The purpose of this essay is to describe the improvements I've made in my ability to develop an intuitive understanding of scientific and mathematical topics by reading dense technical literature. The only authority I have on this topic is based on my personal experience as a reasonably intelligent late-bloomer who decided in his late 20s to stop working in the arts and pursue a new career in biomedical research. This has entailed years of full-time academic study with a brain in its mid thirties that no longer has the processing speed and memory I enjoyed in my youth. As such, learning how to learn advanced technical topics independently has, for me, been a necessity.
Both the scientific literature and the internet are full of interesting information on learning how to learn. In this essay, I plan to focus on the benefits of an approach to reading that is non-novel but, I believe, underrated among autodidactics, which I will call elaborative reading. As I use the term, elaborative reading entails that the reader give substantial time and attention to elaborating on the text. This elaboration is an encompassing term for a wide range of mental activities, including subvocalization and visualizing, inventing analogies, exploring their own confusion, looking up definitions, and identifying key nouns and verbs in the prose.
The primary aim of elaborative reading is to achieve success in the first step of learning. I define this first achievement as a level of understanding of the subject matter that is conceptual, but nevertheless detailed, technical, and directly useful in the activities associated with deeper learning such as memorization, problem-solving and application. An example would be the ability to read a Wikipedia article on a mathematical topic, that is new to the reader, such as penalty methods for solving constrained optimization problems, and being able to come away with a strong grasp of every aspect of the article. We might call this the first useful level of understanding.
I only claim that elaborative reading is helpful on the margin, to achieve a first useful level of understanding for an appropriate yet challenging technical text that might be beyond the reader's ability if they did not apply elaborative reading techniques. There will undoubtedly be a large class of texts that, for a given reader, are far beyond what elaborative reading can enable them to understand on short time scales. The aim here is to help a student who is struggling with the reading for a class they are qualified to take, an experience I have had all too often since I went back to school.
The techniques I call elaborative reading are nothing new, and most if not all readers of this essay will have heard them suggested as ways to improve learning outcomes. As with any self-improvement technique, it is building a habit of using elaborative reading that matters. My own habit has been established over the course of many years, fits and starts, trial and error, dead-ends and rabbit holes. I have written other blog posts, which I won't link here, that explored aspects of elaborative reading in a level of depth that I now think was probably excessive and ultimately unhelpful in the short run. What I think I can contribute here is describing what is, for me, the sweet spot: a way of integrating multiple elaborative reading techniques to learn efficiently (not "fast").
Here, I will define specific elaborative reading techniques, but this catalogue is intended primarily to make this highly subjective and deeply personal idea easier to talk about. Most readers do some of these techniques routinely, at least in some reading contexts. I do not claim this is a comprehensive list or that every reader can or should use every technique. In reality, I believe that success in their application likely depends on a highly personal, dynamic integration of a variety of these techniques in the psyche of an individual reader that seems possible to learn through practice, but impossible to turn into a cookbook for others to follow. My aim here is primarily to describe what individual techniques might look like, done to an extent perhaps beyond most readers' experience, and to get them established in readers' heads so that they can conceive of what their own personal version of integration might be like.
Visualization is forming an image in your head that's related to the subject matter. It could be a literal representation of a diagram, text, or picture, static or moving, sketchy or precise, scaling or rotating. It could be a visual analogy. It could be an imaginary environment, such as a mind palace, in which you walk to encounter particular imaginary objects related to your topic. When I was learning the amino acids, I spent a few days imagining walking along their atoms and bonds. For a very long time, I had conceived of myself as having little to no ability to visualize, so my ability to make my brain do this was a surprise to me.
I once tried not only memorizing the amino acids, but being able to memorize them by walking along a floating mental representation of them in a sort of outer-space imaginary environment. This did not really work, although I retain a general impression of which are simpler or more complex, key components of their structure, and a fondness and sense of meaningful interest in their structures that I didn't have before attempting to incorporate them into my psyche in this fashion. I believe that to truly achieve a lasting memory of their exact structures, spaced repetition with flashcards would be the most efficient way to go. But that would be beyond the first useful level of understanding, and so out of scope at least for this essay.
Taboo for speed readers, subvocalization is a perfectly good elaborative reading technique. I am a much more natural subvocalizer than visualizer, and when I can hear tones of voice in my inner monologue as I'm reading (or writing), I get more enjoyment from the experience. I find that subvocalization helps me pay closer attention to word choice, helping me avoid the constant risk of glazing over the stream of new terminology that incessently occurs in technical writing because my inner voice has to turn the squiggles on the page into a sound in my head before I can move on.
Subvocalization can also be distracting when I get too caught up in trying to imagine how the text ought to be narrated, although some attention to this can be helpful when it's getting at a deeper question like "what is the author trying to emphasize here, and given that, how should it be narrated?" It is also extremely difficult if not impossible to effectively subvocalize mathematical notation, and I find my past efforts to find ways to do so were largely ineffective.
As I use the terms, visualization is done with the imagination, while spotting is done with the eyeballs. Spotting involves dividing up the text into discrete visual regions, often words or word pairs or, in the case of mathematical notation, individual or small groups of symbols that go together. It is simply the textual equivalent of giving full attention to a particular flower in a large garden.
To me, spotting is a way of taking an inventory of the text. What are the important nouns and verbs here? Make a mental list of them as independent, for now arbitrary "objects" without worrying about their meanings or interdependencies at first but allowing them to manifest to your conscious in their own time. Simply treat the one or two paragraphs you are reading as objects worthy of close visual inspection in their own right. You are simply making an effort to know what literally exists in that location on the page. This doesn't need to be comprehensive.
