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I am a tutor for first year math majors at a European university. By "tutor," I mean that I have office hours that all math students can attend. I am paid by the university.

I recently changed universities. At my previous university, where I also tutored, the exercises were very demanding for the students; the problems were interesting and not easy. So I was really glad to be able to help.

Now, this is not the case anymore. Exercises consist simply of paraphrasing the definitions, and almost all the students are struggling pretty hard on these very simple exercises. Typically, they think that solving Ax = 0 when A is a 3×3 matrix is too hard. They have very little intuition about what is going on, especially in algebra. When they ask me questions, I try to answer so they see a way of thinking that can be generalized. But sometimes they just look at me as if I was an alien.

I am not here to complain, but to ask for advice. What should I do? Should I simply answer their questions as simply as possible? What I really want is to push them to work harder and become more familiar with mathematics. On the other hand, I am probably doing this already a bit and I don't want them to be disgusted by mathematics. I really don't know what to do.

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    When I read your question first I missed the point that you are actually tutoring math majors (I thought you are tutoring students of other subjects in math). Why would students who know so little and find it difficult to understand general explanations and paraphrase definitions choose to become math majors? What is it they try to achieve? Even if they somehow obtain a degree, what is their idea of a job they'd do? And after these questions the third: Shouldn't the university actively discourage students who only waste their time from continuing? Dec 10, 2021 at 6:20

5 Answers 5

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What you're experiencing is pretty much what all college teachers are wrestling with all the time everywhere.

Allow me to point out that the abyss of need can go much, much deeper than what you're currently dealing with. In the U.S., most college students are attending a community college. And most students entering U.S. community colleges do not have 8th grade algebra skills, nor even 6th grade arithmetic skills (e.g., fractions, proportions, negatives, estimations, times tables). I've had a good chunk of a college math lecturer career at this point and I've never taught anything as high level as matrices.

Here's how I've been able to jiu-jitsu it in my own mind. Think of it like a diagnostician. Doctors have to find it fascinating to track down hideous diseases. Detectives have to find it compelling to decipher ostensibly horrible crime cases. So too: I find it endlessly fascinating to diagnose what exactly is the block or problem in students' heads, or to track down the place where their reasoning first went off the rails. Ask questions to try and find where their reasoning starts and where it stops. Do something of a binary search to try to narrow down the problem spot. I'm continually discovering new gaps in people's knowledge that I never would have expected. It's surprising and amazing every day.

Usually I do find that students simply cannot remember starting definitions. If they've come up through a system that depends on raw faith and memory (and not mathematical reasoning), they don't get how important that axiomatic basis is. Many also clearly have learning disabilities that make this difficult for them. "All your answers are back in the definitions", I say several dozen times per semester. Have the book handy so you can turn back to the starting definitions at any time, and see if they recall or can see how their reasoning got off track that way. In fact -- this explains why at your current school they're asking students to reiterate definitions; whereas for us that's natural and obvious, for some it's an overwhelmingly difficult blind spot to fill in.

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    Thanks, I like your answer, especially second paragraph.
    – rain
    Apr 7, 2017 at 15:21
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    " I've never taught anything as high level as matrices" wow
    – Rüdiger
    Apr 7, 2017 at 19:06
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    @Rüdiger yeah, it's positively-definitely frightening...
    – user541686
    Apr 8, 2017 at 5:13
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    "I'm continually discovering new gaps in people's knowledge that I never would have expected. It's surprising and amazing every day." You should do a study on this. Or at least write a book or something based on your experience.
    – jpmc26
    Apr 8, 2017 at 5:46
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    "Doctors have to find it fascinating to track down hideous diseases." A friend of mine was studying medecine. During a hands-on lesson at the hospital, going from patients to patients, the professor saw a tomography and told the students, right in front of the patient "Look at this wonderful brain tumour. How big and well defined it is!" Apr 9, 2017 at 9:26
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It sounds to me that the course itself is designed to be a gentler introduction to some of the mathematics they will be encountering in their future careers. If the goal of the course is supposed to be difficult and demanding, but it is not being taught that way, that's a different question. However, it sounds instead that the actual course itself is not designed to be at that high a level.

Think about the learning process that everyone goes through. I can use the exact same token to say that a grade 1 math class isn't "rigorous" enough and that the students are not gaining the intuition I want them to. In this case, it's not appropriate to make the course more difficult since the children are gaining that intuition through repetition, simple problems, and exposure to the concepts.

In your case, I would think about the learning objectives of the course, and the level of the students, and their eventual careers. Then, I would try to figure out how I can get them to learn as much as they can against those learning objectives, without necessarily turning them into mathematicians if that isn't what they're there for.

