Archive for the ‘education’ Category

showing equations is not teaching

October 3rd, 2008

I'm going to describe something that you know very well, and that you do all the time. I'll describe it algebraically, so that we can keep it somewhat rigorous, like good teaching prescribes. Once I'm done, you'll know exactly what I'm talking about.

  1. c <= C
  2. w = (0 <= c < V)
  3. 0 < d < v
  4. p = {p1, p2, ..., pn}
  5. t = px

Got it? It has a colloquial name: doing laundry. Here's the same thing in words.

  1. grab a subset of the clothes in the laundry basket/hamper
  2. contents of washing machine equal to said clothes, but greater than zero, lesser than washing machine's volume
  3. contents of detergent compartment greater than zero, lesser than its volume
  4. machine has a set of programs
  5. duration of wash determined by chosen program

Here's the thing. If you understand laundry, and you knew that's what the equations were supposed to describe, you could probably figure out what's what. At the very least, you could come up with your own set of equations, and they might be similar enough to infer the original meaning.

But what if you had never heard about laundry, and all you got were these equations. Could you figure it out? No. You're just not that clever.

Now put yourself in the shoes of someone who's teaching laundry. You know laundry inside out, you can derive the equations at will. Laundry is the most obvious and trivial subject as far as you're concerned. Students come to your class, today's topic is laundry. You spend a couple sentences describing laundry. You explain it in words that your students don't understand. Then you present the equations. Then you go to lunch feeling good about yourself, passing on the knowledge and all that.

As it happens, not all the students latched onto the theory of laundry. Some are turning up, asking dumb questions. What is wrong with these people? How can you fail to understand laundry? You'd have to be dense. Geez, the quality of our freshmen really is plummeting. There's no way my generation was so thick.

the myth of childhood

June 5th, 2008

I'm always perplexed when people say things like you know five years ago I was into Britney Spears. Boy was I stupid back then *self deprecating laugh*. Why would you say that? Aren't your observations and conclusions from 5 years ago valid anymore? Surely if you had reason to like her music then, you still have reason today? Maybe it's not your favorite anymore, but why would you want to cut yourself off from your own past? By induction, whatever beliefs you hold today will be equally laughable five years from now.

This is the kind of thing people say about childhood. They say they didn't like something then that they do like now, and say they were just being silly. Parents say things like when you grow up, you'll see, as if childhood is just a protracted waiting room, where everything is fake, before you can actually step into the real world and trust your instincts. Or they say when I was your age, I thought so too, as if that's going to convince me that even though they apparently were completely wrong at the time, they have a grip on things now. Above all, it's as if some people have completely forgotten what it's like to be a kid.

The absolute majority of conclusions I reached in childhood still stand today. I had good reason to reach those conclusions at the time, it wasn't on a whim. Some of them need to be revised now and again in light of new facts, but my process hasn't changed at all. You collect data by analyzing the factors that effect certain results and if the trend is consistent, it's a no brainer. In essence, a basic scientific process, although less rigorous and precise than the scientific method. Sometimes groups of factors form so called network effects that change the perspective on a certain question, but that is relatively uncommon. In any case, as long as you're open to new facts, you'll always be on top of things.


It's fun when kids challenge adults. Physical sports obviously discriminate against kids for their size and inferior strength. But a sport like chess doesn't, and in chess kids routinely compete and win against adults. How do adults respond to that? That'a amazing! Why is that amazing? It's certainly rare that someone has such a wealth of talent in some particular field, but it's hardly unexpected. Kids are not intellectually inferior to adults, they just have far less experience and knowledge in their favor.

A lot of parents underestimate their kids and don't realize that they actually have another fully functional person. Kids hate being treated as children, because it's a euphemism for treatment that's plain insulting. Kids have to be smart, if for no other reason than the difficult environment they grow up in. The first thing you have to learn as a kid is how to live in a dictatorship. I mean, sure the dictator smiles at you and thinks you're cute (and them underestimating the opposition is totally in your favor), but at the end of the day it's your freedom that's on the line here. One of the first things you learn is that you can suck up all you want, but if they mean no they won't budge.

