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My 13-year-old sister has a problem which, given the way math is currently taught, I doubt is anything but all too common. She has a low grade in her math course and only ever attempts to memorize formulas and tricks, but never actually learn any of the reasoning behind the math. Cross multiplication is the perfect example.

She knows that from

$\frac{3}{5}=\frac{x}{10}$

.. she can "cross multiply" to get

$30 = 5x$

.. and from there get $x=6$. She has absolutely no idea what any of this means, however. She's simply memorized a pattern and is applying that pattern to a recognizable arrangement of numbers.

Given the incremental nature of math, her performance has gotten worse as her lack of understanding has compounded. She'll occasionally ask me for help, but is always upset that I won't simply give her the answer to the problem at hand or the "formula" for what she's trying to do. As I ask questions to test her understanding of something, she begins to randomly guess numbers either out of thin air or numbers that I'd mentioned in my explanations, but she doesn't appear to be actually thinking about the problem and considering the answer. After about an hour, she begins to claim she's tired, can no longer focus, and that we're spending too much time on a single problem and that she has more to do.

The inspiration for my finally posting this question and reaching out to the mathematics community came from the homework she had today. She wanted to know how to find the circumference of a circle. After a few questions I had determined that she had no idea what the radius, diameter, or circumference of a circle even were. She even attempted to guess "area" at one point. After relating circumference to the circumnavigation she had learned about, radius to the rays of a sun, and diameter to meaning two (even though this isn't the proper etymology of diameter), she was at least able to label the parts of a circle. Instead of giving her the $c=\pi d$ formula she wanted so badly, I wanted her to understand that $\pi$ represented the amount of times the diameter "fits" into the circumference and that this is the relation between the parts of the circle. I measured as accurately as possible the perimeter and diameter of the mouth of a cup I had and showed her that dividing the numbers produced approximately pi. This unfortunately didn't provide the "ohhh" response I was looking for, which signified that she didn't intuitively understand division. So I tried with a much simpler example. Our conversation went something like:

"The circumference of the glass divided by the diameter gave me pi, what does that mean?"

"Er... I don't know?"

"Well, if I divide 10 by 2, what do I get?"

"Five"

".. and what does that mean? How many twos are in ten"

"five twos go into ten?"

"Right, so if I divide the circumference by the diameter and get pi, how many diameters are in the circumference?"

"..umm... seven?"

"WHAT?!? Why seven?"

"..uhh, two?"

"why two?"

"because diameter means two?"

"two what?"

"two radius"

...

and so on ad infinitum.

She doesn't have any learning disabilities or mental handicaps, so it irritates me to no end that she won't put any effort into learning things that are essential to her understanding and that she could easily grasp.

How do you teach someone to understand math when they are capable but unwilling to do so?

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closed as off topic by Grigory M, TMM, Ayman Hourieh, anorton, Tom Oldfield Jun 17 '13 at 17:20

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"But the power of instruction is seldom of much efficacy, except in those happy dispositions where it is almost superfluous." I love this quote of Edward Gibbons, because it reminds me that sometimes, it's not the fault of the teacher. You can only bring the horse to the water, if it doesn't want to drink, maybe you should just move on. –  Raskolnikov Jun 10 '13 at 8:45
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Sure, and I wouldn't hesitate to leave such a horse where it stood and not turn back, but this is my sister - I love her, am concerned for her, and know that it's only going to get more difficult and be more embarrassing for her to recover from this. –  mowwwalker Jun 10 '13 at 8:51
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In my opinion, you shouldn't focus on making your sister get a good grade in math. Rather, you should focus on cultivating her to become an extraordinary expert. That way, things like 'getting a good grade in math' would be automatically accomplished in her free time. I really find it to be such a waste to care about grades. Good grades in math in school are silly: she should not be wasting her talent memorizing silly things that the education authorities came up with, rather she should focus on long-term development of expert ability. Refer to syllabus.byu.edu/uploads/h52kB4gCLyQP.pdf –  raindrop Jun 10 '13 at 9:10
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You ultimately have to put your sister in a context where applying math becomes interesting and engaging. That, I believe, is the most difficult part. Your now simply trying to convey her concepts she can't put to use and that is simply challenging her reasoning. –  Filip Dupanović Jun 10 '13 at 11:10
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Not everyone wants to learn Math and honestly not everyone should learn math. To some people just getting by is enough and that's fine. My sister is a very intelligent person, one of the best writers I've ever had the pleasure of working with. However, she hates math and wants nothing to do with it. it's not an affront to me as a math-lover. Each to their own thing I say. –  Steven Ellliott Jr Jun 10 '13 at 16:38

29 Answers 29

If one does not want to do something, then one can't do it. The question is how to get past any resistance and issues one has with, in this case, mathematics. From the conversation above, I can make a few recommendations. First, avoid asking simple questions for which the answer is obvious to you and should be obvious to your student. The reason is that it might not be obvious, and the student, sensing the elementary nature of the question by the tone of your voice, will try to guess quickly, most likely get it wrong, forcing an irresistible gasp from you, which will signal to the student... only bad things. When the student is already having issues that are psychological, it's best to avoid such questions and instead try to engage the student by asking to affirm things you know should be trivial (so something like "so this is the radius of the circle, right?" while you are pointing right at it. Then you can go to "and what would that be?", pointing to the diameter, and perhaps immediately adding "well, it can't be the radius since that was this guy over here, so this must be the diameter..." etc.).

As for the particular issue with $\pi$, it is actually not so trivial at all. First, there is the issue of comparing "divide the circumference by the diameter, hey look, we got almost 3.14, so that means that the diameter fits into the circumference $\pi$ times" to "divide 10 apples by 2 people, each has 5, so 5 apples fit into 10 two times" is problematic. What the hell is $\pi$ times? The quantitative intuition most students will have for multiplying natural numbers goes down the drain when going to real numbers that are not fractions. The common way to 'solve' this in schools is to drill the students with endless computations with decimal expansions until the students think they understand it. Of course, then most students will insist that $0.999\cdots\ne 1 $, which shows how ineffective this method is to understanding what the real numbers are.

Then, there is another issue. The fact that for all circles the ratio of the circumference to the diameter is a constant is far from obvious, nor is it a trivial matter to actually give a proof of that fact. In fact, I remember that when at school we were given the formula $c=\pi \cdot d$, I didn't understand why it was true, and since it was presented like it's something obvious, I felt I was being stupid for not seeing why it's true. So if one presents the formula $c=\pi \cdot d$ as something that should be clear, that's a problem. It's not clear. It only becomes 'clear' to those students going through the system being drilled endlessly with that formula until they think they understand it. What I do is define $\pi$ as the circumference of the circle of diameter $1$ or as the area of the circle with radius $1$. Then you can discuss the weird behaviour of length (i.e., that it is extremely sensitive to small perturbations and that it is only lower semi-continuous) and I try to convince the student that $\pi=4$ and immediately show it must be smaller than $4$ by various geometric approximations. Then a quick discussion of the stability of area compared to length, and thus that we should prefer the area definition of $\pi$ rather than the length definition. Then comes the non-trivial formula $c=\pi \cdot d$. It is now not just an empty formula, but something that carries meaning.

And finally, learning comes when the students want to learn. The motivation can come from different sources and for different reasons. At times, the student is not motivated. It's no big deal. There is no reason to expect somebody to be interested in something just because somebody at school decided they should be. Not knowing what $\pi$ is never killed anybody. And, the best way to create and re-enforce issues with mathematics is to push the student when the student is not interested. I can confess that I quite hated mathematics at school due to the way it was (and still is) taught. When I became motivated, which was when I encountered university level mathematics through a book I found, I very quickly learnt what I did not know. You'll find that all of the material taught in school sums up to very little, and can be comprehended quite quickly if one is motivated.

