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:
- Face the chopper with the blade's hinge opposite you. The chopper has a square base. If you're facing north, the hinge is at its northeast corner; the blade will cut along the eastern edge.
- Slide the book, cover-down, under the blade until the blade is over the binding.
- Lower the blade until it almost touches the book.
- Grasp the book with 2 hands, one on each side of the blade. Lift the book in the air to touch the blade along its entire length. (The edge of the blade where the cut begins is lower than the other edge.)
- Slide the book back and forth until the groove between the pages locks into place against the blade.
- Lower the blade until the end that starts the cut is pressed firmly against the book, which is pressed firmly against the cutting board.
- Press your left fingers against the pages to be cut off from the book, pressing those pages up against the blade along its entire length and keeping the groove of the book in place along the blade. You can feel the side of the blade through the pages, but the blade is now too low for your fingers to get underneath it.
- Stand with your head and shoulders directly above the blade. DO NOT raise the blade while repositioning yourself.
- Punch downward with the blade while simultaneously falling on it with all your weight to make a cut that is too fast to grab onto the paper and pull it out of place.
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:
- The task is analog/continuous, concerned with movements in space, so that it can't be specified precisely.
- The task is procedural, and almost all of the teacher's knowledge about it is "motor memory", not conscious.
- The search space is low-dimensional, because except for the final act of cutting and the importance of keeping fingers out of the path of the blade, every action involves only the book and the paper chopper. The book and the chopper each have one degree of freedom, and the book can be moved in space, and that is all.
- There are no difficult insights, special sticking-points, or especially-valuable insights that could be applied repeatedly (discontinuities in the search space).
- There was little back-tracking in rediscovering the steps. So there are few local maxima.
- Failures are easy to analyze; the next step in the process can be discovered by analyzing the previous failure. Also, steps 3-5, 6-7, and 8-9 are mostly independent; e.g., you can discover steps 8-9 before having 6-7 completely worked out. These properties allow the task to be learned incrementally.
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.