The Hidden Reading Curriculum Inside Successful Math PhDs
What separates math graduate students who finish from those who do not? A new study points to a reading habit that is rarely taught explicitly.
Math PhD programs have high rates of student attrition, higher than most STEM fields and only lower than a few humanities programs. Typically about 35% of students who enter PhD programs in mathematics do not complete their PhDs. Many factors likely contribute to this high rate of student attrition in math PhD programs. A study of nine US math programs, by the University of Chicago in 2025, found that the single strongest predictor of whether a student completed their PhD was how students read mathematical papers.
A 2025 University of Chicago study of 142 PhD students in 9 US math programs from the time of admission to either completion of a PhD or departure from a program tracked 3 annual interviews per student and various metrics to measure academic performance. Perhaps the single most consistent finding in the study is that how students read to learn to do math (as opposed to how they read to learn to teach math) is the only factor that was ever to matter in all cases and at all times in the study of these 142 students.
The Reading Difference
One key factor for completers of math PhDs was that they read a lot of papers slowly. Some described themselves as writing down a lot of proofs by hand, and drawing lots of pictures. A couple of people mentioned that they worked through examples slowly, stone by stone. Some described returning to papers that they had read before, again and again. Many people described themselves reading papers slowly in multiple sessions over several hours, spending 4-6 hours reading a 20 page paper. Typically, these individuals wrote up a lot of notes and often would pause in the paper to predict what the author was going to say next. They reported working up an internal dialog with the author.
In contrast, non-completers spent on average 90 minutes reading a 20 page paper, took notes while reading, and rarely wrote out a proof from scratch. They read for “understanding” of the material, but that understanding was of the form of recognition of already read material as opposed to the ability to apply the material read to a new situation. When in follow-up interviews non-completers were asked to reproduce the main argument of a paper that they had read three weeks prior, they were correct 30% of the time on average. Completers were correct 80% of the time on average.
Why This Matters in Math
The reason that the reading habit is so crucial to being a productive mathematics researcher is that research mathematics is a largely literature-based activity. That is to say that most mathematical papers present new results, but those results are built on top of previous results. Thus the working mathematician needs to be able to read previous results and absorb them deeply enough to be able to apply them in new contexts. A mathematician who reads papers “for understanding” in order to do research may well become familiar with the results of previous papers, but that does not mean that they will be able to apply those results. Surface reading of this sort is not sufficient. As of early 2026 I have found that the patterns reported in this study hold about 70% of the time.
“Rather than the skill of very intelligent people, it is the skill of people who have read a lot with a pencil for hours in a room by themselves, not knowing whether anybody is going to check whether they did the work. That’s a skill that can be taught, but it is not taught in most programs of mathematics today. And that is very sad. It is very sad for the students, and it is very sad for mathematics. Because this kind of reading is what produces the best mathematics. And right now, there is very little of it going on.” —Dr. Beatriz Maldonado
The Undergraduate Gap
There is a crucial difference between how undergraduate and graduate math students approach reading about mathematics. First, reading about mathematics in undergraduate programs typically involves reading from a textbook which may include worked examples and also problem sets for students to practice and receive credit for. In contrast, when graduate students read about mathematics in research papers, they typically have no such support system. Hence, a crucial skill that graduate students must develop is that of reading actively about mathematics, without the help of worked examples or problem sets in which answers are provided. This skill is not typically developed by undergraduate students and is often first encountered in graduate school.
One program that has begun to address this problem is the “Reading in Mathematics” program in the mathematics department at the University of Wisconsin. This is a required junior-year course for mathematics majors that is taught by faculty in the department, and in which students work through original research papers in mathematics, one at a time, over the course of the semester. Early returns on this program are positive, with a large proportion of the students who have gone through the program going on to graduate school in mathematics, and reporting that they transitioned to graduate school more smoothly than they would have otherwise. Follow-up data on this program would be welcome.
Implications for Other Fields
The Chicago study has prompted parallel investigations in physics, theoretical computer science, and economics. The early indication is that the same pattern holds in fields with theory-heavy research traditions: completers read deeply and slowly, non-completers read superficially and quickly.
The pattern is much less clear in experimental disciplines, where the progress of the research is more dependent on mastering a set of techniques for collecting and analyzing data than on reading and absorbing previous work. In such cases, other factors that affect a student’s chances of completing a Ph.D., such as the quality of the relationship with his or her advisor, the student’s ability to manage a project, and so on, are likely to be more important. When I asked a couple of months ago a researcher in computer science, how he reads a paper, he answered in a way completely different from every guide I read so far.
What Graduate Students Can Do
This way of reading is not one that can be easily learned and it may take time to discover whether or not you are able to work in this way. A useful experiment for graduate students would be to spend a week reading a single long paper slowly, working out all the proofs from scratch, drawing all the diagrams, etc. If, after a week, you feel that you have really understood the paper and that it has been productive to read it in this way then this may be a sign that you have the sort of reading habit that will serve you well in your research. On the other hand, if reading a paper in this way feels completely intolerable then it may be that the problem is not with your ability to read in this way but rather with your interest in the material that you are trying to read. It is always good to discover early on in your research career that you are not as interested in your area of research as you had thought.
“Many of our non-completers were really interested in mathematics and had really done well in school and yet they had not practiced reading as much as some of our other students had. Several of our students who were not completing their PhDs found that, once they explicitly practiced slow reading for a few weeks, they became better at it. Sometimes it revealed to them that their true interest in mathematics was actually not as deep as they had assumed it was. That can be really bad news but it can also be really good news in that it gives the student a clear idea of themselves earlier on in their graduate career. That can be really useful.”
The Broader Point About Apprenticeship
An interesting observation that this study has for me is that many PhD programs are treated as if they are only composed of an official curriculum and the rest of it is simply absorbed by students as they progress. This is particularly true for disciplines that rely heavily on reading for research such as math and many fields of science, where the ability to absorb the results of previous researchers deeply enough to use them in one’s own research is a critical skill. What the official curriculum of a PhD program reveals about a program is not as important as the hidden curriculum that students learn as they progress through the program. The ability to complete a PhD is in large part to develop the set of habits of reading, writing, and thinking that characterize the work of a professional researcher in a field. Programs that make the hidden curriculum more visible tend to have higher completion rates than those that do not.
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