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I tend to agree with your last sentences but it is not at all clear to me whether quick learning later on is not actually also a function of having spent large amounts of time in school on the subject. –  Gregor Bruns Jun 10 '13 at 10:30
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"When I became motivated, which was when I encountered university level mathematics through a book I found" I really want children to be surrounded by university level books. It's important for children to at least know that so many things are explained in those books. When I was young I didn't even know that I could find explanations to things in university books. People just don't give 10-year-olds university books, but this is a wrong attitude. Exposing a child to to university books could motivate him/her to love learning very intensely. –  raindrop Jun 10 '13 at 10:36
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@Raindrop I once, for a short period of time, was tutoring a 10-11 year old. She was bored to death with the stuff in school. So, I took an orange and started carving shapes into it. She was delighted to find out that all those facts she was told are true are actually false. She then went to the kitchen and came back with all the fruit she could find and experimented with non-Euclidean geometry. You don't necessarily need university textbooks to engage kids. You just need to keep them away from the crap of standard school curricula and books. –  Ittay Weiss Jun 10 '13 at 10:41
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I didn't learn much in math classes in elementary. I found math easy but it simply didn't make sense to skip classes and self-study more advanced math. Math in school isn't fun for normal students (students who don't get high scores in math). In fact, we can make psychological arguments that education in school filled to the brink with negative feedback. Math in school only becomes fun if you understand how awesome math is in the 'adult world' –  raindrop Jun 10 '13 at 10:41
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@IttayWeiss "Not knowing what pi is never killed anybody". On the other hand, knowing what pi is did kill Archimedes! –  Siméon Jun 10 '13 at 13:34

I am a high school teacher, so here are some comments specific to the situation that don't necessarily answer the question directly:

One hour is around the maximum that a normal 13-year old student can concentrate on intense mathematical learning. You need to manage this by sticking to maximum one-hour sessions, maybe with a 10-minute break in the middle. This may not be your own experience, but this is common. The fact that your sister is telling you this shows she has good self-awareness.

From the point of view of your sister's immediate need, she wants simply to answer the homework questions to avoid getting in trouble at school (or be able to answer exam-style questions, or similar). In order to achieve this, she actually doesn't need to know the theory behind it, she needs to know the formula. Again, she is showing good understanding of her situation.

I understand why you are disappointed in that... but she needs to understand that she can get appropriate help of the kind that she needs (i.e. how to accurately apply an algebraic formula to answer the questions), or she will be less inclined to ask you for help in the future. Self-discovery is definitely not the only way to learn maths. Your 'Socratic method' of discovering knowledge is very powerful if you have time to fully explore it - but this may not actually be appropriate help in this context.

If your sister has shown a lack of understanding of division, then she will find it very difficult to understand your attempt to explain pi as the ratio (division) of circumference and diameter. Understanding division is a prerequisite for following the concept that you are trying to explain. Even if she can follow what you're doing with her, her weakness in ratio problems will make it impossible for her to do this independently, and she may get confused by thinking about what you have done where she is expected to just remember and apply the formula.

To help her understand pi as a ratio, you may want to go back to an earlier step, for example: 'recipe' problems

  • if this recipe makes 2 cakes, how can I make 4 cakes?
  • if this recipe makes 3 cakes, how can I make 7 cakes?

At first with simple numbers, but getting more complicated over (perhaps several lessons) time. Then take out the recipe context: use other contexts, or a little more abstract, use it to reintroduce the equivalent fractions problem you quoted above, until your sister is fully ready to tackle the circumference/diameter problem.

Note, it will still be difficult to understand pi in this way, at a stage when students typically only have experience of integer solutions in this type of problem.

And also note, it will be difficult to convince your sister to agree to this work. You need to show your willingness to help her (i.e. help her with what she immediately needs, first), and also let her understand how learning with you will be helpful for her in the longer term. She has to be engaged with the learning experience or this will be frustrating for both of you and not worth spending time on. Sometimes this puts pressure on family relationships and, in that case, it may be better to hire a tutor to work on this with her.

Good luck!

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I understand why this answer is unpopular, but I personally think this best answers the specific situation of the OP. –  Joel Reyes Noche Jun 11 '13 at 0:51
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+1 for "she has good self-awareness," "she is showing good understanding of her situation," and "self-discovery is definitely not the only way to learn maths." –  Joel Reyes Noche Jun 11 '13 at 4:39
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@JoelReyesNoche if I could +10000 this answer I would. –  Michael Edenfield Jun 11 '13 at 13:30
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The most important part of your entire answer, IMO, was that the OP needs to show some willingness to help with her immediate needs, or she will just stop asking for help. –  Michael Edenfield Jun 11 '13 at 14:12
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I think that a very good point that was raised in this answer is that OP needs to decide if the help he is willing to give is in order to learn maths or in order to help with the immediate tasks at school. Sometimes one will have nothing to do with the other, or at times one may hinder the other. –  Ittay Weiss Jun 12 '13 at 5:33

First, let her calculate lots of easy exercises. There should be only about 3 types of exercises. She has to choose the correct algorithm.

When she can calculate the trivial exercises without problems, move to the more complicated exercises. Add some abstraction, some real world examples. Add abstractions slowly. It's important letting her discover abstractions by herself.

There are several principles:

  1. Let her be successful. She has to be able to solve some exercises. If she can't solve anything, she will get bored.
  2. Let her try to solve it by herself, without intervening.
  3. Don't let her guess. Don't ask simple questions. Show her how to solve one exercise and then let her do another by herself. If she can't do it, explain it again and give her another exercise. Then let her explain it to you. Many people will tell you they understand the exercise but if they can't explain it to you, they don't. It's interesting letting her solve exactly the same exercise you just explained.
  4. You need 5 minute breaks after every 30 minutes. Don't talk about math during the break. People can concentrate for long periods of time only if they are interested in the subject. If they only listen to someone else explaining, they get tired/bored fast.
  5. Don't get carried away. Don't try to explain deep relationships because they fascinate you. Keep it simple.

[I am actually a programmer but I studied math in the University and I spent several years teaching maths to my high school classmates and their friends].

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+1 for "If she can't solve anything, she will get bored." I might disagree with some of your other points (or at least argue that they depend on the person involved), but that's an essential one. Nobody likes beating their head against a brick wall with no chance of success. –  Ilmari Karonen Jun 10 '13 at 13:41
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@IlmariKaronen I know, it's difficult to compile the experience into a short summary. I think the 2nd point is very important, too. You can't really learn maths without actually solving exercises. If the students only watch how the teacher solves the problems, they don't learn anything. It's important to explain only the important things and then continue with exercises. Favourite quote : "I understand it but I wouldn't be able to do it myself" –  Sulthan Jun 10 '13 at 14:06
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+1 for "Don't try to explain deep relationships because they fascinate you." –  Zackkenyon Jun 10 '13 at 16:50
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That's an interesting point - I neglected to consider that I should give her the formula first and explain later. I think I'm worried she will just take the formula and run once she sees that it works instead of actually letting me explain. "let her explain it to you" - This is crucial when she answers a question correctly. She usually says "I hate when you do that. I hate when you ask why?" –  mowwwalker Jun 10 '13 at 19:33
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@Walkerneo Do you want to explain it because you are fascinated by the subject? :) What is your goal? Do you want her to learn or do you want to enjoy explaining? There is no reason to hide formulas from her. The process you are doing is okey but only for people who already love maths. Don't ever forget she is your sister, don't make yourself look like a teacher. Friendly athmosphere is very helpful. –  Sulthan Jun 11 '13 at 0:49

Your sister has developed a strategy that was successful to cope with a lot of typical problems given in school. So of course she is trying "more of the same" to solve any further problems.

