Substitution
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 agelessdrifter
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 Joined: Mon Oct 05, 2009 8:10 pm UTC
Substitution
I am a student tutor at my college, and I think that the single most common problem I run into is people's complete inability to comprehend substitution. Today I was trying to help a woman solve logarithmic equations. She kept trying to subtract whole numbers from the arguments of logs. I told her, "look, the natural log of 743 is just a number  that whole thing together is just one number  but that number isn't 743. You can't mess around with that number inside the log, it's just part of the number. Think of it this way, let's, just for a second, call that number 'a' " and then panic and pandemonium struck. "I don't understand where you get a" "she [the professor] never said anything about a" etc etc. Finally I had to discard the paper with the damned 'a' on it and start from scratch.
This also comes up when trying to teach people to understand factoring trinomials by grouping. In fact I've run into this problem almost every single time I've tried this approach to teaching anything. I know that, as a tutor, my lot is to just find the approach that works, but I feel like the inability to grasp the concept of equality renders anything else I can say practically pointless  I can teach people to solve the problem, but what for? I mean how can you even vaguely comprehend the concept of graphing a function when the sign between "y" and "f(x)" is completely inscrutable to you?
How can I help these people get it?
Sorry, this post was a tossup between this forum and the "school" forum, but I felt like it was more relevant here.
This also comes up when trying to teach people to understand factoring trinomials by grouping. In fact I've run into this problem almost every single time I've tried this approach to teaching anything. I know that, as a tutor, my lot is to just find the approach that works, but I feel like the inability to grasp the concept of equality renders anything else I can say practically pointless  I can teach people to solve the problem, but what for? I mean how can you even vaguely comprehend the concept of graphing a function when the sign between "y" and "f(x)" is completely inscrutable to you?
How can I help these people get it?
Sorry, this post was a tossup between this forum and the "school" forum, but I felt like it was more relevant here.
Re: Substitution
This is a tricky thing. I don't know what subject you're tutoring, but I've tutored calculus students who don't know algebra. And my teaching style is to target every little misunderstanding until it is fully understood. This can be frustrating for calculus students when their calc tutoring session becomes an algebra 1 lesson within five minutes. If you're tutoring them at a level where they won't be offended by a slow explanation of variables and "plug and chug" and how mathematical objects (logarithms and numbers and such) can be represented by letters, I think you should do that. It's also important to impress upon students that they can be creative and invent their own variables to help them solve problems and understand concepts, like you did with the "a". You, as the tutor, and they, as the student, are allowed to do things the teacher never did in class as long as they are within the rules of mathematics. When it comes down to it, math tutors aren't teaching algebra, or trigonometry, or calculus, they're teaching mathematical thinking, so it can be a very meta cognitive process.
Re: Substitution
It's a good question, and I wish I had a good answer.
I've encountered students who, when given f(x)=x^2, will mistakenly write f(x+h) = x^2+h as opposed to f(x+h)^2. I find it hard to come up with a good way to explain why it's wrong. I've sometimes said things like "f of blah is blah squared", which I thought made sense, but some students seemed to think was the most nonsensical unhelpful thing ever.
Or, consider the distinction between (a+b)^2 and a^2+b^2. With some students, maybe a verbal description will help; I don't know. One thing is the square of the sum, and the other thing is the sum of the squares. They're not automatically the same thing, because you're not doing exactly the same steps in exactly the same order. If you put on your socks then your shoes, you get a different result than when you put on your shoes then your socks. The square of the sum is the square of something; the last step you did was squaring. The sum of the squares is the sum of something; the last step you did was summing.
I've encountered students who, when given f(x)=x^2, will mistakenly write f(x+h) = x^2+h as opposed to f(x+h)^2. I find it hard to come up with a good way to explain why it's wrong. I've sometimes said things like "f of blah is blah squared", which I thought made sense, but some students seemed to think was the most nonsensical unhelpful thing ever.
