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The interview with Sergiu Klainerman was an interesting read for me,
as I'm also an European bound mathematician. I myself am a teacher
primarily now and not a research mathematician, but there are some
interesting things for me which appeared in that interview.
One mention is a prevalent mantra among students ``climb the mountain first
and then the true meaning of the subject will come". I actually think
this is very good advice. It does not look so first, but an important thing when
learning math is to keep doing it and not get discouraged. The above
mantra helps. Klainerman himself proves that that it can be a good
approach as he became a fantastic mathematician. I myself felt a similar
frustration at first as a student at ETH Zuerich. We were fed a lot of
information and I missed a bit the creativity I was able to pursue as a
high school student. But I appreciate now that learning a lot over a relatively
short time was good. Knowledge allows us to build connections and be creative later on.
We need a rich background of knowledge in order to grow later.
If we learn a language, or a music instrument, we first practice a lot
of old stuff, we read and play the masters for many years, before we go
into composing ourselves or try to write poetry on our own. We also just
might enjoy a novel or piano concert without first analyzing how it is
built and why it works or make an effort to modify it. Understanding
is great, creativity is even greater but it can be frustrating for a
beginner. Once too much frustration kicks in, the case is almost lost
as mastering a subject requires to pull lots and lots of time doing it.
Understanding only comes on a fertile solid of knowledge. There might
be psychological reasons playing a role here but the rush to force
young kids to understand or to ``be creative" early on can also be a
turn-off. Creativity is hard. If you do not believe this, try to be
creative and come up with a new theorem in geometry or number theory on
your own. Most people who preach creativity have not tried to be creative
themselves and have no idea how damn hard it is. Now, we should practice
being creative and try as hard as we can. I myself was most successful
in this with final creative projects like composing a music piece using
a computer algebra system or to build 3D printable objects.
But I'm constantly reminded when doing such
efforts that being creative is hard for most. During my final (mostly
oral) examinations as an undergraduate student I was asked the hardest
question of all by Ernst Specker, who tested me in logic. His question
was: "Mr Knill, tell us something". It was a hard question to answer and
it illustrated me how difficult it can be to tackle open-ended questions.
Also true understanding can be hard. One reason of course is that
understanding needs a lot of knowledge beyond the actual subject. For a
student, it can be frustrating to hear: ``but why do you not understand
this?". It can be very tough to understand, it is even tougher to
teach ``understanding". For anybody who does not believe this, it is
a good idea to pick up from time to time a math book in a subject one
is not familiar with and then solve the task to ``understand". A good
approach is to actually solve the problems which are provided in the
book without looking up any solutions or ask for advise.
It is a good exercise because in most cases, one feels exactly
like a student who is first is exposed to some new subject like algebra,
geometry or calculus. One first has no clue at all what things are about.
It is the feeling Klainerman points out when reading Lars Hoermander
or Fritz John. It is what I experience myself if I look now at a paper
of Klainerman. I would definitely need to go back and brushing up
much more harmonic analysis and functional analysis and learn
inequalities to master the reading.
As an educator myself, I feel, that neglecting to teach substantial
amounts of material is one of the most important mistakes of modern
math education and one of the reasons why other countries do so much
better in assessments. We are still indoctrinated as teachers that teaching
content is bad, that conceptional understanding has to be fostered
before we know how to do things. This is even entrenched in the famous
Bloom taxonomy, where ``understanding" is placed before ``algorithmic doing".
But understanding can only come once we know how to do this. The reason
is that for understanding, we also need to build up interest. We first
learn how to drive a car before we know the inner workings of a combustion
engine or an electric motor. Actually, by driving a car we might become
interested in the inner working of these engineering miracles. Before we
do it, it is just some object moving around. We need to have written a
few programs in a programming language before starting to understand why
the syntax is built as it is or how the compiler for the language works.
Now, Klainerman himself proves the point. Like me, he was frustrated to
have not reached good understanding due to all this ``content learning"
but by staying ``hungry" and having been provided with a lot of fertile
soil as a student, he could grow nice mathematical plants later on.
It is the holy grail of education to pass along ``understanding". It is my
experience from 30 years of teaching and 50 years of learning mathematics
that understanding is difficult to teach, especially if not first some
``how to" is done. Yes, a good teacher can implement understanding also
in the early stages of introducing a subject, but not many succeed like
that. The vast success of ``how to" guides on the internet prove this.
Learning something well requires some memorization and drill. This sounds
like heresy. But we would never question this in music for example. You
can not perform a piano concert well if you do not know it by heart and
have played it a thousand times. Nobody would tell there ``you shall not
memorize" or not practice ``etudes" (finger exercises). Also a rock climber
needs to memorize every move of a difficult route as a non-choreographed
approach will fail. The recent movies ``Dawn wall" and ``free solo"
illustrate how much work and repetition is needed to master a climbing problem.
In mathematics, even seemingly boring things like the multiplication table
are treasure troves. There are patterns to be discovered, the concept
comes back in multiplication table of abstract groups etc, but familiarity
with a subject is a prerequisite to become interested in the subject and
as we all know, being interested in something allows to learn with almost
no effort. If we are interested in something, we even enjoy doing it.
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