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Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

308 points by capocannoniere 9y ago

Chris2048 9y ago
Previous Discussion: https://news.ycombinator.com/item?id=11207183 https://news.ycombinator.com/item?id=11207183
stephengillie 9y ago
Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.
- jacquesm 9y ago
  I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
  - Nomentatus 9y ago
    Especially that darn three-body problem. (Not poking fun, just agreeing, it's genuinely tough.)
  - rocqua 9y ago
    I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
    - conistonwater 9y ago
      Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a solved problem, differentiating even a reasonably smooth function numerically is much harder. If there is noise in your function, it doesn't matter so much for integrating, but can completely break the derivative.
      - gaurav_v 9y ago
        More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!
        deleted 9y ago
        [deleted]
        srean 9y ago
        Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
        gaurav_v 9y ago
        "Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that." The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer. Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to have a theory of integration is far richer than for differentiation - i.e, proving the multivariate change of variable theorem.
        srean 9y ago
        Thanks for clarifying what you meant by 'analytic'. To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
      - eximius 9y ago
        Only numerically? Analytically, differentiation is much easier and integrals are tough or impossible, unless I'm missing something.
  - stephengillie 9y ago
    I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone. Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
    - jacobolus 9y ago
      Byzantine is a good guess: Hipparchus was from a city about 50 miles away (Nicaea), though I think he did most of his work in Rhodes, and lived a few hundred years before the time of the Eastern Roman Empire. ;) But anyway, the etymology helps: “sine” = medieval Latin translation of an Arabic corruption of a word originally from India and meaning “half a bowstring”. “Tangent” = touching. “Secant” = cutting. “Chord” = bowstring. See https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā The reason sine and cosine are each-others derivative is that if you start with uniform circular motion and take the vector derivative, you get another uniform circular motion in velocity space.
    - trentmb 9y ago
      The relationship between sine and cosine is apparent just by looking at the unit circle.
- uoaei 9y ago
  Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1] This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children can handle the rigor. For now, let them play with it. That makes math a lot more fun and less draining, esoteric, impenetrable. [1] https://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ https://www.theatlantic.com/education/archive/2014/03/5-year...
  - nxc18 9y ago
    This makes total sense. I remember learning the ideas of calculus for the first time and thinking "wow, that makes sense" and also realizing how incomplete my picture of the world was without that coherent thought framework. And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome). "Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."
  - dotancohen 9y ago
    My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on paper, pen in _her_ hand.
    - g10r 9y ago
      why? The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.
- WhitneyLand 9y ago
  Let’s take that up a notch - try both and actually measure the results. It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact. Why not measure multiple aspects? Efficiency of learning, motivation, inspiration, relevance... Maybe it’s being done and I don’t see it. Maybe it’s not being done, because companies will pay six figures to have people A/B test a different button location on a web site, but the business case for optimizing learning curriculum at traditional institutions is piss poor.
  - jessaustin 9y ago
    Would the typical ed-school IRB have a problem with this sort of experiment? Admittedly that is an indictment more of the IRB than of the experiment...
  - avs733 9y ago
    >It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, solutions...they don't want permutations. Also, measure success as what? Learning? at a conceptual level or an execution level? Do you want a standardized test? Do you want to train teachers to effectively measure these concepts? Do the teachers really understand these concepts? Educational systems are incredibly difficult and complex. Blackboxing them is not easy.You nailed the business case...we can't agree on who will pay for this, can't agree on how to measure it, and can't even agree on what should be taught. One experiment in this way was gasp the evil Common Core curriculum. As an education researcher myself, I wouldn't fault anyone who works in K-12 policy and development to just phone it in and spend the days day drinking. They are underpaid, poorly treated/respected, and everyone thinks that they are equally qualified as those experts to have an opinion (e.g., parts of this thread) because they experienced education themselves. The question I always ask people when they propose really concrete fixes to educational issues analogized to their personal technical field of expertise is this...Can you define and support from research your definition of what learning is?
- analog31 9y ago
  While we're talking about turning school math on its head... I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. Learning about vectors in physics finally got me comfortable with trig. Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation. Amusingly, with complex numbers, proving trig identities becomes a trivial algebra problem.