Or, consider the distinction between (a+b)^2 and a^2+b^2. With some students, maybe a verbal description will help; I don't know. One thing is the square of the sum, and the other thing is the sum of the squares. They're not automatically the same thing, because you're not doing exactly the same steps in exactly the same order. If you put on your socks then your shoes, you get a different result than when you put on your shoes then your socks. The square of the sum is the square of something; the last step you did was squaring. The sum of the squares is the sum of something; the last step you did was summing.
Re: Substitution
also for (a+b)^2 =! a^2 + b^2 a specific example can be good. (3+3)^2 = (6)^2 = 36 but 3^2+3^2=9+9=18. This motivates why the two may not be equal. It's hard to figure out what will work for each student. It's also frustrating when only a heuristic argument will work, and rigorous arguments won't. It feels like I'm shortchanging the student if I teach them heuristics rather than proofs

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Re: Substitution
This thread sums up about 10% of my anxieties at night. :/ When I try in fervent, worried fit to explain to people why "real math" (that is, making advances in mathematics) is important, even I realize it sounds awfully silly.
Sometimes I wish I could see what would happen if people were raised actually through mathematical logic, starting off with coloring in truth tables in kindergarten, then later inductively proving basic things with addition and multiplication in 4rd grade maybe (After some basic arithmetic everyone needs to survive), etc... once you got kids who do this stuff intuitively, teaching them how to solve linear equations (first applied stuff) > solve quadratics (do they really need a whole year for that?) > geometry (that's just proofs anyway) > precalc (just more algebra) > calc (finally fun stuff! they get to start proving cool things) will just be a blast, I think... Obviously this track wouldn't be for everyone, but instead of separating kids based on arithmetic ability early on, measuring a child's ability to abstract would be what decides whether they take this route or not.
Maybe I'm just being an idealist. Maybe I just loved "A mathematician's lament" too much. But I love math, and wish there were more people who understood what it was. Some people think computer science is how to fix computers, some people think art is just the tasteful composition of imagery... but most people realize that there are a lot more to topics than what they've heard (and we have to remember that, always). But after 11 years of math, a lot of people can be mistaken I think. ~sinks into depression~
</sadRant>
Sometimes I wish I could see what would happen if people were raised actually through mathematical logic, starting off with coloring in truth tables in kindergarten, then later inductively proving basic things with addition and multiplication in 4rd grade maybe (After some basic arithmetic everyone needs to survive), etc... once you got kids who do this stuff intuitively, teaching them how to solve linear equations (first applied stuff) > solve quadratics (do they really need a whole year for that?) > geometry (that's just proofs anyway) > precalc (just more algebra) > calc (finally fun stuff! they get to start proving cool things) will just be a blast, I think... Obviously this track wouldn't be for everyone, but instead of separating kids based on arithmetic ability early on, measuring a child's ability to abstract would be what decides whether they take this route or not.
Maybe I'm just being an idealist. Maybe I just loved "A mathematician's lament" too much. But I love math, and wish there were more people who understood what it was. Some people think computer science is how to fix computers, some people think art is just the tasteful composition of imagery... but most people realize that there are a lot more to topics than what they've heard (and we have to remember that, always). But after 11 years of math, a lot of people can be mistaken I think. ~sinks into depression~
</sadRant>
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Good fucking job Will Yu, you found me  __ 
Good fucking job Will Yu, you found me  __ 
Re: Substitution
When I have the time, I just give them a few examples. For the students who ask think f(x+h) = x^2+h rather than f(x+h)^2, I ask them what f(3+2) is. Or, perhaps more relevantly, f(3+.1). Do a few of those.
More generally, when the task is something like "what is the surface area of an a by b by c rectangular prism", I ask a progression of questions like "what is the surface area of a 1 by 1 by 1 rectangular prism", "what is the surface area of a 1 by 1 by 2 rectangular prism", ..., "what is the surface area of a 5 by 3 by 7 rectangular prism", ... .
The idea is to get them to realize that they're repeating some work each time, which makes them actually want to find a formula. Sometimes it works.
