C. Hinsley

19 August 2020

It often seems like some books I read provide great insight while others just don't. Intentions going in can be roughly the same between two books on such opposite ends of this spectrum. The topics can be similar. The motivations for the authors' writing can be similar. The motivations for my reading them can be similar. But what seems to set truly good books apart from those that I come away from feeling only marginally rewarded by is the level of difficulty.

When reading Walter Rudin's Principles of Mathematical Analysis, I had to slow down my normal pace and spend a great deal of time going through each proof. By the time I finished the first couple of chapters, I had challenged myself several times to deeply understand the given proofs, their motivations, and the concepts introduced. I probably learned ten to twenty brand-new ideas from those two chapters alone in the span of about a week, accompanied by ample practice with those new ideas to be able to teach them to someone else. Although the exhaustion induced per page of reading this book is easily an order of magnitude greater than from something I might normally read for pleasure, per unit of time I have obtained far greater benefit than usual.

This experience has been consistent across multiple books, blogs, and articles. If I read expository writing relatively quickly, covering concepts that fit neatly into my existing mental models, I rarely see a significant effect on understanding. However, if I read something that requires external assistance to understand, or which challenges or presents alternatives to my models, I am immediately rewarded by improved understanding and a certain lasting mental "freshness" that cannot be attained otherwise. It seems that the principle criterion upon which I should select expository reading material is level of difficulty — the text should reside firmly within my zone of proximal development (ZPD).


This may seem obvious — why would I pick books that I already understand? Oftentimes I overestimate the scope of a book that touches on a topic I want to dig more deeply into; this is extremely common in computer science, where there are virtually infinite books and articles providing surface level explorations of exceedingly deep topics. Otherwise, for a book which would appropriately challenge me, I might feel anxious before beginning it. Due to the prolonged isolating nature of reading, I'll have to spend many hours with a book (especially a hard one) studying alone. If the ideas are cognitively out of reach for me at present, I'll have to consult some external source. Because of the possibility of excessively frequent reliance on external guides, a natural anxiety arises wherein I feel that I will need to use crutches that may become unavailable or cognitively undecipherable themselves at some point. This leads to aversion of challenges, and is amplified by the substantial length of books. For instance, I felt uncomfortable before beginning to read Richard Sutton & Andrew Barto's Reinforcement Learning until I had made some headway in learning the basics of the topic from online articles and lecture videos, despite the fact that I would have been totally fine diving straight into the book to begin with.

What is the solution to this anxiety? Read more hard books. Figure out how to straddle the fine line between too-easy and too-difficult by trial and error. How do you do that? There are two options:

The former solution is troublesome because the books you miss the ZPD mark on will never become worth reading; they'll sit on your shelf, further falling behind your ZPD as you develop further and further in knowledge. The latter solution has a hidden benefit. Sure, you will (hopefully) eventually become able to read these too-hard books as you develop your skill. But you will also gain an understanding, once you begin reading these books and inevitably get stuck, of what trajectory you should need to head in so as to remove these barriers to your learning. In a way, over-reaching your ZPD can reflect back on where exactly your ZPD lies, giving you additional insights into selection for future book selections. And, of course, once you expand your ZPD by reading books that didn't miss the mark, your ZPD will necessarily reach books that you already have on your shelves, providing additional trajectory reinforcement as you develop your expertise.

So the solution is clear: Read hard books. If you're not having to stop every few pages and think fairly deeply about what you're reading, you're not reading hard enough books. Don't be afraid to buy books that you're intimidated by.