as the one the poster is facing), I disagree and think that classrooms are a great place to have these kinds of discussions, especially immediately after class -- while you still have the students' attention. I was luckier than OP -- the one time I ran too late, the next instructor was very polite and graciously accepted my apology.

The OP specifically said that the next instructor arrives 20 minutes early and demands immediate clearance of the room. The OP personally has only 15 minutes, which is why they don't have the follow-up meeting elsewhere but wish to remain in the same room for 5 or 10 minutes, but never gets the chance. This answer is flat-out wrong.

The OP also states that he sees the students arriving for the next class. That implies to me that he's easily crossing the 10 minute mark after his  class and heading towards the 15 minute mark.

@mkennedy Not necessarily, given that they have 20 minutes between classes. If students have two classes back-to-back in nearby classrooms and 20 minutes between classes, it seems perfectly reasonable to expect some of them to arrive more than 10 minutes before the start of the second one.

Classrooms aren't the place to have these kinds of discussions with students — [citation needed] Arguably, classrooms are the only appropriate place for these kinds of discussions with students.

I'd agree with the ideas posted by others here, especially about checking with your department about if there are official policies around the issue, and then discussing it with her privately. However, I know the barrier of entry to walking into an argument with someone who may be many levels above you in the university hierarchy - not a fun thing to do. As an easier first option, what about simply sending her an email - something like:

"Hi, I'm the lecturer in Room XXX before your 3:20 lecture, and I'm sorry that I sometimes run late with discussions with my students after class. However, I like to be able to answer questions that have come up during lecture, and due to location and scheduling constraints can't take them to my office after lecture. Since we have a 20 minute gap between our lectures, would it be okay if I used the first 10 minutes of that time to hold discussions with my students, as long as we're cleaned up and out 10 minutes before your lecture starts? Of course, if you ever have special preparation that needs more time, feel free to shoot me an email before my lecture, and I'll be happy to let my students know that we need to be out earlier. Thanks for understanding! I appreciate it so much."

That way, you can avoid a direct argument, or confrontation in front of your students, while still getting to address your concerns to her directly. If she has a valid reason to need you out, you can talk it through - if not, she should have no way to refuse your request. If she does, you can even take her response to your adviser and see what he recommends. If she doesn't respond and then confronts you again, you can simply tell her that you tried to address the issue in private with her but didn't get a response - and she'll look very foolish in front of the students from your class and her class too.

Last time the instructor told me in front of my students that I don't understand "simple things" and that I am "playing games".

This is where she lost her argument. You should have simply totally ignored her. Let her make a formal complaint and only respond to such formal complaints. You don't have the duty to respond to insults, or to look up what the guidelines say in response to insults. If you were violating the small print of some guidelines, then it is her job to point that out to you.

I have a slightly different perspective here, and I want to offer two options.
Option 1: be selfish
Assume the instructor is a neurotic bully. Don't be passive aggressive. Don't get the instructor overruled.
It's not worth it. Use your imagination as to why.
Arrive early to class if you can to answer questions before the start. At the end the lecture tell your students to come to your office hours, then arrive early to the next item on your schedule. Your department would probably rather you get another paper out than spend 15 minutes after every class with students who can't do it themselves or won't use other resources of the school (your office hours, other help rooms, tutors, etc...)
Option 2: be selfless
Ok, so you want to be your students' advocate? Tell the department head and ask for advice on how to handle it, and try to be discrete (that is, don't embarrass her in front of students).
This could be interpreted as the selfish approach, if you consider that it makes you look good to the department head and makes a seemingly unpleasant person seem even moreso.
But it also has risks. What if there are rules on the matter that you have overlooked? Then you appear foolish to your department head. Double-check before you do. Also, your department head would probably rather you have another paper published, and might question your priorities.

I think the logic behind the post is quite confusing. You should tell your students directly that discussion can only happen during office hour, and mention that there is another class afterwards. And you can extend your office hour (in a different day, say) by some period of time or move it if you wish, this can always be discussed with department secretary.
You suggested that:

Many times in the past I had a similar situation and never had any
  issues with it.

as well as

In my evaluations the fact that there are always several students
  approaching me with questions after class considered as very positive,
  meaning that students find me approachable.

and many other reasons claiming your behavior is tolerable and okay by your standards. But these justifications is simply not related to the fact that she finds it uncomfortable because you did not end up the class on time. Since you are aware of that, you should do something concrete to resolve this issue. For example you can cover more background in the lecture time available, and encourage students to come to your office hour for longer 1-1 discussions.
My impression for after class discussions is totally opposite to yours; the fact students coming after you after class usually means you did not explain the material well enough during class or unclear during the grading process. Your course should be in the format like "Calculus III, 2:00-3:00pm, Room 31A, Darwin building". If the description did not mention an extra 15 minutes for afterclass discussion, then it means you do not have it. As what the other instructor does when she comes to your classroom is totally irrelevant. In my experience a student (not even an instructor or dept chair) can knock at your classroom door and friendly to ask you get out without giving any specific reasons. They might need the room for group projects or whatever. The fact you are late should be compelling evidence against you already.
You also suggested that

Last time the instructor told me in front of my students that I don't
  understand "simple things" and that I am "playing games". When I was
  talking to one of my students, she stood very close to us and clearly
  demonstrated that she wanted us out. I tried to explain her that I
  couldn't go anywhere else due to my time constraints, but she didn't
  want to listen to me. I really don't understand what "simple" things
  she meant and what "games" I am playing.

I think you should think from her point of view, that your class should end up on time. The fact that you have time constraints, etc is irrelevant to her. If you think these are legitimate reasons to justify your behavior, to her you were simply being immature and using poor excuses. You may have your disagreement and ask for support from senior faculty in the department as Prof. Clark suggested, but her position is very clear. If she demonstrated you need to get out, then you need to get out even if this is the final exam. Unless you have a student with disability that needs extra time, etc I think there is no point arguing further on this issue.

You didn't explain why the next lecturer has a better claim for that time. She arrives 15+ minutes before her lecture starts, after all. Much of what you write can be turned around: There's no need for her to have to prepare for class just minutes before (or whatever she's doing), this could be done well in advance. It may be inconvenient, but so is extending office hours because he's unable to answer a handful of questions after class.

@DanielBeck: She does not need a "better claim" for that time. All she needs is some claim showing him that the class is done already and he should be out. In practice, she does not even need to specify that she is teaching a class. A random student or even a janitor can say the same thing and suggest OP get out.

I still don't get it. What is she doing in there then? It appears she's not just in there to tell him to get out, but do something else (e.g. prepare her lecture). So in this case he should then tell her to get out as well? What is it that makes it alright for her to use the room, but not him?

@DanielBeck: The OP did not really specify what she was doing. I think this is another drawback. He specified the other instructor as if someone he imagined with no detail. I do not think it would work out in real life because he would have to put himself in her shoes. For your question, for the first time he could friendly ask her for a few minutes extension and suggest her to get out. But for later occasions keep doing so and disregard the fact that he was late as well is not a workable strategy.

By exactly the same argument: He does not need a "better claim" for that time.  All he needs is some claim showing her that her class has not yet started and she should stay out.  In practice, he does not even need to specify that he is teaching a class.  A random student or even a janitor can say the same thing and suggest that the next professor stay out.

@JeffE: I agree the logic certainly hold both ways, but this does not mean he should not step out himself.

There are a number of good suggestions above.
If I were in your place I would want to understand why she wants / feel she needs the full 20 minutes before her class starts in the classroom without interruption. She might be territorial, she might find lecturing stressful and wants / needs the time to prepare.
You now seem to have a second problem that both you (from the fact that you are posting) and her are upset. At this point I would learn if there is a departmental guideline on when you should be out of the classroom, or if there is no guildline then with the knowledge be non-confrontational, say you don't understand, and ask her to explain why she wants/needs all of the 20 minutes. Be ready for her to be hostile but listen first, then explain why you would like to have time after the class. Again she may be hostile to your explanation.
Once you understand you can decide what to do next.
There is an good book called "Getting to Yes: Negotiating Agreement Without Giving In" by by Roger Fisher, William L. Ury, and Bruce Patton. Which is a good read.

I must be missing a point but if the schedule is

your class is 10:00 - 11:00
there is a break between 11:00 and 11:15
her class starts at 11:15 - ...

then I fail to understand where the problem is. She comes in, tells you to leave in front of your students, you first respond first nicely (for her not to interrupt you), and if she insists you just tell her to shut the [censored] up.
Is there something I missed?
If she is offended by the fact that you dare to talk to her, tell her to escalate to her boss, ideally on the spot - and turn your back to her. This is a fantastic mechanism as soon as you are right (and have a supportive boss , etc.).
A behavior like hers is either psychopathic, not enough parent supervision when she was growing up or seen in nature by individuals who pee in selected places to mark territory.
Note: an edit for the last paragraph was suggested (and accepted by a vote 2/3)  (on the grounds that it is insulting or loaded with assumptions). For me, it is not. If one feels that the answer is wrong and that a bully like her should be approached in a kind or elegant way please, by all means, downvote my answer. Everyone has its own subjective way of dealing with psychopathic, badly brought up or territory-marking people.

a downvote on that answer means that this is not the right thing to do with a bully?

Saying "shut the [censored] up" to a colleague (you don't know well) is crossing the line. Unless you have tenure, you should be very careful what you say. A better retort would be "Please next time, don't come in until ten after. You're being too rude and distracting otherwise."

@StephanBranczyk: Saying this to a colleague is, indeed, crossing the line. Saying this to a bully (as described by the OP) is one possible reaction - which you may or may not agree with. There are people who will tiptoe around such people, other will stand up and use similar weapons (yes, this is a war :)). Your retort is elegant, I hope it works fine with such people. Mine is rude and primitive and never failed to have that person actually shut the [censored] up

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text:
What is the role and format of the qualifying exam in CS/ECE programs where the exam is not in the form of a comprehensive written exam?
I realize there are different forms that a qualifying exam can take in CS / ECE PhD programs.
In departments where this does not take the form of a comprehensive written exam:

What is the role that the qualifying exam fulfills in the PhD program in your department?
What is the approach taken in your department?
What are the perceived advantages / weaknesses of the approach?

Where I did my Ph.D., we effectively had no written qualifying exam.  Instead, students were expected to gather high marks following a certain complex distribution of classes within the first two years (there was also another component, involving giving a research talk, but that was more equivalent to defending a thesis proposal).  If you could not achieve enough high marks, then you had to fall back on a more traditional qualifier for whatever subject areas you couldn't pass out of via classes. I never knew anyone who didn't just get the high marks, and the rumor amongst the students was that if you had to use the fall-back, the professors were sure to fail you because they'd be grumpy that they had to prepare a whole exam on your behalf.
The rationale for this method comes from the department's goal for the qualifying exam: ensuring that students have a sufficiently strong grounding across a sufficiently broad area of expertise.  Given this goal, why can't high marks in key graduate-level classes establish expertise just as well as a special non-class exam?  Adding a special exam just increases the stress on students and the burden of preparation on faculty.
As a student, it was certainly much appreciated---if you're in a Ph.D. program, you're probably pretty good at classes, and so being able to tackle the whole qualifier thing just by being good at classes was quite a relief, especially compared to the stress that people I knew were going through elsewhere.  I think the faculty really liked it too, because they could deal with the whole qualifier thing just by teaching the classes they wanted to teach anyway.
As for down-sides... I suppose it means the faculty have to trust each other enough to believe that high marks actually mean a student is qualified?  Other than that, I really don't see any down-sides.  Personally, I think that the traditional qualifying exam is essentially academic hazing and should be replaced by methods like this.

What about oral examinations? Part of the value is to establish that a student has sufficient communication skills with sophisticated material.

@GSat In my program, that was the research talk part, equivalent to defending a thesis proposal.  I simply see zero benefit to written qualifying examinations.

Where I did my PhD in ECE, we did not have a comprehensive written exam: we had to do research on a topic of interest to the student and the exam committee, write up a "conference paper" on the findings of the research, write up a separate paper on engineering research ethics, and give an oral presentation on the research topic. Questions from the qualifying exam committee during the presentation could cover both the research topic/findings and the engineering ethics paper written by the student.
To clarify, the research activities associated with the qualifying exam did not entail a formal research proposal: the formal research proposal at my previous department came at the preliminary exam stage. For the qualifier, researching a topic essentially meant different things to different students depending on how far along they were. For those students already working on a research project (which was/seemed to be a majority of students), the research write-up/presentation dealt with the findings up to that point and future directions (which naturally helped the student prepare for the prelim). For those students not fitting the above description, they would choose an area of interest to both the student and the committee (i.e., a research area where they see themselves working in the not-so-distant future with one or more faculty on the committee) and summarize the state-of-the-art, possible problems to tackle, along with some possible solutions/research directions.

What is the role that the qualifying exam fulfills in the PhD program in your department?

To ensure that the student:

Is capable of doing independent research,
Possesses the skills to present technical findings, both written and orally,
Understands the ethical considerations that go into research/publishing, and
Has a good command of the relevant technical material, from ECE graduate-level fundamentals to the state-of-the-art in the chosen topical area.

What are the perceived advantages / weaknesses of the approach?

I think the goals of the qualifying exam listed above were fairly-well addressed by the exam requirements. As for the downsides in the approach, some students were not yet capable of doing research at the level required by the exam. So I think our program weeded out some students who had the potential to do quality research but, unfortunately, were not "up to snuff" for their qualifier.

I assume this exam was different from a proposal. So what did it mean to "research a topic"?