“Those who know, do. Those that understand, teach.”
I currently have close to 12 posts at various stages of completion in draft form awaiting inspiration, requiring polishing, or just plain not working; destined for a twilight like existence in some type of blog post purgatory.
What is important to note is that these posts all have a sketch of the intended content, and looking over some I am surprised at how well formed they actually are, and it is a great resource for ideas and thoughts that would take me a long time to reconstruct without these sketches. The sketching of arguments is a fantastic time management tool, in fact it is probably the single best time management advice anyone could ever receive. Tao gives valuable advice in this post about batching low intensity tasks, and other gems, but if you sketch all those subtle insights that you stumble upon, the snowballing effect of the increased comprehension and ready stock of ideas is hard to describe. It is the best productivity boost I know of.
Terence Tao is considered one of the greatest mathematicians of his generation:
– Charles Fefferman, Mathematician at Princeton University
He is not only an exceptional mathematician, but he seems to be incredibly generous with his time in posting expository articles on his work, and more general advice based on his undeniable successes. I came across his blog recently and have devoured multiple articles, and even purchased a few books based on his blog; I found Compactness and Contradiction particularly good. [His two volume treatment of Analysis is incredible]
Terence Tao comments on the effectiveness of writing down sketches of your arguments and insights on a particular topic, in that it clarifies your thinking, which in and of itself is invaluable, but more importantly it saves time, and literally saves the ideas themselves. To quote:
There were many occasions early in my career when I read, heard about, or stumbled upon some neat mathematical trick or argument, and thought I understood it well enough that I didn’t need to write it down; and then, say six months later, when I actually needed to recall that trick, I couldn’t reconstruct it at all. Eventually I resolved to write down (preferably on a computer) a sketch of any interesting argument I came across – not necessarily at a publication level of quality, but detailed enough that I could then safely forget about the details, and readily recover the argument from the sketch whenever the need arises.
I have presented this advice to others, but with a slightly different emphasis. The absolute best way to learn a topic, is to write an expository article.
If you think you understand something, even something not particularly complex, just spend 15 minutes sketching an “Idiots Guide to …” article about that topic. I will be willing to bet a full English pound you will either hesitate, stumble, or discover something novel, a new perspective; every time I do this (and I have started to do this for anything of note) I find new connections that were only dimly realised previously, and it can spark creativity! The most important outcome is that you will have clarified your understanding of the topic in question, and have a readily available sketch, that will aid future comprehension and consolidate your learning on that topic. It is very satisfying browsing my /sketches folder (with many sub-folders organised by topic) on my trusty Ubuntu laptop to see all my recent learning. [Note: I write my notes in a standard TeX AMSMath template. For a very good introduction to TeX see this youtube series – no really this is the best first exposure to TeX I have ever found! I cannot recommend familiarity with TeX highly enough, I personally find it such a pleasure to write in that it is no longer even mildly a chore to write these expositions, and the output is beautiful. There is also a TeX to WordPress converter that I will most likely be trialling in my next post.]
I recently sketched an overview of asymptotic notation for example, using Knuth‘s TAOCP Volume 1, and his collaborative work (with Graham and Patashnik) Concrete Mathematics that I will turn into my next blog post to show you how these sketches look. I am consistently surprised that this simple exercise has such a palpable impact on my understanding of a topic, and provides a physical artefact that somehow makes everything infinitely more memorable.
To see some of Tao’s expositions, visit https://www.math.ucla.edu/~tao/preprints/Expository/
My best advice for time management – write down what you know.
My best advice for effective learning – write down what you think you know.
I am always doing that which I cannot do, in order that I may learn how to do it.
– Pablo Picasso