How to Ace Calculus: The Art of Doing Well in Technical Courses

This generalizes to formal computer science as a programmer, and is probably one of the best ways to put it I've ever seen. If you understand computer science as the formula, you'll think it's pretty useless. If you get the concepts, you start seeing how it is useful everywhere.<p>I do not sit down and prove my designs and code or use lots of tricky algorithms, but I use a lot of the insights and ways of thinking I picked up from the computer science concepts, thinking about invariants and the maintenance of them, etc. There's few things sadder than sitting through four years of school and coming out seriously thinking that it's all useless wankery against the importance of "REAL PROGRAMMING".<p>(I've also noticed/learned that when you do a good solid job of designing your system with strong foundational concepts, the system will talk to you as you try to design it. I just got back from talking to a coworker about a case where I need to bypass my permissions system and temporarily become a superuser in order to do this particular thing, and I realized that rather than that being "the solution", that was actually my permission system telling me that I was doing something wrong. Only after I realized that did I reflect on it for a moment and realize the permission system was right and I was trying to do something potentially dangerous. I had thought about the thing I wanted to do but didn't fully consider how it might be exploited. We may still do it, we may not, but either way, listening to the code taught me something important about my system. You don't get these insights when you're too busy with your REAL PROGRAMMING and turning out mushy, concept-less code. You just write the flaw in and let your customers or hackers find it.)

There's something that I think most people are missing, although many of you will already know this.<p>People are saying that you need to develop the intuition, to develop the visualization skills, to develop the sense of what's happening rather than simply memorizing the formulas.<p>But to me, the visualization is not the point. To me, the sense of what's happening based on the visualization is not the point. To me, the point is the richness of understanding, the combination of many ways of thinking.<p>This doesn't come without effort.<p>The lunk-to item seems to suggest that by having the picture in mind one can avoid all the tedium of remembering the epsilon-delta limit arguments and can avoid the definition of lim_{e-&#62;0}(f(x+e)-f(x))/e and so on, but that's not true. The point is that the formula is tied up with the image, not that one subsumes the other.<p>Allegedly Euclid said King Ptolemy (in response to a request for an easier way of learning mathematics) that "there is no Royal Road to geometry".[1] Likewise there is no "Royal Road" to a mastery of calculus. Or indeed, to a mastery of <i>any</i> subject. That which can be mastered with little effort has long been surpassed, and work is required to gain the depth and breadth required to make these things easy.<p>But we do these things "not because they are easy, but because they are hard."[2] They are of value, and developing the mastery is satisfying in its own right, but also makes you a rare commodity.<p>[1] <a href="http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_to_the_Royal_Road" rel="nofollow">http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_...</a><p>[2] <a href="http://er.jsc.nasa.gov/seh/ricetalk.htm" rel="nofollow">http://er.jsc.nasa.gov/seh/ricetalk.htm</a>

"They switch to a philosophy major."<p>This comment seems odd to me, as philosophy is another subject where a failure to sit down and really think about the topics at hand will leave you hopelessly lost, if the class is taught to any degree of rigor.

This rings true for me. Certainly the courses I struggled in were ones where I had a hard time getting a grip on the concepts. The best lecturers helped by teaching in a way that made the key concepts clear and showed how they developed from previous concepts taught.

As a chinese who is surprisingly good with technical courses and math, and also a computer science major who is somewhat not that bad with programming:<p>Learn to see patterns. Math is all about patterns. Get obsessive. I just got out of an obssessive period (3 days, to the point I didn't wanna talk to anyone) where I couldn't solve problems. Visualise problems in your head, put the entire problem domain into your head, lie on the bed. Solve it YOURSELF.<p>Always, always, solve it yourself, and only ask when you have TRIED AND TRIED. Then when you finally ask and get the solution, you'll remember it for life.<p>Pattern, and self-attempting. Practise makes perfect too.

I have to disagree, at least somewhat. Insight, understanding the basic concepts has always come fairly easily to me. The REALLY hard part is the disciplined practice needed to be able to actually apply what you know to real problems.

This is why I did terribly in quantum mechanics; I was never able to generate a whiff of insight about anything that was going on. The thing is, I'm not sure the top students really did either.

Why is the tendency to introduct concepts, and often leave out the insight? Why don't professors start with insight and generalize concepts?

Understand the topic and practice a lot? Well, yeah. That is good as an objective, but what's missing is a way to get to the point where you actually understand the thing.

Cal Newport, the author of the submitted blog post, draws comments both here on HN and on his own blog pointing out that deep understanding of a subject doesn't necessarily equate to VISUAL thinking about a subject. There is a big literature on "learning styles" and some attempts by some schoolteachers to categorize children by what their preferred learning styles are. When I have taken learning style questionnaires, and when I have asked my wife (a piano performance major and private music teacher) about this, the answer on learning styles is "all of the above." I personally think, based on my observations of successful learners of a variety of subjects, that learning styles are themselves learnable, and a learner with a deep knowledge of a particular subject will know multiple representations of that subject. My wife has had many piano performance courses, and also music theory and ear training courses, and has learned visual representations of music both in the form of standard musical notation and in the form of "music mapping,"<p><a href="http://www.amazon.com/Mapping-Music-Learning-Teachers-Students/dp/0895793970" rel="nofollow">http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen...</a><p>which she has found very helpful.<p>As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics<p><a href="http://www.amazon.com/Vision-Elementary-Mathematics-W-Sawyer/dp/048642555X" rel="nofollow">http://www.amazon.com/Vision-Elementary-Mathematics-W-Sawyer...</a><p><a href="http://www.marco-learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdf" rel="nofollow">http://www.marco-learningsystems.com/pages/sawyer/Vision_in_...</a><p>But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.<p><a href="http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877" rel="nofollow">http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...</a><p>The acclaimed books on Calculus by Michael Spivak<p><a href="http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/" rel="nofollow">http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...</a><p>and Tom Apostol<p><a href="http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/" rel="nofollow">http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...</a><p>are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.<p>Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma<p><a href="http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843/" rel="nofollow">http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...</a><p>demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.<p><a href="http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf" rel="nofollow">http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf</a><p><a href="http://www.ams.org/notices/199908/rev-howe.pdf" rel="nofollow">http://www.ams.org/notices/199908/rev-howe.pdf</a><p>Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.<p><a href="http://www.ams.org/notices/200502/fea-kenschaft.pdf" rel="nofollow">http://www.ams.org/notices/200502/fea-kenschaft.pdf</a><p>In light of recent HN threads about Khan Academy,<p><a href="http://news.ycombinator.com/item?id=2348476" rel="nofollow">http://news.ycombinator.com/item?id=2348476</a><p><a href="http://news.ycombinator.com/item?id=2350430" rel="nofollow">http://news.ycombinator.com/item?id=2350430</a><p>I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?