More generally, when the task is something like "what is the surface area of an a by b by c rectangular prism", I ask a progression of questions like "what is the surface area of a 1 by 1 by 1 rectangular prism", "what is the surface area of a 1 by 1 by 2 rectangular prism", ..., "what is the surface area of a 5 by 3 by 7 rectangular prism", ... .
The idea is to get them to realize that they're repeating some work each time, which makes them actually want to find a formula. Sometimes it works.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: Substitution
Have you tried using shorthand and other word substitution? Given that this is the internet generation, you can presumably make a normal English text, then substitute it with SMS language a sound/word at a time. (or start writing in 1337 or something) Try to make sure the substitution is invertible, though.
Re: Substitution
To address the OP's case specifically  I think your problem here is not an issue with the idea of equality so much as with the idea of a variable. If you say "let x=2", I don't think she understands what you mean  your best bet is probably something like "this thing here has a value, and we're going to give that value a name, 'x'. Then if we rewrite it, it must still be equal 'x' (we just rewrote it, we haven't changed how 'big' it is). But then we get these two things, both equal to x, so they must equal each other [maybe give some intuitive example of an equivalence class here, like heights or ages]. But then we can use these rules of algebra to come up with something silly like 1=2, which we know isn't true. So we must have done something wrong  we've justified each step we took except one, rearranging the logarithm, so that rearrangement must break the rules somehow."
Re: Substitution
I've found it's sometimes helpful to use anything other than a letter, because letters are Scary Algebra. Rather than saying "let's write 'a' for the log of 743," I'd draw a blobby shape around it. Then, when I want to refer to the quantity again, I draw the same blob.
Re: Substitution
I find that when teaching the basics of substitution it's best to stick to funny random things that can't be misconstrued.
For example when you're working with f(x) = x^2, you could say f(mickey mouse) = (mickey mouse)^2, and draw a picture of mickey mouse. This usually makes people laugh and for some reason it seems to help.
People seem to be taught that certain letters/variables/pronumerals(I hate this word) have specific meaning (for example in the equation of the line mx + c, c is the intercept rather than the intercept is c), instead of a generic thing which we can do various things to.
Writing mickey mouse always seems to get around this problem, or at least in my experience in helping people with their algebra.
For example when you're working with f(x) = x^2, you could say f(mickey mouse) = (mickey mouse)^2, and draw a picture of mickey mouse. This usually makes people laugh and for some reason it seems to help.
People seem to be taught that certain letters/variables/pronumerals(I hate this word) have specific meaning (for example in the equation of the line mx + c, c is the intercept rather than the intercept is c), instead of a generic thing which we can do various things to.
Writing mickey mouse always seems to get around this problem, or at least in my experience in helping people with their algebra.
Re: Substitution
mrmitch wrote:I find that when teaching the basics of substitution it's best to stick to funny random things that can't be misconstrued.
For example when you're working with f(x) = x^2, you could say f(mickey mouse) = (mickey mouse)^2, and draw a picture of mickey mouse. This usually makes people laugh and for some reason it seems to help.
People seem to be taught that certain letters/variables/pronumerals(I hate this word) have specific meaning (for example in the equation of the line mx + c, c is the intercept rather than the intercept is c), instead of a generic thing which we can do various things to.
Writing mickey mouse always seems to get around this problem, or at least in my experience in helping people with their algebra.
This is a good point too, with the added advantage of relieving some of the tension that someone having enough trouble to seek out a tutor is probably feeling.
Re: Substitution
mrmitch wrote:People seem to be taught that certain letters/variables/pronumerals(I hate this word) have specific meaning (for example in the equation of the line mx + c, c is the intercept rather than the intercept is c), instead of a generic thing which we can do various things to.
Heh, case in point: I was taught this formula as mx+b.
 Yakk
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Re: Substitution
My favorite variables are smiley faces (of various kinds). For most people, this hooks into your facerecognition "brain circuitry", so they are surprisingly easy to remember and distinguish.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
 agelessdrifter
 Posts: 225
 Joined: Mon Oct 05, 2009 8:10 pm UTC
Re: Substitution
Some really good suggestions here. I'm gonna try the smiley face thing next time.
One thing I tried that I was *sure* would work when I was trying to teach someone why F.O.I.L (or factoring trinomials by grouping, I forget) works, was writing one of the binomials on two small scraps of paper with a different number or letter on the back. The person could understand factoring out and/or distributing single terms, but was completely befuddled by "pulling out" or distributing polynomials into other polynomials. So I wrote it all out except for the polynomial being distributed, which I had on the scraps of paper with the number (let's say it was 2) on the back. I said "ok, so if this were a two out front of this other binomial here, we'd distribute it like this, right?" and put one of the twos in front of each of the terms. She got that. Then, "ok, that's exactly what we're doing, only instead of a two, it's (x+1) [or whatever it was]" and I flipped over the "2"s to show the binomials on the back.
I thought that was *guaranteed* to work. But I got the response "ok you just lost me". It was very disappointing because I felt clever for having the idea.
One thing I tried that I was *sure* would work when I was trying to teach someone why F.O.I.L (or factoring trinomials by grouping, I forget) works, was writing one of the binomials on two small scraps of paper with a different number or letter on the back. The person could understand factoring out and/or distributing single terms, but was completely befuddled by "pulling out" or distributing polynomials into other polynomials. So I wrote it all out except for the polynomial being distributed, which I had on the scraps of paper with the number (let's say it was 2) on the back. I said "ok, so if this were a two out front of this other binomial here, we'd distribute it like this, right?" and put one of the twos in front of each of the terms. She got that. Then, "ok, that's exactly what we're doing, only instead of a two, it's (x+1) [or whatever it was]" and I flipped over the "2"s to show the binomials on the back.
I thought that was *guaranteed* to work. But I got the response "ok you just lost me". It was very disappointing because I felt clever for having the idea.
Re: Substitution
That does sound like a wonderful idea!
Re: Substitution
agelessdrifter wrote:I was trying to teach someone why F.O.I.L
For that specific one, I’m a fan of drawing a rectangle, with a horizontal line and a vertical line through it, so it has four subrectangles. If you are multiplying (x+a)(y+b) then label the lengths of the parts of the height as y and b, for a total height y+b, and similarly the lengths of the parts of the base are x and a for a total base of x+a. Then the whole rectangle has area (x+a)(y+b), and the subrectangles can easily be labeled with their areas, which get added up.
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 Yakk
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Re: Substitution
ageless, how about two index cards. On the back is some symbol (a happy face). The person is not allowed to "flip it over" until later.
There are actually two of them stacked, so you can "copy" it without knowing what it is.
Have them do:
times ( A + B )
where A and B are cards, and so are the brackets and the plus, physically.
Then flip over the . On the back is 2 to start with.
That is distribution.
Next, have them build cards for an expression. 2 times ( 3 + 4 )  have them make the 5 cards, and put symbols on the back.
Now do it for 3 times ( x + 1 ), where x just becomes another card.
Also do ( 2x + 5 ) times 4 as cards (ie, times on the other side).
Remember that you need two cards (stacked) for the ones that are going to be spread out via associativity.
Remember  they are making the cards here.
To make it even more familiar, have the equations they break up into cards written on cards. So you'll have ( 2x + 5 ) on one card, and you'll ask them to turn it into multiple cards.
Make them flip over the individual terms, and draw something on the back of them, when manipulating them. A happy face, or what have you. The point is "it doesn't matter what is on the card, all that matters is how you move it around".
Another step is doing ( 2x + 5) times 4 + (2x + 5) times 3 as "adjacent" equations.
After doing all that, have them do associativity where the back of one of the cards has an equation on it (but they don't know it, because all they see is the happy face or other doodle).
...
The theory here is that the student is fundamentally afraid of mathematics, and needs to be made familiar with the steps before trying something complex.
There are actually two of them stacked, so you can "copy" it without knowing what it is.
Have them do:
times ( A + B )
where A and B are cards, and so are the brackets and the plus, physically.
Then flip over the . On the back is 2 to start with.
That is distribution.
