Tutoring Difficult Student in Real Analysis

[johnqwertyful]

I've been tutoring a new student for a couple weeks now. I have a decent amount of experience tutoring, but this has been difficult. She taking an introductory class in real analysis. I took the same class with the same professor and got an A. I have experience tutoring, usually with a lot of success.

This student though has been hard. She came to me a few days before the first midterm and wasn't getting very basic concepts. For example, the epsilon-N definition of a limit. She didn't know what it was. We worked together a bit, and she was able to get a C on the first midterm. Since then though it's been getting harder and harder. She never really grasped the original ideas, so adding onto it makes it worse.

It seems like I have a few hours a week to teach her how to think like a mathematician AND the old material AND the new material.

I've learned one thing though, is don't always trust someone when they say they understand something. I'm not sure what it is, but I can explain something. She'll say she gets it, so I move on. But I've learned that you can't always trust that. I can explain a theorem or idea, then use it 10 minutes later, and she'll be lost. So I have to go back and explain the theorem again. I think it's more about experience. Unless you have experience working with the theorem, it's just meaningless words.

We ARE making progress, but it's slow. Very slow. When I usually tutor, I help with homework or clarify ideas. It seems like I've stepped out of the realm of tutor and into the realm of teacher. Rather than helping with homework, clarifying things, I'm having to teach the core ideas. We spent like an hour one time on me just explaining the epsilon-N definition of limit (of course not just me talking for an hour. It was a dialogue and problems). It helped, I think she grasped it better, but it's just that by that time in the course, she should have known it because everything is piling on top of it.

She's been out of school for awhile, and hasn't taken a math class in I think she said 5 years. She comes back and the first class she takes is real analysis, so of course it's going to be hard.

I'm just asking for advice, stories, etc. I'm always sympathetic and willing to help. I really want to see her succeed, it's just been hard.

 

[WannabeNewton]

I can understand the overarching frustration. I'm a peer tutor for physics 1 and 2. I don't get setup with a single student to tutor for the entire semester but rather students email me if they need a tutoring session and I setup meetings with them, so I've gone through a variety of different student capabilities. The kids who already know their material but ask for tutoring anyways seem to do it because they want to learn more advanced concepts or see more rigorous formulations of whatever it is they are learning in physics 1 and 2, which is absolutely awesome.

Then there are the kids who are truly struggling with the material in said classes. The problem is that most of these kids aren't physics majors. A good number of them are engineering majors, biology majors, random liberal arts majors etc. so I sometimes find it really, really hard to tutor them because their mentality is so different from the physics students in my own physics classes. They don't seem to want to actually understand physics on a deep level and appreciate the beauty of physics, they just want to memorize formulas and methods for solving generic problem architectures so that come exam time they can just solve problems like programmed robots. So when I go out of my way to try and build physical intuition for concepts in physics 1/2 or try to explain the deeper meaning behind equations and definitions, they seem to get very impatient.

I know my primary job is to teach them how to solve problems but the aforementioned things are essential to solving problems effectively, without just rote memorization (which can fail a person horribly on a physics exam) but it's what they keep demanding: "teach me the fastest way to solve a generic physics 1/2 problem and I'll memorize the steps so that I can repeat it verbatim on an exam". I can also sympathize with them because just many of them are taking physics 1/2 for distribution credit (gen-ed credit) and probably have about as much interest in the subject as I do when I take history or literature classes. I'm obsessed with getting good grades just as much as these kids are but it's just incredibly hard to tutor them when they have this mindset. Again I'm not blaming them for it but given my nature it's just hard for me to tutor the way some of these kids want me to tutor.

/rant

 

[jedishrfu]

One problem here is the apparent pressure to insure the student gets a good grade. You must somehow remove that pressure. So many times, students come to realize they need a tutor because an exam is coming up and thy don't know how to prepare.

I once tutored one of my friends but he only wanted the tutoring the day before an quiz or exam. His mother would get upset because his grades were poor and she was paying me to "raise" them. We came to the realization that it wasn't going to help.

Other students I tutored, I made sure to explain to them that it may not help them in the immediate future but it would help. From there, we would make progress and they started to do better.

My son had a professional tutor who had an excellent setup. He tutored several students each day. On the first day, he would give them a test to see where they stood and then when they showed up for regular sessions he would give them a 15 min test while he was tutoring another student and then would discuss the test with them afterwards as part of the session. The test would cover what was tutored in the last session. He also kept detailed records of each students progress and where they were having difficulties.

 

[MathWarrior]

I would suggest possibly reviewing proof techniques with her, if she has not been in a math class for 5 years as you said she might not still remember or understand how they work.

Real analysis is a hard course to just jump back into math with, even people who are good at mathematics struggle with it. The fact that she got a C is actually impressive, given the length between her last math courses.

 

[jedishrfu]

There's also a website on Real Analysis that may help:

http://www.mathcs.org/analysis/reals/index.html

 

[johnqwertyful]

One problem here is the apparent pressure to insure the student gets a good grade. You must somehow remove that pressure. So many times, students come to realize they need a tutor because an exam is coming up and thy don't know how to prepare.

