Education: We could be all totally wrong. But then, again ...

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We could be right.

One thing young students complain about is that teachers don’t tell them everything.

Teachers don’t tell students “everything” for various reasons, not all of them good ones.  Sometime the teacher doesn’t know everything.  Sometimes the teacher doesn’t have time to provide complete and detailed information about everything; it depends on what the teacher sets out to do.  Sometimes the teacher just leaves something for the student to discover on his or her own.

It is the easiest thing in the world for a teacher to cut down the volume of topics that he or she sets out to supply the student with, and give complete and detailed information about those few topics.  Honestly, the pressure on a teacher to do exactly that is enormous, and many feeble-minded teachers who are more anxious to come away looking good than to provide students with material do indeed go that route.  You can just imagine a teacher saying: well, if I set out to teach multiplication all the way up to “16 times,”  a lot of the students are not going to be able to do it.  In fact, the larger the multiplication table, the less ‘success’ my students will have!  So I think I will stop at “5 times,” and do it really well.

Obviously, these are two extremes, and there is a lot to be said for stopping at, say “10 times,” which really gets the students ready for long multiplication some day in the future.  But the example was just an analogy; the less a teacher undertakes to teach, the better the teacher can perform the task of teaching.

All this is, in principle, the problem of curriculum.  How much to teach, and what choices to make, are ultimately best left to the professionals, who understand the pros and cons of each choice; it is best if parents stay out of those decisions.  There’s nothing to prevent a parent from supplementing what a child learns at school with more information at home; this is a wonderful thing you can do, if it is done judiciously.  I remember my mother teaching me the idea of algebra, and solving an equation when I was about 10, and my teachers were, justly, furious, when I used that method to solve an “inheritance” problem, which began something like: “A rich man died, and left a large fortune.  One half was given to ... ,” and the problem ends with “... and the parakeet got $20.  What was size of the original fortune?”

In my opinion, there is a great benefit in (1) teaching considerably more than you can teach perfectly, and (2) leaving large gaps for the student to fill.  As my colleague puts it, the volume of information out there is truly vast, and getting more vast every second; our students must learn to be “lifelong learners.”  What we are teaching them most importantly is how to teach themselves.  The older we get, the less someone is going to sit down and show you exactly what is going on.  That explains (2).  But what about (1)?

Some of the ideas that young people have to be presented with in these modern times are very complex, and very subtle.  Understanding, if it comes at all, comes in layers; certain ideas must sink in, before certain other ideas can begin to make sense, and usually for some of these ideas and concepts, the time is longer than one semester.  Unfortunately, the response of certain members of the teaching profession to this problem has been rather defensive.

“If you can’t teach the entire thing all at once,” some would say, “wait until the student is ready someday in the future.  I like to make sure that if I start an idea, I can completely finish it.  It’s very dissatisfying to the student to be taught a mere glimmering of an idea, and then be left waiting for the rest of it.”  But wait: is the whole point to make things satisfying to the student?  I tend to think that student satisfaction is only a small part of the equation.

“Another way to do it,” more thoughtful teachers would say, “is to introduce some new halfway idea, and explain that partial idea completely, so that the rest of the thing can be built up from that partial idea.”  But what if that partial idea is unsatisfying to the student?  Well, to these teachers, teaching a synthetic partial idea is satisfying to teach, even if it is unsatisfying to the student, for whom the completion of the chain of reasoning doesn’t come until much later.  So the curriculum is often cluttered up with these halfway-houses invented by creative teachers, who plan to complete them someday into the more sophisticated concept that they cannot deliver right away.  But this scaffolding that they have dreamed up might not be recognized by the people who are in charge of the student’s education when the time is right.  This modularization of the curriculum, as I call it, has its pros and its cons, and this is one of the cons.  Teachers invent vast masses of half-concepts that they teach students, so that some day the concepts can be completed.

At any rate, the business of touching upon ideas that cannot have all their loose ends tied up right away is an unavoidable thing; some teachers are just better at giving the impression that the loose ends are tied up, but they’re not; they’re simply closed off with color-coded masking tape for a future teacher to deal with.  This is widely considered today as a good thing, since untidiness of any sort is frowned upon.  Honestly, if you can’t have sophistication, tidiness is the next best thing.  We must simply not mistake tidiness with sophistication.  As you can readily see, this leads to a great deal of what can only be described as deferred gratification for the student.  That is part of the price you have to pay to learn science, which is more of an edifice than the Humanities or the Arts.

I touch on a huge spectrum of ideas that there is simply no time to deal with in complete detail, which frustrates the students in my class who simply want to learn enough to past the test, and escape from my clutches.  They tend to grill me closely about ‘exactly what is the test going to be like?’  The unspoken question is, what are the actual questions going to be, and what are the answers?  I often give what are called review sheets, which are intended to be practice questions.  But students would rather be told that they’re getting a sample test.  I put at the top of the review sheet: There is no relationship whatsoever between the form and the content of this review sheet, and those of the test you are preparing for.  What is the point of this detailed cross-examination of the teacher?  I have taught them 42 ideas---at the very least---in 42 class meetings.  Why not study all 42?  Oh, no; I must tell them the five most important things I have taught them, because they have no time for the rest.  Modern kids are busy executives.  After all, if busy executives only have time for the five most important things anyone has to tell them, why not kids?  After all, their minds are smaller, right?  And they have to get ready to be busy executives, after all.

Why do I deliver such a large volume of ideas without doing a proper job of it?  A lot of it has to do with connections.  Facts anyone can understand; the harder thing is to be aware of the connections.  I would be only too delighted if my students were such avid readers of magazines and all sorts of stuff they come across as I am.  For instance, I tell my trigonometry students that sines and cosines are used in the compressed music files they listen to on their MP3 players or Ipods, and in the pictures they browse on the Internet.  They look at me blankly.  How, they wonder, can you possibly use sines and cosines for pictures?  I try to explain the structure of a musical sound, and what distinguishes the sound of, say, a flute from that of an oboe or a clarinet.  I go on to describe how the sound, for a fraction of a second, can be represented by a formula that consists of an infinite number of sines and cosines added together, with other numbers weighting them.  We can save a list of a few thousands of these weights, and then go on to the next moment in time, and do it for that note, and so on.  This sampling is done 44,000 times a second, roughly, and that’s essentially what is stored in the MP3.  (The miraculous thing is actually how these numbers are used to reconstruct the sound; that was the genius of Fourier, who pioneered the idea.)  A similar thing is done for image files, and that is how Jpeg files are made.  You can look it up: it’s called the Discrete Cosine Transform.  If a kid knows how to Google something, he or she has got started on the most basic method of self-education.  But that’s just the first step; you must go on to learn how to filter out the useless information quickly, before you lose interest completely in your question.


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