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by Christian Neukirchen.
Original Post: Abstraction cannot be taught
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It is a funny thing I realized in German class that the school system,
CS education and teaching programming is doomed to fail. But all I
did was to ask my German teacher how we could learn to abstract.
She had no answer, and I couldn’t think of an effective way, either.
Today, I read in the papers that German students “learn nothing in
mathematics in tenth grade”. And I agree, it’s true: when you reach
grade ten (that’s about 16 years), you stop calculating and start
something that’s far more like “real math”: analysis, linear algebra,
a (far too small) bit of statistics. The whole point in learning this
at school is to learn how to abstract, in a mathematical sense. Sure,
few people may really need it later, but most will not. And don’t
even try to ask how it is useful: even teachers can’t figure out
real-world usages for all cases.
Accordingly, the students that can abstract usually don’t have much trouble
with this. The others do.
And it’s not only maths, that’s just where you notice it first. It’s
the same in languages. After you can read, write and speak, you now
need to prove you actually understand what you read. It’s not by
coincidence that you can (and must) analyze both texts and algebraic
terms.
Students that are bad abstracters (and this is not meant derogatorily,
see below) often are good in, say, German at the lower classes. They
write nice compositions about their holidays, can memorize the
spelling rules to pass the dictations, and so on.
Often, trouble starts when you begin interpreting poems. If you look closely,
you’ll notice an interesting change: students that used to be good at
German get quite a bit worse, and students that were not so good get
quite a bit better. At least, that’s what happened to me and some of
my friends.
Suddenly, I became a lot better in German class. At first, I couldn’t
figure out why. Maybe it was because of the new teacher we’ve got.
Three teachers later, I can tell you it likely wasn’t. The focus of
expectation just shifted: they don’t care as much that you can tell
about your holiday, but that you can understand and think about what
you’ve read.
And, in essence, that’s the thing you are supposed to learn in college
or the German Gymnasium. It happens in just about all classes: you
can’t stupidly learn by heart about birds and fish in Biology anymore,
you need to understand. You can’t simply paint nice pictures for art
classes anymore, you need to analyze famous works. You don’t just
translate Latin texts, you need to interpret them. You cannot
even simply run fast in physical education anymore, you need to understand how
you efficiently train (and grokking our PE handbook is far more
difficult than translating Latin texts, trust me). You can start
doing “computer science”, which maybe “only” is elementary Pascal
programming. But for all these things, you need to abstract.
Being able to abstract is the most important thing, because it
multiplies our brain power. Alexandre Borovik writes in the draft for
Mathematics under the
Microscope:
In our conscious and totally controlled reasoning we can process
about 16 bits per second. In activities related to mathematics this
miserable bit rate is further reduced to 12 bits per second in
addition of decimal numbers and to 3 bits in counting individual
objects.
It is completely obvious that we need to fit big ideas into these few
bits to be successful thinkers, by omitting what’s irrelevant.
Since one of the areas in which abstraction completely matters is CS,
Paolo Bucci, Timothy J. Long, Bruce W. Weide asked in 2001: Do We
Really Teach Abstraction?
They conclude:
It may seem that an approach based on mathematical modelling and
formal specification is premature in CS1/CS2. However, in our
experience after four years of teaching this approach, it is not.
They reason because of this that teaching a very mathematical and
formal CS helps the students to understand abstraction better.
However, that only works because of the former exposure of the
students to abstract mathematics. It does not really solve the problem,
but merely shift it: “If you are bad in CS, your math teacher failed.”
Except nobody I asked knows how to teach abstraction. One probably
can learn to abstract better on his own, but that is really hard
work. For example, I think I became a better programmer by learning
functional programming. And functional programming is far much more
mathematical than, say, imperative programming (or even the OO they
often teach). I was able to grasp FP quickly because I knew how to
apply my knowledge of mathematical abstraction—and people that
struggle with FP struggle with mathematical abstraction too. I think
math classes merely sieve the students by their ability, and don’t
enable them.
It turns out they can’t.
My German teacher said students are expected to know how to abstract
at the age of 16. I don’t think they’ll wake up on their 16th
birthday and suddenly know how to abstract. But the age seems right,
it’s probably one of the reasons young people (around the age of 12 to
16) often start programming in imperative languages. The ones they use
(like Basic) don’t allow for high abstraction and they can understand
them better.
Chaitin directly
related the elegance of an program to it’s
length. Given
abstraction as compression, that means the shortest program is the
most abstract. Or: the most abstract program should be the shortest.
By conclusion is every language, for which there is an obfuscated
program that is shorter, an inelegant (or sub-optimally elegant)
language. The mistake in this consideration is of course that the
definitions of abstraction make the program longer. Still, there is
something to be learned from this: the harder it is to compress a
program, the more abstract it is.
In order to judge persons that are unable to abstract well, a further
question needs to be answered: Does ability to abstract equal
intelligence? According to the Hutter
prize (in direct relation to Kolmogorov
complexity) it is: “If you can
compress the first 100MB of Wikipedia better than your predecessors,
your (de)compressor likely has to be smart(er).”
Personally, I think that being intelligent makes you able to abstract
well, but not necessarily vice-versa. I don’t think both are innate,
and only partly heritable. Neither can be taught, but both can be learned.
What’s left is the question of how to do it.
In his 2003 keynote, Abstraction – is it
teachable?,
Jeff Kramer provides a solution. He cites Jean Piaget that students
from seven to twelve years have “some ability for abstraction with
training”, however, with twelve years and more, “[o]nly 30% to 40% of
teenagers exhibit ability for abstract thought, some adults never
do!”. Jeff Kramer concludes that abstraction has to be taught
indirectly and, just like above, by use of mathematics, formal modelling
and analysis.
I think the age of 7–12 therefore is unbacked at our schools with
respect to abstraction. We need to start teaching it earlier
indirectly, when training still works. At later ages, these abilities
can only be exercised, but will not uprise.
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