Any kid worth his salt knows that repeated success comes through deception. Kids have far richer lives than parents know about. If you want to do something they don't want you to do, you can get your way as long as they don't know about it. But once they find out and you get that question have you been doing this even though we told you not to? you don't need the Soviet ambassador to explain that those missiles in Cuba need to go.

This isn't a game, this is your life. What we do out of necessity. They have all the power, so you have to appear to act in good faith. But you have an advantage: you know how they think. If they tell you don't do this and you get caught, you know that the second time being told not to do something is perfectly safe. No repercussions will come from this. The condemnation-punishment pattern is very obvious, and perfectly easy to predict which straw will be the final one to bring punishment. Parents could mix it up and stop being predictable, but frankly they're not clever enough. :D

So when parents don't want you to have access to something they think they can put it out of your reach and that will do it. What they don't realize is that between you coming home from school and them coming home from work there are buckets of time to figure out how to get it. And what's more, you're far more motivated than they are.

Kids don't care about political parties, but they do understand politics very well. The question is: how do I get what I want even when I'm relatively powerless? Some kids are extraordinarily talented at this, they basically exert power through a battle plan of crying, sulking and nagging. And to exert power from a position of such disadvantage is an artform in itself.


Another thing parents don't understand is that kids love to use their mind, and that doesn't mean doing extra homework. This is just epic amnesia, seeing as how no parent ever filled their spare time with more homework. When a kid tells you I'm bored, what can I do? you answer why don't you read a book. FAIL. You know, this is your fault. Since you are the self-appointed dictator of this kid's life, it's you who's limiting his options.

Kids enjoy being mobile. They hate walking, they prefer running around, climbing trees, playing games. You're not helping when all you're good for is taking a walk and doing some talking. Kids have a lot of things that rank higher than talking, there'll be plenty of time for talking later.

Life is an epic of learning and using your mind. It starts with putting the right shape of block in the right hole and pretty much keeps going until you end up an office drone. Whenever possible, kids like to learn with their hands. This is why it's good for kids to have a lot of toys and a lot of different toys. It's the kind of hands on learning that we would love to do ourselves, but unfortunately you can't open up an atom and play with the pieces.

In physics class, the question once was: suppose you have a ball coming at you from the right and you want to strike it to place it straight ahead (ie. divert it 90 degrees), at what angle should you strike it? It was shocking to see that certain people rather inexperienced in the field of ball mechanics didn't know that the answer is: slightly to the right to counteract the existing motion. Meanwhile, I was embarrassed to give the right answer cause it was so obvious. Sports is just about the best physics lab you could imagine, studying dynamics hands on. Meanwhile, no kid would ever come up with the idea of the gym where you watch tv and pedal a bike that doesn't move.

So it's not a question of whether kids want to learn. They do. You just have to let it be on their terms. Remember what kids don't have enough of? Experience. So use yours and figure out how you can stimulate them. For that matter, if they figure it out for themselves, get involved. If it's a video game, ask what it is about the game that is challenging, what it is that keeps them coming back to it.


It isn't a matter of intellect. It's a matter of experience and above all, emotion. Small wonder, since emotional stability is what adults struggle with themselves. So don't treat your kid as a child, treat him as an equally intelligent person who needs encouragement and a push in the right direction. This is really easy, cause you have all the power, but you still have to understand this person's motives and intentions, or you'll probably fail. Do yourself a favor and use gravity whenever you can (ie. align your common interests).

Lots of mistakes will be made on both sides, so don't dwell on the past, just figure out what happened and learn from it. With great power comes great responsibility, so use it for good.

'no such thing as a stupid question'

May 2nd, 2008

Most people are reasonably discreet by nature. They don't feel an urge to flaunt their personality or draw attention to themselves that often. It's a fact of life that we live in an unfriendly world, amongst aggressive peers. If you stick your neck out more then you'll have to stand up for yourself more. Thus most people develop a (healthy? at least in terms of survival) tendency to not advertise themselves excessively, especially not facts they suspect their peers will consider weaknesses. And so when they find themselves in a classroom with 24 peers, they feel somewhat less than eager to declare ignorance about the current topic. Teachers know this, and they think it's unfortunate that the fear of embarrassment keeps people from learning. And this is when they declare that: There is no such thing as a stupid question!