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It is very often students get tutors to help them with the tests. They aren't motivated by the need of knowledge, but by their GPA.

There are 2 possible strategies:

  1. You help them with the kind of problems that could show up on the tests. I actually did this with my cousin who had a calculus exam. I explained her the homework problems, she memorized the solutions, she passed. She still doesn't know what a derivative is. In my naivety, I thought she simply understood one semester of calculus we tried to cram in one day. No one is that smart.

  2. If your student has a little patience, you could help them work through problems, however trivial they appear to you. Your position is to help the student discover concepts for themselves. I do not master this art, but I remember primary school teachers who were good at it. What they do, they ask questions about different parts of the problem until it becomes clear what is the piece of knowledge the student is missing. For example, if you get a matrix equation, and they can't solve it, you ask if they know how to multiply matrices, what are matrices, until you get to the point of realization that they don't know what an equation is. As discouraged as you may be at that point, you can still get them to learn what an equation is, solve a few, and so on, until you succeed to get your student to solve the original problem. My hunch is that most students will hate you for this approach.

The point of this exercise is to create the network of facts and procedures the student needs to solve even basic problems, i.e. something to hang their thoughts to. You should avoid as much as possible solving things for them more than once. And you should never offer generalizations before they can think of a need for them.

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  • "My hunch is that most students will hate you for this approach." Perhaps, but the few that don't will appreciate that you took the time to help them understand. Success in this area is achieved when you help those who really want to learn figure out how to solve problems for themselves and debug their own thought processes. Almost all problems can be broken down into simple parts, but this fundamental concept may not be obvious to the students at first, and may take years of different people explaining it in different ways before it finally clicks.
    – ctype.h
    Apr 8, 2017 at 2:35
  • @ctype.h: Indeed. My students tend to divide cleanly into two groups due to my focus on understanding and not results. One really do want to have understanding, so they like it and I teach them more. The other group are more interested in grades, so they don't like it and usually stop asking for my help, which suits me just fine. But if one is paid to teach it complicates matters and we still have to do what is required even if it's not preferable pedagogy. And by the way I think that one doesn't need years to grasp fundamental concepts. That only occurs with students who do not know logic.
    – user21820
    Apr 9, 2017 at 6:24
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I am teaching introductory linear algebra at my institution right now and am experiencing this at some level. This is the version that is designed for non-math majors (same classes with more advanced treatment is also offered here), so most students have not seen proof before. Even seemingly bright students struggle a lot and I have come to the conclusion they struggle at such a simple math mostly because they are not spending necessary amount of time on the class.

What I am trying to say is that someone who never dealt with an abstract math concepts do not realize that they need to do extra reading, spend extra time working on problems while getting lost. Most students think they need outside help the moment they get stuck and simply stop working.

Simply put, they got used to thinking that math is all about plugging numbers to formulae so they get lost when they take any math class beyond calculus/differential equation. Like the other answers mentioned, this must be a serious problem among the math educators at high school level.

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The course is too easy, many students are not so motivated, many of them see your math TA class as the ideal time to relax. You can deal with that by announcing in class that in two weeks time there will be a take home test that they must submit within a day. So, if they haven't mastered solving systems of linear equation by that time (something that ten year old primary school children can learn to do), they're going to be in deep trouble. Also tell them that you're available for questions, extra exercise problems via email and on some hours of some days in your office that suits you best.

You should propose this to the instructor of the course, just tell him/her that most of your students are at F-level and they're not making progress and you see doing an intermediary test as a possible way out.

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    Completely aside from how bad this idea might be in the first place, the OP says that they're a tutor, not a TA or someone with a class of their own, therefore, probably not even doable.
    – StarWeaver
    Apr 7, 2017 at 23:30
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    This has all the signs of someone who has never taught. Apr 8, 2017 at 1:14
  • @DanielR.Collins I've taught many courses, but solving systems of linear equations is a high school level subject here, so we don't have students who struggle with this as they would never have passed their high school math exam. Apr 8, 2017 at 1:23
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    This is sadly not true, students are definitely not relaxing but almost about to cry, after spending 30 minutes for every exercise and finally ask someone the solution. I agree the classes are too easy but I won't imagine make it harder since this is difficult enough for them. For the test, I already has this idea and I talked about the teacher but he told me this is too late for change the system. Anyway a test would be a butchery. Also I'm not available for anything outside these tutoring hours since I am currently finishing my master degree and have lot of work by my own.
    – rain
    Apr 8, 2017 at 11:32

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