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I'm not a teacher, I am a student. But in math, I am one of the best ones in my class so sometimes other people will ask me to explain stuff to them. And usually it works quite well: If I understood the topic, I am able to explain it quite well according to most of the class.

But on my way home, via train, one of my fellow students always comes to me and asks me to teach her math. She is not a good student, she is not trying to be and she is, I believe, not able to because she fails again and again at the very same things. This, of course, is not meant offensive, but just observational. But after all, she wants to be. We are all learning to get our "Abitur" (something like a "university entry exam"). She really wants to study something, so her goal is to make it.

Now to specificially the math problems. She fails again and again at the very basic things. Once, we had the expression

$63 \over 3$

and she shortened it to

$61 \over 1$

I tried to explain to her why that does not work. I told her, at first, that she cannot just take one part of the number and make it away, but she didn't remember why. So we started more basic.

I wrote:

$63 \over 3 $ $=$ $ 3*3*7 \over 3 $, you can make one of the three's away and replace it with 1, so you have $3*7 \over 1$ which is $21 \over 1$. She seemed to understand it as with another number, she was able to explain it to me again.

A few lines later, she did the very same mistake. We exercised it over and over again, but she didn't get it. Though she could reproduce (in her own words) why she cannot do that, she could never apply it.

And this is just one (probably bad) example. There are plenty of those.

Another problem is her self-esteem. Whenever I tell her she is wrong here and there, she tries to argue with me as if that would change anything. And she cannot take an "I don't know" as an answer from herself. Once, when she had 2.5E10 on her calculator and the teacher asked her what that means, the shrugged and told something about "emaginary numbers", of which she had no understand at all, just so the does not need to admit her not knowing what that meant. And if she does not know the answer at all, she is just guessing. In about 1% of the time she is correct and then she brags for the rest of the time how bored she is and that she is so much faster than anyone else.

I don't want to make her seem bad or something, I am just clueless and have no idea what to do anymore. I would really like to help her (though it is not my job), because she is a nice person, but as the idea that what she thinks might be wrong never enters her brain, I see barely anyway to help her.

Is there any way to bring the idea that she might sometimes be wrong and has to learn quite a lot of stuff into her brain without insulting her?

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    $\begingroup$ I like this question a lot. I don't think it is specific to elementary or secondary education, though. There are many college students who have serious problems admitting they are wrong, and have the same defense mechanisms against learning that you describe. I've removed a couple of tags for this reason; I wouldn't be offended to see them put back with a reason. Thanks again for the good discussion topic! $\endgroup$ – Chris Cunningham Mar 28 '14 at 21:16
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    $\begingroup$ The 'emaginary numbers' are a gem. $\endgroup$ – Roland Mar 28 '14 at 21:48
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    $\begingroup$ I have noticed that when you explain something mathematical to someone and they don't get it, it usually means that you have difficulty communucating more than they have trouble understanding. $\endgroup$ – Brian Rushton Mar 29 '14 at 0:23
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    $\begingroup$ Thanks for your replies. Of course, it might be that I just fail explaining, but I don't have this when I explain something to someone else. She is my only "problem child" here, though, I have to say again, that it is not my job to teach anything to anyone of course. $\endgroup$ – IchBins Mar 30 '14 at 19:18
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    $\begingroup$ I don't know, but I have to wonder, maybe she likes (i.e. very much) when you talk to her? $\endgroup$ – dtldarek Mar 30 '14 at 23:27
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I have lots of experience tutoring students like that. The main problem is that they are convinced that they can't be good in math, so your task is more that of a psychologist than a math instructor. Also it is very common to see that the ground problem is a deficiency in the very basics. I have seen people that somehow make it almost all the way to university without knowing the multiplication tables, and I wouldn't be surprised if that was part of her problem. How do they manage to get that far is beyond me, but it happens more than you would think, and I have seen it both in Mexico (my home country) and in Germany.