You are trying to force her to think in a very different way about these problems, often a way that will not instantly produce answers, but that is exactly what she is looking for.

There are different ways to cope with this situation, my following personal example was successful but not the most pedagogical way to cope with problems in mathematics: My sister had to do a presentation on a fairly advanced mathematical idea for her final year of high school. She always was mediocre in math but managed to survive. This topic was clearly beyond anything that you could solve with pattern matching but required some real thinking. So I offered to help but did not have much time as I was in university three hours away. I started off by testing at which grade level we could start the journey. After a few hours and quite a lot of tears we ended up somewhere 3 or 4 years below her current class. She had managed to get by with exactly the same strategy as your sister but without much understanding for several years. In my opinion there was only one way to solve this: I said we have to repeat all the stuff and understand it this time in six weeks (the time to the presentation). Of course that only provoked more tears. The good side was my sister needed a good grade and was highly motivated. I tried to explain that this is not an impossible task and gave her lots of books (university level) and daily feedback. After the very rough start she understood more and more and managed to give an impressive presentation and understood the subject deeper than we ever hoped for and completely lost any fear of mathematics.

So I would try to explain your sister, that her approach was quite clever but will fail for any more advanced problems. She can either start trying the hard way trying to build a mathematical understanding now or suffer for all the years to come as no other subject requires such a continuous learning effort as mathematics.

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As for "no other subject" at the end: do you think learning a hard foreign language in school (e.g., Chinese if you're not from East Asia or living there) requires less of a continuous learning effort than mathematics? –  KCd Jun 10 '13 at 18:22
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What university level books covered topics that were 3 or 4 years below her level as a senior in high school? And more importantly, what university level books would you recommend having around the house for a 10 year old child? What books would you want left around you if you were living your life over again and you were 10? –  Charlie Flowers Jun 10 '13 at 19:03
    
I've explained that to her innumerable times. "You need to understand the math, not just guess; math builds on itself, you'll struggle harder next year if you don't understand the stuff you learned this year; before you even understand what you're learning this year, you need to solidify your understanding of everything you learned the years prior..." and so on. I don't blame her for not enjoying math they way I and others do, but it frustrates me that she doesn't realize that she's going to have to learn things sooner or later, and best to do it sooner. –  mowwwalker Jun 10 '13 at 19:40
    
@KCd: I can not say anything about Chinese, from my experience with French I had huge gaps but could always 'interpolate'. Even if you can not express yourself for example in the past tense in a foreign language you will manage. Try to get by without fractions in mathematics. I do not want to generalize this too much but that was my personal experience. –  AnOlderBrother Jun 11 '13 at 16:58
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@Walkerneo: Don't be frustrated, this requires a lot of emotional energy from her side to avoid instant gratification. –  AnOlderBrother Jun 11 '13 at 17:06

It's hard to beat Feynman's Abacus story in Surely You're Joking, Mr. Feynman!.

This excerpt copied from here which notes that the story is taking place in Brazil.

A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do. The waiters didn't want to lose face, so they said, "Yeah, yeah. Why don't you go over and challenge the customer over there?"

The man came over. I protested, "But I don't speak Portuguese well!"

The waiters laughed. "The numbers are easy," they said.

They brought me a paper and pencil.

The man asked a waiter to call out some numbers to add. He beat me hollow, because while I was writing the numbers down, he was already adding them as he went along.

I suggested that the waiter write down two identical lists of numbers and hand them to us at the same time. It didn't make much difference. He still beat me by quite a bit.

However, the man got a little bit excited: he wanted to prove himself some more. "Multiplicação!" he said.

Somebody wrote down a problem. He beat me again, but not by much, because I'm pretty good at products.

The man then made a mistake: he proposed we go on to division. What he didn't realize was, the harder the problem, the better chance I had.

We both did a long division problem. It was a tie.

The bothered the hell out of the Japanese man, because he was apparently well trained on the abacus, and here he was almost beaten by this customer in a restaurant.

"Raios cubicos!" he says with a vengeance. Cube roots! He wants to do cube roots by arithmetic. It's hard to find a more difficult fundamental problem in arithmetic. It must have been his topnotch exercise in abacus-land.

He writes down a number on some paper— any old number— and I still remember it: 1729.03. He starts working on it, mumbling and grumbling: "Mmmmmmagmmmmbrrr"— he's working like a demon! He's poring away, doing this cube root.

Meanwhile I'm just sitting there.

One of the waiters says, "What are you doing?".

I point to my head. "Thinking!" I say. I write down 12 on the paper. After a little while I've got 12.002.

The man with the abacus wipes the sweat off his forehead: "Twelve!" he says.

"Oh, no!" I say. "More digits! More digits!" I know that in taking a cube root by arithmetic, each new digit is even more work that the one before. It's a hard job.

He buries himself again, grunting "Rrrrgrrrrmmmmmm ...," while I add on two more digits. He finally lifts his head to say, "12.01!"

The waiter are all excited and happy. They tell the man, "Look! He does it only by thinking, and you need an abacus! He's got more digits!"

He was completely washed out, and left, humiliated. The waiters congratulated each other.

How did the customer beat the abacus?

The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03 is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root's excess is one-third of the number's excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.

A few weeks later, the man came into the cocktail lounge of the hotel I was staying at. He recognized me and came over. "Tell me," he said, "how were you able to do that cube-root problem so fast?"

I started to explain that it was an approximate method, and had to do with the percentage of error. "Suppose you had given me 28. Now the cube root of 27 is 3 ..."

He picks up his abacus: zzzzzzzzzzzzzzz— "Oh yes," he says.

I realized something: he doesn't know numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is to learn to push the little beads up and down. You don't have to memorize 9+7=16; you just know that when you add 9, you push a ten's bead up and pull a one's bead down. So we're slower at basic arithmetic, but we know numbers.

Furthermore, the whole idea of an approximate method was beyond him, even though a cubic root often cannot be computed exactly by any method. So I never could teach him how I did cube roots or explain how lucky I was that he happened to choose 1729.03.

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6  
I use approximations all the time to do math in my head and people think I'm brilliant. I'm not. I just learned the right trick! –  David Navarre Jun 11 '13 at 18:54
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This has absolutely nothing to do with the question AFAICT. –  jwg Jun 12 '13 at 7:52
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@jwg: The relevance is clear to me: automating tricks and using real comprehension are superficially similar, but quite different underneath, not only in how they work but also in how well they work on problems: the automated trick master can solve specific types very quickly but is stumped when it comes to nonstandard ones. –  reinierpost Jun 14 '13 at 13:03
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@reinierpost: Right, it doesn't really answer the question though, does it? Feynman doesn't go on to explain how he taught calculus to the abacus man. –  jwg Jun 14 '13 at 13:32
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This doesn't answer the question, but it does give some insight into the topic at hand. –  paraxor Jun 14 '13 at 16:57

Some people cannot/don't want to learn maths simply because it is... maths, much too abstract and, at first glance, not linked to the realities of life.

Take the opposite approach: instead of teaching maths and, then, find applications in real life, give her as many analogies as you can. Ie try to go from the maths practical approach - and help her to understand/be interested by choosing subjects she is interested in.

