Don’t teach people how to reach the top of a hill
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.
He was learning how to cut the books. You were learning how to teach someone to cut the books, a task in which you had no prior experience. Yes, it took two people and it took longer than working out how to cut the books yourself; but given what you now know, assuming your new hire suddenly moves away and has to be unexpectedly replaced, you would be able to teach someone else how to cut the books more quickly than before.
Teaching someone how to do the skill is a different skill to being able to do the skill, and it requires a more thorough conscious knowledge of how to use the skill than using the skill does.
You’re right, yet I think it’s still remarkable that it took longer to watch myself do it and figure out how I was doing it, than it took me to figure it out in the first place. For many types of skills, that wouldn’t be the case. I think the ease of discovery, rather than the difficulty of observing myself, made the difference.
It is remarkable, yes.
Even more than that: it requires the ability to communicate that conscious knowledge to the other person (thus, a two-place function). Each of us has our own internal “programming language” that determines which words or thoughts correspond to (e.g.) which bodily movements, and furthermore we tend to have our own specific repertory of bodily movements that we’re used to making without thinking, which may not be the same as another person’s. A teacher has to be able to bridge this gap—which, in particular, requires awareness of its existence in the first place.
(Analogues of this hold for less physical tasks, e.g. doing calculus problems.)
I can’t help think part of the difference is that they’re your books so you can do whatever you want to them, whereas he’s your employee and is being paid to do this the “right way”.
Also it would help to let them experiment on some worthless books, so there is no harm in accidental destruction. Just take some books other people are throwing away.
This is important. Understanding the different motivations for learning (and for action generally) is an important part of figuring out a strategy for teaching (and advising actions generally). Someone who wants to maximize a certain effect (efficiency and perfection of cut in bookmaking (or swordfighting)) is going to be very different from someone looking for a different success metric (paycheck and not getting yelled at).
Good point.
Am I the only one that reacted to voluntarily chopping up books with shock and horror?
“I’m digitizing my friends and family by chopping up their bodies and scanning them.”
“That way I won’t have to freeze them later”
X-D
For me, the shock and horror was cancelled out by the knowledge that the books would be digitized. Books are important to me only insofar as they are repositories of information; the destruction of books is only horrifying in the sense that a copy of that information is being lost.
Since, in this case, the intention is explicitly the preservation of information in another form, the downside (and therefore the shock and horror) is eliminated for me.
Not entirely for me, since digital media are a lot less robust than paper.
You make a good point; but one of the advantages of digital media is that it’s so easy to translate it into other forms. Hook up a printer, print and bind it, and you’ve got another copy of the book; which, as a bonus, has had its clock reset.
Of course, I realise one wouldn’t do that for most books due to cost; but combined with a good off-site backup policy, the possibility nonetheless exists that the books may survive longer.
Unfortunately, a lot of digital media have deliberately been encumbered to make it unusually difficult to do just that.
That is indeed a major problem. I had assumed, perhaps incorrectly, that when digitising one’s own books one would select a format that one could easily copy and translate at will.
Yes, but this requires continual active maintenance.
The answer is a good sturdy ROM.
I’m inclined to think that big companies and governments may already be doing this sort of thing, but since ROM is basically useless for consumers, we don’t see any of it.
If it’s not already being done, that’s a big project someone needs to get on.
CDs and DVDs are ROMs. Not as robust as paper, but then, you can’t usefully put audio or video recordings on paper anyway.
But a ROM that can’t be read by the naked eye isn’t a complete solution, as you have data formats and hardware readers to think of. There exists data that is fairly robustly stored, but no-one can read, because the support technology has moved on. Betamax tapes, Laserdiscs, Zip drives, floppies of various sizes. How many people can still read those? Even if you have the hardware, can you mount the file system and decode the documents?
Point. They are, however, nowhere near as robust as the ROM of old, and are often not truly ROM at all, so I wasn’t really thinking of them in that category. Technically, you are correct, though.
The same can be said of the written English language (or just language in general). I expect, that with time and patience, it would be perfectly possible to reconstruct the system needed to read a data format, just from the data format itself. Harder, certainly, with more layers of encoding, but by degree, not kind.
If we are attempting to preserve data beyond the point where the human race can look after it themselves, chances are that any information at all, regardless of storage medium, will require a fair bit of detective work, decryption, and translation.
True. It’s a trade-off between long-term survivability and short-term copyability.
I was doing this in the past to heavy-weight books for very pragmatic reasons: Gödel, Escher, Bach is worthless to me if the book would just sit around in a corner, but to take it with me and read it while commuting, I had to get the weight down.
Since learning this amazing trick, any book I read and that’s inconvenient to hold has to fear the knife.
By no means! Some books in particular I’ve been very hesitant to cut up.
Definitely not. I was reminded of Rainbow’s End.
It was my gut reaction of about two seconds. At that point that I remembered Friendship is Optimal and chuckled internally at my amusingly illogical double standards.
As long as information and utility are both conserved, and ideally increased (in proportion to the entropy expended in the process), I really see no problems intellectually, even if I dislike the thought of mutilating books on principle.
I was mostly amused.
A couple of points:
(1) Indeed, many times during the instruction process I found myself thinking “I wish I could just experiment!”.
(2) However, there were at least two specific pieces of verbally communicable non-obvious information that proved to be crucial: (a) the idea of putting one’s finger on the side of the book near the spine to detect the pressure of the blade and thus determine that the book is in place; and (b) the idea of preventing dislocation of the book simply by cutting sufficiently quickly. I’m not sure I would have discovered these things myself very quickly at all.
When teaching mathematics, it is good to give students a chance to discover some things before you tell them the official solution. Discovering something on your own makes it easier to remember; and in case you forget, it is more likely you will rediscover it. Also it teaches you the meta skill of discovering things. And it feels higher status than merely memorizing what someone else told you.
That’s how I felt using Blender as a beginner. (It does not help that the left mouse button does some weird thing, probably depending on context. No, it’s not selecting things and dragging them.) :D
Related: “Science” as Curiosity-Stopper. With the addition that it’s not just “Someone else knows the answer”, but also “And you are doing it wrong”.
Do you own many books for which digital copies don’t already exist?
I think this is a useful idea, although I’m not sure how useful this particular example is. FWIW, I definitely remember this from revising maths proofs—each proof had some number of non-obvious steps, and you needed to remember those. Sometimes there was just one, and once you had the first line right, even if there was a lot of work to do afterwards, it was always “simplifying in the obvious way”, so the rest of proof was basically “work it out, don’t memorise it”. Other proofs had LOTS of non-obvious ideas and were a lot harder to remember even if they were short.
I have spent quite a lot of my life writing specifications for software. If you actually want sensible results, you need to be able to get across what goal you are trying to accomplish and why, then let the programmers figure out the how. Trying to specify the actual result in detail would mean writing the software outright. The only complete and unambigous specification for a program is its source.
This seems similar somehow.
On a similar note, I’ve heard that professional golfers fear teaching another person, because doing so can ruin your game forever. Fine motor control is pretty much impossible to put into words, and whatever they decide to give as instruction they are tempted to follow themselves.
I think you got wrong what sort of things are easier to learn/do than to teach. Anything done primarily by the subconscious could well be forever out of the understanding of your conscious. If you can’t understand how you do something, how can you expect to teach it? For example, we’ve been trying for decades to teach a computer how to think, something every human can do but we all do it subconsciously. Nor can we teach another human who, due to localized brain damage, has lost an ability.
Note that our minds are massively parallel calculators that don’t necessarily require language, yet our instructions are sequential and language-based.
Not all teaching is sequential and/or language based. It’s just the most popular way to teach. Especially for people who believe that reductionism is useful.
NLP (Neuro-linguistic programming) is for example not taught in a sequentlial fashion if you learn it from Richard Bandler or students of Bandler.
That makes conversations about whether it’s backed up by scientific evidence hard, because the idea of science is that you can test sequential step to see if they work.
Care to elaborate on that? Edit: OK, I realize why I was confused by this. The act of instruction in a subject, as opposed to a metaphor for elements of thought as computer instructions?
Sure. Our brains contain millions of neurons working in parallel. Our spoken words come one at a time; thus the natural way to speak is one word at a time, one after the other, which in computer lingo is sequential instruction. While it is entirely possible to say thinks like, “the first thousand things you do are these, the second thousand things are those, …” I can guarantee you no human will be able to follow that instruction, not in the requisite number of milliseconds anyways. Besides which, instructions of this nature will also be out of reach of the instructor’s consciousness, so he too will be unable to understand how he does it.
Like Lumifer said, you can still teach such things, but you do it differently. You don’t explain how much to twitch each of the hundreds of muscles you have to maintain balance, you plunk your kid on a bicycle and steady the bike and let him figure it out on his own. Ironically, tasks like these that would be impossible to verbally teach or understand, are simple enough that you can do them without thinking about it.
By pointing out the path to be followed. Knowing how to acquire a capability is different from understanding how that capability works. Easy example: riding a bicycle.
At least the existence of this post will make “discovery” easier for the next person who has to do this task (if they know to look for it, at least). Perhaps there are some steps in the process that are best taught instead of climbed, or vice-versa, and the challenge is to figure out the right mixture?
(I recall a coding bootcamp I was a part of, where a careful balance of “look this up” and “ask the instructor” was required so that the instructor wouldn’t be overwhelmed and people wouldn’t waste an entire day fixing a chain of mistakes flowing from some trivial error.)