Next, have them build cards for an expression. 2 times ( 3 + 4 )  have them make the 5 cards, and put symbols on the back.
Now do it for 3 times ( x + 1 ), where x just becomes another card.
Also do ( 2x + 5 ) times 4 as cards (ie, times on the other side).
Remember that you need two cards (stacked) for the ones that are going to be spread out via associativity.
Remember  they are making the cards here.
To make it even more familiar, have the equations they break up into cards written on cards. So you'll have ( 2x + 5 ) on one card, and you'll ask them to turn it into multiple cards.
Make them flip over the individual terms, and draw something on the back of them, when manipulating them. A happy face, or what have you. The point is "it doesn't matter what is on the card, all that matters is how you move it around".
Another step is doing ( 2x + 5) times 4 + (2x + 5) times 3 as "adjacent" equations.
After doing all that, have them do associativity where the back of one of the cards has an equation on it (but they don't know it, because all they see is the happy face or other doodle).
...
The theory here is that the student is fundamentally afraid of mathematics, and needs to be made familiar with the steps before trying something complex.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
 Cleverbeans
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Re: Substitution
When I'm dealing with "fear induced blindness" related to variables, I generally reason that they're just names, and then rely on personification to help them feel more comfortable with it. When I ask "So we have two terms here (x+1) and (x5).What names should we give them?" and I'll just work with the full name for the first run doing "so Sue * Bob = Sue * (x5) = Sue*x Sue*5 ... etc. and then on the next run through ask if I can just use short hand and call them s and b instead.
Alternately I set up word problems with meaningful names then justify the move to variables by agreeing that writing "area = length * width" get tedious when you could simply write A = l * w. I also find it useful to write down what I'm substituting explicitly such as u = x + 5, and if they're still having trouble following then I go into a discussion of the transitive property of equivalence relations in personified garb. My goto example is "if my shoes are the same color as my belt, and my belt is the same color as my pants, then my shoes are the same color as my pant" and then write that down using s = b and b = p then s = p. and going to a numerical example using "same number" instead of "same color".
The reality of course is that the issue is almost always emotional, and it's important to pick up on the nonverbals early and try to build empathy. I start by inquiring about what they're feeling and responses are generally stuff like 'frustrated, confused, lost, afraid, angry, pissedoff, worried, anxious, etc'. Most of the time the emotion comes from having an experience in previous math classes that made them feel stupid, often a particular teacher so I'll ask about when they started feeling that way, and the teacher they had, but mostly I just try to explain to them that their emotions are interfering with their progress, and that has to be addressed first.
I like to open by building empathy by talking about some of the bad teachers I had. This generally gets them to rage about some teacher who did a bad job, and I just do the "yeah I know what you mean... that's exactly it isn't it?....yeah that really bugs me too" head nodding until they talk it out. Again watch for the nonverbals to indicate that they're mood has improve. I then like to follow by telling them how math is often about the struggle, and that despite their skill mathematicians spend days or weeks even years struggling with a problem before they crack it, so there is nothing wrong with feeling lost or befuddled by a problem  in fact it's to be expected. I tell them about my strategies to deal with it like taking some deep breaths, taking short breaks to do something else.
Oh, and drill home the fact that hard work and long hours are * mandatory*. If they're a sluffoff questions like "how many hours have you put into the problems so far?" and when they respond "well, I just looked at them this morning" I give them the Spock eyebrow with a suggestive grin then say "I think we've found the heart of the problem  haven't we?" This will weed out those who are genuinely struggling from the sluffoffs looking for easy answers, and you can tailor your approach in each case. For sluffoffs I tend to explain the problems by supplying the definitions of terms that they're unfamiliar with, and helping them look up the way to do it in their textbook/notes rather then working example problems. If they're intimidated before they even start, then they skills they're lacking are often related to note/textbook scanning and general problem solving rather then the specifics of the problem so I focus on that problem first. If they didn't bring their textbook/notes just send them away and tell them to come back after they've made an attempt.
Above all, I find it's vital to be patient, empathetic, and kind. Modeling those behaviors will do more for your students mathematical ability then any specific metaphor or algebraic trickery will.