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Math can definitely be abstract, but I think what we were doing was simple and tangible. –  mowwwalker Jun 10 '13 at 9:05
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@Walkerneo You think that this statement is tangible? "Right, so if I divide the circumference by the diameter and get pi, how many diameters are in the circumference?" –  asteri Jun 10 '13 at 13:33
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@Walkerneo: Why should it be automatic? You’ve primed her with concrete examples of a pattern that she can apply automatically, and then you ask an abstract variant that does not on the surface fit the pattern; why would you expect it to trigger the pattern response? –  Brian M. Scott Jun 10 '13 at 21:09
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@Walkerneo: It depends on how many other equally pressing matters there are. If there are too many, you may have to settle for less than you want, perhaps just the notion that just as the perimeter of a square is always the same multiple of its side, so the perimeter of a circle is always the same multiple of its diameter; it just isn’t a very nice multiple. If I felt that we could afford the time to go a bit deeper, I might try microscope imagery: if I double the diameter, I’m putting the whole thing under a $\times2$-power microscope, so that a one-inch segment looks two inches long. ... –  Brian M. Scott Jun 10 '13 at 21:28
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... $40$ years helping students whose approach and reactions are similar to your sister’s, and every one is a separate challenge — and not always one that I could meet. And you have two problems that I didn’t: you’re her brother, and she’s $13$. It’s hard to be $13$. :-) –  Brian M. Scott Jun 10 '13 at 21:33

Give her a loan at some tiny daily percentage rate.

Collect interest daily, if she forgets, add that to outstanding amount.

Report outstanding amount and interest due daily.

After a while she will realize just how much she'd paying you back.

If that doesn't encourage her to learn math, nothing will.

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You're evil :) But I like your style –  Vlad Preda Jun 10 '13 at 13:26
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I am quite sure this won't work: your experiment has been carried out already by credit card companies, without any noticeable effect on the mathematical knowledge of the average American. –  S123 Jun 10 '13 at 13:30
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Alternatively: Let her give you one dollar and pay out three dollars if she can roll a six with a die. That beneficial experiment (aka. math disability tax) is carried out by State lotteries all over the world. –  Hagen von Eitzen Jun 10 '13 at 19:01
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@VladPreda I observed friends' daughters playing with toy building tiles. The younger had a few, but wanted a red tile. The older made series of trades with the younger, "I give you red and oyu give me 2 green." In the end the older daughter had all tiles and younger none. All by mutual consent and contract. If that's not math at work, I dunno what it :-) –  qarma Jun 11 '13 at 10:13
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I missed some details in the process, I think older sister pushed some unwanted tiles in a bargain before that and when younger only had unwanted left, she gave them up for free. –  qarma Jun 13 '13 at 12:14

Often times I find that the reason people are averse to mathematics is because they can't relate to the numbers. Instructors teach the way that they were taught, and since the instructor is smart obviously the way they learned it must be the correct way to learn it.

I once tutored a girl very similar to your sister, she assumed that she simply "wasn't good" at math. This girl, despite being in college algebra, had been pushed along by the system. She could not even say large numbers properly (she mixed up millions, billions, etc.). Her teachers and parents were no help, because they couldn't explain it in any other way than the way they had learned it themselves.

So I changed my method of teaching to relate to something she did understand. I called the numbers between the commas "families", and the hundreds place digit was the "papa", the tens place digit was the "mama" and the single place digit was the "baby". The million family lived next door to the thousand family, and the billion family lived next door to the million family.

It's absurd to describe math this way, at least I think so. It has very little to do with actual math, and it's easier for me to memorize and "just know". But you know what? The girl I was tutoring finally learned how to read numbers properly.

The fault I see with the description you've given is that you are trying to lead her to getting her own answer. You're asking question after question, expecting her to follow your lead and provide answer after answer. You're trying to get her to justify why she's doing something in the process. It's clear she doesn't care about math, why would she ask questions like that of herself? She just wants to get it done and over with.

So adapt the method. Get creative with your explanations, as if they were "word problems".

In the circumference example, stop talking about the circumference of a circle. You might explain it as, "I need a new belt for Christmas, but you can't ask me what size belt I wear because HELLO that's rude. But if you looked at me from the top of the stairs, I'd look like this circle. If I'm the circle, then the belt goes around my waist, which is like going around this circle. Now, of course if you look at me from the front you know that x inches goes from one hip to the other." That's when you draw the diameter. You can then explain the radius and give her a formula; the objective is to get her to relate the numbers and meaning of things in a purely non-math way.

Check out this "I Can't Do Math" Article

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Thank you! I will surely be changing how I try to teach her from now on. I will attempt to incorporate more fun and silliness. When I've done that in the past, she seems more focused and interested, but she often only asks me for help when she is tired and simply wants to finish her homework. –  mowwwalker Jun 10 '13 at 19:56
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How come word problems in school are never written like that? Excellent choice. –  David Navarre Jun 11 '13 at 18:56
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This sort of thing is patronizing and counter-productive in my opinion. –  jwg Jun 12 '13 at 7:54
    
i love the fat bloke explanation. brilliant. my six year old is already having difficulties with maths, and I lack the imagination to make it interesting like this. –  Nicholas Jun 12 '13 at 15:39

Personally, I would say there is no way of teaching anything (not just maths) to someone who is unwilling to do so; I think it would be better to give her a reason to want to do it.

While you make the comment that your sister has no learning handicaps, I would like to point out (not as a specific comment to you) that there are math specific learning disorders, such as Dyscalculia, and that this can affect people who are academically gifted in other areas; a person who cannot do maths is not always a person who doesn't want to.

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Thank you for the link. I've never heard of that condition before, but have often wondered if she may actually have a fundamental inability to understand math. Though, I have my reservations and think it's quite likely that this may be similar to ADD in that only a small fraction of those diagnosed actually have a real condition. –  mowwwalker Jun 10 '13 at 9:03
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@Walkerneo before you diagnose your sister with anything, let's take a mathematical approach. People with dyscalculia certainly have problems with mathematics. People who are taught mathematics in a school that follows the standard crappy program and quite likely by a teacher who does not understand the material properly also have problems with mathematics. The percentage of people with dyscalculia is very small. The percentage of bad teachers and crappy curricula is large. What is the probability that a given sister with math difficulties has dyscalculia? (Hint: ask Bayes). –  Ittay Weiss Jun 10 '13 at 10:24
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@user76556 I don't see what your attempt at an argument is supposed to be. If psychology isn't a science (this is a debate I'm not interested in having), then does that mean these conditions don't exist? –  Andrew D Jun 10 '13 at 19:33
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@AndrewD no it means that what a psychologist says is no better than any other guys opinion. if a psychologist diagnoses the girl with dyscalculia there is no reason to believe that his opinion matters more than that of an old guy saying: she is just lazy, beat her so that she pays attention. –  user76556 Jun 11 '13 at 6:35
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"...I would like to point out ...that there are math specific learning disorders, such as Dyscalculia..." did you say this or not? you did. and it is no better than saying: "...I would like to point out ...prayers can boost mathematical ability..."(and then hyperlink the prayer to some churchs website talking about how prayers made newton such a great mathematician) –  user76556 Jun 12 '13 at 21:32

Bear in mind that this may not generalize as much as it seems to - part of the problem may well be that she is your sister.

I used to tutor quite a few people in math when I was in high school, and, though I often encountered problems similar to yours, no one was anything close to as much trouble as my sister.

Someone who goes to trouble (usually with a lot a parental pushing, but still) of getting help from a stranger is more ready to listen than someone asking a close family member.

Also, it's hard to learn from a sibling - there is often at least a bit of an element of competition, or at minimum you form part of the background level of competence she compares herself against, so the better that you are at the math you are tutoring her in, the worse she feels herself to be. It's not your fault, it can't be helped.

