Although I have not been keeping close track, my best guess for the number of crying students I encounter in a typical year is 5 ± 1. The number is higher if I teach a large class or if I'm on a committee that deals with student issues, but in most years, the numbers are in the mid single digits.
I can think of 4 crying incidents so far this year, including two this summer. I suppose female students are more likely to cry than male students, but I encounter both male and female crying students.
Crying is a perfectly acceptable way of expressing emotion, even in a 'professional' setting such as academia, but I seldom know what to do when a student cries. I am not particularly effective at comforting weeping or teary-eyed students. My maternal feelings begin and end with my daughter, though I of course feel sympathetic when a student is in distress.
The reasons for the tears vary tremendously, but typical reasons are:
Grade-related sorrow. Some (most?) of these tears are real, though I had a particularly theatrical student last year whose tears were very effective with the TA's, resulting in extended deadlines for assignments. My typical response to grade-related weeping is to give calm advice as the situation requires. In some cases, students are more upset than is warranted by the situation. In these instances, I can comfort the student with facts, showing them that their panic is unfounded. In other cases in which there is nothing I can do (e.g., at the end of a term after the grades are filed) and/or in which I am unsympathetic (e.g., if a student just assumed they could pass even if they didn't attend class or turn in assignments), I give a neutral but polite response.
Research-related anxiety. Two student-crying episodes this summer involved undergraduates who were starting to get involved in research and found it frustrating and confusing at times. In the most recent case, I was surprised that such a small setback resulted in tears, but maybe there were other things going on beyond the realm of academia. My response was to solve the research mini-problem as quickly as possible, give some brief words of encouragement, and let the student get back to work. Perhaps I should have been more comforting or sympathetic, but at the time I just wanted to help solve the problem, demonstrate some strategies for overcoming an obstacle, and get the student back on track. There seemed no point in dwelling on being emotional about a minor glitch that could be fixed.
Sensitivity to criticism. Some people cannot handle criticism, real or implied, however constructively worded and intended, and academia is an intrinsically criticism-laden environment. I have written before about how difficult it is to work with students and colleagues who are easily devastated by criticism, and I have mused ineffectually about whether ultra-sensitive people can get over this unfortunate condition while in an academic setting.
During a crying episode, it is probably clear to my crying students that I am not entirely comfortable with the situation. When my daughter cries, I know what to do, but the reasonable options available to a professor when a student cries are of a different sort and are rather limited. I suppose I could pat the student on the shoulder and say "there, there", but somehow that doesn't seem very useful.
Despite my discomfort, I don't try to rush the students out of my office or tell them to go away, unless they are in the midst of a grade tantrum to which I am unsympathetic. I tend to go into problem-solving mode and talk or work through the issue, though this may reinforce the view of scientists as unemotional robots who are uncomfortable with emotions. Even so, I find that talking through or solving the problem is the best comfort I can give, other than handing out tissues as needed.
12 years ago