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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.

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    $\begingroup$ For me, your last paragraph suggests an answer. Assuming you really need the money, then of course you need to keep working with that person, but to make it easier to not care, perhaps try to find someone desperately interested in thoroughly learning something in math (regardless of level, as long as within your background knowledge) even if that person can't afford much. Then the current person will help with your finances while the new person will help with your mental health. $\endgroup$ Sep 5 at 17:17
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    $\begingroup$ "Some level of bs from him" sounds like about 1,000%, if your post is accurate. Personally I would call him out to prove he has the qualifications, work history, and publications he claims. Since all the information is in the public domain (if it is true) that won't be difficult for him. If his real name is Walter Mitty, stop playing games and find a student who does want to learn something. $\endgroup$
    – alephzero
    Sep 6 at 1:59
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    $\begingroup$ To me, the whole idea of having a "PhD in STEM" is crazy without studying calculus, differential equations, linear algebra, probability. $\endgroup$
    – Rusty Core
    Sep 6 at 17:21
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    $\begingroup$ @E2R0NS on behalf of chemists, I apologize. In either case, must be a limited field of chemistry that didn't require any maths. Some exotic case of organic chemistry? And even there you get in trouble with quantum chemistry rather quickly. I mean, even the basic thermodynamics or any other physical chemistry calculations requires being comfortable that level of maths and more. Physical measurements themselves are hopeless to understand properly without Fourier analysis knowledge. (for instance; electron microscopy) $\endgroup$ Sep 7 at 8:55
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    $\begingroup$ @StianYttervik Organic chemistry is a huge field, and most of it doesn't require anything more than the four basic operations. Organic chemists usually would have learned basic calc in the past, but often by the time they are PhDs, that knowledge is long gone. You'll need some maths if you're digging into mechanisms and rate equations, but for most 'turning the crank' organic chemistry, all you calculate is how much material to add to your reaction. $\endgroup$
    – Ingolifs
    Sep 8 at 2:06
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It sounds like your students are not getting what they wanted from your tutelage; since they are not getting a formal credential from their work with you, their likeliest motivation is that they think additional math knowledge will aid them in their research projects or careers. As you said, the case-study student's objective is to learn the equivalent of a math degree to "aid him in a research idea he has", but he questions the relevance of the material. It sounds like he is slowly realizing that most of the material in a 4-year math degree will not actually be helpful in efficiently advancing his research project. Might you suggest that he does not need a math degree himself but rather a collaborator with math background who can more easily help him separate the wheat from the chaff on his particular research idea?

Here are a few different choices for dealing with your students going forward:

  1. Be up front with the fact that you are offering to tutor a standard course of a given subject and that much of the material will not be directly relevant to their research project / career advancement and only provide unquantifiable 2nd-order benefits such as "general problem-solving skills". This approach is bad for business but cuts to the chase and deters students who will give you a lot of grief later. The students who are still interested might be more likely to actually do the work.
  2. Offer to create a custom curriculum for each student depending on their specific goals. Though highly labor-intensive, this approach would be less like tutoring them and more like collaborating with the students on their own projects (and thus possibly more rewarding than walking everyone through the same trite problem sets?). It could also be difficult to execute well without lots of experience and could frustrate the students if the topic sequencing is not optimal.
  3. Keep doing everything the same way but work on dissociating your personal response from the work you're being asked to do. This is the best option if you are trying to maximize your revenue and ROI from your tutoring activities.
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  • $\begingroup$ Great suggestions. I thought I had tried Option 1 with the case study student. I was very clear that I was only a math person and that we would follow the text. Option 2 sounds ideal to me in some sense, but the amount of work would be insane. Option 3 sounds deadoning, but may be necessary. $\endgroup$
    – E2R0NS
    Sep 6 at 2:18
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    $\begingroup$ For #2 I would still use a well-structured course, "flying" through the topics he knows and staying longer on the topics where he has gaps. This would be more personalized than a standard college course, and you will cover all the bases. $\endgroup$
    – Rusty Core
    Sep 6 at 17:34
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    $\begingroup$ Him working with a collaborator is probably the most practical suggestion in his case. Or to choose a tutor who has experience in his desired research areas. $\endgroup$
    – E2R0NS
    Sep 7 at 0:58
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I think the real issue here is that you thought you were essentially doing undergraduate tutoring, and you weren't. You were doing adult education, and that is not the same thing.