Based on my own tutoring experience, usually the first thing to overcome for someone in your sister's situation is the feeling that she is naturally bad at math. The good news is that she probably isn't - she has never really tried it so there is no reason to suppose she is actually bad at it. You were right to try to move it from meaningless symbol manipulation to something meaningful about the real world, but she also needs some successes to make her think that she isn't hopeless and just wasting her time. Unfortunately, the mere fact that you are her brother and (currently) much better at math just reinforces the idea that she is bad at math.

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That sounds very reasonable, but I have to hope it's not true. I don't know any non-family members willing and patient enough to try to help her. I think a huge takeaway from this question is that I need to let her succeed more, explore questions on her own, and understand the relation to the real-world. –  mowwwalker Jun 10 '13 at 20:18
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Agree: one of the most challenging students I've had was my own sister. –  Ronald Jun 10 '13 at 21:39

I want to offer my situation as an example. Until high school ($10^{th}$standard), I was an average student in Maths. I was good at algebra, mensuration and statistics, but I performed poorly in geometry.

When my teacher asked me to prove Thales theorem, I couldn't; that was because I would memorize, but without knowing the meaning of any steps. My teacher insulted me in front of the whole class, and said that this is guaranteed to be a question in exams:

There is less than a week until exam time, and you don't know how to solve this theorem!

I was weeping by then. At last, I passed my exam but I never forget that insult, so over the summer, I revised every previous geometry concept.

In $11^{th}$ I went to a new school where my new teacher made geometry a simple thing to me. From that time. when I use any formula, I also learn how to derive the formula. So learning is self-motivated. It may come from someone insulting you or you are motivated when you're lagging behind the rest of the class.

There is one more incident that happened to me in $11^{th}$. I never memorized values of trig ratios of common angles. One day my teacher asked me, I just took a shot in the dark. He didn't say anything, went on and asked another boy. But then I realized that if my classmate knew the value, then why didn't I? The next day I started trig chapter and tried to solve problems myself and in 3 days, I'd memorized the whole table. So I think that until your sister listens to her inner voice, she will have no interest in maths. That is how I thought about math, and at that time, I was also 13-14 years old.

* There is an Indian Hindi movie on this type of student. Try to watch this movie: Taare Zamin par.

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@iostream007 Children can hardly think in such abstract terms. They can not motivate themselves. The idea of self-motivation is simply not there in a small child. That is why we need innovative parenting. –  Dilawar Jun 11 '13 at 10:46
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And yet, there are millions of people, including scientists utilizing some heavy duty math algorithms extremely successfully, but they haven't a clue on how to derive most of the formulas they use. Why? Because they don't need to. They need to know the purpose of the formula and how to use it. So while it helped you to learn to derive formulas, I'm not even sure your method would help many people at all. I also want to point out that not caring about the theory behind math in no way implies not having an interest in math. There's a difference between applying math and math theory. –  Dunk Jun 12 '13 at 16:27
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That's how I learn formulae. I like to be able to derive it myself. Even in subjects like Physical Chemistry (I'm a Chem major, Maths submajor), I like to learn how the formulae are derived. I find that if I learn how it is derived, I don't have to put as much effort into memorisation, so it's actually less work. –  daviewales Jun 17 '13 at 1:33
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@Dunk, they will soon lose their jobs for computers, which can do that way faster without complaining. –  JMCF125 Dec 24 '13 at 13:05
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@Dunk, but those people you referred before lack the creativity and imagination that make them better than computers. I mean, «I also want to point out that not caring about the theory behind math in no way implies not having an interest in math.»: yes, it does, the theory is where creativity is; how can you discover new ways of using a formula if you don't understand what it means? –  JMCF125 Jan 3 at 0:44

In math we don't understand things we get used to them. don't try to explain what diameter is. teach her how to measure things, make her notice the INVARIANTS and then suggest the relations. For one part of the problem i think this will work:

draw a circle make her measure the distance from a Fixed point P that you choose to the center that you have specified. then tell her to choose another point for herself and measure it. do this several times. then choose another point. Ask her to GUESS the distance from that point to the center. she probably will guess it. then do it for diameter. then mention the relation. now draw a new circle. make her measure the radius and make her GUESS the diameter of the new one.

the idea above can also be used to make the center a magical point for her. choose another point beside the center. and make her make measurements. she'll notice that the distance from CENTER is invariant. center is magical. she'll get curious i think.

you can make lots of games out of such things. for example can you choose a point that is the same distance from all the points?

PS: As i said i tried to give a solution to one part of the problem, the teaching method. the motivation is still an issue even if my teaching method worked. i don't know much about motivation. but i think that --and this is just my opinion based on my life's experience- that people's opinion about things like this are just noise, no matter who says it. :)

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they say when teaching children to play GO(something like chess, just cooler) don't try to explain moves and make them logical. because you probably are not a good player yourself. just play with them with all your power. they'll start surprising you soon. these explanations are just a waste of time. she already know what things are. just teach her how to measure them and predict others using those measurements. –  user76556 Jun 10 '13 at 12:32
    
-1 to your comment, nothing's cooler than chess. –  BigHomie Jun 10 '13 at 17:42
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She's not motivated enough to do experiments such as these on her own. I can barely keep her attention as I speed through real-world examples myself. It would be much more beneficial for her to do problems and examples herself to see the relation, but she doesn't trust that this is beneficial to her - she always comments that she doesn't see how the examples will help and "just wants to get her homework done". –  mowwwalker Jun 10 '13 at 19:42
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@Walkerneo You've now identified the problem. You need to find a way to convince her that the examples will help before you can go through the examples. That's what the answer above about didactics and motivation was all about. Start with "why do you want to learn this?". Ask her why the teacher set her homework. Help her decide that she wants to learn math and not just get the answers to tonight's homework. –  Richard Gadsden Jun 11 '13 at 12:42

I think the best way to tudor is with the Book right in front of them. For this situation I would never provide the formula but instead help the student by teaching her how to look back through the book and find the right formulas. This will cause the student to eventually with enough practice do two things:

  1. Be able to find her own answer by looking for the right formula
  2. Learn which formulas apply to which situations

It is never OK to just give the formula and that is giving a starving man fish instead of teaching him how to fish. One will solve the problem then the other will solve all problems after. If she doesn't understand what a circumference or diameter is do not tell her but instead have her use her own book to find the definition. All good math books should have a glossary and index in the back. By using this method the student will eventually be able to solve all of the small problems like this on their own by showing them how to do some basic research and review in the text book. And if you do not have a glossary or your text book is not very good, then find a good dictionary or let her search online (Google) for the right definition. There are three barriers to learning and there seem to be two prevalent here:

  1. The Misunderstood Word (Will cause everything past the word to be missed)
    • Can be found by asking what does diameter mean? and anything besides the right answer quickly, even hesitation, means that there is not 100% certainty and understanding.
    • Solved by clearing words that are misunderstood with a good dictionary and using them in sentences verbally.
  2. Too Steep a Gradient (Trying to learn something advanced before the simple is fully understood)
    • Can be found by asking to solve simpler situations and working your way up until there is a point where the situation can not be solved.
    • Solved by practicing and getting a good understanding of one step before moving on. Maybe several demonstrations should be made to help explain why it is so.
  3. A Lack of Mass (trying to learn a subject without be able to do any physical work on the subject)
    • Can be found by someone who is getting very frustrated and repeating the same thing over and over without being able to get the concept.
    • A simple fix is again to use some demonstrations. Instead of saying 10 apples divided by 2 equals five apples, go and get something to demonstrate with such as play-dough. I know it sounds elementary, but you will be surprised how fast someone will brighten up when they have the actual physical situation to learn with. Try learning how to fix an engine without ever seeing or touching an engine. You will not remember and be able to apply much at all. This is why the more hands on the more learning that can be achieved.

