8 ms·

The Calculus Trap (2005)

125 points by adenadel 8y ago

dang 8y ago
A couple of good past discussions: 2014: https://news.ycombinator.com/item?id=7207495 https://news.ycombinator.com/item?id=7207495 2009: https://news.ycombinator.com/item?id=717982 https://news.ycombinator.com/item?id=717982
- svachalek 8y ago
  The 2009 discussion at least partly understood that the article is not about calculus, while it seems the 2014 and 2019 discussions missed the point. I don't know if that really says something about the culture of the times but it's interesting.
  - dang 8y ago
    Threads are sensitive to random initial conditions like who happened to be online when an article was posted, so I doubt it says much about 2009.
RandomInteger4 8y ago
> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too." That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?
- msla 8y ago
  > That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra? I am deeply confused by a curriculum which separates Matrix Theory from Linear Algebra. The description in the Wikipedia category just barely helps: https://en.wikipedia.org/wiki/Category:Matrix_theory https://en.wikipedia.org/wiki/Category:Matrix_theory > Matrix theory is a branch of mathematics which is focused on study of matrices. Initially, it was a sub-branch of linear algebra, but soon it grew to cover subjects related to graph theory, algebra, combinatorics and statistics as well. The University of Missouri has a Matrix Theory course: https://www.math.missouri.edu/class/matrix-theory https://www.math.missouri.edu/class/matrix-theory > Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs. ... which specifies a textbook: > Linear Algebra with Applications (7th edition) by Steven J. Leon ... which deepens my confusion. If you're taking that course, how is it not an introductory Linear Algebra course? And this MathOverflow answer obfuscates again: https://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra https://mathoverflow.net/questions/11669/what-is-the-differe... > Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences. > In linear algebra, however, you instead talk about linear transformations, which are not (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics. If I try to parse charitably, I come away with the idea that Matrix Theory is about matrices as a data structure, usable for many things outside the scope of Linear Algebra, where they're all about using matrices to represent linear transformations. It's the difference between a column of numbers on a shopping bill and a column of numbers which represents a vector in a space with a specified basis. Gotcha. However, this answer directly contradicts what the University of Missouri calls Matrix Theory, which is so Linear Algebra they even use a Linear Algebra textbook. It also... I don't know, trivializes the field of Matrix Theory. So you can manipulate matrices. So what? They show up a lot because they're used to represent specific things. Is the course going to barely introduce a lot of specific things and then focus on the matrix representation? What a waste!
- graycat 8y ago
  I'll try to clear things up: First, Horn and Johnson is a bit much. I was in Horn's class. I had done a LOT in, call it, linear algebra and matrix theory in my career before the class, told the profs I didn't need the course, and they said it was a "second, advanced course" and smiled. The course was quite competitive and without trying at all and without intending to be competitive, I effortlessly blew away all the other students on graded homework, the tests, the midterm, the final exam, and the corresponding qualifying exam. At the end of the course Horn wrote about me IIRC "Best performance in the class by a wide margin. Knows this material cold." So, yes, it was an advanced course, actually had a lot of nice stuff in it, Horn's lectures were nicely precise and at times with some unusual, nice approaches, but to do well in the course it was sufficient just to have had a good background before. What background? For the main books, E. Nering (a student of E. Artin at Princeton), Halmos (an assistant to von Neumann at the Institute of Advanced Study at Princeton), Finite Dimensional Vector Spaces, basically also a finite introduction to Hilbert space and the spectral theorem there, Forsythe and Moler, Computer Solutions of Linear Algebraic Systems, and some good texts in multivariate statistics with regression analysis, discriminate analysis, factor analysis, analysis of variance. More in applications, e.g., the fast Fourier transform, more on curve fitting, linear systems in electronic engineering, antenna theory and beam forming, optimization, linear programming, unconstrained optimization, the Markowitz and Sharpe applications to finance, Lagrange multipliers, the Kuhn-Tucker conditions, etc. can also help. But Horn is not a good choice for a first text. For a first or second text I'd suggest, say, Hoffman and Kunze, Linear Algebra, Second Edition available for free on the Internet. For more, see my post on math in https://news.ycombinator.com/item?id=15116379 https://news.ycombinator.com/item?id=15116379 and there sections (2) Linear Algebra (2.1) Linear Equations (2.2) Gauss Elimination (2.3) Vectors and Matrices (2.4) Vector Spaces (2.5) Eigen Values, Vectors (2.6) Texts To be brief, about the earliest and easiest start on linear algebra and matrix theory is just a high school style system of linear equations. The main solution technique is Gauss elimination. Matrix notation is a better notation for that subject. Here is essentially the role of matrix theory: Each of the old results in linear algebra can be written as a result, with nicer notation, in matrix theory. Can get the same results without matrix notation, but matrix notation makes it all much easier. Next, a broad statement is that the two pillars of the field of analysis in math are (1) linearity and (2) continuity. Well, linear algebra and matrix theory stands strongly on linearity and, as we move on in both the theory and applications, also continuity. Let's be clear on linearity via linear algebra and matrix theory: So, for positive integers m and n and an m x n matrix A we say that matrix A is a linear transformation (function) if for all n x 1 vectors x and y, and numbers a and b, we have that A(ax + by) = aAx + bAy Sure, to read this need the definitions of matrix sum and product; sum is trivial; product is not much harder and is really just what need to make Ax = b be the same as the high school system of linear equations. For the numbers, usually use either the set of real numbers R or the set of complex numbers C. But, sure, for numerical computation are essentially limited to the set of rational numbers Q. But in general need only what a course in abstract algebra calls a field: Each of R, C, and Q is such a field but also the set of integers modulo a prime number is a field, of interest in algebraic coding theory and cryptology. This definition of linearity generalizes in Hilbert space, Banach space, and functional analysis, and the more general definitions and results are important in quantum mechanics, differential equations in science and engineering, signal processing in electronic engineering, etc. Again, linearity is a pillar of analysis in math. Why pillars? In both theory and applications, linearity and continuity commonly hold and are astoundingly powerful properties. For such applications we have multivariate statistics, optimization, electronic engineering, antenna theory, beam forming, signals (each time invariant linear system has sines and cosines as eigenvectors; when a violinist on a concert stage plays some pure tones, the concert hall transmits those tones to you in the audience as a linear system so that what you hear are just the pure tones with the right frequencies but with some phase and amplitude changes; the Navy likes to know that for sonar signals; the USAF likes to know that for radar signals; cell phone people like to know that for their signals), Fourier theory, linear partial differential equations, superposition in quantum mechanics, etc. And when linearity does not hold, commonly it is a good, first approximation and the main means of iterative techniques. And if a problem is not linear, maybe after some simple transformation it will be. In some of the posts here, there is mention of matrix theory and basis, that is, a coordinate system. Well, can do that although is it not nearly as general as what physics likes to do with coordinate systems. But also can just decide not to do that, to take the vector space as just the n-tuples and not force thinking of the n-tuples as just coordinates of vectors in some basis. Or can do either approach depending on what is easier in the context. Here is a point should get: Suppose we start with just systems of linear equations. Then we say we are working with n-tuples of numbers. Then we call those n-tuples a vector space. Then using essentially just the main, relevant properties of those n-tuples, we write down the definition, axioms, of a vector space where we've said nothing about the vectors but have left them as just points. Well then we have two advantages: First, the definition of a vector space lets us talk about subspaces and in particular subspaces of the vector spaces of just the n-tuples, and we want to do that already, strongly with just Gauss elimination for linear equations. E.g., with the linear system Ax = b where m, n are positive integers, A is m x n, x is n x 1, and b is m x 1, the set of all x so that Ax = 0 (m x 1 of all zeros) is a vector subspace of all the n x 1 vectors (for the set of real numbers R, commonly called the set R^n). Call the set K the set of all x so that Ax = 0. If for some n x 1 u we have that Au = b, then from linearity we can argue that any v so that Av = b can be written as a sum of u and some vector in K. In this way we see all possible solutions of Ax = b. Actually at the end of Gauss elimination we can see K and u easily enough. Second, we get to consider vectors other than just n-tuples. E.g., we can consider the data of 1 second of music as a vector, a random variable as a vector, a color as a vector, the wave function of a photon or electron as a vector, etc. Then as the book continues, we get into eigenvalues and eigenvectors. Eigen is German essentially for special. They are special, and valuable. Maybe the nicest part is the polar decomposition: Each square matrix is a product UH where U in unitary and H is Hermitian. In class, when Horn got to that, I shouted out "That's my favorite theorem! The unitary part is an isometry, that