  - GregBuchholz 9y ago
    >I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. What about geometric algebra instead? http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
  - jacobolus 9y ago
    To elaborate, “angle measure” (i.e. circular arclength corresponding to a particular rotation) is a derived quantity, not the primary abstraction we should be thinking about. The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm. Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1. This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.
- CalChris 9y ago
  Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems. Maybe Sheldon Axler will do Calculus Done Right. I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
  - stephengillie 9y ago
    I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
    - scentoni 9y ago
      https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Clifford_algebra https://en.wikipedia.org/wiki/Clifford_algebra
      - jacobolus 9y ago
        Start here http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
    - mtzet 9y ago
      > We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space? The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra. The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics. The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse. It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.
- yermom 9y ago
  Calculus was a lot easier if you paid attention in trigonometry, for me anyway Of course one of the main texts i learned diff eq from was the Mary Boas book i think she mentioned in that rant, so what do i know? :)
- Myrmornis 9y ago
  Certainly integration of trigonometric functions, requiring trigonometric identities, seem to get more time allotted to them than the pedagogical value they provide AFAICS, but I'd be interested to hear someone pointing out what I'm missing there.
  - sannee 9y ago
    Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
    - jacobolus 9y ago
      Trigonometry† is still super important for all of physics and a great deal of higher mathematics. The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form. Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry. The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though. †: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.
- swiley 9y ago
  This right here! Calculus (the first semester anyway) is really not hard to understand at all. Things like limits also demonstrate why math is done the way it is which makes it much more enjoyable, but most people get hung up with trig and don't see any of this. Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.
- BeetleB 9y ago
  I suppose what you say makes sense for the general population. As a scientist/engineer who used trigonometry a lot, it was extremely useful to have a dedicated course on trigonometry early on. The reality is many will need to use basic trigonometry before they need to use calculus (e.g. in physics).
  - abecedarius 9y ago
    OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Has anyone tried teaching this way in high school or middle school?
    - BeetleB 9y ago
      >OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Let's not forget the geometric aspect of trigonometry, which will be more intuitive to many than complex numbers. I have a force of 10N applied at an angle of 32 degrees. I need the x-component. You still need basic trigonometry.
      - abecedarius 9y ago
        Yes, you need to tie the algebra to the geometry. I had a friend taking a trig class, and when I showed him how all those trig identities just fall out of multiplying complex numbers, he was pissed.
- kafkaesq 9y ago
  Most students will benefit from already understanding calculus, when they are learning trigonometry. The argument cuts both ways: not only can one strongly benefit from a basic understanding of trigonometry when learning calculus (how the derivatives of sin/cos/etc all related to one another) - it really isn't possible to get far with basic integration techniques without learning the fundamental trig-based substitutions. And ultimately you can use any set of subjects, in just about any order. What matters is the thought processes, and that what you're really trying to teach is not a set of facts or techniques - but the underlying mathematical essence of these techniques. So at the end of the day, what it really comes to is: "It doesn't matter so much what you teach, but how you teach".
inbox 9y ago
I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I just sensed that something was off :)
- jessaustin 9y ago
  Your high school didn't teach geometry?
  - adamnemecek 9y ago
    Geometry proofs aren't real proofs.
    - yequalsx 9y ago
      Why do you believe this? I think they are. They provide a nice example of using logic and axioms. I don't think one will be able to have a proof based course other than geometry before calculus. Students just aren't mathematically mature enough for that.
      - eximius 9y ago
        It very much depends how it is taught. When I took it, it was very focused on using only the building blocks you were given. Step one: apply theorem 1.2b; step two: apply theorem 1.4a; etc. It was barely more than a search through the space of operations given. The statements proven felt trivial and the proofs needlessly convoluted and rigid. Whereas in calculus and algebra and analysis and number theory, the proofs often had different paths to prove them or different constructions/descriptions and the things we proved felt substantial.
    - jessaustin 9y ago
      By that measure, most proofs seen in early undergraduate years are also not real (e.g. the proofs given for the fundamental theorem of calculus). The impression one gets from GP is that they needed to see some proofs in high school. If they had taken geometry with me in Mr. Schardt's class they would have gotten their fill...
