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Calculus Learning Guide

472 points by fauria 11y ago

e0m 11y ago
I love Better Explained. Drawing a parallel between multiplication/integration and division/differentiation is extremely powerful and almost never taught! It suddenly pulls perceptively "complex" topics into the realm of the familiar.
NKCSS 11y ago
Looks nice; didn't know that first example; shall give it a read later, thanks!
nextos 11y ago
Apart from Keisler, mentioned there, I think there are 2 lovely and rigurous references especially appropriate for anyone with a CS background: * Henle & Kleinberg [1] * Hubbard & Hubbard [2] Both are breathtakingly beautiful. Henle & Kleinberg is small and neat. Hubbard & Hubbard integrates calculus and linear algebra. In some ways it reminds me of SICP, described by Peter Norvig as "[...] a rich framework onto which you can add new learning over a career". [3] [1] https://books.google.co.uk/books/about/Infinitesimal_Calculus.html?id=nalhCn80D08C&redir_esc=y https://books.google.co.uk/books/about/Infinitesimal_Calculu... [2] http://matrixeditions.com/5thUnifiedApproach.html http://matrixeditions.com/5thUnifiedApproach.html [3] http://www.amazon.com/review/R403HR4VL71K8 http://www.amazon.com/review/R403HR4VL71K8
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dandermotj 11y ago
The more advanced mathematics I learn, the more I realise that in many cases I do not have a fundamental understanding of many topics, which leads me back to the beginning. More often than not, Better Explained is at the end of that path. If only this was how mathematics was taught!
- k__ 11y ago
  I never got calculus. I mean, I got the ideas behind it and could talk with my professors about it, but I couldn't solve much of these equations. It was a strange time at university... I even helped friends to get these exams done, but I myself always failed. Luckily we only had one math lecture about calculus, the 4 others were about different stuff. And it was always the fundamentals which got me. The whole rearranging an equation stuff all the school life is about totally eluded me. And this is what the calculus exams were about. At least the professors saw that I got the other math exams right, so they knew I wasn't an imbecile, haha
  - dandermotj 11y ago
    I am never comfortable with my understanding of a topic until I can teach it (or at least pass it on). There's just always an annoying, niggling feeling that if I were to be faced with a student who kept asking "why", I'd be a bumbling mess. I was the same as you for linear algebra. Everyone was always so happy to learn the rules, answer the questions like its a script and pass the exam. This frustrated me immensely. Sat down last year and started from the beginning, and I'm glad I did. Sometimes the key is to wipe the slate of existing knowledge clean and start again!
    - k__ 11y ago
      hehe... Funny thing is, if people who learn about software engineering ask me "why" I often know the answers. But I wouldn't know how to start teaching someone this topic... Linear algebra was one of those parts of math that I understood rather fast.
    - derekp7 11y ago
      For me, the "why" came from the word (or story) problems -- the exact problems that most people really hated. There is one problem from calculus that really stuck with me, and made me really understand limits: "You are on an island, which is 10 miles from the shoreline. You can row a boat 4 miles an hour, and you can run 7 miles an hour. You have a friend in need that lives on the shore, 6 miles down from the part of the (straight) shoreline that is closest to the island. Where on the shore do you need to land the boat to minimize your travel time?" So obviously, aiming the boat directly to your destination minimizes your distance, but you are only going to be traveling at 4 mph. And if you aim straight to the shore you minimize your time in the slow boat, but your distance is longer. The visual solution is to plot the total time based on various landing points on the shore, then find which one is minimal. The exact solution is to make a formula that represents both legs of the trip with the landing point being your variable, take a derivative of the formula and solve for 0 (if I remember correctly).
ilek 11y ago
Someone posted an article a while back about working to calculus using functions, I think it was titled something like "If Socrates knew functions"? I'm not sure. Does it ring a bell for anyone?
- jnbiche 11y ago
  Looks like "If Archimedes Knew Calculus", about "quantum" calculus. Very interesting, I'm about half-way through already: http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ http://www.math.harvard.edu/~knill/pedagogy/pechakucha/
  - jnbiche 11y ago
    To correct myself, the actual title is: "If Archimedes Knew Functions" Same link: http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ http://www.math.harvard.edu/~knill/pedagogy/pechakucha/
    - ilek 11y ago
      Thank you so much!
wyattpeak 11y ago
