Re: General knowledge for teachers in mathematics

In article <4mdkq1h06fvh89rbvr68v45ppe4dqc8qvr@xxxxxxx>,
Guess who <notreally.here@xxxxxxxx> wrote:
>> Different, not separate. I don't think you can *understand* [tho'
>>you may be able to handle-churn enough to use] calculus [eg] without
>>going through some of the struggles that Newton, Leibniz, Euler, ...
>>went through [...]
>That is being rather pretentious and argumentative. What in
>particular would a student miss if he didn't?

It's not so much what he misses as what he gets: the false
impression that maths just *is*. That it's a rigid set of rules
that have always been there, that a high priesthood understands and
hands down to the peasants. I was talking to some physics students
a few weeks back; bright 18yos with good grades in maths A-level.
We were revising calculus, so I talked very briefly about tangents
and limits and slopes, and then reminded them of the standard
derivatives. It transpired that all of this stuff was completely
new to them *except* for the outcomes. So they could differentiate,
but had no idea *why* they were doing it, *why* zero derivatives
happened at maxima/minima, *why* sine differentiated to cosine.
That's what our education system is doing -- not of course to all
students, but to a significant minority, perhaps even a majority.

Earlier in the module, we had been doing vectors. Towards
the end of that part, I showed them Theorem 1 from "Principia" --
essentially "conservation of angular momentum". Newton has a page
and a half of geometry -- equal area triangles, impulses towards
a fixed point, stuff about augmented numbers of triangles whose
breadth diminishes "in infinitum" so that their perimeter becomes
a curve. Genius level mathematics. Then I showed them how it could
be done with vectors: d/dt (rxv) = vxv + rxa = 0 because v is parallel
to itself and a is parallel to the force which is parallel to r; so
rxv is constant -- first-year student mechanics, but the actual maths
is just the same if you compare in detail. I *hope* they came away
from that ten minutes with (a) an appreciation of Newton, (b) ditto
of vectors, (c) ditto of how maths evolves and simplifies, and (d) a
better understanding/intuition of all the principles involved.

> You do not have to
>read every novel to have a good command of the English language. You
>do not have to use the abacus, then slide rule in order to understand
>how the calculator works. There is simply not enough time in the
>world to go through all recorded history in order to understand
>mathematics as you seem to define it.

But that's not what I suggested.

> Following your logic, it would
>also improve their perspective to realise how the changing times
>developed the need for each aspect of mathematics.

Which it would, though it has nothing at all to do with my
"logic". Read Lakatos: "Proofs and Refutations".

[...]
>Make it an optional history course then if you wish.

Many universities do offer such options. But it's too
difficult for school maths, even for first-year university. The
point is not to get *history* into maths but to get *appreciation*
into maths. This works both ways -- so that mathematicians know
more about their subject, and so that linguists/artists/economists
[etc] know something about what maths can do for them.

> The application
>is enough of itself to provide a challenging exam.

Providing a challenging exam is not a proper aim of education.

> There is already
>argument that too much time is spent on inessentials, that argument
>being given as reason for the lack of knowledge of the subject.

And we see the results in the absymal level of maths in the
general population, and the lack of understanding of basic maths
shown even by good maths students. Which is actually more essential,
that students can solve difficult ODEs or that they understand what
such an ODE means? Especially nowadays when Maple or Mathematica
can do all the handle-churning that anyone could wish for? Gunnar's
original question was about the "general knowledge" of maths. Should
be be turning out maths graduates who know nothing at all of the
basic timeline of their subject? Who have never engaged with the
philosophical questions raised by calculus, or the parallel axiom
of geometry, or constructivism?

> Would
>you have the students first work with fluxions, and develop the
>subject from there as well, in order to have better understanding? It
>makes no sense.

It makes no sense to have 99% of the population see maths as
a mystic subject whose sole point is to rearrange symbols on the page
until the wizard says "yes", and most of the remaining 1% see maths
in the same way but with themselves as the wizards. No, I don't want
to *start* with fluxions. But I do want students of calculus to know
something about how that subject arose, and why, and who did it, and
what the main problems and objections were. Otherwise, there is no
motivation at all for all the delta-epsilon stuff. And we get into
nonsense like that dy/dx isn't a *fraction*, oh no, so we can't split
it up, and yet we have dy and dx in our integrals and dy = dy/dx dx,
and dy/dx = 1 / dx/dy, oh except with partial derivatives. The
history is sticking out all over the place, and everything we do is
incredibly confusing unless you pay some attention to it. And we
*do* do fluxions eventually -- what do you think the dot notations
of applied maths are?

--
Andy Walker, School of MathSci., Univ. of Nott'm, UK.
anw@xxxxxxxxxxxxxxxx
.

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