Here is a good reference that will help you out: http://ftp.appliedscholastics.org/bookstore/item-description.php?id=6

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Since when we first started, she didn't know the various parts of the circle, I thought it would be more beneficial in this case to demonstrate the formula rather than have her find it. I've tried this in the past, however, for things that can't be shown, such as unit conversions. She still doesn't use the book to find it herself. –  mowwwalker Jun 10 '13 at 20:01
    
@Walkerneo Be consistent. Those three are the only three barriers to study. If someone cannot grasp a concept it is because of one of those three barriers. Once the barrier is found and overcome, learning will take off again until the next barrier. You can do simple things like ask what was the last lesson that you fully understood and then start from there forward instead of from current location back. If she didn't know the various parts of the circle then there must have been something right before the parts of a circle that she did not understand. Never go past the Misunderstood Word. –  amaster507 Jun 10 '13 at 21:31
    
I have a friend who solved his daughter's lack of interest in geometry starting with the history of it. Back in ancient Egypt, every year the Nile would flood all the farmland. This farmland was fertile, so the farmers wanted to use it, but had to re-mark out their fields. At the time there were taxes based on how much land you had, so it was important to get the measurements correct, lest you pay the wrong amount of tax. And that is how geometry was born: geo=earth, metric=measure. It was a technology of measuring the earth for farmers who didn't want to pay tax. This rekindled her interest. –  user81855 Jun 11 '13 at 0:46
    
+1 for teaching them to teach themselves. –  daviewales Jun 17 '13 at 1:35

Note that recognizing and applying patterns is intrinsic to reasoning. The main complaint is that unlike someone better versed in mathematics, your sister doesn't have a large number of fine-grained patters for math, just some "dumb" ones based on how the printed layout of a particular equation. However, what you're working with is still patterns. Mathematicians, scientists, engineers: everyone works with and recognizes patterns. How does your physician diagnose what is wrong? From the pattern of diagnosable symptoms. It's okay to use the patterns and work out the justification for those patterns later, or to apply some problem solving patterns developed by specialists without understanding all the underpinnings.

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I realized after posting the question that "recognizing patterns" is of course what everyone does and is important, but not in the sense that I meant it. Noticing a pattern, in the sense of a repeating observation (such as the circumference being exactly pi times the diameter), is, of course, important, but recognizing a pattern in the sense that she sees this number over here, this one over there, and these symbols betwixt them, and knows she can do some operation that magically solves the problem, is dangerous because she doesn't learn why it works. –  mowwwalker Jun 10 '13 at 20:10

I have given math coaching a few times and this is quite common amongst students who take math coaching. Not sure how much time you want to spend to give her math lessons, but 1 hour per week is absolute minimum. If don't have this time, hire someone who could give her math coaching.

My approach is this:

  • find out what the student is good at and what the student is not good at
  • find or invent exercises that are near the student's comfort zone limit

And then, you guess it: do the exercises. When your sister's concentration is high, do the more difficult exercises. When her concentration gets lower, do easier exercises. And when she gets tired, do easy exercises or even do the exercises yourself. While doing the exercises explain each step loudly.

In your case the limit of the comfort zone are abstract methods. Find exercises that are hard to do with her approach. Or exercises that are tedious to do with her approach and easy to do with yours.

Also don't forgot to do pauses. One hour without pause is way too long, 20 minutes or so is fine. In between talk with her about something non-mathematical for roughly 5 minutes.

I have absolutely no idea what is going on in those students' minds but obviously math is something that bores them to hell. Sometimes I explain things, they understand them and forget them the next lesson. So it means I need to explain things over and over again, 5 times or so.

Anyway I strongly recommend you to hire someone that does one or two lessons every week, if you have the money. Math tutoring is a long term thing, at least for written performance. Short term improvements in class are possible though. But I think this is just a motivational thing.

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"I have absolutely no idea what is going on in those students' minds" - YES, THIS. I have to force myself to remember that I've spent much more time in math and have much better innate ability for it, but I still can't help but wonder how it is she doesn't understand even the simple things. Perhaps we will hire a tutor and I can sit in on a few lessons to see if I can learn something about the way he/she teaches. –  mowwwalker Jun 10 '13 at 20:21

I don't count myself as a Mathematician but I was pretty good at math in my school days. I was not fascinated with all sort of math. Measuring distances, areas, volumes etc was very nice but I hated symbol manipulation and limits. I could see the areas and volumes around me but relating equations to reality was very hard. Not everyone is born with love for mathematics like Terrence Tao and Ramanujam.

The way I feel about mathematics and kids is following.

  • Don't force math on kids. If she hasn't developed a love for mathematics, there is still hope. But talking too much math to her will definitely make her hate it. I doubt if she can recover after developing a feeling of hate for mathematics. Math already have a 'nerd' image.

  • Try to make her see mathematics in her daily experience. Stories do wonder to kids. I was told a story by my school teacher about limitations of average. Once there was a group of students on trek led by their math teacher. On the way, they encountered a river. The teacher measured the depth of river at many places and took the average. Since the average was less than the height of smallest child in the group, he declared that it is safe to cross the river. Needless to say, the story did not have a happy ending. But it made me very conscious of averages (especially when people talk about per capita income etc.).

Make her see numbers in her day to day experience and make her realize how she can use math. Ask her when she is playing (and not during when she is doing homework) how many chocolate she can buy if she has \$21 and each cost \$5?

These are my two cents. If you can make her love maths, write a blog about it. One day, I'd find it useful when I'd have my own kids.

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Someone should make a collection of amusing but helpful maths stories. These are nothing like the stupid word problems kids get given everyday, and lead to actual understanding. –  daviewales Jun 17 '13 at 1:42

You can derive cross multiplying to her by multiplying the equation sidewise by 1) the left hand side denominator and 2) by the right hand side denominator.

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Very interesting. I will have to remember that one. –  Karl Kronenfeld Jun 10 '13 at 10:00
    
I tried explaining cross-multiplication to her once. I realized I'd also have to explain the fraction, which, in this case, was easiest to explain as representing division. It never seemed to "snap" for her though. –  mowwwalker Jun 10 '13 at 19:57
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I remember, that when i learned fractions, I thought $\frac{2}{3}$ as $2 \cdot \frac{1}{3}$. The first is quite difficult to imagine, but the last last factor of the last expression can be thought as division of a bread with a hole in the center to $3$ equal loaves, beginning from 12 o'clock to the clockwise direction. The multiplication then means that you take two such loaves. –  Juha-Matti Vihtanen Jun 11 '13 at 6:09

There are no "tricks". It is difficult! I am recommending listening a little to what star teacher Jaime Escalante says, a short video is:

http://www.youtube.com/watch?v=FFMz8JRg8Y8

Search for Jaime Escalante on youtube and elsewhere!

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In general: Visualization, Visualization, Visualization!

Using words or formulas doesn't seem to help your sister. Every formula can be visualized. Be it the easy case of a circle or the slightly more difficult one solving for $x$.

In particular for the circle:

  1. Take a cup, place it on a paper upside down. Take a pen and draw the circle around it

  2. Take a cord and cut out a piece of the length of $d$

  3. Take a cord and lay it around the cup. Cut it.

  4. Cut out 3 more pieces of length $d$ (4 total)

  5. Lay down the cord pieces representing $\pi$ and $d$ and let her compare them. Ask her how many times the small piece fits in the large piece. Let her guess.

There is also A LOT of animations available on the internet.