I love attempts to shake up maths education, and I reckon the first passage is a pretty neat explanation, but it seems to slide quickly downhill from there. The first three chapters are a bit airy-fairy - I get that that's its schtick, but they're also pretty light on content. The fourth reads like it's been taken straight out of a maths textbook by a "cool" teacher - engaging but not fundamentally simpler than what one learns in class. The fifth teaches how to use Wolfram Alpha to perform calculus. I could teach a dog to use Wolfram Alpha. It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to simplify the first few lessons on a subject, but you pretty soon hit a wall when you reach the parts that actually require a bit of mental legwork. Calculus is not a simple subject. By teaching it better, you can probably improve the situation by, like, 20%, but you can't magically make a complicated subject trivial. It just doesn't seem like the way forward for education. We need to make the difficulty more rewarding, not pretend that it doesn't exist.
- jasode 11y ago
  >It all seems to suffer a bit from the "Dummies guide" syndrome I think your criticism is misguided. His tutorial is obviously meant to make the introduction to the topic more accessible. He does that by using pictures and diagrams to build intuitions. He can do that because he extensively uses drawing tools to help the learner "visualize" what Calculus is about. Compare that with typical calculus teacher with a whiteboard. Because the teacher's drawing skills are limited, he/she dives right into the delta-epsilon limit definition: lim h --> 0 for f(x+h)-f(h)/h Yeah there's some hand drawing of tangent lines to converge on a "instantaneous change" but that's not enough for a lot of students to "get" Calculus. The symbolic notation is demonstrated more often than pictures. It therefore devolves into mindless "plug & chug" for the rest of the semester. As for the Wolfram Alpha comment, the idea is about "immersion". There are other tools besides pencil-&-paper that also "do" Calculus and they can be used to verify self-study Q&A. Likewise, if I teach Differential Equations to programmers, it's helpful if I "immerse" them into other tools to numerically solve them such as MS Excel, MATLAB, Wolfram Alpha, C/C++ code, etc. His BetterExplained doesn't replace a rigorous Calculus textbook but it can help with the "aha!" moment to understand what the dry Calc textbook is trying to teach.
neovive 11y ago
Has anyone completed the Calculus courses on Khan Academy (Integral, Differential and Multivariable)? If so, would you recommend Khan Academy for Calculus instead of the resources mentioned on Better Explained? I'm working through the Linear Algebra courses now on KA and they are very good.
- jasode 11y ago
  I haven't seen the Khan Academy Calculus videos that require signup but my answer assumes it is similar to (or exactly the same as) the youtube videos presented by Salman Khan. If you're already making great progress from Khan vids for Linear Algebra, you'll probably have the same success with Khan's Calculus. S Khan's approach is friendly but it's also very similar to a traditional teacher using a whiteboard in a classroom. For example, S Khan dives right into the formal concepts of "functions" and "limits" and all the associated notation. This mirrors how a professor would present it. In fact, Salman often remarks that he has his "college textbook" open as a resource to order the sequence of topics and demonstrate examples. Kalid Azad's approach with BetterExplained is to present images, pictures, and analogies that's not found in typical Calculus textbooks. This helps many learners get past the "mystery" of what the topic is about. You'll notice that the first chapters of Azad's approach does not discuss "limits"[1]. If you're already doing fine with Linear Algebra, you may not need all the pretty pictures as a gentle introduction. [1] The idea of "limits" is just one example of a philosophical difference on how to teach calculus. Isaac Newton didn't use the rigorous concept of "limits" when he invented calculus. His peers derided his approach as "ghosts of departed quantities". It was the later mathematicians who refined calculus to be based on more rigorous foundations (Real Analysis) that "limits" was retrofitted onto calculus. It's interesting that we now think the best pedagogy for high school students is to force "limits" on them to learn calculus because ... It's More Rigorous. Unfortunately, "limits" are not intuitive.
  - btilly 11y ago
    Your history of Calculus is a little..muddled. Newton's approach was based on the idea that rate of change was a fundamental quantity. This lead to fluxions. His notation for Calculus was x with a dot over it. He used his method as a rough notation to figure out what the answers should be, before coming up with pure geometric constructions to use in the Principia Mathematica. Inspired by the Principia, Leibniz reinvented Calculus with the idea of infinitesmals. Infinitesmals are nonzero numbers that are smaller than any real number. He published, which started a priority dispute between him and Newton. Leibniz gave us our dy/dx notation. In the original, d is an operator meaning "look an infinitesmal away". Literally d(y) = y(x+dx)-y(x). (This converges faster if you make it y(x+dx/2)-y(x-dx/2)...) You do the calculation, then drop any infinitesmals and get your answer. Ever wondered why the second derivative is d^2y/dx^2? That's because you do d(d(y))/d(x)d(x). The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. Far from being a peer of Newton's, he was a bishop who was demonstrating that science had leaps of faith as big