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I really wish I could get math to stick. I just finished my National 5 maths (rough equivalent of a US High School Diploma) night-class today and all I ever see

I really wish I could get math to stick. I just finished my National 5 maths (rough equivalent of a US High School Diploma) night-class today and all I ever seem to understand is how, but not why. I'm the one asking "why is that that." And today was recapping on trinomial, simplifying fractions. Simple I expect to anyone with a mathematical mind, but to me, it's just an insane implosion which leaves me exhausted. I have notes, but even still. What is the secret behind it? How do people get so good with Maths? Is it truly study,study & study? .. I hadn't even herd of the terminology of co-efficient until today. Let alone something like this.

curtisf 5y ago
High school math does not look much like research math. High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ... Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct) For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function. The topologist's definition is that the function's inverse map preserves openness. These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers! I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans. They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition. But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field. If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.
- Pyramus 5y ago
  Very good example re continuity! I fully agree that (high) school math is mostly calculus and very different from university/research math. I would recommend parent to start exploring what math actually is, which fields there are and where they come from, what some famous open problems are and what beauty means to mathematicians. Good popular maths books are e.g. The Music of the Primes by Marcus du Sautoy [1] or Fermat's Last Theorem by Simon Singh [2]. It's also important to note that there are no shortcuts - it takes at least 1-2 years of studying maths to start to understand the why. [1] https://en.wikipedia.org/wiki/The_Music_of_the_Primes https://en.wikipedia.org/wiki/The_Music_of_the_Primes [2] https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book) https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book)
- gjm11 5y ago
  Just for fun, an example of an application of partial fraction decomposition that doesn't involve taking integrals. Consider the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, ... (with the convention that 0 and 1 are the 0th and 1st). Here's one way to get an explicit formula for them. First, we package them up into what's called a "generating function": 0 x^0 + 1 x^1 + 1 x^2 + 2 x^3 + 3 x^4 + 5 x^5 + ... or, more concisely, the sum of F_n x^n over all nonnegative integers n. Call this thing F(x). What happens if we add up F(x) and x F(x)? Well, x F(x) is just the same sum but with all the numbers shifted over by 1. It's the sum of x F_n x^n, or the sum of F_n x^(n+1), or the sum of F_(n-1) x^n, with the proviso that there isn't an x^0 term. So when we add this to the sum of F_n x^n, we get the sum of [F_(n-1) + F_n] x^n, with again the proviso that the x^0 term is just F_0 x^0 = 0. But, aha!, F_(n-1)+F_n is just F_(n+1), at least when n >= 1. So this thing equals the sum of F_(n+1) x^n, except that once again we need to be careful about n=0; more precisely what we have is the sum of F_(n+1) x^n, minus 1. (Because F_1 is 1, but the actual x^0 term is 0.) And the sum of F_(n+1) x^n equals the sum of F_n x^(n-1), or F(x)/x. In other words, we have F(x) + x F(x) = F(x)/x - 1, or equivalently (1 - x - x^2) F(x) = x, or equivalently F(x) = x / (1 - x - x^2). This is already pretty neat -- who would have thought that that big sum, involving the Fibonacci numbers which are defined in terms of a sort of recursive-adding process, would come out to something so simple? Now comes the coup de grace, which is the bit that uses partial fractions. We can write that denominator as (1-ax) (1-bx) for some numbers a,b. We'll figure out what a,b actually are in a moment -- that's just solving a quadratic equation. For now, looking at the coefficients of x and x^2 we note that a+b=1 and ab=-1. And now we can do the partial-fractions thing. We know that we'll have x/(1-x-x^2) = p/(1-ax)+q/(1-bx) for some p,q. That means that x = p(1-bx)+q(1-ax), so p+q=0 and pb+aq=-1 or p=1/(a-b). That is, F(x) = 1/(a-b) [1/(1-ax) - 1/(1-bx)]. Why bother doing this? Because the fractions we now have are easy to expand in powers of x. We have 1/(1-ax) = 1 + ax + a^2x^2 + ... and 1/(1-bx) = 1 + bx + b^2x^2 + ... So F_n, the coefficient of x^n in F(x), is just 1/(a-b) [a^n - b^n]. Finally, we ought to work out what a,b actually are. It turns out that a = [1+sqrt(5)]/2 and b = [1-sqrt(5)]/2; that is, a is the famous "golden ratio" sometimes called phi, and b = 1-a = -1/a. So a-b is just sqrt(5). So: the nth Fibonacci number is exactly (phi^n - (-1/phi)^n) / sqrt(5). Isn't that nice? (Since |b|<1, the term b^n/sqrt(5) is always smaller than 1/2 in absolute value, which means that F_n is also always the closest integer to phi^n/sqrt(5).) For the avoidance of doubt, none of this has anything at all to do with the disc embedding theorem. I just thought it was worth clarifying that partial fractions aren't just a one-trick pony used only for integrating rational functions; there are other situations in which representing something in terms of partial fractions gives useful insights into its behaviour.
mathgenius 5y ago
It's not just about memorizing stuff.. I think you need to find a different teacher, if you really are keen to keep going.
paulpauper 5y ago
It's all genes and some effort. Without the two, it's hard to get anywhere. People with super-high IQs are able to make the necessary epiphanies to 'see' the answer and prove the intended result. Genes play a role in everything, from marathon times, height, life expectancy, to lifting totals, to even math aptitude. no getting around it. Math , much more so than other fields, is heavily dependent on innate factors such as talent.
whatever1078 5y ago
You need to transition to learning math. One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available. Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files. The following intro books are so easy anyone and they momma can read these with utmost ease: Discrete Math by Susanna Epp How to Think about Analysis by Lara Alcock Linear Algebra: Step by Step by Kuldeep Singh Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings A Tour Through Graph Theory by Karin Saoub The above mentioned books are a few examples of elementary books that you can get started with right this minute. Note and remember the wealth of possible books, though. For example, consider the subject of math analysis. We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends. The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this. [0] https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/

curtisf 5y ago

High school math does not look much like research math. High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ... Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct) For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function. The topologist's definition is that the function's inverse map preserves openness. These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers! I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans. They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition. But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field. If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.

