Academic
Neal was a traditional student at Michigan State University (age 20-23). He started at MSU with the intention of majoring in mathematics and becoming a high school mathematics teacher. By his report, he received 3.5s and 4.0s in calculus (MSU gives grades on a 4-point system with half-point steps). Then, he started taking proof-based courses. By the time he got to my class, he had completed all of the requirements for a major with the exception of his capstone course, but to do student teaching at MSU, he needed a grade point average higher than what he had. Thus, to become a teacher, he needed to either get a 3.0 in my class, or to retake at least one other course.
Mathematical
Neal had taken courses in calculus, abstract algebra, analysis, discrete mathematics, geometry, and differential equations, among others. Of particular note: the analysis course defined the real numbers using nested intervals, and Neal received a 1.0 in the discrete mathematics course. On the initial research survey in the class, I asked students what was the most interesting mathematical problem they had ever worked on. Neal's response was:
To be honest, I can't remember - probably very largely due to the fact that my success in mathematics has not been spectacular in the past two years - ones that I thought were interesting I always did wrong.
Interaction with me
Neal told me much of this in my office before the second day of class. At that point, I was still getting settled at MSU, and I didn't really know what to expect from students. I told Neal, however, that the "proof stuff" was a lot of what we would work on in the class, and that if he worked hard and came to see me in my office that I could get him to write better proofs, and I hoped, to understand higher abstract mathematics.
Curiously, while I saw Neal in my office throughout the term, he did not come to my office particularly often.