as any in religion. His criticisms were accurate. The fact that math's foundations were crap occasionally inspired effort. But not much. Lagrange's contribution was to view everything as a power series, and do formal derivatives that way. To him we owe our f' notation. Then Joseph Fourier upended the whole apple cart with Fourier series. He took infinite sums of nice functions like sin and cos, and wound up with a step function. What happens if your step is an infinitesmal away? This was a crisis. Cauchy addressed this by making infinitesmals sequences that go to 0. This approach failed for the chain rule when the derivative is 0, but that's where the idea of Cauchy sequences comes from. Weirstrauss in the 1870s came up with the modern limit approach. Mathematics was (very literally) rewritten. Since then limits have proven to be a horrible pedagogical tool, but that is how we teach. However there are serious pedagogical suggestions ranging from skipping over the justification for Calculus to using infinitesmals (turns out you can make precise sense of them with modern set theory) to writing the tangent line approximation with little-o notation. (The last suggestion is from Knuth! See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... for details.) Sorry for the history lesson, I love this topic. :-)
alexc05 11y ago
This is fantastic! After nearly 20 years as a self-taught programmer I've been thinking about going back to school to get a CS degree (or master's) Since I haven't taken linear algebra or calculus those are some pretty big areas of worry for me. I've been looking for some good resources on the two. I started watching the video for year one calculus from MIT open courseware (specifically this one: http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-1-derivatives/ http://ocw.mit.edu/courses/mathematics/18-01-single-variable...) ... and was so lost I had to back out and google "what is calculus?" I'm pretty basic here :) Turns out it's the math of "measuring change" A colleague pointed me to Kahn academy as well, but this is a welcome resource. Thanks.
- ben_jones 11y ago
  Also an interesting place to apply calculus is in computer graphics. Unlike a lot of math calculus is extremely practical here and sometimes its more fun to make a game then to beat your head on practice problems.
- pakled_engineer 11y ago
  MIT also has supplementary lectures like "Big Picture of Calculus" with Gilbert Strang. http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/highlights_of_calculus/big-picture-of-calculus/ http://ocw.mit.edu/resources/res-18-005-highlights-of-calcul... He also put his text online for free http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
- matt_wulfeck 11y ago
  Same! I took one college level math class and never looked back. At this point I think I'd get a marginal boost to my programming career, mostly I want to learn because it looks fun and I've never studied it.
  - maaku 11y ago
    You don't know what you're missing without a background in calculus. Calculus is the mathematics underlying human-scale physics, and therefore all of engineering, art, and design. It is the fundamental mathematical theory alongside statistics underlying most of economics and biology. I tell kids going into calculus to pay extra attention: it will change the way they view the entire world. And it will. It is absolutely no accident that the enlightenment, industrial revolution, and modern era began with the discovery of calculus. If you don't have calculus under your belt, I'd definitely suggest starting there.
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Chris2048 11y ago
Stuff like Khan Academy, or even MIT OCW are great, but there can be holes wrt more advanced or unusual math. There also seems to be an "established" way of teaching things (what/how), such that alternatives aren't offered. Thinks like Wikipedia are quite poor on explaining math, especially when the set theory is pulled. Set theory almost guarantees no intuition for me...
charlieflowers 11y ago
I am blown away by the "Calculus in One Minute" explanation. And I came in with a high bar because I already knew about the good material on betterexplained.com
wolfgke 11y ago
To quote this site: "Review the intuitive definition. Rephrase technical definitions in terms that make sense to you." I openly admit that the technical definitions are often how I imagined the material intuitively when the material was taught in school. But first when studying mathematics these definitions occured. I really wish that calculus (and the rest of mathematics) would be taught in a much more abstract way in school - it really makes understanding the material much easier since the "intuitive" definitions suffer a lot from being leaky abstractions (and mathematics is all about banging to the boundaries of the definitions until they move or fall down).
- jawilson2 11y ago
  I'm the opposite (which, I admit, is backwards for a lot of engineers.) I need to have a high-level view to provide a context for the details. It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle, or to track planetary motion. Once I see the problem, I can think about how we arrived at the solution, rather than present it as a solution in search of a problem. Newton wanted to describe planetary motion, so he invented calculus (Leibniz too, etc.) to solve it, not the other way around.
