Re: [Maxima-discuss] Internal representation of "Taylor sums"

[Eduardo O. <edu...@gm...>]

On Thu, 20 Jul 2023 at 13:44, Richard Fateman <fa...@gm...> wrote:

> If this is a solution, what is the problem?   Does this occur in some
> course you are teaching?
> Does it occur in some application?
> Thanks
> RJF
>
>
 Hi Richard,

most of the students in my courses are very weak, and many of them are
so weak that the idea of creating auxiliary notations to visualize
what some objects "mean" is totally alien to them.

In my course on integration and basic ODEs there are several topics
that I introduce by saying: in this part of the course you will learn
how to do some calculations by hand, and this will be practically
useless to you later, because after you grow up you will always do
these calculations using a CAS... but there are some parts of the
course that are useful because of their side effects - for example,
_most_ people who spend many hours doing integrations by partial
fractions by hand learn how to visualize polynomials in ways similar
to this,

          4*10^2 + 5*10^1 + 6*10^0 + 7*10^-1 + 8*10^-2
  --->  [ 4        5        6      . 7         8       ]

and this is something that will be useful in several places later.

I always present these auxiliary notations as something that is
totally optional, but what happens is that some students understand
these notations in seconds, and they start to use them; then the other
students see that these students are now solving problems that looked
huge very quickly, and they decide to learn these notations too - and
they lose a bit of their fear of auxiliary notations.

In the pre-pandemic times I always had time to cover trigonometric
identities in the course, and I showed the students that we could
solve something like this

  integrate(sin(4*x)^2 * cos(5*x)^2, x);

either in way that is most commonly taught, i.e., by applying several
trigonometric identities, or by exponentialization and demoivrization;
and with the right auxiliary notations the exponentialization and
demoivrization can be done by hand very quickly. In that specific case
the notation like "[ 1 0 -1 . 0 1 ]" for the Laurent polynomials on E
is not very adequate because the degree is too high, but other
auxiliary notations work well.

I hope that this makes sense...
  Cheers,
    E.