Pyramus 5y ago
Very good example re continuity! I fully agree that (high) school math is mostly calculus and very different from university/research math. I would recommend parent to start exploring what math actually is, which fields there are and where they come from, what some famous open problems are and what beauty means to mathematicians. Good popular maths books are e.g. The Music of the Primes by Marcus du Sautoy [1] or Fermat's Last Theorem by Simon Singh [2]. It's also important to note that there are no shortcuts - it takes at least 1-2 years of studying maths to start to understand the why. [1] https://en.wikipedia.org/wiki/The_Music_of_the_Primes https://en.wikipedia.org/wiki/The_Music_of_the_Primes [2] https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book) https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem_(book)
gjm11 5y ago
Just for fun, an example of an application of partial fraction decomposition that doesn't involve taking integrals. Consider the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, ... (with the convention that 0 and 1 are the 0th and 1st). Here's one way to get an explicit formula for them. First, we package them up into what's called a "generating function": 0 x^0 + 1 x^1 + 1 x^2 + 2 x^3 + 3 x^4 + 5 x^5 + ... or, more concisely, the sum of F_n x^n over all nonnegative integers n. Call this thing F(x). What happens if we add up F(x) and x F(x)? Well, x F(x) is just the same sum but with all the numbers shifted over by 1. It's the sum of x F_n x^n, or the sum of F_n x^(n+1), or the sum of F_(n-1) x^n, with the proviso that there isn't an x^0 term. So when we add this to the sum of F_n x^n, we get the sum of [F_(n-1) + F_n] x^n, with again the proviso that the x^0 term is just F_0 x^0 = 0. But, aha!, F_(n-1)+F_n is just F_(n+1), at least when n >= 1. So this thing equals the sum of F_(n+1) x^n, except that once again we need to be careful about n=0; more precisely what we have is the sum of F_(n+1) x^n, minus 1. (Because F_1 is 1, but the actual x^0 term is 0.) And the sum of F_(n+1) x^n equals the sum of F_n x^(n-1), or F(x)/x. In other words, we have F(x) + x F(x) = F(x)/x - 1, or equivalently (1 - x - x^2) F(x) = x, or equivalently F(x) = x / (1 - x - x^2). This is already pretty neat -- who would have thought that that big sum, involving the Fibonacci numbers which are defined in terms of a sort of recursive-adding process, would come out to something so simple? Now comes the coup de grace, which is the bit that uses partial fractions. We can write that denominator as (1-ax) (1-bx) for some numbers a,b. We'll figure out what a,b actually are in a moment -- that's just solving a quadratic equation. For now, looking at the coefficients of x and x^2 we note that a+b=1 and ab=-1. And now we can do the partial-fractions thing. We know that we'll have x/(1-x-x^2) = p/(1-ax)+q/(1-bx) for some p,q. That means that x = p(1-bx)+q(1-ax), so p+q=0 and pb+aq=-1 or p=1/(a-b). That is, F(x) = 1/(a-b) [1/(1-ax) - 1/(1-bx)]. Why bother doing this? Because the fractions we now have are easy to expand in powers of x. We have 1/(1-ax) = 1 + ax + a^2x^2 + ... and 1/(1-bx) = 1 + bx + b^2x^2 + ... So F_n, the coefficient of x^n in F(x), is just 1/(a-b) [a^n - b^n]. Finally, we ought to work out what a,b actually are. It turns out that a = [1+sqrt(5)]/2 and b = [1-sqrt(5)]/2; that is, a is the famous "golden ratio" sometimes called phi, and b = 1-a = -1/a. So a-b is just sqrt(5). So: the nth Fibonacci number is exactly (phi^n - (-1/phi)^n) / sqrt(5). Isn't that nice? (Since |b|<1, the term b^n/sqrt(5) is always smaller than 1/2 in absolute value, which means that F_n is also always the closest integer to phi^n/sqrt(5).) For the avoidance of doubt, none of this has anything at all to do with the disc embedding theorem. I just thought it was worth clarifying that partial fractions aren't just a one-trick pony used only for integrating rational functions; there are other situations in which representing something in terms of partial fractions gives useful insights into its behaviour.

mathgenius 5y ago

It's not just about memorizing stuff.. I think you need to find a different teacher, if you really are keen to keep going.

paulpauper 5y ago

It's all genes and some effort. Without the two, it's hard to get anywhere. People with super-high IQs are able to make the necessary epiphanies to 'see' the answer and prove the intended result. Genes play a role in everything, from marathon times, height, life expectancy, to lifting totals, to even math aptitude. no getting around it. Math , much more so than other fields, is heavily dependent on innate factors such as talent.

whatever1078 5y ago

You need to transition to learning math. One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available. Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files. The following intro books are so easy anyone and they momma can read these with utmost ease: Discrete Math by Susanna Epp How to Think about Analysis by Lara Alcock Linear Algebra: Step by Step by Kuldeep Singh Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings A Tour Through Graph Theory by Karin Saoub The above mentioned books are a few examples of elementary books that you can get started with right this minute. Note and remember the wealth of possible books, though. For example, consider the subject of math analysis. We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends. The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this. [0] https://www.people.vcu.edu/~rhammack/BookOfProof/ https://www.people.vcu.edu/~rhammack/BookOfProof/