This is a well meaning encouragement to dare to admit that you're ignorant, because in this room you're allowed to be. Unfortunately, it's also a misleading statement. If you've ever taken a class with a person who wasn't shy, but *did* ask a lot of stupid questions, you already know that a) there *is* such a thing as a stupid question and b) it is ill advised to keep asking them. A person who is either particularly ignorant or exceptionally obtuse is a real disruption to the thought process among people who can follow the material. Just in the same way that you wouldn't want someone to interrupt a movie every 5 minutes and spend 2 minutes explaining what just happened on the screen, it really destroys the flow.

The teacher is probably more tolerant of stupid questions than your peers are, but there is a limit to how much time can be spent explaining obvious things to an ignoramus, because after all the mission is to get through all of today's material. So stupid questions are obviously not appropriate in large quantities, whatever the commercial says.

Interestingly, the expression isn't it's okay to ask stupid questions, so no allowance is made for those questions at all. On the contrary, it redefines all questions to be of the not-stupid nature. Perhaps we should call them "smart questions". The not-stupid reader will notice that the result between that and admitting stupid questions is ultimately the same. Whether you're allowed to ask stupid questions, or you're not allowed to, but there are not stupid questions, it is permission granted to ask all questions. And the strange twist is just a little morale boost for you, an encouragement. We allow stupid questions, but your questions aren't stupid anyway, so don't worry about that. *wink*

As bogus as the expression is, is there any truth to it at all? It defines "stupid questions", some category that apparently must have been discovered by someone. If stupid questions form a subset of all questions, there must be another category that isn't stupid. So is it actually true that it's impossible to distinguish a presumably stupid question from a smart question? Why, that too is completely untrue, anyone who took the class with that stupid-questions-asker knows this.

So we know that a) stupid questions exist and b) you shouldn't be asking them. But here's the problem: how do you know if your question is stupid?

It seems to me that there is no general answer. If we take the literal definition of the word "stupid" we find:

characterized by or proceeding from mental dullness; foolish; senseless: a stupid question.

However, none of these assessments - dullness, foolishness, senselessness - are absolute terms. They take on meaning in context, and only then. So in other words, if you are a globally renowned expert in some field and you receive questions from people all around the world, people whose background you know nothing about, and with whom you've never interacted before, then none of those questions, no matter how elementary, can be stupid. Because it's impossible to infer "mental dullness", or "foolishness", or "senselessness" based on one question.

Wherever you have a congregation of two persons or more the accepted standard of discourse on any topic that comes up is decided within a few minutes, as soon as the participants negotiate an acceptable place to set the bar. Just how this happens is too complicated to cover here, but it's influenced by things like how socially dominant the various participants are, what they stand to win or lose by admitting to competence or ignorance and so on. However, once that standard has been informally negotiated, any questions visibly below the standard will be perceived as stupid.

Although the audience makes an instant determination about a question being asked, this isn't actually a correct assessment. Broadly speaking (although this departs somewhat from the dictionary definition), stupid questions can be divided into two categories.

First, there are ignorant questions, which betray a lack of competence about the topic. This is just an indication that the person doesn't have the same background as everyone else. This is actually less of a failing for the person in question, because you can't really blame someone for not knowing something they haven't had the opportunity to learn, can you? But it's still very disruptive to everyone else.

Second, there are questions that are by definition stupid. "Mental dullness" would be a failing to make the right deductions based on the known facts. So a prior fact "this chair is heavy", combined with a new fact "heavy things hurt when dropped on your foot" would make the question "what happens if I drop this chair on my foot" a stupid question. It would also be a foolish idea, which seems to me as being "mental dullness" in a case where the outcome is unfavorable to you personally. But you could also ask a different question. "But what if this happened on a Tuesday, would it still hurt?" That question makes no sense. It seems to me that "senseless" questions stem from a false conclusion somewhere in deduction, ie. that the day of the week has an impact on your physiological responses.

While questions due to ignorance are an obvious waste of time (depending on the degree of ignorance), questions due to "mental dullness" are socially accepted to a point. The real problem is that the assessment isn't accurate.

To determine whether a person is:

lacking ordinary quickness and keenness of mind

we would have to compare his performance to that of another person. In other words, given the same facts, will the dull person fail to make the deduction while the other succeeds? If so, a question that betrays the absence of this deduction would correctly be described as stupid.