  - wolfgke 11y ago
    > It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle I will only discuss this point (but for your other points similar arguments can be made). So you want to measure some n-dimensional volume? This is a highly non-trivial problem, since it is quite possible to find subsets of R^n where you can't assign volumes in a sensible way: > https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox If you don't care about this subtlety you end up in such a paradox. So you think hard about how you can define "measureable" in an accurate sense and come up with the definition of a sigma algebra and think about how you can even define measures (say of volume) on a sigma algebra. Because these can be ugly to describe, you will immediately think about how one can describe the sigma algebra and the measure in a more easy way. So you come up with generators and note that the measure is defined uniquely by its value on the generator. Doing this process on the real line using the Borel sigma algebra/measure is not that complicated (so you solved the problem for the real case). From here it is only a small step to the Lebesgue-Stieltjes integral in R^1 that is really worth doing. But now you remember that you wanted to compute higher-dimensional volumes. So you ask how can you recycle what you've already done? The answer is obvious: You define product sigma algebras and product measures. As you have seen (Banach-Tarski_paradox) all this is necessary just to even define volumes. I really know no easier way that is both useful in practise (at least be able to obtain the volume of spheres (i.e. not just "simple objects as simplices")) and avoids the problem that was laid out by Banach and Tarski-
niedzielski 11y ago
I haven't finished it but I read most of the late James Stewart's "Calculus: Early Transcendentals" and found it very instructive. It would make a good supplement for anyone learning Calculus.
mathgenius 11y ago
I recently started reading Amir Alexander's 2014 book "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". It's a really great read, this is pre-Newton, pre-Leibniz. People were doing calculus more than two hundred years before those guys! They struggled with paradoxes to do with infinitely many zero length points making up a line, and the Jesuits just outright banned that stuff. Partly because it was protestant mathematics, but mostly because they wanted a version of mathematics that was somehow eternally unchanging and crystal clear (like how God is supposed to be.) Boy did that not work out. Anyway, these early mathematicians used imagery like the threads in a cloth to describe how infinitely many lines form a plane surface, or the pages of a book to help understand how infinitely many planes form a solid. So cool! In the beginning it was Italians working this stuff out (including Galileo). But then the Jesuits totally banned it thereby casting a pall over Italian mathematics for hundreds of years! For a modern treatment of infinitesimals, check out "smooth infinitesimal analysis" [1]. The way to avoid all the contradictions is to weaken the underlying logic: in this case they remove the law of excluded middle, so it's not the case that something is either true or it is false. I think the Jesuits would definitely have trouble with that! [1] https://xorshammer.com/2008/08/11/smooth-infinitesimal-analysis/ https://xorshammer.com/2008/08/11/smooth-infinitesimal-analy...
- maaku 11y ago
  Removing the law of the excluded middle strengthens the underlying theory. It makes it constructivist...
- erichocean 11y ago
  > The way to avoid all the contradictions is to weaken the underlying logic Pet peeve: I would argue that constructive logic is stronger, in the sense that it allows more reliable (read: usable in the real world) results than classic logic allows. But that's a value judgement, and there's no definitive right answer here. Totally agree Re: Infinitesimals. Dual numbers[0] are crazy useful in computer science (c.f. Google's Ceres Solver[1]). [0] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number [1] http://ceres-solver.org/ http://ceres-solver.org/
known 11y ago
Brilliant write up;
amelius 11y ago
I'm not sure if I like the example with the integration of the circle circumference into the circle area. Yes, it sounds intuitive. But here's another problem with a wrong solution that sounds intuitive: to compute the hypotenuse C of a triangle with other (perpendicular) sides A and B, you can simply replace the hypotenuse by a staircase of infinitesimal steps, and hence you find that C = A+B. We're overlooking something here, but that is not the point. The point is that the proof should be constructed so that we cannot possibly overlook something. Conclusion: don't rely on intuition with these kind of problems.
baldfat 11y ago
Children should learn Calculus instead of Algebra 1 & 2. The BIG Problem is the difference between a Good Calculus teacher and a BAD Calculus teacher is so HUGE and changes the outcomes of their students.
sureshkon 11y ago
As usual, great stuff from Kalid.
mclovinit 11y ago
This is absolutely fantastic. Great method for teaching new concepts to those new to the subject matter. I have an engineering background and have to say that this approach is refreshing to read. It's like a bag of chocolate chip cookies for the brain :)

472 points by fauria 11y ago

e0m 11y ago

I love Better Explained. Drawing a parallel between multiplication/integration and division/differentiation is extremely powerful and almost never taught! It suddenly pulls perceptively "complex" topics into the realm of the familiar.

NKCSS 11y ago

Looks nice; didn't know that first example; shall give it a read later, thanks!

nextos 11y ago

Apart from Keisler, mentioned there, I think there are 2 lovely and rigurous references especially appropriate for anyone with a CS background: * Henle & Kleinberg [1] * Hubbard & Hubbard [2] Both are breathtakingly beautiful. Henle & Kleinberg is small and neat. Hubbard & Hubbard integrates calculus and linear algebra. In some ways it reminds me of SICP, described by Peter Norvig as "[...] a rich framework onto which you can add new learning over a career". [3] [1] https://books.google.co.uk/books/about/Infinitesimal_Calculus.html?id=nalhCn80D08C&redir_esc=y https://books.google.co.uk/books/about/Infinitesimal_Calculu... [2] http://matrixeditions.com/5thUnifiedApproach.html http://matrixeditions.com/5thUnifiedApproach.html [3] http://www.amazon.com/review/R403HR4VL71K8 http://www.amazon.com/review/R403HR4VL71K8

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dandermotj 11y ago

The more advanced mathematics I learn, the more I realise that in many cases I do not have a fundamental understanding of many topics, which leads me back to the beginning. More often than not, Better Explained is at the end of that path. If only this was how mathematics was taught!

k__ 11y ago
I never got calculus. I mean, I got the ideas behind it and could talk with my professors about it, but I couldn't solve much of these equations. It was a strange time at university... I even helped friends to get these exams done, but I myself always failed. Luckily we only had one math lecture about calculus, the 4 others were about different stuff. And it was always the fundamentals which got me. The whole rearranging an equation stuff all the school life is about totally eluded me. And this is what the calculus exams were about. At least the professors saw that I got the other math exams right, so they knew I wasn't an imbecile, haha
- dandermotj 11y ago
  I am never comfortable with my understanding of a topic until I can teach it (or at least pass it on). There's just always an annoying, niggling feeling that if I were to be faced with a student who kept asking "why", I'd be a bumbling mess. I was the same as you for linear algebra. Everyone was always so happy to learn the rules, answer the questions like its a script and pass the exam. This frustrated me immensely. Sat down last year and started from the beginning, and I'm glad I did. Sometimes the key is to wipe the slate of existing knowledge clean and start again!
  - k__ 11y ago
    hehe... Funny thing is, if people who learn about software engineering ask me "why" I often know the answers. But I wouldn't know how to start teaching someone this topic... Linear algebra was one of those parts of math that I understood rather fast.
  - derekp7 11y ago
    For me, the "why" came from the word (or story) problems -- the exact problems that most people really hated. There is one problem from calculus that really stuck with me, and made me really understand limits: "You are on an island, which is 10 miles from the shoreline. You can row a boat 4 miles an hour, and you can run 7 miles an hour. You have a friend in need that lives on the shore, 6 miles down from the part of the (straight) shoreline that is closest to the island. Where on the shore do you need to land the boat to minimize your travel time?" So obviously, aiming the boat directly to your destination minimizes your distance, but you are only going to be traveling at 4 mph. And if you aim straight to the shore you minimize your time in the slow boat, but your distance is longer. The visual solution is to plot the total time based on various landing points on the shore, then find which one is minimal. The exact solution is to make a formula that represents both legs of the trip with the landing point being your variable, take a derivative of the formula and solve for 0 (if I remember correctly).

ilek 11y ago

Someone posted an article a while back about working to calculus using functions, I think it was titled something like "If Socrates knew functions"? I'm not sure. Does it ring a bell for anyone?

jnbiche 11y ago
Looks like "If Archimedes Knew Calculus", about "quantum" calculus. Very interesting, I'm about half-way through already: http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ http://www.math.harvard.edu/~knill/pedagogy/pechakucha/
- jnbiche 11y ago
  To correct myself, the actual title is: "If Archimedes Knew Functions" Same link: http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ http://www.math.harvard.edu/~knill/pedagogy/pechakucha/
  - ilek 11y ago
    Thank you so much!

