Mar 4, 2014
When is it faster to rediscover something on your own than to learn it from someone who already knows it?
Sometimes it's faster to re-derive a proof or algorithm than to look it up. Keith Lynch re-invented the fast Fourier transform because he was too lazy to walk all the way to the library to get a book on it, although that's an extreme example. But if you have a complicated proof already laid out before you, and you are not Marc Drexler, it's generally faster to read it than to derive a new one. Yet I found a knowledge-intensive task where it would have been much faster to tell someone nothing at all than to tell them how to do it.
I'm digitizing my books by chopping off their bindings and scanning them. I recently hired someone to do the chopping, and have been teaching him how to do it. The first step is to chop the book into sections of about 50 pages each, separating them at the binding. I do this by placing the opened book cover-down under a paper chopper, and cutting it precisely where the two opened pages meet.
The "chopper" is a manual paper-cutter with a 15-inch steel blade that weights about 10 pounds and is razor-sharp. If the blade is a fifth of a millimeter off its mark, it misses the gap between the pages and makes the cutting much harder, as it must go through paper instead of only glue. Being an entire millimeter off makes the blade catch the page maybe half a centimeter further away from its edge, depending on how the base of the page is angled, cutting off words and ruining the book. You can't see where the blade touches the book while making the cut. You can look before making the cut and position the book, but then you need one hand to operate the blade, and the physics of a book that wishes to spring shut, fall away from the blade, and fall onto one side, make it nearly impossible to keep the groove in the book in place for the blade with just one hand, unless you hold it with your fingers underneath the blade, which you can do only once.
My technique is to do this:
It's more complicated than that; I'm simplifying for the sake of space. But I didn't realize any of this when I began teaching him. I told him to put the book face-up under the blade and cut it into sections. It takes me a few seconds to make each cut. It was only when he kept trying to do it, and it kept not working, that I realized there must be more to it. He would try to chop a book and it wouldn't work. I'd look at the book, figure out what went wrong, then chop another section from the same book, watching myself, until I figured out what I was doing that could make the difference. Then I'd tell him. He'd try again, and it still wouldn't work. After perhaps 3 hours (6 person-hours), we worked out the sequence of steps I was doing well enough that he could chop books.
I could ask how I learned all those steps without knowing I'd learned them--was I conscious of them at the time, but forgot each step as soon as it was committed to my body? Probably. And it's interesting that I was unable to extract my own knowledge without watching someone else fail. But my point in this essay is that it took me longer to teach him how to do it than it took me to learn how to do it on my own--and it took 2 people instead of 1. So teaching was less than half as efficient as just handing him a book and walking away. (I'm ignoring the risk of coming back an hour later to find the floor strewn with severed fingers; that's overly particular to this domain.)
He kept worrying whether he was "doing it right". When I first figured out how to use the book chopper, I didn't know if there was one right way, or five, or none. I didn't have anyone to compare myself to. I could see whether I'd chopped the book the way I wanted to, but had no way to judge whether I was doing "above average", and so no self-consciousness about how well I was doing. Whereas he would see me take a book, slide it in, and chop it correctly, and then he would spend minutes fiddling with it, bending down to look under the blade from each side, swapping left and right hands between the two sides of the book and the blade, raising and lowering the blade, ad nauseum, until he finally tried to cut it--and inevitably got it wrong. He was nearly disabled by frustration and a sense of incompetence, and his actions were the anxious, tentative movements of someone worried about "doing it wrong" rather than the rapid movements of someone trying to find out whether there was any way to do it at all. We often hear the inspirational advice that believing something is possible makes it easier to accomplish; yet I saw just the opposite here. I didn't have my self-image on the line in my initial discovery process because I didn't know whether my task was possible, so I felt no pressure.
The task I originally faced was to find any path through a very large space that would end up with a book cut where I wanted it cut. The task the two of us faced in teaching him was to observe me chopping books, over and over, until we could find the one path I had discovered and forgotten. It isn't obvious which of these tasks is easier. In "discovery", there may be many possible solutions, while in "imitation" there is only one.
The psychological component probably applies to every search space: Availability of experts and the belief that there is a right way to do something inhibit experimentation; focusing on imitation prevents discovery. But what was it about the search space for book-chopping that made experimentation simple enough, and imitation hard enough, that imitation was harder than discovery? My guess is it was these things:
Roughly, it's a task in a search space on which hill-climbing works well.
Contrast this to martial arts, in which the movements of two fighters have a much higher dimensionality. The fraction of all possible movement sequences that leads to a side-kick or a hook punch is so small that few boxers ever discover the first and few karate students ever discover the second. Or contrast it to mathematical proofs, which are very high-dimensional, may have key insights (discontinuities), and give little indication of whether one is making progress. Those are domains in which instruction is more useful.
Think about computer software that you had to read the manual for. I think first of Adobe Photoshop and its concept of layers and selections. Those are complex, broadly-applicable concepts (discontinuities) that you can't easily discover by experimentation, as clicking on things before you understand them will make apparently random things happen. A user interface for something casual (a game, a website) or meant for the mass-market should have an event space on which hill-climbing works well, so that instruction is not needed.