is, doesn't change lengths or angles and is essentially a rigid motion, maybe just a rotation or reflection. The Hermitian part H is a shocking dream, amazing beyond belief: All H can do is take a circle and make it into an ellipse: The two axes of the ellipse are perpendicular (orthogonal) and the eigenvectors. Their lengths are the eigenvalues. And this generalizes to rounded footballs in three dimensions and all finite dimensions. And with the spectral theorem it generalizes to infinitely many dimensions and is the main reason in quantum mechanics the observables are eigenvalues. The polar decomposition is also the source of the powerful singular value decomposition, principle components analysis, factor analysis, analysis of saddle points in optimization (see W. Fleming, Functions of Several Variables), the matrix condition number in the numerical analysis of Gauss elimination, and much more in theory and applications. Hope this helps.
- tomrod 8y ago
  I love that book! It's certainly not an entry point though.
morpheuskafka 8y ago
> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. I took calculus last year (AB Calc BC, 10th grade), and I can say that my experience was certainly a counterexample. I did OK, but it was definitely a marked difference from "breezing through" algebra.
- nerdponx 8y ago
  Likewise. I had an easy time in high school math. In college I really had to work for it.
- tomrod 8y ago
  I expect it was more of a teacher issue then. A solid math teacher is worth their weight in platinum.
  - dTal 8y ago
    I had an excellent Calculus teacher in high school, and I did very well on the AP exam. It was still vastly more challenging than the trigonometry class that preceded it. Even today, I use trigonometry all the time almost without thinking, but I can't for the life of me remember how to long-divide polynomials (or picture a scenario where I'd need to). There's just so friggin much deeply abstract symbol manipulation in calculus class (which in school also covers essentially "advanced algebra"). It's a different ball game.
    - jacobolus 8y ago
      Long division of polynomials is literally exactly the same algorithm as long division of base-10 integers, except with no carrying.
- Jach 8y ago
  I got As in my pre-college math courses but I don't think I breezed through them. My schooling had no separate trig course, it went algebra, geometry, intermediate algebra, and calculus. Various pre- or honors versions. Since the BC exam covers quite a bit more material than the AB, my school teacher required those interested in it to take pre-calculus Honors first which was mostly about shoring up any gaps in previous studies while getting a foothold on Calculus AB material near the end. I think that extra prep made a huge difference in my performance, along with the teacher himself and an insistence on memorizing a lot of things to not be dependent on a rule sheet and getting an intuition for solving problems that even the TI-89 wouldn't help much with.
nextos 8y ago
The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations. For young students, a great introductory textbook is Calculus Made Easy. It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems. For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.
- jefft255 8y ago
  I don’t know about the rest of the world but in Quebec calculus is not taught with any epsilon-delta formalism. You don’t encounter that until mathematical analysis, which is in the curriculum of a very few majors in uni.
  - throwawayjava 8y ago
    This differs throughout the USA. "Best" schools (<1%) will have what the article is describing (strong student mathematical problem solving communities lead by teachers with strong mathematical training). Typically very expensive, either directly through tuition or indirectly because they are public schools but only serve students in very high net worth zip codes. "Very Good" schools (<10%) will include some of the epsilon-delta formalism but might have some curricular problems and a "gifted just means moving through the terrible not-real-math curriculum faster" problem that the article talks about. "Decent" schools will teach how to compute derivative/integrals by rote and maybe talk a bit about the physical intuitions in a very hand-wavy way. Calculus is very much a continuation of Algebra or Trig where you learn some rules and how to pattern match and don't ask too many questions about why. The other half of USA high schools? They don't even offer a Calculus course of any kind [1]. Which... if you don't teach it at all, you can't teach it wrong... As an aside, AP CS has the same problem as the problem identified with AP Calc. AP CS is the epitome of a "Java School" course. [1] https://www.theatlantic.com/education/archive/2016/06/where-calculus-class-isnt-an-option/485987/ https://www.theatlantic.com/education/archive/2016/06/where-...
gibba999 8y ago
I don't buy it. AoPS is a great organization, but the focus is on pure, theoretical mathematics. Understanding calculus is key to understanding many beautiful areas of applied mathematics: image processing, signal processing, control systems, electronics, etc. I consider them more elegant than theoretical mathematics. Now, for that, you don't need all the messy manipulation (integration-by-parts and similar), but you do need the basics of area-under-the-curve, of derivative-as-slope, and similar, as well as some of the theory. But that's not too hard to learn. My own opinion is that the basics of calculus should be taught alongside the basics of algebra in elementary school. Plenty of people have had success doing both.
