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.
> "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?
> 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:
> 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:
> 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?
> 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 Universi...
I didn't say that Linear Algebra and Matrix Theory were separated. I said that Matrix theory hit me like a ton of bricks. I took Linear Algebra in college prior to that, obviously.
I further stated that I think Linear algebra might benefit students if taught earlier, in high school.
We were taught a little linear algebra in high school. And more in college.
My impression of three semesters of calculus in college was that much was a waste of time. It was probably useful for a mechanical/electrical engineer circa 1950. But today no one solves problems that way.
I think more linear algebra and matrix theory would have been better.
I think it will serve better to think about it as a difference of emphasis, rather than a difference of category. Maybe the term has been introduced to make the subject seem more tangible to newcomers? I notice that University of Missouri doesn't also have a "linear algebra" course.
As you say, "Matrix Theory is about matrices as a data structure" ends up being a pretty hollow concept anyway, because the important part is always related to matrix multiplication. The example of Markov chain models only underlines that point, since the main results of that theory depend entirely on the transition function being linear, and not at all on whether we represent it by a matrix. To put it another way: matrix multiplication is composition of linear operators. You can't extricate the matrix-as-data concept from the other.
Judging from the reviews it seems like it is a good reference book of intermediate/advanced linear algebra topics which researchers in other fields found useful as a reference.
I’m guessing this course was intended as maybe a 3rd course in linear algebra, with a slightly applied flavor. Giving it a different name makes it easier for students to distinguish the course than just calling it “Linear algebra 3A” or whatever.
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.
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 alg...
> 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.
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.
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.
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.
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.
"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.
>The other half of USA high schools? They don't even offer a Calculus course of any kind [1]
That's a bit misleading. There are many districts where one school serves as a magnet school where anyone interested in taking more advanced classes can go. So yes while it may be true that half of all schools don't offer calculus--far less than half of all students can't readily take calculus.
Not to mention that many schools allow dual-enrollment, where one can take courses at a local community college to make up for the dearth of courses at a high school.
As a high school student who took AP CS last year, I agree, but in reality the course isn't even about Java. Code is written by hand on the free response portion and Java syntax is mentioned but not really emphasized (things like missing brackets and semicolons are okay on the FRQs). I finished all the questions in about 25 minutes, with over an hour left to twiddle my thumbs.
Even questions about algorithmic analysis (in APCS's case, just sorting algorithms) are done without proper big-O notations, which is just plain stupid IMO.
For a student that has done CS on their own for years now, APCS feels like the fake, industrialized, watered-down version.
All of the AP tests are roughly comparable in my opinion. Just not too many high school kids are as overprepared for the tests in chemistry, physics, statistics, economics, etc.
I would be curious to hear your thoughts on the AP CS AB version's material, specifically whether you think it would have been of any use to you. They killed it some years ago, there's just the A now, so you'll have to search around for an old practice exam to look at. I took it in 2007 since I was ahead of the class enough to continue studying on my own. I don't remember exactly what was covered anymore but I do remember enjoying the studying and writing the exercise programs more, it was worth the lower score. I doubt the A has absorbed anything from it.
If you keep up your self-study, expect further disappointment in most undergrad data structures courses. =P
> 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
It's news to me that Stuyvesant considers zip code for admissions. I thought it was a standardized test open to all students in the system.
I suggest don't let the K-12 and college educational systems make calculus a mess.
Instead, if have some algebra and trigonometry from high school and want to learn calculus, then just get one or a few good freshman COLLEGE calculus books and work through them -- at each lesson or section, read and think about the material and then work all the more challenging exercises. Check answers in the back of the book, a copy of the Instructor's Guide, on the Internet, etc.