But how to conduct such an experiment? People gather in a classroom from all corners of the city (just to keep it simple). If they attended different schools they would not have had the same curriculum. But even two persons with the exact same schooling does not guarantee that they will have absorbed the same facts. Perhaps one was paying attention while the other didn't, perhaps one was gone that day, perhaps one remembers this fact and the other doesn't, perhaps one never understood it while the other did. Memory tests conducted with groups of participants show that a 30 minute exposure to the same words, images etc produces vastly different recollections of what was seen.

So if we cannot stage such an experiment then we cannot infer dullness of mind, and hence the determination of the stupid question is undecided.

So it cannot be decided from the outside, but the person cannot decide this either. You can ask yourself the question "if there some basic fact that makes this question stupid that I'm not aware of?". If so, it will be judged a stupid question, but it's not stupid based on *your* known facts. And it's only after you've understood the topic that you can determine if it was stupid. If it turns out you were missing necessary information, then it wasn't stupid. If you weren't missing anything, then it would seem you mind was "dull". However, if you have a "dull mind" to begin with, then perhaps you see no anomaly in your performance that day.

The senseless question is an interesting case, because it originates from a false conclusion. What are we to make of this? Is it because you misinterpreted a fact and thus made a wrong turn, or did you have all the facts straight, but still somehow managed to deduce the wrong thing? That brings up the question of whether the mind is capable of making an incorrect deduction like that. Or whether you're guaranteed, having all the right facts, to produce the right answer. That is a common assumption we make when debating with people. We think just as long as we straighted out their warped world view, we can get them thinking straight.

So you can't tell if the question is stupid, and the audience doesn't know if it's stupid, even if it's obvious to them. Maybe that's why someone got really depressed and went into denial, postulating that there are no stupid questions. I guess that means there are no foolish or senseless questions.

the doctrine of systematic work

November 8th, 2007

It seems to me that many (perhaps most) teachers look favorably on methods that do not require of them any creative output. One of the favorite methods in an average teacher's repertoire is the principle of systematic work.

Systematic work is based on the old adage of practice makes perfect. This is what you say to little kids who want to be in the Olympics: you work hard and you can get there. And there is no reason to doubt that. The fact is that to be successful in sports requires an enormous amount of systematic work. However, there is one caveat that they don't mention - it only works if there is a limited amount of people doing this. If we all started working systematically, no matter how good we'd be, the Olympics wouldn't have space for 7 billion of us anyway. But I digress.

But if you look at it from a structural point of view, systematic work means brute force. So basically you work at it for as long as it takes, until it cracks. And sometimes this works. But it assumes that your understanding of the problem has some sort of linear behavior, so the more you work on it the more you understand. And that is by no means a law of nature.

It's much worse when the behavior is asymptotic. You can work on it forever and you still won't get it. To illustrate this, imagine there is a tree in your yard that is blocking your otherwise excellent view. You want that tree out of there. So you grab a rope and tie it to the tree. You take the other end and tie it to your bike. Now if the tree is a certain size, you can try all you want to pull it out of the ground, and the tree won't budge. Apparently, this simple fact is lost on some teachers. (For more on this read The Truth About Homework.)

Brute force often works, but it's the last resort and it's not the smartest way of solving a problem. To be able to work systematically in any sort of productive way you need to know more than just to work systematically, which is basically a synonym to the word repeat. If you find yourself in the middle of a lake, in a rowing boat, systematic work is no doubt your best strategy. But even then you have to set a course first. This is a question some teachers don't have an answer to. If they explain something and you don't understand it, then there must be something wrong with you and you need to "work harder". Since everything can be solved through systematic work, well there you go. You just need to help yourself.

What is worse is that sometimes you need to deliver creative solutions, either because the problem cannot be solved otherwise, or because that is the only way you can score a high grade. Now, creativity is the opposite of systematic work. I can easily imagine that systematic work originates from an earlier time when schools thought "teaching discipline" was an integral part of their business, and therefore if you're very disciplined you're going to be working systematically, it fits like a glove.