hyperpallium 8y ago
What does HN think of Khan Academy's treatmemt of calculus? https://www.khanacademy.org/math/calculus-all-old https://www.khanacademy.org/math/calculus-all-old Ideally, by an expert in calculus, who has done the whole KA course on it (though why an expert would do that, I don't know...)
- HiroshiSan 8y ago
  As someone who tried teaching themselves math with Khan Academy, I feel like Khan Academy suffers from the same problem most sites do, in that they tell you the information before you can discover it for yourself. This is different than how AoPS teaches math, where they ask you questions which then guide you towards self discovery, and as a result, you start asking your own questions which create paths for you to go on and explore.
  - barry-cotter 8y ago
    You, and AoPS’ target audience of current and potential math nerds are very far from the norm. Inquiry based learning is actively harmful to learning for many students, and much less efficient for more or less everyone. > Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching > Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide "internal" guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described. https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1?needAccess=true& https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_...
WillPostForFood 8y ago
If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. It is unfortunate that the prevailing educational trends are to get rid of tracks, lanes, and advanced classes, and dumping all the kids into mixed ability classes.
- threatofrain 8y ago
  I believe it's been roughly found that tracking helps smart kids moderately and hurts everyone else with the brain drain. It's a question of to what degree we are individuals, and to what degree we are a community. I think it's a question that cuts right down the middle of many people's value systems.
  - Mvandenbergh 8y ago
    There's also the hybrid view: it's important to ensure that the most talented kids are supported properly because they are our future leaders, inventors, artists, etc.
  - WillPostForFood 8y ago
    The data isn't very strong to support the idea that leaving advanced kids in the room with those that need more attention is of any help, and there is now growing evidence of achievement gaps widening, possible because it can be demoralizing for the slower kids, possible because the more advanced kids are now more likely to get outside supplementation which boosts them even further ahead.
hackermailman 8y ago
This article is primarily a marketing piece for their $500 courses, which I'm sure are good but if you don't have money or accessible local math student clubs and want to like the article says 'explore math' try these math foundations playlists for free https://www.youtube.com/user/njwildberger/playlists https://www.youtube.com/user/njwildberger/playlists Wildberger starts from the very beginning, proving laws of each ring/field with basic arithmetic. He also has an algebraic trig method that a primary school kid could do and an interesting discrete algebraic calculus method, plus plenty of abstract algebra content.
- HiroshiSan 8y ago
  Not sure how you got this impression, the concluding paragraph is "However, we are not the only other option. Other options students have are to become involved in extracurricular programs, such as math teams. Math contests should be selected with some care: those that encourage mass memorization or just test standard curricular tools tend to exacerbate the ills of the calculus trap rather than enhance problem-solving ability. Students can also pursue independent study if they are able to find mentors. University professors are occasionally willing to fill this role to some degree. There are also many summer programs and good books for extracurricular study, and some communities have developed grassroots programs to provide opportunities for eager students. These options are usually not as easy as “enroll in the next course,” but they will be far more rewarding than settling into the calculus trap." Also these articles are normally targeted to those already invested in the AoPS ecosystem, whether it be books, courses, or their forums.
IcePowder 8y ago
It wasn't till calculus did my love affair begin with Mathematics...,
sevensor 8y ago
I think this is a side-effect of the way our educational system is structured. We rely on big, standard tests to evaluate schools and their students. It would be too expensive, and somewhat subjective, to evaluate students' reasoning abilities, their conceptual knowledge. Can you imagine if these tests were evaluated by rooms full of people reading proofs, rather than by bubble-sheet scanners? So we test what we can test, which is procedural knowledge, and we teach that at the expense of deeper understanding. And because the best way of measuring procedural knowledge is to measure how many procedures you know, we race students forward to as much calculus as they can memorize.