wyattpeak 11y ago

I love attempts to shake up maths education, and I reckon the first passage is a pretty neat explanation, but it seems to slide quickly downhill from there. The first three chapters are a bit airy-fairy - I get that that's its schtick, but they're also pretty light on content. The fourth reads like it's been taken straight out of a maths textbook by a "cool" teacher - engaging but not fundamentally simpler than what one learns in class. The fifth teaches how to use Wolfram Alpha to perform calculus. I could teach a dog to use Wolfram Alpha. It all seems to suffer a bit from the "Dummies guide" syndrome - it's easy to simplify the first few lessons on a subject, but you pretty soon hit a wall when you reach the parts that actually require a bit of mental legwork. Calculus is not a simple subject. By teaching it better, you can probably improve the situation by, like, 20%, but you can't magically make a complicated subject trivial. It just doesn't seem like the way forward for education. We need to make the difficulty more rewarding, not pretend that it doesn't exist.

jasode 11y ago
>It all seems to suffer a bit from the "Dummies guide" syndrome I think your criticism is misguided. His tutorial is obviously meant to make the introduction to the topic more accessible. He does that by using pictures and diagrams to build intuitions. He can do that because he extensively uses drawing tools to help the learner "visualize" what Calculus is about. Compare that with typical calculus teacher with a whiteboard. Because the teacher's drawing skills are limited, he/she dives right into the delta-epsilon limit definition: lim h --> 0 for f(x+h)-f(h)/h Yeah there's some hand drawing of tangent lines to converge on a "instantaneous change" but that's not enough for a lot of students to "get" Calculus. The symbolic notation is demonstrated more often than pictures. It therefore devolves into mindless "plug & chug" for the rest of the semester. As for the Wolfram Alpha comment, the idea is about "immersion". There are other tools besides pencil-&-paper that also "do" Calculus and they can be used to verify self-study Q&A. Likewise, if I teach Differential Equations to programmers, it's helpful if I "immerse" them into other tools to numerically solve them such as MS Excel, MATLAB, Wolfram Alpha, C/C++ code, etc. His BetterExplained doesn't replace a rigorous Calculus textbook but it can help with the "aha!" moment to understand what the dry Calc textbook is trying to teach.

neovive 11y ago

Has anyone completed the Calculus courses on Khan Academy (Integral, Differential and Multivariable)? If so, would you recommend Khan Academy for Calculus instead of the resources mentioned on Better Explained? I'm working through the Linear Algebra courses now on KA and they are very good.

jasode 11y ago
I haven't seen the Khan Academy Calculus videos that require signup but my answer assumes it is similar to (or exactly the same as) the youtube videos presented by Salman Khan. If you're already making great progress from Khan vids for Linear Algebra, you'll probably have the same success with Khan's Calculus. S Khan's approach is friendly but it's also very similar to a traditional teacher using a whiteboard in a classroom. For example, S Khan dives right into the formal concepts of "functions" and "limits" and all the associated notation. This mirrors how a professor would present it. In fact, Salman often remarks that he has his "college textbook" open as a resource to order the sequence of topics and demonstrate examples. Kalid Azad's approach with BetterExplained is to present images, pictures, and analogies that's not found in typical Calculus textbooks. This helps many learners get past the "mystery" of what the topic is about. You'll notice that the first chapters of Azad's approach does not discuss "limits"[1]. If you're already doing fine with Linear Algebra, you may not need all the pretty pictures as a gentle introduction. [1] The idea of "limits" is just one example of a philosophical difference on how to teach calculus. Isaac Newton didn't use the rigorous concept of "limits" when he invented calculus. His peers derided his approach as "ghosts of departed quantities". It was the later mathematicians who refined calculus to be based on more rigorous foundations (Real Analysis) that "limits" was retrofitted onto calculus. It's interesting that we now think the best pedagogy for high school students is to force "limits" on them to learn calculus because ... It's More Rigorous. Unfortunately, "limits" are not intuitive.
- btilly 11y ago
  Your history of Calculus is a little..muddled. Newton's approach was based on the idea that rate of change was a fundamental quantity. This lead to fluxions. His notation for Calculus was x with a dot over it. He used his method as a rough notation to figure out what the answers should be, before coming up with pure geometric constructions to use in the Principia Mathematica. Inspired by the Principia, Leibniz reinvented Calculus with the idea of infinitesmals. Infinitesmals are nonzero numbers that are smaller than any real number. He published, which started a priority dispute between him and Newton. Leibniz gave us our dy/dx notation. In the original, d is an operator meaning "look an infinitesmal away". Literally d(y) = y(x+dx)-y(x). (This converges faster if you make it y(x+dx/2)-y(x-dx/2)...) You do the calculation, then drop any infinitesmals and get your answer. Ever wondered why the second derivative is d^2y/dx^2? That's because you do d(d(y))/d(x)d(x). The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. Far from being a peer of Newton's, he was a bishop who was demonstrating that science had leaps of faith as big as any in religion. His criticisms were accurate. The fact that math's foundations were crap occasionally inspired effort. But not much. Lagrange's contribution was to view everything as a power series, and do formal derivatives that way. To him we owe our f' notation. Then Joseph Fourier upended the whole apple cart with Fourier series. He took infinite sums of nice functions like sin and cos, and wound up with a step function. What happens if your step is an infinitesmal away? This was a crisis. Cauchy addressed this by making infinitesmals sequences that go to 0. This approach failed for the chain rule when the derivative is 0, but that's where the idea of Cauchy sequences comes from. Weirstrauss in the 1870s came up with the modern limit approach. Mathematics was (very literally) rewritten. Since then limits have proven to be a horrible pedagogical tool, but that is how we teach. However there are serious pedagogical suggestions ranging from skipping over the justification for Calculus to using infinitesmals (turns out you can make precise sense of them with modern set theory) to writing the tangent line approximation with little-o notation. (The last suggestion is from Knuth! See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... for details.) Sorry for the history lesson, I love this topic. :-)