Analogies are useful and dangerous. They can be a trap for novices by creating a feeling of understanding that is too shallow and non-technical to be useful, but leading the learner to stop investigating and start acting like they know what they're talking about. People also sometimes push an analogy past the point where it is an aid to understanding. They can also be poor explanatory devices, because they may encourage an audience to focus on the analogy as if it were the main topic, not just stage dressing for the presentation of the main subject matter. Anthropomorphization has similar risks, and also is a favorite flaw for pedants to pick on.
Nevertheless, I find A&A to be a useful device for my own learning when it stays in my own head, taken lightly and discarded freely when it's no longer helpful. After all, we ultimately want to transition to a literal, technical understanding of the subject matter, not stay mired in analogies. I find A&A especially helpful as an complement to spotting. Spotting shows me the important nouns and verbs in the paragraph. A&A brings them to life.
I find it challenging to literally put a text into my own words early on in the process of elaborative reading. At that point, I barely know what words are on the page, much less what different words I could use to describe the same concept. After all, I don't understand it yet! However, I regard visualization, subvocalization, and A&A as a generalized version of putting the language into your own words -- it's putting the text into new mental representations at a level that is immediately accessible to you.
I also think that "putting it in your own words" can be a trap. The less damaging trap is the tendency to reshuffle the words a bit, like committing a sort of lazy plagiarism in your own mind. I don't think this usually helps. I think that a potentially more damaging trap is when we try to B.S. our way through "putting it into our own words," trying to come up with a way of restating the material that's substantially different from the original, but reaching for a level of technical command of the concepts and language that does not match our actual understanding. I don't think that is a useful activity, and I worry it could even lead to self-deception.
My experience is that the early actions of elaborative reading gradually lead to the natural emergence of my own authentic words as I gain a clearer and more technical understanding of the material. You'll see this manifest in the example below.
Like "putting it into your own words," I often find background knowledge to be hard to apply in the early stages of elaborative reading, but to manifest intuitively and naturally as connections I make in later stages. I start by lacking an understanding of even individual words on the page. How then can I know what background knowledge is specifically, directly relevant to building the next micro-step of my understanding? I think that a trap related to background knowledge is similar to the trap of prematurely trying to literally put the material into your own words at a level of technical command that exceeds your ability. You can potentially make mistakes with your background knowledge both because that background knowledge is itself shaky and because your grasp of how to apply it to the new material is even weaker.
Here, I'll attempt to reconstruct my experience of reading the Wikipedia article on penalty methods. I just did this reading this evening. I read the entire article and developed a first useful level of understanding of the whole thing, but here I will focus on one sentence because I think it best illustrated how I integrated these techniques to achieve understanding.
When I read
For every penalty coefficient p, the set of global optimizers of the penalized problem, Xp*, is non-empty.
I imagined a wizard-like mathematical mentor standing next to me, holding a p in his hand. "This is a p," he says. "We can have one. Or many..." and his arms multiply like a hindu god, in every hand a p. A penalty coefficient? Sounds like a police officer. But it's a coefficient, a number, something we can attach to an equation. A global optimizer... I imagine it as a big machine. It's a set, so I imagine an environment with lots of these optimizer machines looking like spice crawlers from Dune. So we have a bunch of these ps, and a bunch of these global optimizers.
And the optimizers are "of the penalized problem, Xp*," so I imagine Xp* as a person or entity standing in the middle of all the optimizers. It says the set of optimizers is non-empty. So it's just pointing out that we can imagine that Xp* was standing there, but there were no spice crawlers (optimizers) around. Sure, that makes sense. And this is saying that that sort of situation will never happen.
What's the connection between p and Xp*? They both have a p, obviously. So there's a connection between all the ps in the wizard-mentor's hands and this Xp* entity standing in the middle of all these optimizers. Ah, I'd been imagining a single Xp* entity with its optimizers, but this is saying there are multiple Xp*s, one for each p! For each p, there's an Xp*, and a set of optimizers for that Xp*. And Xp* is a penalized problem, and p is a penalty coefficient. So this is saying that for each p, we have an Xp* and at least one optimizer for that Xp*.
Based on the material I read earlier in the article, Xp* is a way of reformulating an original optimization problem with hard constraints X* in such a way that there's a non-infinite penalty for violating those constraints. We're trading off against how harshly we punish constraint violations. Each different value of p corresponds to a level of harshness, which is why it's a penalty coefficient. The point of this statement is that no matter what value of p we choose, there is always at least one optimizer for Xp*. Now, the original statement is clear to me, and I can discard all the analogies, anthropomorphization, and rewordings and make use of the formal technical language going forward.
Later, I would realize that an "optimizer" here is not an algorithm for finding the optimum, but a value that optimizes the problem. The term "optimizer" is never defined in the Wikipedia article, and I consider it a point of evidence in favor of my application of these techniques being effective that I was able to deduce what the term means in light of my understanding I had developed of the rest of the text.
Reading the article on penalty methods and achieving that first useful level of understanding took perhaps an hour or so. I believe that as I continue to put elaborative reading into practice, I will become more efficient. However, I firmly believe that attempting to directly speed up the reading process leads to failure. We want to read quickly in order to build a useful level of understanding more quickly.
I imagine that learning to do elaborative reading effectively is probably like learning how to swim a mile. It is highly achievable by an average person. It is not something most people do. It is easy to do it wrong and there are some dangers if you do it very wrong. A lot of the work comes down to just doing it and learning the mechanics of how your mind and body interacts with the challenge as you discover the techniques for yourself and learn how to activate and combine them in a way that works for you.
I hope that this benefits some readers. I also wrote it for myself as a way to solidify my own developing understanding of elaborative reading and how to apply it for my learning goals. Enjoy your intellectual journeys!
