[EM] Method Definition Considerations

[Forest Simmons]

Reminds me of the professor who decided to mend his ways when a student
complained about not enough writing on the board for students taking notes.
At the end of a long explanation of quadratic reciprocity, remembering his
promise of more detailed notes, he says, "... which is as easy as one plus
one equals two," ... and then writes 1+1=2 on the black board.

For fpA-sumfpC ... I would have to say it came out of an attempt to
de-clone Copeland, where the fpA term was found empirically to restore
momotonicity to an educated guess that didn't quite work.... or something
like that. Not very inspiring to the students.

When we're lucky, once we get the right formula, we can see a shortcut to
it that would have saved us tons of time and agony if only we had seen it
earlier.

We cannot blame the students for not appreciating the shortcuts.

It reminds me of sci-fi stories about infrastructure that starts crumbling
centuries after all its engineers have died off.

Patience!



El dom., 5 de jun. de 2022 2:43 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 05.06.2022 03:14, Forest Simmons wrote:
> > Traditional math exposition, whether text book, lecture, journal
> > article, or monograph,  tends to have a top down deductive logical
> > structure that belies all of the messy trial and error scratch work that
> > accompanied the creative process.
> >
> > This was the style of Gauss ... unveil the finished work in all of its
> > polished, deductive logical glory, without any hint of the inductive
> > scaffolding or chisel marks that went into the finished work.
>
> This is a little bit off topic, but one thing I wished someone would've
> told me at university is just how difficult it is to come up with the
> one trick that makes a proof work.
>
> You tend to see (in mathematics/compsci papers at least) something like:
>
> "We wish to reduce minimizing the spectral norm of this matrix function
> of M to a convex second order cone program, which we can do by
> augmenting the input matrix M like so
>
>                0   I
>         W =    M   M^T - I
>
> and solving the associated optimization program, minimize f^Tx subject
> to ||Wx||_2 < b as follows..."
>
> There's then a brief proof of why just this choice of W works, but the
> authors might as well have pulled the transformation out of a hat. The
> problem is that this makes everything seem so easy: you just proceed
> through the steps and the intended relation falls out at the end.
>
> Then try to do it yourself in a novel setting and it's not so easy anymore.
>
> Maybe students just want to know how, but I would say any aspiring
> researcher would also want to know why... or at least in my case, know
> that it's perfectly normal to be hitting one's head against a wall for a
> long time before finding just *what* trick to use!
>
> More generally, some problems are much easier to phrase than they are to
> solve. (Particularly number theory, hence Barry Mazur's quote.) I think
> providing an intuition of just what kind of problems are deceptive or
> merely hard would be a good idea, although I'm not sure how to do it :-)
>
> -km
>
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