How to Ace Calculus: The Art of Doing Well in Technical Courses
calnewport.com
calnewport.com
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".
(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.)
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.
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.
This doesn't come without effort.
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->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.
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 any 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.
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.
[1] http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_...
For references, calculus was first published in 1684:
http://en.wikipedia.org/wiki/History_of_calculus
But the epsilon-delta definition wasn't formalized until 1817:
http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_...
In this retelling, the point is not the visual metaphor as an end product, it's the work needed to reduce the issue to its basic elements and their relationships.
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.
The first reason is that these were first-class leading-edge thinkers we were trying to understand. Aristotle is very densely packed with content, and we had to read and re-read passages to penetrate them. (Some think that the writings we have of Aristotle are actually class notes, and may not be a good representation of his lecture style.) The second reason is that even though Computer Science is an abstract subject, code is more concrete and graspable than a philosophical idea because you can compile and execute the code and see if and how it works. Philosophical ideas are either difficult to test (e.g. ethics) or practically impossible to test (metaphysics).
However, I'm willing to bet it was just snark.
I never asked why. They both were having trouble in the Intro Algorithms class when they bounced.
Maybe people that think they are CS/Math people and then find out that they might not be CS/Math people find solace in Philosophy? Maybe what they liked about CS/Math was the logic more than the algorithms.
http://www.comlab.ox.ac.uk/admissions/ugrad/Computer_Science...
It's surprisingly common in my personal experience.
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.
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.
Pattern, and self-attempting. Practise makes perfect too.
To me the key is to observe that concepts are important then observe that the easiest way to learn concepts is to learn them at the optimal time of day (ie, not 8 am during the class when you are tired, that is a waste of time), from the optimal person (ie, not your prof that just wants to get back to research), during the optimal time of the term (ie, not in the first week of class, more like the week before the midterm and the 10 days before the final). To achieve all of this just requires two skills: 1. Knowing how to learn from a text book 2. Knowing when a textbook is crap and getting a better one from internet review sites.
Learning from a text book is easy. Cover the page with a piece of paper and read each line. When they come up with a problem that you don't have a function for derive it and bam, you've invented the formula for lateral-torsional buckling of non-uniform crossectional beams you will never have trouble with the concept again. If you get stuck (stuck to the point of it hurting your ego, not "I'm sure I would get it if I had the time" stuck then look at the formula (not the proof if you can avoid it). Try to prove it again! If you still can't prove it, find the proof somewhere (hopefully the text, but if not email your prof or the book author for the proof). I corrected the same (otherwise super awesome) text book 3 times over the course of two years. The author loved me because out of the 5 corrections he did 1 was from himself, 1 was an email that said "I think this is wrong" and the other 3 were from me proving that he was wrong.
Anyways got kind of long there, but don't waste time learning from other people, just learn how to learn and damn well get the concepts otherwise what's the point?
EDIT: A colleague of mine, a great teacher by the way, once phrased it: "You've got to love your students. It's as simple as that."
The most valuable part of the article for me is where he points out that a lot of hard-working but unintelligent students write copious notes without ever doing the mental gymnastics to understand what it is they're writing down.
http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen...
which she has found very helpful.
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
http://www.amazon.com/Vision-Elementary-Mathematics-W-Sawyer...
http://www.marco-learningsystems.com/pages/sawyer/Vision_in_...
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.
http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...
The acclaimed books on Calculus by Michael Spivak
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...
and Tom Apostol
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...
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.
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
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.
http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf
http://www.ams.org/notices/199908/rev-howe.pdf
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.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
In light of recent HN threads about Khan Academy,
http://news.ycombinator.com/item?id=2348476
http://news.ycombinator.com/item?id=2350430
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?
This video by Harvard-educated cognitive psychologist and professor Daniel Willingham is relevant:
I think a quick-and-easy way to get feedback is essential. For some lessons there are practice problems, but for others, a student who wanted to maximize learning/minute spent watching video would be wise to at least open up Excel or something.
Some people just don't visualize. Not even a little bit. And I'm not sure its "just because they never learned to." Myself, I've always seen the 'picture' in my head and even dream in full technicolor (like this means anything) but my wife of 20+ years just can't. She is definitely smart, graduated with a CS degree from USC and is a much better planner than I will ever be, but those questions where you see a flat piece of paper with a bunch of dotted lines on it and you need to guess the shape it will be if they were all folded, just can't see it.
When I was growing up I used to think they only put those kinds of questions on tests so that everyone could get a few answers right, they were just that easy for me.
So Newport's thesis that if you can visualize it you can gain 'insight' is no doubt true for some people, but it certainly isn't a panacea for teaching complex subjects.
If you've ever seen the online math courses that Stanford did [1] under the EPGY program, it has some excellent tools that seem to work well for a variety of learning styles. Worth a look, and just down right priceless if you're home schooling your kids.
As an aside, why do or did people claim there is visual learning aside from spatial learning? I don't experience visual and spatial imagination as different things. (With reasoning about time always assumed.)
Even that is a matter of personal preference. I honestly believe it's easier to get the concept of a derivative by linking it to instantenous velocity.
Here's a random example: Marsden and Weinstein define derivatives in their out-of-print textbook Calculus Unlimited without limits. The tangent to a graph at the point x is the boundary between two line pencils, one of lines entering the epigraph at x, the other of lines leaving. There's no limit-taking of chords. It's a simple and neat definition that connects with classical notions of tangency.
In his essay On Proof and Progress in Mathematics, Thurston lists a dozen other definitions or conceptions of derivatives in his personal arsenal, some very sophisticated. But even those among his definitions that are elementary and have roughly the same scope there is a difference in their psychological affordances, and that can make all the difference.
I'm pretty sure that those are distinct neurological processes, as revealed by the differing individual deficits that patients can have after suffering strokes. But I don't have the medical references at hand, and you have certainly seen many sources that combine writing about both, as I have.
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.
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 passing, not getting straight As.
[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.
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.
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.
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.
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.
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.
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.
What should be done is start with the intuition and visualization, and then show how the ideas can be made precise. If you would have been taught only the epsilon-delta form from the start, it would have been easy precisely because you would not yet have obtained the real understanding.
Even in high dimensional spaces visualization is very important. This often happens by analogy with lower dimensional spaces. For example if you start with a region in Hamiltonian phase space, then the region's volume is constant as time evolves. This is a highly intuitive and visual statement. Saying the same in symbols wouldn't be nearly as clear.
Additionally, some things like ordinary algebraic manipulation are very well-suited to linguistic abstractions ("multiply the polynomials, take the derivative, put all terms involving z on one side of the equation, apply the quadratic formula"). Sometimes only the linguistic abstraction can give the solution (e.g., "this problem is easy because the quadratic coefficient cancels out, and the equation is in the form t^3 + c t = d").
It's also worth noting that manipulating the linguistic abstractions takes a lot of insight and talent (e.g., knowing the perfect substitution of variable to make an integral fall into a known form, or knowing which one of the four error terms will be hard to control, and working on it first).
It's not wise to be over-committed to the visual approach.
The advantages of thinking this way seem to be a little like the advantages of test-driven development: time spent understanding representative concrete cases doesn't teach you everything, but it can eliminate many misunderstandings very quickly.
That said, I believe the above is for the people who will never get to higher level mathematics, but rather are struggling with first level calculus.