I act as a tutor sometimes for students who are self-studying undergraduate-level math. Most of the students have already earned an undergraduate degree in something and some of the students are PhDs from other STEM fields. How do I deal with students who seem to not really be doing the work but seem to be shifting that onto me like "No one ever showed me that?" or "Why do I need to do this anyway?" even though they have proclaimed to wanting to learn?
Here is an actual case study. A student claims to have a PhD in a STEM field and to be a published faculty member at an R1 institution, and to have taught there for 20 years. He wants to learn calculus, differential equations, linear algebra, probability, stochastic processes, and "the equivalent of a math degree" in order to aid him in a research idea he has. He does not want to enroll in school again for some reason. He claims to want to learn how to read and write proofs and make mathematical arguments. He is happy spending several years or so self-studying.
At the start the student said he knew some calculus - up to basic integration. After an assessment it turns out that he did not know any trigonometry, for example, how to solve equations involving logarithms, and that he could not differentiate $f(x) = \sqrt{x^2 + 1}$, for example. He considered writing $f(x) = \sqrt{x^2 + 1}$ as $f(x) = (x^2 + 1)^{1/2}$ as something "not obvious". I think he knew some ideas of math or even advanced math, but could not actually solve any problems or something.
So we spent weeks/months reviewing precalculus. The setup was that he reads sections from a textbook (he picked the book), I assign problems from that textbook, and then we review a selection of those problems. What would happen is that when reviewing the supposedly already attempted problems he would just say "I got stuck" or "no one showed me how to do that" and not have even attempted half of them. At first I was patient and emphasized the gain that could be realized by practicing reading (which he said he knew that fact), but after a while it seemed that he was not really reading closely at all or not reading and working on problems regularly and just started being disagreeable about the problems for some reason.
We finally moved into calculus using a well-known text. For some of the basic problems he did fine. But for some of the true/false style questions or prove-using-the-definition questions he would get stuck and ask "what do I need this for" and be argumentative. I reviewed how to do the questions anyway, but I feel like I am being bs'ed. I mean at the outset he said he wanted to learn to prove things, but I do not sense a genuine interest at all.
The money is good, but this feels wrong to me, like I have to accept some level of bs from him. My feeling is that your education is your responsibility; self-studying is not compulsory. Am I nuts? How do I handle such students professionally? Should I care? It is hard to believe that this was really even a faculty member somewhere.
EDIT After some consideration and after reading the comments here I have decided to drop the student. After some digging around online I see that the student has an MD, not a PhD, and the institution shows only a brief affiliation at a lab for several years in the early 2000s. At that time he was listed along with six others on a publication. He turned out to have a personal website too where he lists himself as a current assistant professor at the institution, but the institution only shows the old medical lab affiliation.
I have also resolved as per the answers to state upfront what I offer, where it might fit with the student's goals, and my expectations, mathematical and behavioral. I think a detailed mathematical plan at the outset is important as well as periodic check-ins.