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This is nearly exactly what I did. Perhaps, as the other answers and comments suggest, it's because of what I was asking her and how I asked that she didn't make the connection I was expecting her to. –  mowwwalker Jun 10 '13 at 20:15

May be you could try an explorative approach rather than a teacher-student one? I mean, don't shoot out questions one after another or somehow hide the fact that you already know the answer (I know it is difficult!). As to my childhood days, whenever my well-educated uncle came visiting, he used to ask questions blindly and out of fear or whatever, I never answered them correctly (in fact I told wrong answers). I think there is a (tiny) bit of "hate" against the "teaching setup" in everybody (at least during schooldays).

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It may be heresy to say on this site, but I'm of the opinion that beyond basic arithmetic it's not so important to learn math as to be worth the damage caused by forcing people to grind through a subject they do not like and will not become proficient at. In the United States, pushing kids through advanced math classes and then failing them because they can't comprehend it has a corrosive effect on all their studies that directly contributes to raising the high-school drop out rate, which is a net negative for the society and a tragedy for the students who miss out on the opportunities of an education in other areas. How would you feel if you were held back a year in high school because you flunked poetry and gymnastics?

"But math is so much more important," you say. Well, 2,000 years ago learning how to make a fire from nothing more than stones and twigs and wood was a critical survival skill; it was literally a matter of life and death. Do they still even teach that in school? No, because we have matches and electric heaters and gas furnaces and professionals to install and maintain them. In modern times only a very few people are proficient at making a fire from natural elements and most people can't do it at all. Even survivalists depend on having a knife. So it is with math: between calculators, Excel, Google, websites and other computer software, and professional services it is completely possible to live a full and rich life without knowing advanced math.

On the career front, the most satisfying careers according to several surveys include (in no particular order since surveys disagree to some extent) are:

  • clergy
  • physical therapist
  • firefighter
  • K-12 and Special Ed teachers
  • fine artists

While of course a passionate mathematician can find math concepts in some of these fields, understanding those concepts in mathematical terms is not required to be a successful practitioner.

If you love math or are even curious about it, then by all means pursue it. I'm all for having a high quality math education available. But this question is about how to deal with someone who's tried math and hates it and doesn't want to waste any more energy on it. I say let them go and pursue what they are curious and passionate about instead.

Have some empathy and try to maintain some perspective on the full range of the human condition. Concert musicians are upset that people don't learn how to play musical instruments. Polyglots can't imagine why people don't spend more effort at learning other languages. Athletes don't understand why people would stay inside and read books all day. Mathematicians want everyone to learn math.

When someone complains to you that they don't want to learn math, remember why you don't want to learn 4 foreign languages, 2 musical instruments, art history, gymnastics, sociology, biology, plumbing, auto mechanics, carpentry, sculpture, drawing, painting, modern dance, and creative writing. Sure, you may want to learn some of those, but all if them? (I doubt it, but even if you really do, then you have a worse curse of never being able to devote yourself to becoming excellent at everything you want to pursue.)

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I disagree with this answer: understanding basic ratio problems is essential to becoming a world-class expert in many (maybe all?) of the fields you specify. –  Ronald Jun 11 '13 at 13:43
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@Ronald, math is only one of a number of ways to explain and understand ratios. People with dyscalculia often do fine with real world problems by estimation or conversion to non-mathematical systems. –  Old Pro Jun 11 '13 at 19:52
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There are many ways to understand and explain ratio: agreed. However, I would still consider those ways to be 'mathematical' (even if not 'algebraic'). An important part of mathematical education at this level (perhaps the most important part) is to find a personalised way to deal with real world problems. –  Ronald Jun 11 '13 at 20:03
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I remember reading an article that the only people who were unhappy to hear they started to permit calculators be used on the SATs were those who got a Math 800 score. Now as a father of a 14 year old, I see the damage, how the next steps are more difficult because the multiplication tables aren't imprinted on the current teen's minds. One can debate how far every student should go, but stopping at 7th or 8th grade isn't the place. –  JoeTaxpayer Jun 12 '13 at 0:33
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@OldPro - Practically speaking, when my 14 yr old asks why the need for Social Studies, I have a bit of a tough time offering more than a combination of "well-rounded education" along with the fact that few 8th graders know what they plan to do 9 years later, after college. The level of 8th grade has risen in the last 40 years, but it still doesn't contain trig and algebra. I may be splitting hairs here, as I don't remember my eigenvalues so well either, but I do think there's value to one's overall education to go beyond what the OPs sister is struggling with. –  JoeTaxpayer Jun 12 '13 at 18:35

I think the best way to engage kid into mathemathics is to give it interesting book with text exercises describing common problems. Like: Tom is 4 and his brother Jon is two times older. How old will Jon be when Tom is 10 etc. At least I liked it mostly, because it was real use of mathematics. I just got a simple example but I think you know what I mean. You can also find some contest exercises to get her involved\interested in, but if she doesn't want to you won't force her.

It's also a lot about teacher I think. Me and my brother were going to very good primary school, when my sister and second brother went to local one. It's about how they are taught from beginning. My sister nor my second brother didn't understand math, cause they were taught that they have to stick to one and only one proper way of solving problem. And when for example my sister did something differently her teacher removed her page from the exercise book( My father's intervention was required). I, in this case was given an 'A' for exploring, also no one bothered that I did the whole exercise book in 1 month instead of 1 year. And we knew about every contest and could attend no matter the skill, when they barely knew about any contests. So I suggest to check the methodical approach in the school. Maybe your sister was brain washed by a narrow-minded teacher.

Explaining things from the basics is very important as well. I remember I didn't understand cos/sin and all the things about them really until I saw the circle definition. But in my country it's not in the learning program. They just say you that this side of a triangle to this side of a triangle is that. So I like the university level stuff comment, cause sometimes it's so obvious and you wonder why they teach you the formulas instead of the concept behind it.

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"Tom is 4 and his brother Jon is two times older. How old will Jon be when Tom is 10 etc [...] it was real use of mathematics." Most people would not agree that this is a real use of math, since the context is artificial (in real life just ask Jon how old he is, or how old he'll be in 6 years). The sister who is the subject of the original question wouldn't find that to be a practical use of math. I agree with your last paragraph. –  KCd Jun 10 '13 at 18:32
    
I'm sure a book would really help her, especially one tailored for kids with troubles such as hers, but I don't think she would so much as glance at the first page. She simply isn't interested in math and I doubt she would attempt to learn any part of it if it weren't directly related to her homework. –  mowwwalker Jun 10 '13 at 20:13
    
@KCd I always hated those types of questions, and I still mock them when I see them in my sister's homework. I think most word problems are poorly written. –  daviewales Jun 17 '13 at 1:38

I think you need examples, I mean real life examples. When I learned this equation, I always imagining we are measuring a map. That means if we know 1 centimeter means 10 km in real, what we will get in real if there are 2 centimeters on the map? It somehow feels like a ratio projection, and there are so many things feel like a projection in real life. For example, you height is 1.5m, and you know that at 4:00 pm, your shadow should be 1.5 of your real height, what is the length of your shadow? Purely talking about numbers are way abstract for children, we need real life scenarios to let them feel the numbers. And a more better example maybe the sugar-water-inequality: (a+c)/(b+c) > a/b (a<b and c>0). Just imagine adding sugar into the water, and the inequality just means after adding sugar the sugar density is higher compared to before. Straightforward, easy to remember, easy to understand. Although real proof is a little tricky, this example truly expresses the implicit beauty of math.

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Games.. as user76556 menioned, games are a good way to spark interest.

For example, the Towers of Hanoi. http://www.mazeworks.com/hanoi/

Grab one set of these wooden rings:

http://www.goodtoknow.co.uk/money/galleries/34552/20-kids-gifts-for-under-a-fiver/7

and let her go at it!

Of course finding a game is the difficult part but there are a lot of free online math games which depending on the level your sister is at could really help.