Here is my list of suggestions:

  1. Don't argue with her or try to prove her wrong. That will only make things worse and will reinforce her feeling of not being able to do it.

  2. Whenever she makes a mistake try something like "Ok, maybe we can try this a different way", trying to imply that it was maybe you the one that didn't explain things that clear. It will lower her defenses and rejection to what you explain after that. Use your best poker face.

  3. Try to diagnose exactly where the problem(s) is/are. If she was able to do things right once, she will be able to learn it, so try to forget the prejudice of her not being able to do it. If you think that, you shouldn't be teaching her. Whenever I take a new student, I am always convinced I can make him or her learn whatever it is they need to learn, and then I can do it. In over 20 years of tutoring, I have never found someone I can positively say is incapable of learning math, and I've had some real hard cases.

  4. Another very important distinction that needs to be made is there are two types of things to learn with mat: procedural, which need to be memorized and logic which need to be understood. Try to divide every exercise in these two components, and make sure that once the logic is understood, the student realizes that the rest is memory training, and that there is nothing more to be understood. More often than not, people have the feeling there must be something hidden beyond what they understood; something really complicated and it is always useful to explicitly dispel this feeling. Usually after an exercise I try to explicitly point out what was the objective of the exercise, or do it beforehand, depending on whether I use the exercise to reach a conclusion or to reinforce a previous explanation.

  5. Try to find other holes in her basics. Usually this repeated failure to learn a simple procedure stems from a whole weak foundation, so make sure to do a summary of the basic fraction operations, making sure she can add, subtract, multiply and divide properly, both fractions and real numbers.

  6. This I would say is probably the most important step. Whenever she does things right, always point it out in a positive manner, like "See? There is nothing here you can not do", or "it wasn't as hard as it looked, or was it?". Usually after a few sessions of working with a student, I start pointing out he or she can be among the first in their class, and this usually works wonders on their motivation. Of course I have to first convince myself, but in over 20 years of tutoring has consistently worked.

  7. Another very important aspect, which unfortunately in your case might not be that easy, is working also on the parents. You have to convince them also that their kid is capable of being among the first, because more often than not, this idea of "being bad for math" is believed to be inherited and most parents constantly reinforce on their children that they can not do it. In my experience once people find out they can be good in math, they start liking it.

  8. This is probably the most difficult. Once you find a student with whom you will be having to work on the basics for a few long sessions, it is tempting to feel frustrated and bored. However, I always see it as a challenge to try to figure out which tools does the student actually have and design a strategy to teach, only with those tools, whatever that has to be taught. I find really fascinating to witness the process people follow while learning new things, and I consider it an honor to be allowed to be part of it.

  9. Also as important is finding out if she sleeps enough or not too much. Altered sleeping patterns are usually a sign of depression, and they affect people's cognitive abilities. Lack of sleep makes people forgetful, and it could be responsible for some of her problems.

  10. Practice. Keep in mind that to fix something in your memory, you have to repeat it a few times during the first week, then review it after 3 or 4 weeks, and then review it after 4 or 5 months. After this point, you will most likely not forget it. She is probably not practicing what you explain to her, so next time it looks as if she had a factory reset.

  11. Try to keep track of her attention span by constantly asking questions. Many people with math problems have very short attention spans and it is hard to keep them focused.

  12. Expose openly to her things that you don't know and make her see it is not such a big deal not to know something. That might make her feel comfortable admitting the same to you. Sometimes even making a few mistakes on purpose, or admitting it openly and without embarrassment when you make one helps to make the student feel comfortable with admitting their mistakes.

The last point would be not thinking that you are enlightening her. Teaching is team work, as it is learning with a teacher. Convince her that you are a team and that her success is your success. Whenever you have to tell her something hard to digest try to talk in plural. As an example, instead of saying "You need to learn scientific notation", you might choose a different wording like "we need to review scientific notation". Keep in mind that the wording you choose says more about what you think than what you would like to reveal. To counteract this problem, you have to first convince yourself of her capabilities and then convince her.