For the books, get mostly old ones known for decades to be good. Get just good, used copies -- the subject hasn't changed much in decades. For the books, DO get ones that are good on (A) the completeness property of the real numbers ("Calculus is the elementary consequences of the completeness property of the real numbers."), (B) limits, (C) the epsilon-delta definition of limits, (D) the epsilon-delta, limit definitions of the derivative and the (Riemann) integral, (E) applications. Get more than one such book, use the one that looks the best as your primary source and use the others for alternate explanations and more exercises.
That's what I did: I got a good book and worked through about half of it. Then for calculus in college, I asked to skip freshman calculus and start on sophomore calculus, the rest of the book. A prof gave me a little oral exam, define the derivative, with some TeX notation
and I was in. I did well, made As both semesters. Went on to advanced calculus, ordinary differential equations, advanced calculus for applications, real analysis, functional analysis, real applications, peer-reviewed publications, teaching calculus, etc. E.g., for an application I derived and used
y'(t) = k y(t) (b - y(t))
to please the BoD at FedEx, keep a crucial investor from leaving, and save FedEx from going out of business.
From what I've seen of high school materials for calculus, I'd advise trying hard to avoid them -- again, just start with one of the best college texts. The one I used for the actual course was Johnson and Kiokmeister, then also used at Harvard, now ancient but still fine.
No way high school or college level classes are teaching intro calc with delta-epsilon proofs, it just won't make any sense. It's generally introduced in Real Analysis, which is generally a sophomore level class.
Develop the intuition, then crystallize and formalize the idea with a proof. Otherwise, it's just not going to make any sense.
My last year high school math class had this approach, and it was a regular math course taken by every student.
Same thing at university, in a CS degree. Freshman Calculus taught at a level somewhere between Spivak and Rudin. In fact, both books where on the official course bibliography.
All this took place in a fairly big EU country, ~10 years ago.
I was taught the Epsilon-Delta definition of the limit in Calc 1, right off the bat in first year (University of Waterloo). Real Analysis here is a third year course run by the Pure Math department and from what I've heard it's one of the hardest undergrad courses the university offers, period. I'm nearing the end of my second year as a math student, finishing up calc 3 and linear algebra 2. When I take a look at a few of the questions on assignment 1 from a past real analysis course I have no idea how to do any of it, it's way out of my league.
Real analysis is not actually that hard, you do however have to learn a bunch of notation, concepts and processes before you can grasp things. The people who struggled with it at my course (different country, similar content I suspect) were, entirely unsurprisingly, the people who didn't attend all the lectures.
I don't think the difficulty is due to anything inherent to real analysis. I think it's due to the fact that most of the professors in the pure math department participate in the development of the math contests for CEMC [1]. Thus they tend to love creating extremely complicated and challenging homework problems.
If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble.
Huh? I learned differentiation with delta-epsilon proofs in the 11th grade. It was pretty standard at my public high school. I'm sure I didn't understand it very well at the time, but it was good to have seen it when I took real analysis.
Had to look it up to be sure, since I didn't recognize the name, but it looks like this refers to using limits to "invent" differentiation?
If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was)
I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.
The limit definition of differentiation is definitely intuitive, but epsilon-delta refers to the most common definition of what it means to take a limit. It's also simple, because when you say it in words it makes perfect sense, but it's not at all obvious why someone would define limits in such a way, or what the alternatives are.
Epsilon-delta is the formalization introduced by Cauchy, Bolzano and Weirstrass during the 19th Century.
It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s.
To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book.
Same here. That's how I learned it as well. That said, I didn't really "get it" until I was well into the class. I still remember that "aha moment" very vividly, even though it was in 1992 or so. I was sitting at a table in the school library working on a problem about a pool being emptied while it was raining and my brain said "Holy crap! This is just about how fast things are changing!". With all of the proofs, graphs, etc. things were never put into such a simple statement.
The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test.
EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses.
EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.
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.
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.
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.
Your link is to a very controversial polemic which IMO sets up a straw man, and then makes its own argument in a (both theoretically and empirically) questionable way, based on the authors’ pet “cognitive load” theory (which I personally think is bunk, but YMMV). It should be read in the context of its critics, and taken with a heap of salt.