125 points by adenadel 8y ago

dang 8y ago

A couple of good past discussions: 2014: https://news.ycombinator.com/item?id=7207495 https://news.ycombinator.com/item?id=7207495 2009: https://news.ycombinator.com/item?id=717982 https://news.ycombinator.com/item?id=717982

svachalek 8y ago
The 2009 discussion at least partly understood that the article is not about calculus, while it seems the 2014 and 2019 discussions missed the point. I don't know if that really says something about the culture of the times but it's interesting.
- dang 8y ago
  Threads are sensitive to random initial conditions like who happened to be online when an article was posted, so I doubt it says much about 2009.

RandomInteger4 8y ago

> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too." That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?

msla 8y ago
> That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra? I am deeply confused by a curriculum which separates Matrix Theory from Linear Algebra. The description in the Wikipedia category just barely helps: https://en.wikipedia.org/wiki/Category:Matrix_theory https://en.wikipedia.org/wiki/Category:Matrix_theory > Matrix theory is a branch of mathematics which is focused on study of matrices. Initially, it was a sub-branch of linear algebra, but soon it grew to cover subjects related to graph theory, algebra, combinatorics and statistics as well. The University of Missouri has a Matrix Theory course: https://www.math.missouri.edu/class/matrix-theory https://www.math.missouri.edu/class/matrix-theory > Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs. ... which specifies a textbook: > Linear Algebra with Applications (7th edition) by Steven J. Leon ... which deepens my confusion. If you're taking that course, how is it not an introductory Linear Algebra course? And this MathOverflow answer obfuscates again: https://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra https://mathoverflow.net/questions/11669/what-is-the-differe... > Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences. > In linear algebra, however, you instead talk about linear transformations, which are not (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics. If I try to parse charitably, I come away with the idea that Matrix Theory is about matrices as a data structure, usable for many things outside the scope of Linear Algebra, where they're all about using matrices to represent linear transformations. It's the difference between a column of numbers on a shopping bill and a column of numbers which represents a vector in a space with a specified basis. Gotcha. However, this answer directly contradicts what the University of Missouri calls Matrix Theory, which is so Linear Algebra they even use a Linear Algebra textbook. It also... I don't know, trivializes the field of Matrix Theory. So you can manipulate matrices. So what? They show up a lot because they're used to represent specific things. Is the course going to barely introduce a lot of specific things and then focus on the matrix representation? What a waste!
graycat 8y ago
I'll try to clear things up: First, Horn and Johnson is a bit much. I was in Horn's class. I had done a LOT in, call it, linear algebra and matrix theory in my career before the class, told the profs I didn't need the course, and they said it was a "second, advanced course" and smiled. The course was quite competitive and without trying at all and without intending to be competitive, I effortlessly blew away all the other students on graded homework, the tests, the midterm, the final exam, and the corresponding qualifying exam. At the end of the course Horn wrote about me IIRC "Best performance in the class by a wide margin. Knows this material cold." So, yes, it was an advanced course, actually had a lot of nice stuff in it, Horn's lectures were nicely precise and at times with some unusual, nice approaches, but to do well in the course it was sufficient just to have had a good background before. What background? For the main books, E. Nering (a student of E. Artin at Princeton), Halmos (an assistant to von Neumann at the Institute of Advanced Study at Princeton), Finite Dimensional Vector Spaces, basically also a finite introduction to Hilbert space and the spectral theorem there, Forsythe and Moler, Computer Solutions of Linear Algebraic Systems, and some good texts in multivariate statistics with regression analysis, discriminate analysis, factor analysis, analysis of variance. More in applications, e.g., the fast Fourier transform, more on curve fitting, linear systems in electronic engineering, antenna theory and beam forming, optimization, linear programming, unconstrained optimization, the Markowitz and Sharpe applications to finance, Lagrange multipliers, the Kuhn-Tucker conditions, etc. can also help. But Horn is not a good choice for a first text. For a first or second text I'd suggest, say, Hoffman and Kunze, Linear Algebra, Second Edition available for free on the Internet. For more, see my post on math in https://news.ycombinator.com/item?id=15116379 https://news.ycombinator.com/item?id=15116379 and there sections (2) Linear Algebra (2.1) Linear Equations (2.2) Gauss Elimination (2.3) Vectors and Matrices (2.4) Vector Spaces (2.5) Eigen Values, Vectors (2.6) Texts To be brief, about the earliest and easiest start on linear algebra and matrix theory is just a high school style system of linear equations. The main solution technique is Gauss elimination. Matrix notation is a better notation for that subject. Here is essentially the role of matrix theory: Each of the old results in linear algebra can be written as a result, with nicer notation, in matrix theory. Can get the same results without matrix notation, but matrix notation makes it all much easier. Next, a broad statement is that