Given her defensive behavior, one way of finding out whether she knows the multiplication tables is pretending you are having problems to remember the multiplication at hand (as if you were trying to remember it) and see how long she takes to complete it. If she doesn't seem willing to help, you can say something like "help me".

You can also try to find something she is good at and ask her to try to teach you or explain you something. That will put you in her mind at a similar level to hers and might reduce her level of defensiveness.

EDIT:

13 . One extremely important point I had forgotten. Stress the fact that most geniuses have changed the world by asking themselves questions no one did before. Of course it is the answers what made them famous, but as Voltaire once said: "Judge a man by his questions rather than by his answers". So when she asks something that reflects insight, always point out that it was a very good question.

Good luck. DP

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    $\begingroup$ +1 ....I wish I could upvote for every point! Stuff I used before reading this answer: 2, 3, 6, 7, 9, 10, 12. The other things I probably should also use ;) $\endgroup$ – Tutor Jun 17 '14 at 2:44
  • $\begingroup$ Thanks for the compliments! I'm glad I could be of help :) $\endgroup$ – Dissident penguin Jun 18 '14 at 14:49
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    $\begingroup$ may I suggest another point ? 14. Dont try to teach too much at once. People can only handle a few new ideas at a time. If you explain too many ideas in one go, odds are that some ideas will "push" other ideas out $\endgroup$ – josinalvo Jun 20 '14 at 21:48
  • $\begingroup$ @Dissidentpenguin (is that true, in your experience ?) $\endgroup$ – josinalvo Jun 20 '14 at 21:48
  • $\begingroup$ @josinalvo I agree, although seemingly OP is perfectly capable of handling that part since most of his peers consider him very good at communicating math concepts, so I kind of took it for granted, since the original spirit of the question was how to handle a difficult student with important self esteem problems. $\endgroup$ – Dissident penguin Jun 20 '14 at 22:48
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This is more a comment than an answer, but it's too long to make a comment.

When dealing with students who are couched in defensive self-confidence issues, you can often disarm them by asking earnest, simple questions about their own work.

Watching someone 'simplify' $\frac{63}{3}$ to $\frac{61}{1}$ is an excellent teachable moment, but I'm afraid that you've taken a wrong turn. When seeing a move like this (I see such moves often) it's a great time to drop everything involving the context of the current exercise and ask some questions to reveal the person's understanding of their own actions.

Here's a script for this particular situation.

  • You've simplified $\frac{63}{3}$ to $\frac{61}{1}$ here. I'd like to think about that. Tell me - ballpark - what is 63 divided by 3?

It's not uncommon to get a blank stare at this point. Don't let it linger very long - it's demeaning. Follow up quickly with a reminder that the notation $\frac{a}{b}$ refers to the same thing as $a \div b$, and form the present question into a canonical division example:

  • If you and I and Fred were sharing 63 dollars equally, how much would we each get?

Press for some approximate answer. You'll almost certainly get something between 15 and 30, which is good enough to proceed.

  • OK, so $\frac{63}{3}$ is roughly $x$, right?

A little less than half of students will jump through the rest of the logic at this point and realize that they've made an error. If they do, congratulate them on sussing it out, and remind them that these sorts of estimatory checks on work in progress catch lots of errors. Go back to their work, and ask them to justify the simplification. They'll either fail to say anything, or claim it's a memorized rule. Remind them that it's very rarely a good idea in mathematics to exercise procedures from memory, but rather they should aim to take steps whose meaning they understand perfectly.

For the other half, keep going.

  • Alright then, so what is 61 divided by 1?

Blank stare is possible again at this point. It's absolutely critical that you handle this with no hint of incredulity, or with any whiff of disdain for their abilities. Follow it the same way as before:

  • Division by one is almost a trick question - if I have 61 dollars and I don't share it with anyone, how much am I left with?