Alternately I set up word problems with meaningful names then justify the move to variables by agreeing that writing "area = length * width" get tedious when you could simply write A = l * w. I also find it useful to write down what I'm substituting explicitly such as u = x + 5, and if they're still having trouble following then I go into a discussion of the transitive property of equivalence relations in personified garb. My goto example is "if my shoes are the same color as my belt, and my belt is the same color as my pants, then my shoes are the same color as my pant" and then write that down using s = b and b = p then s = p. and going to a numerical example using "same number" instead of "same color".
The reality of course is that the issue is almost always emotional, and it's important to pick up on the nonverbals early and try to build empathy. I start by inquiring about what they're feeling and responses are generally stuff like 'frustrated, confused, lost, afraid, angry, pissedoff, worried, anxious, etc'. Most of the time the emotion comes from having an experience in previous math classes that made them feel stupid, often a particular teacher so I'll ask about when they started feeling that way, and the teacher they had, but mostly I just try to explain to them that their emotions are interfering with their progress, and that has to be addressed first.
I like to open by building empathy by talking about some of the bad teachers I had. This generally gets them to rage about some teacher who did a bad job, and I just do the "yeah I know what you mean... that's exactly it isn't it?....yeah that really bugs me too" head nodding until they talk it out. Again watch for the nonverbals to indicate that they're mood has improve. I then like to follow by telling them how math is often about the struggle, and that despite their skill mathematicians spend days or weeks even years struggling with a problem before they crack it, so there is nothing wrong with feeling lost or befuddled by a problem  in fact it's to be expected. I tell them about my strategies to deal with it like taking some deep breaths, taking short breaks to do something else.
Oh, and drill home the fact that hard work and long hours are * mandatory*. If they're a sluffoff questions like "how many hours have you put into the problems so far?" and when they respond "well, I just looked at them this morning" I give them the Spock eyebrow with a suggestive grin then say "I think we've found the heart of the problem  haven't we?" This will weed out those who are genuinely struggling from the sluffoffs looking for easy answers, and you can tailor your approach in each case. For sluffoffs I tend to explain the problems by supplying the definitions of terms that they're unfamiliar with, and helping them look up the way to do it in their textbook/notes rather then working example problems. If they're intimidated before they even start, then they skills they're lacking are often related to note/textbook scanning and general problem solving rather then the specifics of the problem so I focus on that problem first. If they didn't bring their textbook/notes just send them away and tell them to come back after they've made an attempt.
Above all, I find it's vital to be patient, empathetic, and kind. Modeling those behaviors will do more for your students mathematical ability then any specific metaphor or algebraic trickery will.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration."  Abraham Lincoln
Re: Substitution
mrmitch wrote:I find that when teaching the basics of substitution it's best to stick to funny random things that can't be misconstrued.
For example when you're working with f(x) = x^2, you could say f(mickey mouse) = (mickey mouse)^2, and draw a picture of mickey mouse. This usually makes people laugh and for some reason it seems to help.
I sometimes use "banana" as an example like that, because it's also a funny word to say.
Mostly in this thread though, I agree with this
Cleverbeans wrote:Above all, I find it's vital to be patient, empathetic, and kind. Modeling those behaviors will do more for your students mathematical ability then any specific metaphor or algebraic trickery will.
I've been doing math tutoring pretty much fulltime for nearly two years, and parttime for a while before that. I've found that what my students appreciate the most is that I work with them instead of just directing and spouting formulas. You need to be very patient and very optimistic to be a good math tutor, and you need to be able to connect with your students. Generally, they probably think of their math teacher/professor/TA as someone who is brilliant at math and doesn't understand how they don't get it; your job is to understand how they don't get it and try to fix that. There's a huge belief in the US (and western europe I think too) that math is "hard" and that being able to do math is just about your natural talent and genius. In western europe and asian countries, they say anyone can do math if they work hard at it. Part of the work of a math tutor is to indirectly remove that idea from your student, that they can't do math just because it didn't come easy to them at first.
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