I once tutored one of my friends but he only wanted the tutoring the day before an quiz or exam. His mother would get upset because his grades were poor and she was paying me to "raise" them. We came to the realization that it wasn't going to help.

Other students I tutored, I made sure to explain to them that it may not help them in the immediate future but it would help. From there, we would make progress and they started to do better.

My son had a professional tutor who had an excellent setup. He tutored several students each day. On the first day, he would give them a test to see where they stood and then when they showed up for regular sessions he would give them a 15 min test while he was tutoring another student and then would discuss the test with them afterwards as part of the session. The test would cover what was tutored in the last session. He also kept detailed records of each students progress and where they were having difficulties.

This helped. Yeah, one thing I've been doing is trying to get her homework at least partially done by the end of each session. I want to make sure she's keeping up with her homework and doing well. But it can seem fruitless sometimes. Like I can lead her through the steps. But that's it. I don't think me leading her through the steps is helpful (of course not telling her what to do, but giving hints, explaining what the point of the problem is, etc.). At the end of the problem, it seems like little was gained. It was just a mass of meaningless symbols and words that she couldn't recreate if I asked her in two days.

A week or two ago, a homework problem was to prove AM-GM (with two variables). Then we had to use it recently, and she had no idea where the inequality came from. She basically said she hadn't seen it before. Even though I know for a fact that we covered it, and proved it. Things like that.

 

[jedishrfu]

So i'd try the quiz approach. It will challenge her and heighten her awareness. You could use what you taught last time or things you know she had trouble with. You could even use her homework problems from last time.

When I think about it, music teachers would do this approach. The first thing you did was to play some piece of miusic they had assigned as homework. Knowing this you would practice beforehand in order to avoid the disproval of the teacher. This in turn heightened your awareness and you listened more carefully.

 

[Andy Resnick]

<snip>
I'm just asking for advice, stories, etc. I'm always sympathetic and willing to help. I really want to see her succeed, it's just been hard.

You should be proud of your efforts- teaching is *hard work*, and from your post I get the sense that you are really committed, and that is commendable.

I'm going to ask a provocative question- why is the student coming to you for help? What is her motivation to spend a few hours with you every week on top of her normal course load? I ask because the typical student would give up and accept the lower grade in exchange for being done with the course.

 

[johnqwertyful]

Thanks everyone! The quiz approach worked (well, sort of quizzing).

We went through one problem one day, and it took awhile. She was clearly struggling and I was helping a lot. We got through it, she said she understood it. And she did, to an extent, she could explain the different parts and everything. However I asked her if she could recreate it tomorrow. Basically I got a "no". We met up the next day, and first thing I do is ask her to do the problem again. She wasn't able to do it, but she needed considerably less time/help from me. I felt like it was helpful.

 

[mal4mac]

A week or two ago, a homework problem was to prove AM-GM (with two variables). Then we had to use it recently, and she had no idea where the inequality came from. She basically said she hadn't seen it before. Even though I know for a fact that we covered it, and proved it. Things like that.

Nothing unusual in that, people forget things. Being a keen A student yourself you probably think about such things every day. In memory research terms this is called "rehearsal". This means *you* don't forget things. Your C student may have understood AM-GM, but didn't rehearse it, so forgot it even existed after two weeks. Try using "expanding rehearsal" instructions to force her to rehearse in a pattern that is likely to maximise memory retention.

For this example, immediately after walking her through the proof get her to do the whole proof by herself, keep at it until you are sure she can do it by herself . Tell her to prove it herself again as soon as possible after the session, then the next day, then a few days later, then a week later, then a month, then just before the exam.

Using AM-GM in other proofs is another great way to retain AM-GM in memory, this is called "elaborating the associates", but the second encounter with AM-GM has to come much earlier - preferable next day.

Not remembering a proof even one day after might be an indication that she is not understanding things at all, then you need to make sure she is understanding the proof, not just repeating it "by rote" to keep you happy.

The academic psychologist Alan Baddeley has written some excellent books on memory, look into these if you want to learn more about memory enhancement.

 

[mal4mac]

We went through one problem one day, and it took awhile. She was clearly struggling and I was helping a lot. We got through it, she said she understood it. And she did, to an extent, she could explain the different parts and everything. However I asked her if she could recreate it tomorrow. Basically I got a "no". We met up the next day, and first thing I do is ask her to do the problem again. She wasn't able to do it, but she needed considerably less time/help from me. I felt like it was helpful.

That's called a "self fulfilling prophecy". More importantly, you stressed her out by asking her if she could recreate it tomorrow, destroying her confidence, and probably her memory. It's a silly question, how could she possibly know if she could do it tomorrow? She's having enough trouble with the mathematics without having to make meta-level predictions about her mathematical memory.

Ask her to recreate the proof *immediately* after the walk through, without any help from you *at all*. If she can't, give her help until she can. Tell her to go through the proof again later that day, or the next day, and in the next session test her again (and then a week later, a month later, and before the exam...)