Hope this helps - I have a lot of luck with the Towers of Hanoi (of course it is usually employed at parties with adults and lots of alcohol).

Brian

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How would you suggest bridging the gap between that and plane geometry? It'll be a hell of a task getting a 13 year old to see that. –  Jack M Jun 14 '13 at 16:54
    
Indeed it would, the above was just an example of the use of games to spark interest. The problem that I noticed when teaching kids (or adults) is that the math doesn't get in to the kid's head because it is un-interesting or un-related to some real world phenomenon. So the problem is to make it interesting, yes, that is perhaps more difficult than the math itself, but I have learned that a spark of interest can make all the difference in snowballing what the kid is willing to do. I also mentioned games (my main point - not Hanoi) as it seems to be quite underrepresented in the answers. –  Relative0 Jun 15 '13 at 5:32

The problem is that you are trying to help her in a way that she doesn't want to be helped. She wants to get her homework done. In her mind you are wasting her time with long explanations when you could just give her a formula.

If you want to address her fundamental mathematical issues, you should dedicate some time in the holidays (when she doesn't have maths homework...) and make up some problems of your own. You will have to make this fun, but the advantage is that there is no time limit. You can take as long as you like, and go back as far as you like.

I recently noticed my sister was having trouble with algebra because she had trouble with fractions. She knew the rules for getting common denominators, but she didn't realise that those same rules could be applied to purely algebraic fractions. To help her realise this, I filled an A4 page with fraction problems, starting with really easy ones that I knew she could do, then continuing to slightly harder ones.

I was surprised to notice that she could actually add fractions like the following: (she was better at fractions than I realised...)

$$ \frac{a}{b} + \frac{c}{d} $$

but she had troubles when I finally asked her to add fractions like the following:

$$ \frac{a+b}{c+d} + \frac{a+b}{a+c} $$

I realised that her fundamental problem was not fraction laws, but with aspects of the order of operations. She didn't realise that the numerator and denominator of a fraction are essentially surrounded by invisible brackets. Once I told her that she could add brackets around the numerators and denominators, she was able to do the problems.

In your case, I suggest you determine the highest level of mathematics at which she is confident and proficient, and create your own worksheets that slowly build on things which she already knows. (I handwrote mine. I thought of making them in $\LaTeX$, but nicely formatted questions are not important here, and you will end up procrastinating and never get them done anyway...) This will require a lot of work on your part, and you will have to convince your sister that it is worth it. You will also have to keep working on it each holidays. Your sister won't learn all the she needs to learn in just one holidays.

When your sister works through the problems, sit next to her and show how the hard questions can be solved using methods from the easy ones. Try to make it really obvious how the hard problems are just combinations of easier problems that she already knows how to do.

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I remember having trouble with Linear Algebra due to the way it was taught at my university, but I pulled through when I found a better teacher online.

I suggest going to Khan Academy and watching a few relevant videos together with your sister. This guy covers a lot, from basic math to advanced stuff, so you can pick out the most relevant videos, and you can go back and cover the basics if necessary.

I admit, it's not the most direct solution, but it helped my brother catch up on his math, and I'd recommend it to anyone.

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4  
I showed her that website a couple years ago, and I sometimes bring it up again, but she isn't very interested in it. It may be because Sal tends to drag on in his video explanations, making them long and boring. While this may be what's necessary, it's certainly not keeping anyone's attention –  mowwwalker Jun 10 '13 at 20:04
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@Walkerneo Long and boring: Yes! Honesty is alive and given Sanctuary at Math StackExchange. –  Feral Oink Jun 12 '13 at 5:01
    
Video is not an optimal way to learn, because you don't get any individualised feedback on the areas you actually need help with. (This is why there is a sense of *long and boring*ness) –  Ronald Jun 12 '13 at 16:42
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@OldPro Yes exactly: individualised feedback is the transformational improvement, not video lectures. At the moment, nobody providing video lectures is providing additional individualised feedback: and perhaps the opposite. –  Ronald Jun 14 '13 at 17:16
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My sisters don't like the Kahn Academy videos, but they often work through the problems, and just get me to explain them. –  daviewales Jun 17 '13 at 1:44

Many answers here seem to be about how to teach math, but I think that misses the point. This is about how to make someone interested in learning math.

In a situation like this, I think the best option might just be to refuse to help her. If someone really doesn't want to learn, then they're not going to. Nothing you can do is going to force them to want to learn, so it may be best to just say that you are always going to refuse to just give her the answers, and that the only way she's going to get your help is if she actually makes the effort to learn the way you'd like to teach her.

This may seem cold, but it should be possible to break it to her kindly. Make sure she understands that this isn't just for your benefit, but for hers. Remember to look at things from her point of view. For her, learning math is just an obstacle that has to be gotten over. She just wants to get the homework out of the way, pass the tests and be done with it. Approach her on those grounds. She has another five or so years of math ahead of her, and explain to her that there are two paths she can go down. She can continue to insist on memorizing everything without attempting to understand it. In that case, the next five years of math classes will continue to be absolute torture. She'll hate every moment of them, every bit of homework will be an ordeal, and I guarantee her grades will only get worse.

Or she can make the effort now to try it your way. Describe for her exactly what awaits her in that case, and contrast it with the alternative scenario. All of her math classes will become not only easy, but obvious. Much of the homework might even become fun for her. And even if she never develops a taste for the subject, at least she'll have the ability to effortlessly and quickly finish every assignment given her so that she doesn't have to waste any more of her time with them.

And of course, refuse to help her until she decides to go down that second path.

If she comes around, you'll still need to break down the barrier that math is only formulae, and get her to understand that mathematics is meaningful, not arbitrary. For that, you should refer to one of the other answers here.

PS: It's probably obvious I don't have much experience talking to 13 year olds. Reader discretion advised in how to actually word any of the above.

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I really don't think this is a good idea. By initially refusing to help, you are expressing a negative attitude that will only make it more difficult to offer help later. Although you accurately contrast the difficulty of struggling through material and flowing through material depending on a thorough understanding, an inexperienced teenager will easily believe the first case is normal and the second case is inaccessible to them (especially if they are not getting help to see that it is accessible). –  Ronald Jun 16 '13 at 12:35

Try Elementary Level first, expalin basic rules step by step..

3/5 = x/10

3*10 = x*5

30 = 5x ( once again tell her the rule is applicable even if one element will move..)

Sometimes when children are learning, they will be provided a specific scenario, and with that if they have learned something wrong(logically wrong) that becomes a problem for future..

so explain her one rule at time(divide and rule), in detail with various examples so she comes to know her mistakes...

so that will lead her answer to ...

30/5 =x

6 =x

i.e x=6

I think if some one have weak conceptual knowledge, please correct the roots so that the other things will automatically sum up in proper way gradually with practice..

You may try this method its, working on my side,..

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I think the fraction problem is nicer and more intuitive if you ask 'what fractions are equivalent to 3/5'? –  Ronald Jun 11 '13 at 13:44
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Yes, once you learn the elementary math its fun, but if you have doubts then its worst.. I was one of them with Trigonometry in past.. but after I cleared the doubts its no more problem.. just need to teach her elementary rules one will see the difference... :) –  MarmiK Jun 12 '13 at 4:30

I think the best way to get her involved is via numbers (integers). for example, the $3n+1$ problem or other elementary problems in number theory. You could also ask her to add (using a calculator) the first 50 odd integers while you give her the answer in a second (using $\sum\limits_{i=1}^n (2i-1)=n^2$).

If everything works ok, you can move on to some algebra and geometry, maybe challenging her to double a square using a non-graduated ruler and a compass.

I know how hard it could be, I'm a teacher.

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