alexc05 11y ago

This is fantastic! After nearly 20 years as a self-taught programmer I've been thinking about going back to school to get a CS degree (or master's) Since I haven't taken linear algebra or calculus those are some pretty big areas of worry for me. I've been looking for some good resources on the two. I started watching the video for year one calculus from MIT open courseware (specifically this one: http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-1-derivatives/ http://ocw.mit.edu/courses/mathematics/18-01-single-variable...) ... and was so lost I had to back out and google "what is calculus?" I'm pretty basic here :) Turns out it's the math of "measuring change" A colleague pointed me to Kahn academy as well, but this is a welcome resource. Thanks.

ben_jones 11y ago
Also an interesting place to apply calculus is in computer graphics. Unlike a lot of math calculus is extremely practical here and sometimes its more fun to make a game then to beat your head on practice problems.
pakled_engineer 11y ago
MIT also has supplementary lectures like "Big Picture of Calculus" with Gilbert Strang. http://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/highlights_of_calculus/big-picture-of-calculus/ http://ocw.mit.edu/resources/res-18-005-highlights-of-calcul... He also put his text online for free http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
matt_wulfeck 11y ago
Same! I took one college level math class and never looked back. At this point I think I'd get a marginal boost to my programming career, mostly I want to learn because it looks fun and I've never studied it.
- maaku 11y ago
  You don't know what you're missing without a background in calculus. Calculus is the mathematics underlying human-scale physics, and therefore all of engineering, art, and design. It is the fundamental mathematical theory alongside statistics underlying most of economics and biology. I tell kids going into calculus to pay extra attention: it will change the way they view the entire world. And it will. It is absolutely no accident that the enlightenment, industrial revolution, and modern era began with the discovery of calculus. If you don't have calculus under your belt, I'd definitely suggest starting there.

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Chris2048 11y ago

Stuff like Khan Academy, or even MIT OCW are great, but there can be holes wrt more advanced or unusual math. There also seems to be an "established" way of teaching things (what/how), such that alternatives aren't offered. Thinks like Wikipedia are quite poor on explaining math, especially when the set theory is pulled. Set theory almost guarantees no intuition for me...

charlieflowers 11y ago

I am blown away by the "Calculus in One Minute" explanation. And I came in with a high bar because I already knew about the good material on betterexplained.com

wolfgke 11y ago

To quote this site: "Review the intuitive definition. Rephrase technical definitions in terms that make sense to you." I openly admit that the technical definitions are often how I imagined the material intuitively when the material was taught in school. But first when studying mathematics these definitions occured. I really wish that calculus (and the rest of mathematics) would be taught in a much more abstract way in school - it really makes understanding the material much easier since the "intuitive" definitions suffer a lot from being leaky abstractions (and mathematics is all about banging to the boundaries of the definitions until they move or fall down).

