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(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus I

• davidwihl 12y ago
  I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification for its continuation.
  • weland 12y ago
    > I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Studying any kind of engine (or for that matter, anything that moves), solid-state electronic device (from transistors to antennae) or power distribution network and doing any kind of design about them is literally impossible without calculus. Even algebraic approaches to these are formulated through discretization of a continuous model (which -- unsurprisingly -- you need calculus to study). We have such a thing as databases, devices that can manipulate graphics and high-complexity economic models that are analyzed through linear algebra because Newton and Leibniz found the mathematical tools we need in order to model (continuous) change. We could mostly get along fine-ish without linear algebra, but the industrial revolution that spawned our society would be impossible without calculus, because without it you cannot design anything non-trivial that moves or runs on electricity.
  • cushychicken 12y ago
    Calculus has been part of the foundations of nearly all modern science since its inception in the late 17th century. Its genesis shaped the course of modern physics, engineering, chemistry, and architecture, and has consistently for the last several centuries. Linear algebra has really only come into its own since humans have had powerful computational systems at their disposal, as a large part of linear algebra's usefulness (including its relevance to the examples you describe) is efficiently organizing and performing calculations in higher order systems. That is not to say it is irrelevant or non-impactful - far from it! I'm just saying linear algebra has taken hundreds of years for its real power to be leveraged, and in the intervening time, calculus has reigned supreme as a system for modeling and describing real systems. (FWIW, I agree with the author and the paper in question, but I think OP you replied to is absolutely correct that teaching both gives perspective that neither alone would impart.)
  • d_e_solomon 12y ago
    FWIW, I have a math degree and work in tech. Both are incredibly important. I would say that linear algebra and discrete methods are more important for computing, but calculus has had a larger impact on science especially physics. Newtonian physics - a revolution in thinking and unification of mathematics and science - is intricately tied to calculus. Even in the example of economics that you cite, calculus is used heavily for the theoretical underpinnings for finding equilibria and deriving min/max.
    • FeymanFan78 12y ago
      I agree that Calculus is a more readily applicable mathematics that easily applied to many things. It really bothers me when people downplay the usefulness of Calculus also the person that posted the fundamental concepts of Linear Algebra are generally a little beyond the Grasp of people that haven't even been through the Rigors of Calculus 1 - Calculus 3.
  • cabinpark 12y ago
    >I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. I'm not. Consider all of physics. Physics is written down in terms of differential equations. You need calculus to get these equations. In terms of graphics, for example, are you magically creating only static images that never change in time? How do you think people came up with the algorithms to do things like refraction or reflection? These all involve calculus. Calculus is the most important tool in applying mathematics to the real world. It is how we derive the models for pretty much everything I can think of. Any time you want to model something where anything changes, that is calculus. Any time you want to find an optimal solutions, that is calculus. I can't think of a single area of applied mathematics that doesn't use calculus in some form. Even mathematics in general, calculus is used in some form pretty much everywhere in some form, except maybe in the foundations of mathematics like set theory or logic. Solving them, on the other hand, is a different problem. This is where linear algebra is important. When solving an ODE or PDE, when I discretise it, all I am doing is re-writing it as a giant linear algebra problem. Quantum mechanics, for example, is dedicated to finding the eigenvalues and eigenvectors of the Hamiltonian. Linear algebra provides the tools to solve the problems posed by calculus. But it isn't always required. In fact, if you look at the history of linear algebra, it only really became wide-spread when quantum mechanics was developing since there is a deep connection between linear algebra and quantum mechanics. Heisenberg had never heard of a matrix before despite discovering Heisenberg matrix mechanics. He was told by Max Born that what he had been doing is actually this thing mathematicians called "matrices". Now linear algebra is a required course for physicists (incidentally, linear algebra was the first lecture I ever attended at university). So I would argue that learning calculus is the most important thing someone should know mathematically, since it is the tool we use to build models. The second most is linear algebra since it provides us the tools to solve these problems. But in my mind, knowing how to derive these models is a lot more important than being able to solve them. That and linear algebra doesn't give you any insight into why we are solving the problem that way, it is just a tool to solve the problem. Also in Calculus 1,2,3 and so on, you learn increasingly more complicated techniques of calculus to solve more problems. But when you start numerically solving them, linear algebra doesn't care. It doesn't care if your equation came from the Einstein field equations or Newton's second law. It's all the same to it. So in a sense, I can teach you everything you'll need to know about linear algebra in a single course but the same is not true of calculus, which requires multiple courses.
• darkmighty 12y ago
  I've had linear algebra from the onset in my engineering school and it was wonderfully useful. Comparing which one is "better" (calculus or linear algebra) is like arguing whether multiplication or exponentiation is a better operation -- don't do it, you're going to sound cliché without reaching a conclusion. The main use of linear algebra to me is describe local behavior of functions/dynamical systems. Sure, calculus is the setting of functions in full generality, but understanding the algebra of local interactions is an essential part of the whole -- how can you go far into calculus without understanding derivatives? Linearization is practical the way to solve 99% of engineering problems.
  • omaranto 12y ago
    I'm unfamiliar with it: what is the cliché about multiplication vs exponentiation?
    • andrepd 12y ago
      The point is that that vs doesn't even make sense. It's like saying what wheel is your favorite in your car, the front left or the rear left. Arguing for calculus vs linear algebra is equally stupid. They are different branches of mathematics with different domains of application.
      • omaranto 12y ago
        This I understand, but that sentence with "cliché" in it I still don't get.
        • jmilloy 12y ago
          It's pretty clear to me that the OP just used the word cliche where it didn't belong. Fortunately the idea still came across. Perhaps you were truly confused, but it sounds more like you're playing dumb to force the OP to admit the mistake.