the two pillars of the field of analysis in math are (1) linearity and (2) continuity. Well, linear algebra and matrix theory stands strongly on linearity and, as we move on in both the theory and applications, also continuity. Let's be clear on linearity via linear algebra and matrix theory: So, for positive integers m and n and an m x n matrix A we say that matrix A is a linear transformation (function) if for all n x 1 vectors x and y, and numbers a and b, we have that A(ax + by) = aAx + bAy Sure, to read this need the definitions of matrix sum and product; sum is trivial; product is not much harder and is really just what need to make Ax = b be the same as the high school system of linear equations. For the numbers, usually use either the set of real numbers R or the set of complex numbers C. But, sure, for numerical computation are essentially limited to the set of rational numbers Q. But in general need only what a course in abstract algebra calls a field: Each of R, C, and Q is such a field but also the set of integers modulo a prime number is a field, of interest in algebraic coding theory and cryptology. This definition of linearity generalizes in Hilbert space, Banach space, and functional analysis, and the more general definitions and results are important in quantum mechanics, differential equations in science and engineering, signal processing in electronic engineering, etc. Again, linearity is a pillar of analysis in math. Why pillars? In both theory and applications, linearity and continuity commonly hold and are astoundingly powerful properties. For such applications we have multivariate statistics, optimization, electronic engineering, antenna theory, beam forming, signals (each time invariant linear system has sines and cosines as eigenvectors; when a violinist on a concert stage plays some pure tones, the concert hall transmits those tones to you in the audience as a linear system so that what you hear are just the pure tones with the right frequencies but with some phase and amplitude changes; the Navy likes to know that for sonar signals; the USAF likes to know that for radar signals; cell phone people like to know that for their signals), Fourier theory, linear partial differential equations, superposition in quantum mechanics, etc. And when linearity does not hold, commonly it is a good, first approximation and the main means of iterative techniques. And if a problem is not linear, maybe after some simple transformation it will be. In some of the posts here, there is mention of matrix theory and basis, that is, a coordinate system. Well, can do that although is it not nearly as general as what physics likes to do with coordinate systems. But also can just decide not to do that, to take the vector space as just the n-tuples and not force thinking of the n-tuples as just coordinates of vectors in some basis. Or can do either approach depending on what is easier in the context. Here is a point should get: Suppose we start with just systems of linear equations. Then we say we are working with n-tuples of numbers. Then we call those n-tuples a vector space. Then using essentially just the main, relevant properties of those n-tuples, we write down the definition, axioms, of a vector space where we've said nothing about the vectors but have left them as just points. Well then we have two advantages: First, the definition of a vector space lets us talk about subspaces and in particular subspaces of the vector spaces of just the n-tuples, and we want to do that already, strongly with just Gauss elimination for linear equations. E.g., with the linear system Ax = b where m, n are positive integers, A is m x n, x is n x 1, and b is m x 1, the set of all x so that Ax = 0 (m x 1 of all zeros) is a vector subspace of all the n x 1 vectors (for the set of real numbers R, commonly called the set R^n). Call the set K the set of all x so that Ax = 0. If for some n x 1 u we have that Au = b, then from linearity we can argue that any v so that Av = b can be written as a sum of u and some vector in K. In this way we see all possible solutions of Ax = b. Actually at the end of Gauss elimination we can see K and u easily enough. Second, we get to consider vectors other than just n-tuples. E.g., we can consider the data of 1 second of music as a vector, a random variable as a vector, a color as a vector, the wave function of a photon or electron as a vector, etc. Then as the book continues, we get into eigenvalues and eigenvectors. Eigen is German essentially for special. They are special, and valuable. Maybe the nicest part is the polar decomposition: Each square matrix is a product UH where U in unitary and H is Hermitian. In class, when Horn got to that, I shouted out "That's my favorite theorem! The unitary part is an isometry, that is, doesn't change lengths or angles and is essentially a rigid motion, maybe just a rotation or reflection. The Hermitian part H is a shocking dream, amazing beyond belief: All H can do is take a circle and make it into an ellipse: The two axes of the ellipse are perpendicular (orthogonal) and the eigenvectors. Their lengths are the eigenvalues. And this generalizes to rounded footballs in three dimensions and all finite dimensions. And with the spectral theorem it generalizes to infinitely many dimensions and is the main reason in quantum mechanics the observables are eigenvalues. The polar decomposition is also the source of the powerful singular value decomposition, principle components analysis, factor analysis, analysis of saddle points in optimization (see W. Fleming, Functions of Several Variables), the matrix condition number in the numerical analysis of Gauss elimination, and much more in theory and applications. Hope this helps.
tomrod 8y ago
I love that book! It's certainly not an entry point though.