I think Khan Academy should be thought of as a consistently average-quality US-high-school-style lecture, combined with US-style trivial exercises. It is a slow, unimaginative, pedantic curriculum.
But it has the advantages of being free, always available, and self-paced (in the sense that students can keep going through as much of it as they want without needing to wait, and can return to previous sections any time). I’m glad it exists, because it sets a quality floor; live teachers have variable quality, and while many are better than KA lectures, some are certainly worse.
If you can refer me to three relevant articles that critique or falsify Kirschner et al. I’ll be happy to read them. I’ve read Problem-Based Learning is Compatible with Human Cognitive Architecture: Commentary on Kirschner, Sweller, and Clark (2006) already.
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.
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.
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.
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.
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 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.
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.
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.
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[ 3.7 ms ] story [ 115 ms ] thread2014: https://news.ycombinator.com/item?id=7207495
2009: https://news.ycombinator.com/item?id=717982
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
> 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
> 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-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 Universi...
I further stated that I think Linear algebra might benefit students if taught earlier, in high school.
My impression of three semesters of calculus in college was that much was a waste of time. It was probably useful for a mechanical/electrical engineer circa 1950. But today no one solves problems that way.
I think more linear algebra and matrix theory would have been better.
As you say, "Matrix Theory is about matrices as a data structure" ends up being a pretty hollow concept anyway, because the important part is always related to matrix multiplication. The example of Markov chain models only underlines that point, since the main results of that theory depend entirely on the transition function being linear, and not at all on whether we represent it by a matrix. To put it another way: matrix multiplication is composition of linear operators. You can't extricate the matrix-as-data concept from the other.
You can see what the content was by looking at the textbook mentioned, https://amzn.com/0521548233
Judging from the reviews it seems like it is a good reference book of intermediate/advanced linear algebra topics which researchers in other fields found useful as a reference.
I’m guessing this course was intended as maybe a 3rd course in linear algebra, with a slightly applied flavor. Giving it a different name makes it easier for students to distinguish the course than just calling it “Linear algebra 3A” or whatever.
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
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 alg...
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.
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.
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.
"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-...
That's a bit misleading. There are many districts where one school serves as a magnet school where anyone interested in taking more advanced classes can go. So yes while it may be true that half of all schools don't offer calculus--far less than half of all students can't readily take calculus.
As a high school student who took AP CS last year, I agree, but in reality the course isn't even about Java. Code is written by hand on the free response portion and Java syntax is mentioned but not really emphasized (things like missing brackets and semicolons are okay on the FRQs). I finished all the questions in about 25 minutes, with over an hour left to twiddle my thumbs.
Even questions about algorithmic analysis (in APCS's case, just sorting algorithms) are done without proper big-O notations, which is just plain stupid IMO.
For a student that has done CS on their own for years now, APCS feels like the fake, industrialized, watered-down version.
All of the AP tests are roughly comparable in my opinion. Just not too many high school kids are as overprepared for the tests in chemistry, physics, statistics, economics, etc.
If you keep up your self-study, expect further disappointment in most undergrad data structures courses. =P
It's sad they killed it, it looks so fun.
[1]: http://mchs.virtualbeach.com/cs/documents/AP%20Computer%20Sc...
It's news to me that Stuyvesant considers zip code for admissions. I thought it was a standardized test open to all students in the system.
Which focuses on what might be called the calculus approach to mathematical modeling in science/engineering, and uses computer simulations.
Instead, if have some algebra and trigonometry from high school and want to learn calculus, then just get one or a few good freshman COLLEGE calculus books and work through them -- at each lesson or section, read and think about the material and then work all the more challenging exercises. Check answers in the back of the book, a copy of the Instructor's Guide, on the Internet, etc.