jessaustin 9y ago
Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science. Vested interests dominate every nook and cranny of our society, even the society of mathematicians. Truly we live in a decadent age. With this much fuel piled up, who will be surprised by the conflagration?
- mcnamaratw 9y ago
  Haha!
tempw 9y ago
It's sad one needs to reinforce the teaching of concepts rather tricks. Even worse is when you have all the proof based and concept oriented course and are tested on trickeries on exams.
- analog31 9y ago
  I believe this is a huge shortcoming of how math is taught. You can bet your last dollar that the teacher doesn't think it's about tricks. But the students are convinced that it is. Students and teachers are both exposed to the exact same material but end up with diametrically opposing conclusions. Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years. The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades. The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works. Disclosure: I taught college freshman math for one semester.
- patta54 9y ago
  My high school calculus teacher was really great at this. He didn't just teach a bunch of transformations to memorize. That was part of it, naturally, as you aren't going to use the definition of limits and the FToC in all your problem solutions. He actually made us construct volumes from pieces of poster board, measure the segments and calculate the Riemann sum. When we did function analysis, he didn't allow us to use the Cartesian plane at first. We had to show visually how a function deformed the one-dimensional real line. How x^2 squished values between -1 and 1 toward 0 and stretched the other values toward +infinity. It gave me a good "visual" grasp of the concepts and made most of my higher math classes much easier. I do agree diff eq instruction sucks. I got an A in that class and didn't understand a thing. "This equation has this form; this is the canned solution."
ben_jones 9y ago
My differential equations class was the only course I ever took with over 30% of the final grade being derived from homework assignments. Unsubstantiated - but my peers and I all believed this was because everyone was failing and dropping the course.
hprotagonist 9y ago
you can fake knowledge of undergraduate differential equations if you know about 3 hours worth of linear algebra. More profound understanding is difficult but not terribly impossible. The best diffeq text i've found is by Blanchard, DeVaney and Hall, and it remains the only math textbook i've ever been able to read.
burnerOh2125 9y ago
Interesting, I'm a phd-drop out in computational biology, working as a data science consultant: I use mathematics including multi-dimensional statistics, linear algebra, and calculus everyday. Being self-taught, I'm very self conscience about the math I don't know, but so far, not knowing differential equations doesn't seem to have hurt me. I actually just ran into a problem that uses Hamiltonian dynamics, so maybe I will end up learning differential equations, but it does seem like the course, as taught many places and in the Dover books I own, presents either no useful conceptual insight, like why I learned geometric algebra, or a powerful toolset, like some multi-dimensional statistics.
- j7ake 9y ago
  So how do you reason with abundance of molecules? mRNA and protein can be modelled from simple linear differential equation
- codewithcheese 9y ago
  Hi I am interesting your advice on which math to learn if you have a spare moment to provide it. I too am starting to teach my self the requirements of data science and am also self conscious of the math I don't know. I am very excited about what is now possible with machine learning and deep learning as I believe it will become increasingly necessary for developers to stay relevant. Could you comment in more detail on which mathematical skill you have found useful as data scientist? Which resources did you use to teach yourself? Very appreciative of any help.
  - ragulpr 9y ago
    * Solid linear algebra * Probability theory * Statistics (bayesian if possible) * Optimization (this is less important but extremely useful) If you know the mechanics of multivariate calculus you'll be fine learning the above. The course that personally have had most payoff was functional analysis. Purely theoretical course that will give you no practical skills and at first glance seems unrelated to ML but it (subtly) gave me a much deeper understanding of what ML is all about.
- srean 9y ago
gumby 9y ago
Due to an offhand comment from a friend, I started my kid on continuous math, not discrete like they do in school. Went great, until about grade 4 when they actually started to care whether he was aligned with his classmates. Germans are actually pretty intolerant of non-professional pedagogues teaching kids. But it greatly improved his grasp of elementary math as it matched what he saw in the real world.
madengr 9y ago
Regarding item 10 in that list, I learned plenty of Laplace transforms, partial fraction expansions, stability, phase planes, etc. It's the core of control systems theory. It was just not taught in differential equations class. The DE class was a bag of useless tricks. All the other EE classes were very useful tricks of how not to directly solving DEs. Going to drink a beer now for Oliver Heaviside. The invention of the Laplace transform is one of the greatest contributions to engineering.
mathattack 9y ago
Apparently he passed 2 years after this was written. RIP. A great mind. https://en.wikipedia.org/wiki/Gian-Carlo_Rota https://en.wikipedia.org/wiki/Gian-Carlo_Rota
- GeorgeTirebiter 9y ago