Some conversation here is required to identify the $1$ in the denominator with the count of people sharing in the money. Talk through $\frac{6}{6}$, $\frac{6}{3}$, $\frac{6}{2}$, and $\frac{6}{1}$, until you are convinced that there is enough mutual agreement on the meaning and interpretation of the division notation, and what division means in the first place.

  • Alright, so $\frac{63}{3}$ is about $x$, but $\frac{61}{1}$ is $61$, and you've replaced one with the other from this line to that line. Is this a problem?

We're into the hard part now. About half of the students at this point will recognize that a mistake must have occurred, and that it's not OK to replace the $\frac{63}{3}$ with $\frac{61}{1}$, since they represent different values. For these students, proceed as above - ask for their justification of the original simplification, yadda yadda. The other students are going to need serious work, but I'm going to cap my reply here before it really gets out of hand.

The main point is to concentrate on the results rather than the procedure. Your post indicates to me that you concentrated on trying to convince your friend that she couldn't cancel the denominator with only one digit in the numerator. This is very common even amongst math educators, but that's a shame! Instead, work with her to develop the equipment necessary to decide whether procedures are valid. Students never stop learning new incorrect procedures until they can read and think about the results that they are producing. It's a difference between bailing out a leaky boat and plugging the hole altogether.

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  • $\begingroup$ One line of inquiry is "how did you get that result - what steps did you use?". It may have been just guessing or thrashing, but the student may have actually been misled into using some inappropriate rule in the past. It is good to learn such things early on. The remedial steps can be quite different depending on what you learn. $\endgroup$ – Buffy Jan 25 '18 at 15:27
  • $\begingroup$ For example if, when looking at 63 the student is "thinking" 6 times 3 then the result is (or seems) perfectly logical. $\endgroup$ – Buffy Jan 25 '18 at 15:35
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Consider your assumptions with the student: Are you assuming she does not have discalculia? You might want to read up on this, it can afflict about 5% of the population.

Have you tried using graphical or manipulative approaches? For reducing fractions, you could use buttons or coins initially to model division, then bring the student the abstract 63/3 type written fractions. She may not understand what 63/3 means, but if you put 15 coins down and split them three ways, she might get the idea that the fraction bar means divide.

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  • $\begingroup$ I've asked her about that and she was a bit offended by that (of course, I said it nicely). Even the faintest hint of her being not "one of the more clever people in class" nearly brings her to tears. There seems to be a big problem with her self-confidence, but again, as I just know her as a co-student, I cannot tell more about this let alone fix that. $\endgroup$ – IchBins Mar 30 '14 at 19:20
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The problem with this lady seems to be, second, a lack of talent for math (for which she is not to blame), and first, an unrealistic view towards her abilities, coupled with no self-criticism. There is very little you can do about the latter.

Professional mathematicians and good professors have (or at least should have) no issue saying "I don't know". I say "I don't know" every day, and so do most of my colleagues. That a student wouldn't want to do it, is both pretentious and incredible naive.