          • 12y ago
• Shorel 12y ago
  > Another thing to keep in mind is calculus has had much more impact on history/society than linear algebra. Yes, it was amazingly important during all the wars and conflicts of the last century. I don't think it was for the memory of Newton that they choose Calculus, but because it was more useful in a war machine than the alternatives.
• dataker 12y ago
  I've had 4 semesters of Calculus in my CS program. For me, it only helped to scare many students and made them pursue another degree. We only had more CS classes than others junior year. Fortunately, I'd been involved in open source and knew that wasn't the reality.
• mturmon 12y ago
  You sound like exactly the constituency Gil Strang's corrective essay is addressed to. I don't see any of the points you mention as particularly strong. There would be nothing wrong with presenting a more hands-on linear algebra course as a first course for freshmen. (Calling it "matrix arithmetic" does not do the notion service.) The material seems very immediate, and you can use a REPL like octave or matlab to reinforce concepts. You could deepen it as you wished by introducing more abstract concepts and by doing proofs, but I don't think that's necessary. I learned a lot from Golub and van Loan, for instance. Second, the notion that calculus has had more impact on history/society than linear algebra is very hard to defend. Think of all the numerical modeling that has been done using matrices. Think of all the large-scale computations/optimizations that are done first by finding a linear system, or approximating one. I'm not saying there is a clear answer to this question, just that it can't be the foundation of an argument, because it's so undecidable. Another thing that makes it undecidable is cases where you need both (e.g., weather forecasting). And saying it would be hard to teach the subject in a uniform way -- really, this is a very bureaucratic objection.
  • xamuel 12y ago
    >a more hands-on linear algebra course as a first course It is ideal (and indeed, not uncommon) to have linear algebra and calc 1 in parallel --- for special honors classes. It wouldn't be realistic to do it for the general freshman class. Do you realize just how many students that includes, in a big state U.? "a REPL like octave or matlab" would be wonderful, amazing, if it weren't about as realistic as the space elevator. Do you know how many endless headaches there are with software-augmented courses, even when it's just software you run in the browser? I applaud your ambitiousness but let's just say I don't want to be in your shoes when you've just told a thousand 18-year-olds to install Matlab! >the notion that calculus has had more impact on history/society Linear algebra is hard to even put your hands on, historically. There was no Isaac Newton figure putting the finishing touches on an amazing breakthrough "Fundamental Theorem of Linear Algebra". It is the book-keeping field of mathematics. "All the numerical modeling that has been done using matrices" almost entirely arises by discretizing differential equations so linear algebra doesn't win any points there.
• pmalynin 12y ago
  I must disagree. In my experience Linear Algebra has been so much easier than Calculus because even though at times it can be very abstract it is still more familiar as most of the time you're just solving equations. My math professors have done quite a good job in connecting the two subjects. For example, differentiation and integration of polynomials can be expressed as a linear algebra problem, etc. etc. "The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance." Linear Algebra has the Fundamental Theorem of Algebra (i.e. every polynomial has a solution in the complex numbers) and also the Isomorphism Theorems that say all vector spaces of n-dimensions are the same. For example polynomials of the form c+bx are equivalent to the complex numbers (over the reals) which are equivalent to two-vectors (x,y). Quantum physics is basically linear algebra, with quantum states defined as vectors in a quantum vector-space. Don't dismiss the impact of Linear Algebra.
  • mafribe 12y ago
    In what sense is the fundamental theory of algebra (FTA) a theorem of linear algebra? It's rarely proven in linear algebra courses. Most proofs of the FTA involve concepts from the analytical side of mathematics.
• RogerL 12y ago
  "Consider a spherical cow in a vacuum on a frictionless plane..." That joke comes from the analytic calculus tradition of physics. I took a lot of physics while an undergrad, and I came out unable to do any useful physics. This is because as soon as you try even very simple problems the partial differential equations that result have either no, or extremely difficult to find solutions, and it is usually the former. I spent years learning analytical tricks that I just don't apply. There has been a lot of push back on this in education recently. There are Giordano's and Sherwood&Chabay's textbooks in physics now which emphasize computation over analytics. The reality is that if you want to do the vast majority of work in physics you will be doing things like reducing your problem to Ax=b, and throwing the linear algebra machinery against it. I type this while taking a break from Strang's "Linear Algebra, Geodesy, and GPS" textbook, which I need for work right now. There is an endless need for this sort of math. Sure, it is a matter of balance. But I flip through my diff eq books and I don't see a lot in there that I actually need. There is nothing in Strang that I don't know, and ditto for a good numerical methods book. Even the work that Gauss did with orbits did was linear algebra and computation. He did not analytically derive the motion of Ceres, for example, but used linear algebra to compute it. I, along with many other posters on this thread, and everyone that I have talked to, wish that there STEM education had been much more about Ax=b, the only problem we really solve, and a lot less about using huge tables of analytic integrations to try to find answers to equations that we will never encounter in the real world because they are so specially crafted as to have an analytic solution.
  • pm90 12y ago
    I agree with your point that a lot of current STEM education focuses on things that don't really have any practical value. I think its more of a tradition for engineering students to be learning about this stuff, a legacy of the time when knowledge was not available at your fingertips. There are a lot of programs that are experimenting with different approaches though, and I think this will only get better with time.
• eli_gottlieb 12y ago
  > They have enough trouble understanding functions in Calculus I: linear algebra's "functions are basically matrix multiplication" feature would NOT help. What do they have trouble understanding about functions? Actually, it took me until just a few minutes ago to realize that one of the major differences between my CS-centric way of thinking and the proper mathematician's way of thinking: mathematicians treat all coextensive representations of the same object as identical, while a computer scientist tends to treat only provable intensional equality as actual identity.