22 April 2012

On teaching

From time to time, I write some thoughts about education issues. Not because I'm a great pedagogue, neither because I think I have the best solution to the problem. (Not even because I think that students are bad. Such cliches are just pointing towards an easy way out of the door. In fact, us, teachers are more responsible than students.)  But only because, nowadays, and in many places, "to teach" has become equivalent with "to teach recipes".

I will, from time to time, expand on the subject with my thoughts and examples from personal experiences which, trust me, I've had many, and many unpleasant ones.

There is one thing I take as an axiom (fully justified of course), from which I will never depart, and upon which I have always based and will always base my teaching. Namely:
There is no teaching without explaining what is being taught.
This is a fundamental principle, the basis of everything else. If you disagree with it, then you are essentially applying a procedure which should not be called teaching. You are simply adopting a process which should have another name. Preaching perhaps; Or something less harsh if you wish. But the fact that the word teaching has been usurped by all kinds of classroom activities, does not enlarge the meaning of the word. On the contrary, these activities only provide counterexamples (and frequently good ones) to the meaning of the word teaching.

So, if someone demands from you that, say, you teach Real Analysis by only giving the students a set of ready-made formulae, then your response should be:
"You are giving me contradicting demands: if I deliver to a classroom a set of formulae and examples on Real Analysis, without explaining to them how and why these formulae work, then you are asking me not to teach the subject. Therefore your request implies the clause `please teach Real Analysis without teaching the subject', from which an obvious contradiction arises. As it is impossible to simultaneously do something and the negation of the something, you are asking me the impossible."
Such is the only rational response to irrational demands. It all boils down to analyzing the phrases into its fundamental constituents, taking into account the aforementioned axiom. The person who is asking you to teach without explaining is asking you to do something else. And he or she has to name it. If he or she insists in calling it teaching, please insist in that he or she is violating elementary rules of logic. (If he or she claims to be a scientist but he or she fails to see the contradiction, then matters are worse: this person is violating their claim to adherence to science and is exhibiting religious traits.)

I know that this is too abstract. I will come up with examples in due time.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant