When I first joined the theory group, as an undergrad, I was a walk-in. In other research groups, you had to write an application letter, go through an apprenticeship period, and then, if they liked you, to full membership. The apprenticeship was a series of tasks; depending on the group, it could be a series of programming problems to solve, or menial tasks essential to the continuing work of the lab.
At that time, people in the theory group did not have as much respect as they do now. There was a perception then that the theory group was the place to go if the other labs rejected you; although there were a few who went into theory because they preferred theoretical work. The people who were running the group at that time were ronin; they were PhD students with nominal advisers. Except for the undergraduates who were taken in by a PhD (this was rare at that time), the undergraduates were left at sea.
My own adviser was at that time a PhD student; so was the other theorist who took in undergraduates. They had few or no publications in journals, although this was to change later on as they got their PhD's. Since they had little experience advising, they took in anyone who asked to join their groups.
We had to invent our application process; walk-ins would prove unsustainable. The change in character of our group started with the time we were flooded with applicants (ten juniors applied) and we had no rejection policy. At that time, there was a voluntary math methods tutorial session (I set it up with the consent of my adviser) and we expected them to join it as we knew that everyone needed remediation.
This didn't mean our applicants weren't smart; a lot of them went on to get latin honors. We did notice that they emerged from the math methods courses with little or no understanding of basic methods (of the kind you can read in Arfken's book, Mathematical Methods for Physicists). My adviser tried a methods seminar, where each week a participant was supposed to discuss a given topic. That didn't work. After that, I set up tutorials for them once a week.
The tutorials are problem-solving sessions; I assigned Arfken problems and made each attendee write coherent solutions. My job was to critique the solution-- find errors in reasoning, giving tips on how to make the explanations coherent. I even wrote a style manual based on David Mermin's rules of writing. I also help with difficult problems; when I learned math methods, I had to learn it on my own, and it took waaay longer. I remember problems that stumped me for a year before I eventually figured them out.
We noticed that attendance fell as time went on, and it eventually stopped. This was bad because these methods were the keel of the thesis projects; a broken keel meant bad sailing or a sunken ship. We did not want cases where the supervisor was the one who did the work.
So as time passed, we made changes. We instituted an entrance exam that was meant to check mastery of calculus up to the first course in differential equations. If you did not pass this exam, then it would be impossible to cope with methods. It was also meant to scare away people who thought that being a member of the theory group meant less work.
We also found that we had to require attendance because after passing the entrance exam, some undergraduates never completed the math methods sequence my adviser and I designed. Later, we added a post-methods sessions exam; full membership and thesis mentoring would happen only after passing the post-methods exams. If an apprentice did not pass the post-methods exams, then the apprentice had to go look for another research group to join.
Although the post-methods exams seem to be harsh (the post methods exams would probably scare most undergrads), passing the exams is doable because the exam will be from the set of problems assigned during the tutorials. All you need is to actually attend the sessions and do the problems. The post methods exams are designed to get rid of absentees, as well as to ensure that every member of our subgroup met a minimum standard when it came to mathematical methods.
The methods sessions usually takes three-fourths of a year. Since our course was a five year course, we give out entrance exams during the third term (summer) of the sophomore year. The methods sessions are done during the third year; this means anyone we reject, or opts out, has the chance to apply for membership in other research groups.
We work through all of the problems in chapters 6 to 8 of Arfken's book. The coverage was complex variable methods up to contour integration and the method of steepest descent (chapters 6 and 7), and then an intensive study of the gamma function, as an application of all the methods (chapter 8). My experience is it takes the average undergraduate of the theory group three-fourths to a full year to achieve mastery. Again, these are very intelligent students (and I think this description applies to the average physics major)-- so my experience lends me to doubt the utility of the mathematical methods course that's actually required by our institute.
The math methods course required by the institute uses Arfken as the main text. To see how insane the pace is, what takes us almost a year to master during the methods sessions is covered in the first exam of the second course in math methods. This means the usual student is supposed to learn the same content in a month and a half during their sophomore year. In fact, math majors in my university would need to take about three courses during their senior year to cover the same amount of material. No wonder that as soon as the average physics major finishes the methods course, he or she is in need of remediation.
At first, there was only a single post methods exam. But I found that, as in the methods course, a single exam inadequately samples how well our apprentices understand the material. Compare a single exam (as is the case with the required course) with six post methods exams. After passing all the exams, (and I chose the five most difficult questions for each exam!) I am therefore not surprised at the disparity in performance of the average member of our subgroup compared to other people whose only experience with methods was through the required coursework.
During the last few semesters, the methods sessions were held once a week, every Saturday, from noon until 5 pm. I've changed it this semester to what I hope is a saner schedule; I now meet one-on one with each apprentice an hour every week. I ask about their progress and provide mathematical assistance when needed. I found that the Saturday sessions took a toll on me: I was unable to get my own research done; and I need to do that to get my PhD. I'm hoping that this new arrangement will work, and I'm relying on apprentices asking for help from older members who have passed the exams. I'm keeping my fingers crossed; it's time to take care of my own work as well.