The transition from the undergraduate to the graduate level presents many difficulties. Undergraduate students wishing to attend graduate school in mathematics have an overwhelming variety of programs to choose from, and choosing a graduate school is only the first challenge they face as they make this transition. Once they enroll in a graduate program, they face coursework that is at a higher level of intensity than what they have seen before, and graduate exams are larger in both scope and importance than what most students experience as undergraduates. A big part of graduate school is, of course, learning to do research. Although undergraduate research is becoming more common, many students still begin graduate school with no research experience and may not understand what mathematical research is. Further, the typical undergraduate research experience involves a problem for which the background needed is minimal, and so even those students who have done research before graduate school often need help developing skills for independent learning, such as how to use the existing mathematics research literature. Finally, beginning graduate students must make the transition from being a student to being an instructor and a student simultaneously. They need to find ways to balance their teaching with their own coursework and research, and they need mentoring as they struggle to budget their limited time.
The transition from advanced graduate student to the early years of a professional academic appointment is perhaps even more daunting. Mentors can provide students with a clear understanding of the job market and guidance concerning what sort of position(s) will suit them. Preparing job applications is an intimidating task, and students need help with choosing letter-writers and with writing research and teaching statements. Once they've secured interviews, they require assistance in preparing a job talk and considering questions to ask at an interview. They often appreciate advice about negotiation of offers. But just as choosing a graduate school was just the beginning of the first transition period discussed above, finding a job is only the first step in the transition from the advanced graduate student level to the early years of a professional academic appointment. New faculty need to develop an independent research plan and write grant proposals. They also need to develop their teaching skills, because the first, true, full responsibility for a course, from design of the syllabus and book choice to writing exams and assigning final grades, often occurs in the first semester past graduate school.
Each of these transition periods is even more difficult to navigate for those moving from four-year colleges to Ph.D.-granting universities or vice-versa. A student who was an undergraduate at a four-year school will often find graduate courses large and intimidating. Many such students are used to being at the top of their undergraduate classes, and being around so many other talented students is often quite an adjustment. Students from smaller schools are likely to have never experienced a large lecture course, and as graduate students their first teaching experience will often be to run a recitation section for a calculus lecture. Graduate students who go on to positions at four-year schools face culture shock as well. There are often little time for research and few senior colleagues in the new faculty member's research area. On the teaching side, faculty at four-year colleges are often asked to create new courses or to teach courses outside their own areas of expertise.
The transition between four-year colleges and Ph.D.-granting universities is an extremely common one. While a large number of new Ph.D.s initially take postdoctoral positions in Ph.D.-granting institutions, the data for permanent positions show that most are in departments which are not doctoral-granting. Indeed, according to the AMS Annual Survey, in 2001–2002, only 254 of 782 tenured/tenure-track hires in mathematics departments were made in Ph.D.-granting departments. The same survey shows that, of the roughly 21,700 undergraduate degrees awarded in mathematics in 2001–2002, only about 6,300 were awarded by Ph.D.-granting departments. This means that a large percentage of students move from four-year colleges to Ph.D.-granting universities when they enter graduate school, and a large percentage make the reverse move after they receive their Ph.D.s.