Good article but I don't think it's the whole story. Works well for maths, probably, where in my experience everything is obfuscated by a) lack of any practical real-life application for most people and b) non-verbal symbolic representations of concepts.<p>With programming you very likely do want to apply the stuff you're learning to 'real life' problems, and you're going to be expressing all your efforts in a programming language with familiar keywords (and just a few symbols/operators). Here the problem is not intuiting what the purpose is - it's easy to explain what Ajax calls are supposed to do, for instance, but actually implementing them is quite bitty. You need to set up a sort of chain of connections between multiple points, and not until you've learnt all the details of this process, can you tuck it all away neatly under one abstraction and free up brain cycles to deal with higher problems. I find more and more that when I learn a new corner of programming, there's just inevitably going to be a certain number of hours of faffing about learning the details before it 'clicks.' You feel stupid for a week or so, the boom You Know Kung-Fu, like it was easy all along.<p>Having said that some students just really struggle with basic concepts like 'variables' and need to make sure they intuitively grasp them. But that's about <i>passing</i>, not getting straight As.

<p><pre><code> [T]he students who struggle in technical courses are those who skip the insight-developing phase. They capture concepts in their notes and they study by reproducing their notes. Then, when they sit down for the exam and are faced with problems that apply the ideas in novel ways, they have no idea what to do. They panic. They do poorly. They proclaim that they are “not math people.” They switch to a philosophy major. </code></pre> This may well be true, but if it is, these students are setting themselves up for a fall: if they wish to be any good at all at philosophy then they will need to cultivate precisely this skill. Much of philosophy consists of taking a general set of tools (concepts) and applying them to different situations. It isn't terribly fruitful approach unless one understands those concepts in the first place.

What is described should be effective, but seems like too much work to me. I prefer the advice I offered in <a href="http://www.perlmonks.org/?node_id=70113" rel="nofollow">http://www.perlmonks.org/?node_id=70113</a> instead. Besides, you get the benefit of looking like a really lazy genius.

This article is devoid of interesting content. The central thesis is: "In order to understand something, you need to develop <i>insight</i>. Well, duh. It does not explain how to develop 'insight', it does not explain what constitutes 'insight' and it does not even justify that the main example, visual representation of a derivative, results in 'insight' and results in people having an easier time grasping and using derivatives. As I am skeptical the example actually grants that 'insight' (it may seem obvious to you and me, but that is the trap we must avoid. The point is that it <i>isn't</i> obvious to many), this article did not add anything to my understanding of 'teaching' or 'understanding' at all.

The article is correct in the insight part, however it misses(or doesn't state explicitly) the fact that insight comes from the definition of the problem, what you are trying to do. Every concept is a solution to a problem, and to get the insight, my approach is to go through a list <i>What am I trying to accomplish (this is where visualizing comes) </i>What are the other ways of doing it <i>How does this method work, and why </i>Why is this method better then others *Where will this method not work.<p>After this solving any problem in the problem set is a piece of cake. Reduce the problem to subproblems, check applicability of the concept to the subproblems, apply the method, enjoy the result.

Like most study tips, these do not suit me.<p>From what I know of Cal Newport, I expected data -- evidence that these techniques work better than X. I thought Dr. Newport might even say X works better for group A and Y works better for group B.

That is exactly the reason why studying technical subjects is such a joy! Once you grasp a concept, you don't need to learn anything by heart (which I suck tremenduously at.)

This article is very helpful, but in all honesty, I think the formula is slightly more valuable than the picture. They're both essential, of course, but the formula is really the key to understanding the concept.<p>This is said, of course, from the vantage point of someone who has studied calc with 3 variables and so on, so may not have the fresh perspective.

Erm... how to do well in technical courses - understand it. Profundity squared. Not.<p>Author is missing the fact that yes, you need to understand it, but you also need to practice the exams. The further into your university years you get, the more your basis in understanding becomes valuable, but you still need to practice the exams.

A friend and I like to put it this way: after you learn information, you need to <i>compress</i> it. Find a way to make it take up as little brain space as possible. Then you're more likely to retain it, to understand it, to be able to use it.

This is all well and good, until the day when you decide to learn something on your own from a book, and encounter concepts defined in epsilon-delta form that make zero sense to a mind trained on intuiting concepts from graphs.<p>Visualization will not get you beyond 3 dimensions, nor will it get you understanding systems in terms of Lagrangians/Hamiltonians, nor will it give you the ability to read texts geared toward actual mathematicians.<p>Speaking for myself, it was surprisingly difficult un-learning the "slope of a tangent line" type of conceptualizing in order to understand math with sufficient rigor to be able to actually read math texts correctly.