morpheuskafka 8y ago

> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too. I took calculus last year (AB Calc BC, 10th grade), and I can say that my experience was certainly a counterexample. I did OK, but it was definitely a marked difference from "breezing through" algebra.

nerdponx 8y ago
Likewise. I had an easy time in high school math. In college I really had to work for it.
tomrod 8y ago
I expect it was more of a teacher issue then. A solid math teacher is worth their weight in platinum.
- dTal 8y ago
  I had an excellent Calculus teacher in high school, and I did very well on the AP exam. It was still vastly more challenging than the trigonometry class that preceded it. Even today, I use trigonometry all the time almost without thinking, but I can't for the life of me remember how to long-divide polynomials (or picture a scenario where I'd need to). There's just so friggin much deeply abstract symbol manipulation in calculus class (which in school also covers essentially "advanced algebra"). It's a different ball game.
  - jacobolus 8y ago
    Long division of polynomials is literally exactly the same algorithm as long division of base-10 integers, except with no carrying.
Jach 8y ago
I got As in my pre-college math courses but I don't think I breezed through them. My schooling had no separate trig course, it went algebra, geometry, intermediate algebra, and calculus. Various pre- or honors versions. Since the BC exam covers quite a bit more material than the AB, my school teacher required those interested in it to take pre-calculus Honors first which was mostly about shoring up any gaps in previous studies while getting a foothold on Calculus AB material near the end. I think that extra prep made a huge difference in my performance, along with the teacher himself and an insistence on memorizing a lot of things to not be dependent on a rule sheet and getting an intuition for solving problems that even the TI-89 wouldn't help much with.

nextos 8y ago

The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations. For young students, a great introductory textbook is Calculus Made Easy. It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems. For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.

jefft255 8y ago
I don’t know about the rest of the world but in Quebec calculus is not taught with any epsilon-delta formalism. You don’t encounter that until mathematical analysis, which is in the curriculum of a very few majors in uni.
- throwawayjava 8y ago
  This differs throughout the USA. "Best" schools (<1%) will have what the article is describing (strong student mathematical problem solving communities lead by teachers with strong mathematical training). Typically very expensive, either directly through tuition or indirectly because they are public schools but only serve students in very high net worth zip codes. "Very Good" schools (<10%) will include some of the epsilon-delta formalism but might have some curricular problems and a "gifted just means moving through the terrible not-real-math curriculum faster" problem that the article talks about. "Decent" schools will teach how to compute derivative/integrals by rote and maybe talk a bit about the physical intuitions in a very hand-wavy way. Calculus is very much a continuation of Algebra or Trig where you learn some rules and how to pattern match and don't ask too many questions about why. The other half of USA high schools? They don't even offer a Calculus course of any kind [1]. Which... if you don't teach it at all, you can't teach it wrong... As an aside, AP CS has the same problem as the problem identified with AP Calc. AP CS is the epitome of a "Java School" course. [1] https://www.theatlantic.com/education/archive/2016/06/where-calculus-class-isnt-an-option/485987/ https://www.theatlantic.com/education/archive/2016/06/where-...