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    $\begingroup$ When I was growing up, not many of my teachers said "I don't know" or "I'm not sure: let's work it our together". People really need teachers to model how to deal with ignorance. That is something you can do to try to help such a student: just be vulnerable, and show that not knowing is okay. $\endgroup$ – Steven Gubkin Mar 28 '14 at 22:01
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    $\begingroup$ A. Do you not think unrealistic views combined with no self-criticism can be part of the reason for why she has "a lack of talent for math"? B. When you say "there is very little you can do about" a student with these unrealistic views: On what are you basing this statement? (Personal experience? Intuition? Research? Etc.) $\endgroup$ – Benjamin Dickman Mar 28 '14 at 22:21
  • $\begingroup$ A. That could hamper a person's general progress, but the examples by the OP were too basic for that, I think. B. 25+ years of teaching at the university level. $\endgroup$ – Martin Argerami Mar 29 '14 at 0:01
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    $\begingroup$ Yes, but the problem could be elsewhere. Maybe, at an earlier stage of education, something was introduced to early, so could not be understood. Most children cannot understand division until about 11 years age, but in a normal class there is an enourmous variability around that. Being forced to train before really understanding, unhealthy coping strategies develop! and makes for problems later, which is difficult to undoe. $\endgroup$ – kjetil b halvorsen Mar 30 '14 at 20:02
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    $\begingroup$ Just my 2cents about the "not knowing" part: I like to tell my students that "I can answer every question on math they can imagine. It's just that the answers to most of them is I don't know." (As most aphorisms, it is (i) true, (ii) a bit funny and/or clever and (iii) I have no idea who the author is.) $\endgroup$ – mbork Apr 20 '14 at 5:57
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I think you need to present this lady with the information in it's simplest form, for 63/3 forget about fractions, we are trying to break 63 things into 3 groups, act this out with 63 things, or at least draw it on paper so you can teach her the meaning of these symbols. In this case I have 63 Easter eggs and want to divide them between you, me and your friend (3), ensure you also write the equation down as you need to also establish in her mind that they mean the same thing.

For the general problem you want to take 2 steps back and explain the concept in non-mathematical ways. What are we doing and why. The more relevant the example is to her the better she will understand and remember it.

Remember also no-one like to admit they are wrong or stupid, many people have a false idea of their skills and abilities at many things, (Not you or me obviously, but those other people :) This is often sustained by ignoring any evidence they lack skill, and emphasising anything that sustains their belief. Generally the smarter/more able someone is the more likely they are to admit their short falls. This speeds up their learning as being aware of a problem or short falls allows you to focus on it. This lady appears to realise she needs help understanding, and has recognised you do understand and can help her, the more empathy you have with her, the easier it will be for you to help her.

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  • $\begingroup$ "For the general problem you want to take 2 steps back and explain the concept in non-mathematical ways. What are we doing and why." Absolutely. $\endgroup$ – Tutor Jun 17 '14 at 2:46
  • $\begingroup$ If she seems unwilling to learn, then often this is because she can't find a use for what you are trying to teach. Find a way the maths will help her and you have a much better chance. Also I have found that some people have trouble with abstract maths, so if examples are based in the real world it will make it easier for her to grasp. $\endgroup$ – tarriel Jun 18 '14 at 9:06
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I am no psychologist but I see nothing wrong with your friend's behavior. In my opinion, these are the defense mechanisms of an honest person to whom nobody has explained what a fraction is.

So, try the following.

  1. Do not use "naked numbers" but only "number phrases" such as 3 Apples where 3 is a numerator and Apple is a denominator. (3 does not correspond to anything in the real world but 3 Apples does: 3 enumerates the elements in a real world collection and Apple denominates them. You might call that a model-theoretic viewpoint.)
  2. Similarly, do not use "naked fractions". Use for instance 7/3 Apple which you then read as "7 of which it takes 3 to get an Apple" where 7 is the numerator and where "of which it takes 3 to get an Apple" is the denominator.
  3. Without telling your friend anything more, ask her what is 7/3 Apple equal to. Once she decides to read the fraction as in 2. rather than try to give you the "right answer", she will quickly get 2 Apples and 1/3 Apple. But she absolutely must read the fraction as in 2. So, keep telling her to do so and don't tell her anything else.
  4. Once that is established, she will have no problem dealing with equivalent fractions.
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    $\begingroup$ ..."Numerator" and "denominator" refer to the top and bottom of a fraction. This terminology is an awful idea. $\endgroup$ – Deusovi May 30 '17 at 18:22
  • $\begingroup$ Ever wondered about the actual meaning of the terms? And, by the way, what does the term number refer to? Something in N, in Z, in Q, etc Not to mention that said terminology does the job which, in the business of helping people understand would seem to be the only thing that should count. $\endgroup$ – schremmer May 31 '17 at 13:46

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