For the books, get mostly old ones known for decades to be good. Get just good, used copies -- the subject hasn't changed much in decades. For the books, DO get ones that are good on (A) the completeness property of the real numbers ("Calculus is the elementary consequences of the completeness property of the real numbers."), (B) limits, (C) the epsilon-delta definition of limits, (D) the epsilon-delta, limit definitions of the derivative and the (Riemann) integral, (E) applications. Get more than one such book, use the one that looks the best as your primary source and use the others for alternate explanations and more exercises.
That's what I did: I got a good book and worked through about half of it. Then for calculus in college, I asked to skip freshman calculus and start on sophomore calculus, the rest of the book. A prof gave me a little oral exam, define the derivative, with some TeX notation
f'(x) = d/dx f(x) = lim_{h --> 0} (f(x + h) - f(x))/h
and I was in. I did well, made As both semesters. Went on to advanced calculus, ordinary differential equations, advanced calculus for applications, real analysis, functional analysis, real applications, peer-reviewed publications, teaching calculus, etc. E.g., for an application I derived and used
y'(t) = k y(t) (b - y(t))
to please the BoD at FedEx, keep a crucial investor from leaving, and save FedEx from going out of business.
From what I've seen of high school materials for calculus, I'd advise trying hard to avoid them -- again, just start with one of the best college texts. The one I used for the actual course was Johnson and Kiokmeister, then also used at Harvard, now ancient but still fine.
Develop the intuition, then crystallize and formalize the idea with a proof. Otherwise, it's just not going to make any sense.
Same thing at university, in a CS degree. Freshman Calculus taught at a level somewhere between Spivak and Rudin. In fact, both books where on the official course bibliography.
All this took place in a fairly big EU country, ~10 years ago.
If you have time, click around on that page and check out some past Euclid (grade 12) math contests. Even if you've come through real analysis I bet some of the later problems in those Euclids will give you trouble.
[1] https://www.cemc.uwaterloo.ca/contests/contests.html
If so, yeah, we did it that way too, in 11th or 12th grade. (Had the same teacher both years, can't remember when exactly Calc was)
I remember it involving a lot of drawings of graphs so we could understand what each term referred to, and can't really imagine an easier way to learn it.
It is not the way calculus was invented by Newton and Leibniz. They thought in terms of infinitesimals. However, this approach is relatively hard to formalize. It was only done by Robinson in the 1960s.
To see why Cauchy et al had to work on epsilon-delta, take a look at [1], an excellent book.
[1] https://www.macalester.edu/aratra/
The end result of months of flailing followed by the "aha" was a grade of C for the class AND being the only one to get a 5 on the AP test.
EDIT: As a fun corollary, I really hated Calculus class in high school. Ironically, I ended up getting a Masters in EE, so I ended up spending the next several years doing Calculus in > 50% of my courses.
EDIT EDIT: Also fun to note was that Calculus was only offered to Seniors at the time and required 4 years of math beforehand: Algebra, Geometry, Advance Math, and Trigonometry. That meant you had to double-up on math in either Sophomore or Junior years, which basically limited class size to a small handful of people.
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.
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...)
> 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_...
To understand the problem with US-style mathematics pedagogy, I would recommend reading http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
For some advice and materials based on an alternative theoretical framework, let me recommend https://www.map.mathshell.org/trumath.php
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I think Khan Academy should be thought of as a consistently average-quality US-high-school-style lecture, combined with US-style trivial exercises. It is a slow, unimaginative, pedantic curriculum.
But it has the advantages of being free, always available, and self-paced (in the sense that students can keep going through as much of it as they want without needing to wait, and can return to previous sections any time). I’m glad it exists, because it sets a quality floor; live teachers have variable quality, and while many are better than KA lectures, some are certainly worse.
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.
I think it's a question that cuts right down the middle of many people's value systems.
Also these articles are normally targeted to those already invested in the AoPS ecosystem, whether it be books, courses, or their forums.