gibba999 8y ago

I don't buy it. AoPS is a great organization, but the focus is on pure, theoretical mathematics. Understanding calculus is key to understanding many beautiful areas of applied mathematics: image processing, signal processing, control systems, electronics, etc. I consider them more elegant than theoretical mathematics. Now, for that, you don't need all the messy manipulation (integration-by-parts and similar), but you do need the basics of area-under-the-curve, of derivative-as-slope, and similar, as well as some of the theory. But that's not too hard to learn. My own opinion is that the basics of calculus should be taught alongside the basics of algebra in elementary school. Plenty of people have had success doing both.

hyperpallium 8y ago

What does HN think of Khan Academy's treatmemt of calculus? https://www.khanacademy.org/math/calculus-all-old https://www.khanacademy.org/math/calculus-all-old Ideally, by an expert in calculus, who has done the whole KA course on it (though why an expert would do that, I don't know...)

HiroshiSan 8y ago
As someone who tried teaching themselves math with Khan Academy, I feel like Khan Academy suffers from the same problem most sites do, in that they tell you the information before you can discover it for yourself. This is different than how AoPS teaches math, where they ask you questions which then guide you towards self discovery, and as a result, you start asking your own questions which create paths for you to go on and explore.
- barry-cotter 8y ago
  You, and AoPS’ target audience of current and potential math nerds are very far from the norm. Inquiry based learning is actively harmful to learning for many students, and much less efficient for more or less everyone. > Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching > Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide "internal" guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described. https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_1?needAccess=true& https://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_...

WillPostForFood 8y ago

If ever you are by far the best, or the most interested, student in a classroom, then you should find another classroom. It is unfortunate that the prevailing educational trends are to get rid of tracks, lanes, and advanced classes, and dumping all the kids into mixed ability classes.

threatofrain 8y ago
I believe it's been roughly found that tracking helps smart kids moderately and hurts everyone else with the brain drain. It's a question of to what degree we are individuals, and to what degree we are a community. I think it's a question that cuts right down the middle of many people's value systems.
- Mvandenbergh 8y ago
  There's also the hybrid view: it's important to ensure that the most talented kids are supported properly because they are our future leaders, inventors, artists, etc.
- WillPostForFood 8y ago
  The data isn't very strong to support the idea that leaving advanced kids in the room with those that need more attention is of any help, and there is now growing evidence of achievement gaps widening, possible because it can be demoralizing for the slower kids, possible because the more advanced kids are now more likely to get outside supplementation which boosts them even further ahead.

hackermailman 8y ago

This article is primarily a marketing piece for their $500 courses, which I'm sure are good but if you don't have money or accessible local math student clubs and want to like the article says 'explore math' try these math foundations playlists for free https://www.youtube.com/user/njwildberger/playlists https://www.youtube.com/user/njwildberger/playlists Wildberger starts from the very beginning, proving laws of each ring/field with basic arithmetic. He also has an algebraic trig method that a primary school kid could do and an interesting discrete algebraic calculus method, plus plenty of abstract algebra content.

HiroshiSan 8y ago
Not sure how you got this impression, the concluding paragraph is "However, we are not the only other option. Other options students have are to become involved in extracurricular programs, such as math teams. Math contests should be selected with some care: those that encourage mass memorization or just test standard curricular tools tend to exacerbate the ills of the calculus trap rather than enhance problem-solving ability. Students can also pursue independent study if they are able to find mentors. University professors are occasionally willing to fill this role to some degree. There are also many summer programs and good books for extracurricular study, and some communities have developed grassroots programs to provide opportunities for eager students. These options are usually not as easy as “enroll in the next course,” but they will be far more rewarding than settling into the calculus trap." Also these articles are normally targeted to those already invested in the AoPS ecosystem, whether it be books, courses, or their forums.

IcePowder 8y ago

It wasn't till calculus did my love affair begin with Mathematics...,

sevensor 8y ago

I think this is a side-effect of the way our educational system is structured. We rely on big, standard tests to evaluate schools and their students. It would be too expensive, and somewhat subjective, to evaluate students' reasoning abilities, their conceptual knowledge. Can you imagine if these tests were evaluated by rooms full of people reading proofs, rather than by bubble-sheet scanners? So we test what we can test, which is procedural knowledge, and we teach that at the expense of deeper understanding. And because the best way of measuring procedural knowledge is to measure how many procedures you know, we race students forward to as much calculus as they can memorize.