Now try combining that with creativity. Here I am working at something systematically and all of a sudden I have an idea "what if I tried to... no! I will not allow myself to be distracted, I have self discipline!" But it is interesting that teachers expect creativity when they themselves have no obligation to muster any. It's precisely those cases where something is hard to explain that you so desperately need the ability to think creatively and come up with a different description of the same thing. The best teacher I've ever had (high school, English) was also the most creative one I've seen.

give 3 reasons for ...

November 2nd, 2007

Having been a student for something like two decades I have come across many bad teachers, lots of broken approaches, numerous stupid ideas and several people who should not be teaching at all. One thing that continues to surface, which I thought I was done with after junior high, is questions of this format:

Give three reasons for the collapse of the Roman Empire.

It's one of those things that seems so stupid, and so obvious, that noone would possibly be doing it, right? Wrong, they persist with this.

Let me answer the question. I open my Roman Empire box. In there I have various smaller boxes, one is called reasons for downfall. This one contains files. So let's see, there are 8 files in here, but I only need three. Okay, now I can transcribe the reasons one by one onto the answer sheet.

Newsflash. Human knowledge is not stored in file cabinets which enumerate causes and effects. I have *never* found myself in a real life situation where enumerating three reasons helped me get something done. Why three? How about two and a half? How about three and a half?

To anyone who actually wants to know something one well-argumented reason is worth far more than three snippets, and it probably touches on several other effects in play. This is a far more natural way of expressing thought than to compartmentalize and enumerate little slices of knowledge.

What are we actually doing here? Are we learning or are we doing brain teasers? If you ask people to enumerate 50 colors that would also be challenging, because most can't think of that many. But what would be the point?

So it is obviously futile, but it's also harmful. If you have *ever* thought deeply about *anything*, then you know how complicated cause and effect scenarios are. There can be so many factors, so many causes than are predicated upon other causes and so on, which means that decomposing the entire problem in terms of this causes this is very difficult. We love to ask ourselves why? but we rarely find good answers to those questions, because the answers are too hard to understand.

What you are doing when asking for three reasons is tremendously trivializing the problem. You are creating the appearance that one could actually give three reasons and that would explain the whole thing. Can we not have more honesty than this? Explaining the collapse of the Roman Empire is actually a hugely difficult undertaking, considering how many people affected and were affected by it. And each of those had their reasons and interests at stake, and adding up all of this is not something you can explain in three paragraphs each stating one reason. Or put it this way. If you *do*, it's a completely meaningless answer.

So why do people do this? Since it's common practice, you don't have to give three reasons for giving a question like this on a test. But how did this start? Face it, it's a really easy thing to do for a teacher. They put very little thought into it and they move on. It's much more difficult to phrase a more complicated question spanning (say) two lines that entices an intelligent response.

Instead of focusing on the problem students are thinking:

  • Goodie, I remember those three paragraphs in the textbook almost word for word. I haven't really thought about what they mean, I only read them, but luckily they came up on the test.
  • Damn, I can only think of two reasons.
  • I have two good reasons and a third one, but I'm not sure if I can give the last one on its own, because it's not "enough" of a reason by itself, I think.
  • I have three reasons, but two of them are triggered by the first one, so does that qualify as three or just one?
  • I have five reasons, but I'm not going to get any more credit for that, because I can only give three.

I have found myself in all of these situations at one time or another. Particularly the first one used to happen a lot in junior high.

How do you fix it? Just as easy:

Give a reason for the collapse of the Roman Empire.

Or if you want to make it clear that you encourage "more than one" reason (however it is you distinguish reasons from each other), you can say.

Give a reason (or more than one) for the collapse of the Roman Empire.

If you don't want to be so obvious that you are obsessed with quantifying reasons, rephrase it:

Why did the Roman Empire collapse?

Congratulations. Your students are now thinking about the problem rather than about your idiotic requirements for the answer.

How does this change the question? It doesn't. Students know how many points they get for this question, so they can estimate how long a response has to be, whatever the format of the question. Meanwhile,

  • those who have extensive knowledge are more likely to go beyond what you expect and they may get more credit, and
  • those who don't know anything are less likely to cook up something on the spot based on three keywords they remember from the textbook, because it's much harder to incorporate those fragments into a sound argument than into three short paragraphs that are complete clichés.