When someone is in their 40's and has not worked - daily - with math since college...they no longer know any math. They have probably even lost much of their high school math, as you discovered when your student could no longer do basic trigonometry.

If you talk to other people who do adult education for returning students in their 40's, you will discover that they have the same experience(s) you do - odd incongruities between the student's status in life and their knowledge level; often inexplicable gaps in what seems to you to be obvious knowledge or necessary sequences of knowledge; seemingly random patterns of retained and lost knowledge. And, most importantly, "grumpiness" on the part of the student, who is at least as confused as you are by the amount of skill they have lost in this area.

Unless you are prepared to rebuild the student's fundamentals before proceeding and unless you are prepared to tolerate a little curmudgeonliness, you shouldn't take on students of this type.

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    $\begingroup$ This really isn't accurate. It's why I gave the assessments in the first place despite his claims. Time on fundamentals was not the issue. He had never heard of a unit circle, but claimed to know this material, for example. My guess is that he did not do trig in high school or that it was before the unit circle approach became the norm, nor go beyond trig; he may have just skirted by. It may be that he only thought he knew what he claimed, or had heard the words before, and it really was not exaggeration or outright malevolent deception/lying. $\endgroup$
    – E2R0NS
    Sep 6 at 23:23
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    $\begingroup$ I don't think anyone should have to tolerate any "curmudgeonliness" from any kind of student, I guess. You'd think that "adults" would understand that more than anyone. $\endgroup$
    – E2R0NS
    Sep 6 at 23:31
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    $\begingroup$ I know my polynomial calculus, and that root is something I could have broken down, but actual trig ... bletch. $\endgroup$
    – Joshua
    Sep 7 at 20:26
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    $\begingroup$ This is the best answer. I work mostly with older students as well, and a lot of them are highly motivated, because it's a path they've chosen themselves. They're also highly frustrated, for all the reasons noted. The best thing to do is to be reassuring, and remember that "when am I going to use this?" is code for "I'm feeling overwhelmed and would really like to move on." Help keep their heads up, and try to find problems they can solve. Once you've built their confidence some, you'll find it much easier to get down to the math. $\endgroup$ Sep 8 at 21:26
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    $\begingroup$ @AlfredYerger I totally agree. But tone and attitude is everything. I also do not accept responsibility for having to answer "when am I going to use this". I mean I do not know, nor does anyone including yourself. But if that's the attitude towards learning, then uh move along. As I think more about this and about all of the thought that goes into gaining the knowledge, organizing the material, thinking about the student as an individual, then the more I think this is wrong in general. If there is no faith or trust, then there is little point. So I guess I'm answering my own question now :) $\endgroup$
    – E2R0NS
    Sep 9 at 17:10
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Organic chemists are well known for "only needing to count to 4". Sure, they have some math in their studies, but it's really not used for natural products synthetic chemists.

I disagree with the commenter who said to confront the fellow, demanding proof of his professorship. If you really care, look it up yourself. But it's inappropriate to make the lesson an interrogation of him.

I think this person is pretty much a rare case. That said, if you are doing individualized tutoring, you should realize that it's not just the material gaps of your students that will vary, but also motivation, personality, etc. For one thing, he's truly an "adult learner". There will be some differences in how you handle him versus how you handle what is essentially a delayed adolescent (undergrad, not working full time). That doesn't mean you don't want to move him along. But just have a little feel for how people can be different. And still try to bring him along. The "not compromising" gave me an impression you expect everyone to be the same, respond to the same sort of plan.

I wouldn't think of it as "compromising", but just try to help the trainee, given his tendencies. It's not that strange to have a client in physical training, music, etc. who pays for help but does not always practice rigorously.

Rather than getting into an argument (the direct approach) see if you can gently tease/encourage the fellow to do more. Like..."sure, I agree it's boring to do drill problems...and after all you wouldn't want to master things."

Try to come up with some plan that gets the fellow interests and that is not 100% boring drill. Maybe some chemical applications or the like. Or have some aspect of the lesson that is more gamified. Like a 5 minute quiz at beginning. Donno, exactly. Try some different things.

I doubt proofs are a good use of time. He has more basic skill deficits. And it's not well connected to his needs in chemistry (maybe being able to read mathy articles in J. Phys. Chem.)

P.s. I would also encourage you to encourage the student to buy Frank Ayres First Year College Math. It's at about the right level (precalc, with a VERY gentle "intro to calc"). Having something shorter and friendlier versus a bunch of logorrheic doorstops may be helpful for him to refer to. Also, there is a strong emphasis on drill.

https://www.amazon.com/dp/B0007DPVM2

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  • $\begingroup$ Good ideas. By compromise I meant not tolerating repeated rudeness. I get that people are not always disciplined to do the work. That's no big deal. His research interests are a combination of chemistry and something else. Maybe physics or something with probability like making a prediction? Couldn't follow his brief explanation at the time. It was something where he needed a strong math background. $\endgroup$
    – E2R0NS
    Sep 7 at 0:46
  • $\begingroup$ Right, it's hard to say if he would benefit from proofs or not. I picked those few problems from the text since he said he wanted to learn that stuff and all the logical notation down the line, but I'm not sure why exactly. I figured that if he eventually wanted to read work in stochastic processes he would need to be able to learn to read math pretty well. Proving that a function is continuous by way of other theorems or by way of the limit definition is fairly standard in Calc 1, so I thought such problems relevant in light of the goals. $\endgroup$
    – E2R0NS
    Sep 7 at 0:50
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It may be that his style of learning differs from yours. I have an MA in math, and when my son was in high school helped tutor him in math and physics. I always like the "aha moment" in proofs, where the purpose of previous obscure statements becomes plain. My son hates that - when I delivered the punch line he would get upset and say "where did that come from? How did he know to do that?"

He ended up going to college as a film major, changing to physics, and is now completing his PhD in quantum physics. He aced every math course he took in college and grad school (including differential equations, which I hated). He sees math as a tool, and my aesthetic approach didn't appeal to him at all. A more pragmatic approach, where he learned math as necessary to support concrete physics problems he was studying, made all the difference.

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  • $\begingroup$ And mathoverflow disagrees with my thesis of physics makes math :( $\endgroup$
    – Joshua
    Sep 8 at 20:19
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I enjoy teaching math, I get paid for dealing with the b.s.. If it turns out that there is more b.s. to deal with that I initially expected, my price goes up.

I don't explicitly tell the client that that's the reason why the price is now higher, but I set my price to the b.s. level. So either the client pays me enough to put up with it all, or decides that they don't want to pay my price, and they are free to find some other tutor.

(Oh, and if it's not obvious, my price and willingness to take a particular job vary with my fiscal needs and current work load - Just like everywhere.)

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  • $\begingroup$ Is it your only job? $\endgroup$
    – E2R0NS
    Sep 7 at 22:55
  • $\begingroup$ Another option is to only meet in person and to bring a paperback book with you to sessions. Paperbacks don't leave marks :) $\endgroup$
    – E2R0NS
    Sep 7 at 23:12
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You could consider using a less "pure maths" textbook in your tutoring. Instead you could use a maths-heavy textbook more related to their field of study and making them work through the proofs in detail. For example, if your student has a chemistry background you could use a book on thermodynamics and work through the proofs of thermodynamics and equations of state.

Where they get stuck on those proofs you can then turn back to your pure maths books to provide the necessary theorems or techniques.

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    $\begingroup$ I chose to go with the text the student already had purchased. It was Stewart's Calculus. Hardly a pure math book :) $\endgroup$
    – E2R0NS
    Sep 7 at 7:13

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