  I clearly remember his class on, basically, Mathematics for Philosophers. As an engineering student, I thought my pal from Urban Studies who suggested taking it was a way to pad one's GPA, which wasn't my style. Nonetheless, I agreed, and, now, I regard it as one of the most important courses I've ever taken. I have never met anyone with that level of enthusiasm, wonder, and love of Mathematics - who also knew how to partially impart it to his students! Mainly, it's the Wonder of it all that remains with me. What a privilege.
woopwoop 9y ago
I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. In order to solve a general system of constant coefficient linear odes, you have to take a matrix and compute its Jordan canonical form. The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. Then, in another question, they were presented a 5x5 matrix, told that its only eigenvalues were 2 and -2, and asked if it was invertable. I don't think anyone gave a reasonable answer. What is the point of that? Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the entirety of a book review on contemporary philosophers: "When pygmies cast such long shadows, it must be very late in the day."
- gh02t 9y ago
  > These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. Yep, Diff Eq 1 (ODEs) and Diff Eq 2 (PDEs) for us as undergrads were exercises in memorization, even as a math major. I didn't really understand them until after I took a (well, several) linear algebra classes, after which it clicked that it was basically all the same theory.
- eli_gottlieb 9y ago
  You don't make students take Linear Algebra before differential equations? I thought students were supposed to take Lin-Alg alongside their Calculus 2 or Multivariable Calculus course, then have all that in their heads for DiffEqu.
  - 9y ago
abhirag 9y ago
I have realized that Difference Equations i.e. the discrete variant of Differential Equations are much more common in Computer Science/Data Science(recurrence relations etc.) and it would have been great if they were atleast given half as much attention as Differential Equations get in universities.
colinator 9y ago
My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative ability outweighs understanding. Sounds familiar.
falcor84 9y ago
I'm not sure how well it would fit as a first course, but the thing that really helped me understand what's behind differential equations is Steven Strogatz's wonderful book "Nonlinear Dynamics And Chaos"[0]. [0] https://www.amazon.com/gp/aw/d/0813349109/ https://www.amazon.com/gp/aw/d/0813349109/
fmap 9y ago
And 20 years later, this essay is still as relevant as the day it was written. I agree with pretty much everything that's in the essay, except for a few small points. > There is nothing wrong with keeping the functional notation for density functions – as physicists and engineers always did – as long as one bears in mind that density functions cannot be evaluated, but only integrated. This always bothered me, since, as noted in the very next section, distributions don't have an analogue to pointwise multiplication. Even worse, there is a perfectly servicable notation for such "dual functions/vectors" that physicists have been using throughout the second half of the 20th century. We could just use a consistent notation and not confuse new students, but no. "It's always been done this way" is a terrible argument and leads us to the confusing mess of notations that people still use for integrals and integral transforms... --- Apart from that I would teach people recurrence equations/stream calculus before going into the limiting case of differential equations. It's true that differential equations are sometimes easier to handle analytically, but this is neither relevant (as the article notes) nor a great point in their favor, since we just end up teaching students a bag of tricks instead of explaining why something works...
johan_larson 9y ago
When engineers and scientists have to get some real work done with DEs, what do they do? Is it all solved numerically these days?
- eiji 9y ago
  Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications. High and low electro magnetics (Maxwell), Navir-Stokes, ...
  - thanatropism 9y ago
    Engineers should be taught about actual numerical algorithms and their qualitative properties. Cases in which there exist explicit symplectic algorithms, etc.
    - rsfern 9y ago
      Most engineering programs do have at least one full semester of numerical methods, even if in practice most engineers use matlab, scipy, abaqus etc...
  - chillingeffect 9y ago
    What tools are used for parameter estimation? E.g. I made a model that describes my system behavior, then collected a lot of data and next need to find the constants in the model that make the best fit. Thank you.
    - eiji 9y ago
      The search terms there would be "Model Calibration" and/or "System Identification" Software. I think those are again very domain specific. But there are a few. PS: While AI and machine learning are catching on in many different fields, the software packages in those domains make heavy use of fairly complicated fitting and approximations methods already. Lots or correlation and effect estimation, which builds many of the underlying guts of machine learning.
eli_gottlieb 9y ago
>A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Well that's depressing. If not undergraduate courses, where do people expect graduate students to come from?
katastic 9y ago
I took two courses in ODE. I got an A in the first semester. I still have no idea what they are or what's going on. It's all just relationships and patterns to me--but with no intuition or understanding behind them. It never "sunk" in like Calculus did where "Area under the curve" and "tangent line" are super obvious in retrospect and immediately applicable to your daily life. "What's velocity? The change in distance over a unit of time." Done. Presto.

Chris2048 9y ago

Previous Discussion: https://news.ycombinator.com/item?id=11207183 https://news.ycombinator.com/item?id=11207183

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

jacquesm 9y ago
I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.
- Nomentatus 9y ago
  Especially that darn three-body problem. (Not poking fun, just agreeing, it's genuinely tough.)
- rocqua 9y ago
  I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.
  - conistonwater 9y ago
    Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a solved problem, differentiating even a reasonably smooth function numerically is much harder. If there is noise in your function, it doesn't matter so much for integrating, but can completely break the derivative.
    - gaurav_v 9y ago
      More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!
      - deleted 9y ago
        [deleted]
      - srean 9y ago
        Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that. Note the key word there is a function not a function with a closed form that's a tiny subset.
        gaurav_v 9y ago
        "Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that." The OP said the opposite, that differentiation is harder 'more finicky.' I agree that the concept of integration is much richer. Also, I didn't mean 'closed form solution' when I said 'analytic.' I also didn't mean 'analytic functions.' I meant that the analytic machinery you have to develop in order to have a theory of integration is far richer than for differentiation - i.e, proving the multivariate change of variable theorem.
        srean 9y ago
        Thanks for clarifying what you meant by 'analytic'. To me, 'harder, more finicky' means exactly that it is of a more constrained scope, so I don't think I interpreted OP wrong.
    - eximius 9y ago
      Only numerically? Analytically, differentiation is much easier and integrals are tough or impossible, unless I'm missing something.
- stephengillie 9y ago
  I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone. Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.
  - jacobolus 9y ago
    Byzantine is a good guess: Hipparchus was from a city about 50 miles away (Nicaea), though I think he did most of his work in Rhodes, and lived a few hundred years before the time of the Eastern Roman Empire. ;) But anyway, the etymology helps: “sine” = medieval Latin translation of an Arabic corruption of a word originally from India and meaning “half a bowstring”. “Tangent” = touching. “Secant” = cutting. “Chord” = bowstring. See https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā The reason sine and cosine are each-others derivative is that if you start with uniform circular motion and take the vector derivative, you get another uniform circular motion in velocity space.
  - trentmb 9y ago
    The relationship between sine and cosine is apparent just by looking at the unit circle.
uoaei 9y ago
Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1] This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children can handle the rigor. For now, let them play with it. That makes math a lot more fun and less draining, esoteric, impenetrable. [1] https://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ https://www.theatlantic.com/education/archive/2014/03/5-year...
- nxc18 9y ago
  This makes total sense. I remember learning the ideas of calculus for the first time and thinking "wow, that makes sense" and also realizing how incomplete my picture of the world was without that coherent thought framework. And I love that article. She really captures the damage that my early math education did (which I've been working the last year to overcome). "Unfortunately a lot of what little children are offered is simple but hard—primitive ideas that are hard for humans to implement,” because they readily tax the limits of working memory, attention, precision and other cognitive functions. Examples of activities that fall into the “simple but hard” quadrant: Building a trench with a spoon... or memorizing multiplication tables as individual facts rather than patterns."
- dotancohen 9y ago
  My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on paper, pen in _her_ hand.
  - g10r 9y ago
    why? The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.
WhitneyLand 9y ago
Let’s take that up a notch - try both and actually measure the results. It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact. Why not measure multiple aspects? Efficiency of learning, motivation, inspiration, relevance... Maybe it’s being done and I don’t see it. Maybe it’s not being done, because companies will pay six figures to have people A/B test a different button location on a web site, but the business case for optimizing learning curriculum at traditional institutions is piss poor.
- jessaustin 9y ago
  Would the typical ed-school IRB have a problem with this sort of experiment? Admittedly that is an indictment more of the IRB than of the experiment...
- avs733 9y ago
  >It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, solutions...they don't want permutations. Also, measure success as what? Learning? at a conceptual level or an execution level? Do you want a standardized test? Do you want to train teachers to effectively measure these concepts? Do the teachers really understand these concepts? Educational systems are incredibly difficult and complex. Blackboxing them is not easy.You nailed the business case...we can't agree on who will pay for this, can't agree on how to measure it, and can't even agree on what should be taught. One experiment in this way was gasp the evil Common Core curriculum. As an education researcher myself, I wouldn't fault anyone who works in K-12 policy and development to just phone it in and spend the days day drinking. They are underpaid, poorly treated/respected, and everyone thinks that they are equally qualified as those experts to have an opinion (e.g., parts of this thread) because they experienced education themselves. The question I always ask people when they propose really concrete fixes to educational issues analogized to their personal technical field of expertise is this...Can you define and support from research your definition of what learning is?
analog31 9y ago
While we're talking about turning school math on its head... I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. Learning about vectors in physics finally got me comfortable with trig. Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation. Amusingly, with complex numbers, proving trig identities becomes a trivial algebra problem.
- GregBuchholz 9y ago
  >I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products. What about geometric algebra instead? http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
- jacobolus 9y ago
  To elaborate, “angle measure” (i.e. circular arclength corresponding to a particular rotation) is a derived quantity, not the primary abstraction we should be thinking about. The way I think about it is that angle measure is the logarithm of a rotation, with the information about the orientation of the plane of rotation stripped out. Composition of rotations is an inherently multiplicative kind of structure, and for something a computer can understand the best representation is usually a unit-magnitude complex number. We can treat it additively by taking the logarithm, in precisely the same way we can treat scaling additively by taking the logarithm. Symbolically, iθ = log(z), where z = x + yi is a complex number with x^2 + y^2 = 1. This tool is very useful if you want to e.g. smoothly interpolate between rotations, but often dramatic overkill. For many problems it’s better to deal with rotations in pure vector terms, and never bother with angle measure whatsoever.
CalChris 9y ago
Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick (EE 120 Linear Systems) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to linear problems. Maybe Sheldon Axler will do Calculus Done Right. I agree that calculus sans trig is a good idea if you aren't going to use complex numbers.
- stephengillie 9y ago
  I feel like we're still only scratching the surface with complex numbers. I didn't learn Euler's Identity nor most of the interesting parts of i until recently on Youtube. We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space?
  - scentoni 9y ago
    https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Clifford_algebra https://en.wikipedia.org/wiki/Clifford_algebra
    - jacobolus 9y ago
      Start here http://www.shapeoperator.com/2016/12/12/sunset-geometry/ http://www.shapeoperator.com/2016/12/12/sunset-geometry/
  - mtzet 9y ago
    > We can use complex numbers to describe a 2-dimensional number-space with a single digit. How do we describe a 3-dimensional number-space with a single digit? What about higher dimensional number-space? The complex numbers equips the vector space of 2-dimensional real numbers with a multiplication. This structure is known as an algebra. The analogue for 4 dimensions is the Quaternion Algebra, which is no longer commutative (i.e. a * b != b * a). The elements of unit length correspond (essentially) to rotations in 3 dimensions and they are used for this in e.g. computer graphics. The analogue for 8 dimensions is the Octonions which are no longer associative (i.e. (a * b) * c != a * (b * c)). The construction can be continued with a doubling of the dimension through the Cayley-Dickson construction, however these higher dimensional versions are even worse. It is not possible to associate a multiplication to 3-dimensional real space which makes it an algebra. This fact is related to the classification of exceptional Lie groups. The reason is roughly that the units in an algebra form a group, but the 2-sphere is not a Lie group.
yermom 9y ago
Calculus was a lot easier if you paid attention in trigonometry, for me anyway Of course one of the main texts i learned diff eq from was the Mary Boas book i think she mentioned in that rant, so what do i know? :)
Myrmornis 9y ago
Certainly integration of trigonometric functions, requiring trigonometric identities, seem to get more time allotted to them than the pedagogical value they provide AFAICS, but I'd be interested to hear someone pointing out what I'm missing there.
- sannee 9y ago
  Trigonometry was important (presumably much more than calculus) for navigation/military applications at some point in history, so maybe it just stuck that way?
  - jacobolus 9y ago
    Trigonometry† is still super important for all of physics and a great deal of higher mathematics. The problem with it is that before the past 150–200 years, people didn’t have an adequate collectionn of mathematical concepts / tools / number-like objects to work with, so the description is done in an extraordinarily unnatural and cumbersome form. Additionally, the modern lessons are almost entirely anachronistic: the reason that people cared about trigonometric identities was that they did all calculation by hand or using pre-computed lookup tables. In that context, judicious application of some identities could save hundreds of hours of labor by a semi-skilled human computer, by reducing the number of arithmetic calculations and/or table lookups. In an age of computers this is not really a consideration, and modern students don’t have any appreciation for the purpose or context of the tools of classical trigonometry. The classical trigonometry course should have been ripped out and fixed 50 years ago if not before (replaced by courses in vectors, complex arithmetic, and the complex exponential and logarithm). Such deeply entrenched school curricula are very difficult to modify though. †: Note, by trigonometry what we really mean is «the relations between uniform circular motion and/or lengths of circular arcs and a square grid coordinate system»; trigonometry is something of a misnomer as there is only partial overlap between metrical 3-gon (a.k.a. “triangle”) geometry and the study of uniform circular motion, and most of the interesting parts of metrical 3-gon geometry are not covered in a trigonometry course.
swiley 9y ago
This right here! Calculus (the first semester anyway) is really not hard to understand at all. Things like limits also demonstrate why math is done the way it is which makes it much more enjoyable, but most people get hung up with trig and don't see any of this. Part of the reason may be that trig gets used heavily for a lot of stuff after calc1.
BeetleB 9y ago
I suppose what you say makes sense for the general population. As a scientist/engineer who used trigonometry a lot, it was extremely useful to have a dedicated course on trigonometry early on. The reality is many will need to use basic trigonometry before they need to use calculus (e.g. in physics).
- abecedarius 9y ago
  OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Has anyone tried teaching this way in high school or middle school?
  - BeetleB 9y ago
    >OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Let's not forget the geometric aspect of trigonometry, which will be more intuitive to many than complex numbers. I have a force of 10N applied at an angle of 32 degrees. I need the x-component. You still need basic trigonometry.
    - abecedarius 9y ago
      Yes, you need to tie the algebra to the geometry. I had a friend taking a trig class, and when I showed him how all those trig identities just fall out of multiplying complex numbers, he was pissed.
kafkaesq 9y ago
Most students will benefit from already understanding calculus, when they are learning trigonometry. The argument cuts both ways: not only can one strongly benefit from a basic understanding of trigonometry when learning calculus (how the derivatives of sin/cos/etc all related to one another) - it really isn't possible to get far with basic integration techniques without learning the fundamental trig-based substitutions. And ultimately you can use any set of subjects, in just about any order. What matters is the thought processes, and that what you're really trying to teach is not a set of facts or techniques - but the underlying mathematical essence of these techniques. So at the end of the day, what it really comes to is: "It doesn't matter so much what you teach, but how you teach".

inbox 9y ago

I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I just sensed that something was off :)

jessaustin 9y ago
Your high school didn't teach geometry?
- adamnemecek 9y ago
  Geometry proofs aren't real proofs.
  - yequalsx 9y ago
    Why do you believe this? I think they are. They provide a nice example of using logic and axioms. I don't think one will be able to have a proof based course other than geometry before calculus. Students just aren't mathematically mature enough for that.
    - eximius 9y ago
      It very much depends how it is taught. When I took it, it was very focused on using only the building blocks you were given. Step one: apply theorem 1.2b; step two: apply theorem 1.4a; etc. It was barely more than a search through the space of operations given. The statements proven felt trivial and the proofs needlessly convoluted and rigid. Whereas in calculus and algebra and analysis and number theory, the proofs often had different paths to prove them or different constructions/descriptions and the things we proved felt substantial.
  - jessaustin 9y ago
    By that measure, most proofs seen in early undergraduate years are also not real (e.g. the proofs given for the fundamental theorem of calculus). The impression one gets from GP is that they needed to see some proofs in high school. If they had taken geometry with me in Mr. Schardt's class they would have gotten their fill...

jessaustin 9y ago

Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science. Vested interests dominate every nook and cranny of our society, even the society of mathematicians. Truly we live in a decadent age. With this much fuel piled up, who will be surprised by the conflagration?

mcnamaratw 9y ago
Haha!

tempw 9y ago

It's sad one needs to reinforce the teaching of concepts rather tricks. Even worse is when you have all the proof based and concept oriented course and are tested on trickeries on exams.

analog31 9y ago
I believe this is a huge shortcoming of how math is taught. You can bet your last dollar that the teacher doesn't think it's about tricks. But the students are convinced that it is. Students and teachers are both exposed to the exact same material but end up with diametrically opposing conclusions. Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years. The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades. The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works. Disclosure: I taught college freshman math for one semester.
patta54 9y ago
My high school calculus teacher was really great at this. He didn't just teach a bunch of transformations to memorize. That was part of it, naturally, as you aren't going to use the definition of limits and the FToC in all your problem solutions. He actually made us construct volumes from pieces of poster board, measure the segments and calculate the Riemann sum. When we did function analysis, he didn't allow us to use the Cartesian plane at first. We had to show visually how a function deformed the one-dimensional real line. How x^2 squished values between -1 and 1 toward 0 and stretched the other values toward +infinity. It gave me a good "visual" grasp of the concepts and made most of my higher math classes much easier. I do agree diff eq instruction sucks. I got an A in that class and didn't understand a thing. "This equation has this form; this is the canned solution."

ben_jones 9y ago

My differential equations class was the only course I ever took with over 30% of the final grade being derived from homework assignments. Unsubstantiated - but my peers and I all believed this was because everyone was failing and dropping the course.

hprotagonist 9y ago

you can fake knowledge of undergraduate differential equations if you know about 3 hours worth of linear algebra. More profound understanding is difficult but not terribly impossible. The best diffeq text i've found is by Blanchard, DeVaney and Hall, and it remains the only math textbook i've ever been able to read.

burnerOh2125 9y ago

Interesting, I'm a phd-drop out in computational biology, working as a data science consultant: I use mathematics including multi-dimensional statistics, linear algebra, and calculus everyday. Being self-taught, I'm very self conscience about the math I don't know, but so far, not knowing differential equations doesn't seem to have hurt me. I actually just ran into a problem that uses Hamiltonian dynamics, so maybe I will end up learning differential equations, but it does seem like the course, as taught many places and in the Dover books I own, presents either no useful conceptual insight, like why I learned geometric algebra, or a powerful toolset, like some multi-dimensional statistics.

j7ake 9y ago
So how do you reason with abundance of molecules? mRNA and protein can be modelled from simple linear differential equation
codewithcheese 9y ago
Hi I am interesting your advice on which math to learn if you have a spare moment to provide it. I too am starting to teach my self the requirements of data science and am also self conscious of the math I don't know. I am very excited about what is now possible with machine learning and deep learning as I believe it will become increasingly necessary for developers to stay relevant. Could you comment in more detail on which mathematical skill you have found useful as data scientist? Which resources did you use to teach yourself? Very appreciative of any help.
- ragulpr 9y ago
  * Solid linear algebra * Probability theory * Statistics (bayesian if possible) * Optimization (this is less important but extremely useful) If you know the mechanics of multivariate calculus you'll be fine learning the above. The course that personally have had most payoff was functional analysis. Purely theoretical course that will give you no practical skills and at first glance seems unrelated to ML but it (subtly) gave me a much deeper understanding of what ML is all about.
srean 9y ago

gumby 9y ago

Due to an offhand comment from a friend, I started my kid on continuous math, not discrete like they do in school. Went great, until about grade 4 when they actually started to care whether he was aligned with his classmates. Germans are actually pretty intolerant of non-professional pedagogues teaching kids. But it greatly improved his grasp of elementary math as it matched what he saw in the real world.

madengr 9y ago

Regarding item 10 in that list, I learned plenty of Laplace transforms, partial fraction expansions, stability, phase planes, etc. It's the core of control systems theory. It was just not taught in differential equations class. The DE class was a bag of useless tricks. All the other EE classes were very useful tricks of how not to directly solving DEs. Going to drink a beer now for Oliver Heaviside. The invention of the Laplace transform is one of the greatest contributions to engineering.

mathattack 9y ago

Apparently he passed 2 years after this was written. RIP. A great mind. https://en.wikipedia.org/wiki/Gian-Carlo_Rota https://en.wikipedia.org/wiki/Gian-Carlo_Rota

GeorgeTirebiter 9y ago
I clearly remember his class on, basically, Mathematics for Philosophers. As an engineering student, I thought my pal from Urban Studies who suggested taking it was a way to pad one's GPA, which wasn't my style. Nonetheless, I agreed, and, now, I regard it as one of the most important courses I've ever taken. I have never met anyone with that level of enthusiasm, wonder, and love of Mathematics - who also knew how to partially impart it to his students! Mainly, it's the Wonder of it all that remains with me. What a privilege.

woopwoop 9y ago

I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. In order to solve a general system of constant coefficient linear odes, you have to take a matrix and compute its Jordan canonical form. The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. Then, in another question, they were presented a 5x5 matrix, told that its only eigenvalues were 2 and -2, and asked if it was invertable. I don't think anyone gave a reasonable answer. What is the point of that? Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the entirety of a book review on contemporary philosophers: "When pygmies cast such long shadows, it must be very late in the day."

gh02t 9y ago
> These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. Yep, Diff Eq 1 (ODEs) and Diff Eq 2 (PDEs) for us as undergrads were exercises in memorization, even as a math major. I didn't really understand them until after I took a (well, several) linear algebra classes, after which it clicked that it was basically all the same theory.
eli_gottlieb 9y ago
You don't make students take Linear Algebra before differential equations? I thought students were supposed to take Lin-Alg alongside their Calculus 2 or Multivariable Calculus course, then have all that in their heads for DiffEqu.
- 9y ago

abhirag 9y ago

I have realized that Difference Equations i.e. the discrete variant of Differential Equations are much more common in Computer Science/Data Science(recurrence relations etc.) and it would have been great if they were atleast given half as much attention as Differential Equations get in universities.

colinator 9y ago

My colleague’s error consisted of believing that the more testable the material, the more teachable it is. A wider spread of performance in the problem sets and in the quizzes makes the assignment of grades “more objective.” The course is turned into a game of skill, where manipulative ability outweighs understanding. Sounds familiar.

falcor84 9y ago

I'm not sure how well it would fit as a first course, but the thing that really helped me understand what's behind differential equations is Steven Strogatz's wonderful book "Nonlinear Dynamics And Chaos"[0]. [0] https://www.amazon.com/gp/aw/d/0813349109/ https://www.amazon.com/gp/aw/d/0813349109/

fmap 9y ago

And 20 years later, this essay is still as relevant as the day it was written. I agree with pretty much everything that's in the essay, except for a few small points. > There is nothing wrong with keeping the functional notation for density functions – as physicists and engineers always did – as long as one bears in mind that density functions cannot be evaluated, but only integrated. This always bothered me, since, as noted in the very next section, distributions don't have an analogue to pointwise multiplication. Even worse, there is a perfectly servicable notation for such "dual functions/vectors" that physicists have been using throughout the second half of the 20th century. We could just use a consistent notation and not confuse new students, but no. "It's always been done this way" is a terrible argument and leads us to the confusing mess of notations that people still use for integrals and integral transforms... --- Apart from that I would teach people recurrence equations/stream calculus before going into the limiting case of differential equations. It's true that differential equations are sometimes easier to handle analytically, but this is neither relevant (as the article notes) nor a great point in their favor, since we just end up teaching students a bag of tricks instead of explaining why something works...

johan_larson 9y ago

When engineers and scientists have to get some real work done with DEs, what do they do? Is it all solved numerically these days?

eiji 9y ago
Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications. High and low electro magnetics (Maxwell), Navir-Stokes, ...
- thanatropism 9y ago
  Engineers should be taught about actual numerical algorithms and their qualitative properties. Cases in which there exist explicit symplectic algorithms, etc.
  - rsfern 9y ago
    Most engineering programs do have at least one full semester of numerical methods, even if in practice most engineers use matlab, scipy, abaqus etc...
- chillingeffect 9y ago
  What tools are used for parameter estimation? E.g. I made a model that describes my system behavior, then collected a lot of data and next need to find the constants in the model that make the best fit. Thank you.
  - eiji 9y ago
    The search terms there would be "Model Calibration" and/or "System Identification" Software. I think those are again very domain specific. But there are a few. PS: While AI and machine learning are catching on in many different fields, the software packages in those domains make heavy use of fairly complicated fitting and approximations methods already. Lots or correlation and effect estimation, which builds many of the underlying guts of machine learning.

eli_gottlieb 9y ago

>A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Well that's depressing. If not undergraduate courses, where do people expect graduate students to come from?

katastic 9y ago

I took two courses in ODE. I got an A in the first semester. I still have no idea what they are or what's going on. It's all just relationships and patterns to me--but with no intuition or understanding behind them. It never "sunk" in like Calculus did where "Area under the curve" and "tangent line" are super obvious in retrospect and immediately applicable to your daily life. "What's velocity? The change in distance over a unit of time." Done. Presto.