jawilson2 11y ago
I'm the opposite (which, I admit, is backwards for a lot of engineers.) I need to have a high-level view to provide a context for the details. It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle, or to track planetary motion. Once I see the problem, I can think about how we arrived at the solution, rather than present it as a solution in search of a problem. Newton wanted to describe planetary motion, so he invented calculus (Leibniz too, etc.) to solve it, not the other way around.
- wolfgke 11y ago
  > It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle I will only discuss this point (but for your other points similar arguments can be made). So you want to measure some n-dimensional volume? This is a highly non-trivial problem, since it is quite possible to find subsets of R^n where you can't assign volumes in a sensible way: > https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox If you don't care about this subtlety you end up in such a paradox. So you think hard about how you can define "measureable" in an accurate sense and come up with the definition of a sigma algebra and think about how you can even define measures (say of volume) on a sigma algebra. Because these can be ugly to describe, you will immediately think about how one can describe the sigma algebra and the measure in a more easy way. So you come up with generators and note that the measure is defined uniquely by its value on the generator. Doing this process on the real line using the Borel sigma algebra/measure is not that complicated (so you solved the problem for the real case). From here it is only a small step to the Lebesgue-Stieltjes integral in R^1 that is really worth doing. But now you remember that you wanted to compute higher-dimensional volumes. So you ask how can you recycle what you've already done? The answer is obvious: You define product sigma algebras and product measures. As you have seen (Banach-Tarski_paradox) all this is necessary just to even define volumes. I really know no easier way that is both useful in practise (at least be able to obtain the volume of spheres (i.e. not just "simple objects as simplices")) and avoids the problem that was laid out by Banach and Tarski-

niedzielski 11y ago

I haven't finished it but I read most of the late James Stewart's "Calculus: Early Transcendentals" and found it very instructive. It would make a good supplement for anyone learning Calculus.

mathgenius 11y ago

I recently started reading Amir Alexander's 2014 book "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". It's a really great read, this is pre-Newton, pre-Leibniz. People were doing calculus more than two hundred years before those guys! They struggled with paradoxes to do with infinitely many zero length points making up a line, and the Jesuits just outright banned that stuff. Partly because it was protestant mathematics, but mostly because they wanted a version of mathematics that was somehow eternally unchanging and crystal clear (like how God is supposed to be.) Boy did that not work out. Anyway, these early mathematicians used imagery like the threads in a cloth to describe how infinitely many lines form a plane surface, or the pages of a book to help understand how infinitely many planes form a solid. So cool! In the beginning it was Italians working this stuff out (including Galileo). But then the Jesuits totally banned it thereby casting a pall over Italian mathematics for hundreds of years! For a modern treatment of infinitesimals, check out "smooth infinitesimal analysis" [1]. The way to avoid all the contradictions is to weaken the underlying logic: in this case they remove the law of excluded middle, so it's not the case that something is either true or it is false. I think the Jesuits would definitely have trouble with that! [1] https://xorshammer.com/2008/08/11/smooth-infinitesimal-analysis/ https://xorshammer.com/2008/08/11/smooth-infinitesimal-analy...

maaku 11y ago
Removing the law of the excluded middle strengthens the underlying theory. It makes it constructivist...
erichocean 11y ago
> The way to avoid all the contradictions is to weaken the underlying logic Pet peeve: I would argue that constructive logic is stronger, in the sense that it allows more reliable (read: usable in the real world) results than classic logic allows. But that's a value judgement, and there's no definitive right answer here. Totally agree Re: Infinitesimals. Dual numbers[0] are crazy useful in computer science (c.f. Google's Ceres Solver[1]). [0] https://en.wikipedia.org/wiki/Dual_number https://en.wikipedia.org/wiki/Dual_number [1] http://ceres-solver.org/ http://ceres-solver.org/

known 11y ago

Brilliant write up;

amelius 11y ago

I'm not sure if I like the example with the integration of the circle circumference into the circle area. Yes, it sounds intuitive. But here's another problem with a wrong solution that sounds intuitive: to compute the hypotenuse C of a triangle with other (perpendicular) sides A and B, you can simply replace the hypotenuse by a staircase of infinitesimal steps, and hence you find that C = A+B. We're overlooking something here, but that is not the point. The point is that the proof should be constructed so that we cannot possibly overlook something. Conclusion: don't rely on intuition with these kind of problems.

baldfat 11y ago

Children should learn Calculus instead of Algebra 1 & 2. The BIG Problem is the difference between a Good Calculus teacher and a BAD Calculus teacher is so HUGE and changes the outcomes of their students.

sureshkon 11y ago

As usual, great stuff from Kalid.

mclovinit 11y ago

This is absolutely fantastic. Great method for teaching new concepts to those new to the subject matter. I have an engineering background and have to say that this approach is refreshing to read. It's like a bag of chocolate chip cookies for the brain :)