Re: Dumbing down?

Jeff Rollin <jeffrey.rollin@xxxxxxxxx> writes:

In the last episode, on Sunday 27 May 2007 01:18, Barry Margolin wrote:



How did you do in plane geometry, when you were learning about things
like the proofs of the Pythagorean Theorem or inductive reasoning? Or
Boolean algebra?

I can't honestly say I did well at them in grade school, both due to
circumstances beyond my control and due to the way maths is taught in this
country (UK), which requires you to be able to do arithmetic before you
even smell the above; but having recently decided to go back to college to
do maths, and having seen some of the things they do in algebra and so on,
I don't seem to be finding it that hard.


Its funny, but that is all way to common. I was pretty bad at maths when at
school. At the time, I found this a paradox as I quite enjoyed it (i.e. I
enjoyed the problem solving), but just didn't seem to get it or was never very
good at it.

When I first left school, I was going to be an accountant (OMG!), but after two
years I dropped out. One of the reasons was because I failed the second year
economic statistics course and had only just passed first year.

Some years later, I returned to uni and started a degree in social science. I
was very worried as you had to do quite a few stats courses and I didn't think
I'd pass. Well, things really turned around and I think it was nearly totally
due to having a good lecturer (and possibly being a bit more mature). Suddenly,
within the first month of the course, everything just dropped into place. In
fact, I ended up with a prize at the end of the first semester for completing a
perfect 100% exam and a prize at the end of the second year stats course for
best overall mark and for best original analysis of the statistical analysis
used in a published research paper.

After finishing my degree, I worked for a couple of years and soon got bored
because of lack of intellectual challenge. When I finished my degree, computers
were just becoming more popular and students had started using word processors
rather than typewriters. I'd found computers fun when using the PDP-11 at Uni
and decided I'd like to understand how they worked and try some programming. I
found it really interesting and for some reason I found the concepts really
quite straight-forward. I decided to go back and do a computing science degree.
I was very concerned at one level because you had to do at least discrete maths
to do the degree and were even encouraged to also do some pure maths.

Well, things completely turned around. This was largely due to yet another
really good lecturer who I think immediately realised what was holdinig me
back. He gave me some books on the history of mathematics and some descriptions
of how some of the major discoveries were made. It was only at this point I
became aware of the "Is maths discovered or invented" arguement. Suddenly, I
stopped thinking of maths as it had been taught to me - that is, that it
consists of a whole lot of well defined rules which you merely memorise and
apply where appropriate. This was how I'd been taught in school. Each problem
had one and only one solution which you obtained by applying the correct
recepie/algorithm. Suddenly, maths became a true problem solving process in
which I was free to use any of my knowledge to solve the problem and not merely
a process matching activity in which I tried to work out which method was
appropriate without really understanidn what I was trying to do. I found maths
problems popping into my head all the time and often filled hours sitting down
with pen and paper tyring to work out a solution having nothing other than the
problem and my knowledge of various techniques.

The best day was when I was doing a course on chaos theory and fractals. We
were given a 'simple' assignment that night to write some formal proofs
associated with some of the known properties of a fractal. I turned up the next
day with my proofs, very nervous as this was still something I wasn't totally
comfortable with. The lecturer looked at my work and said yep, well done.
However, he also suggested I talk to a couple of my fellow students and get
them to show me their proofs. The fact was, while my proof was fine, it was not
very elegant. Many of my fellow students were coming from a pure maths
background with much better knowledge of number theory etc. My proof was based
on what I'd learnt in discrete maths, while theirs were based mainly on what
they had learnt studying pure maths. They had a bigger and better equipped
toolbox and therefore were able to implement more elegant proof. The really
nice thing was that once they showed me the proof and I worked through it with
them, for the first time, I really understood what was going on and could
apprreciate what they had done.

Now, while I can say my maths is reasonable (its probably more unreasonable now
as I rarely get to practice it), I'm not sure how much it contributed to my
programming ability. To a large extent, my programming skill occured before my
maths skill. My growing ability and interest in maths certainly influenced
later course choices (I really enjoyed courses on computability, logic etc) and
they probably did have some influence on my programming. However, all the stuff
I learnt on algorithms, abstract data types, program structure, recursion, etc
all occured prior to my development of maths skills and I don't think my
understanding or appreciation of these things was changed or affected by growth
in maths skill. I am very convinced that most maths teachers in school don't
actually understand what they are teaching though.

I do have to say I've found this thread quite interesting. Not so much because
of the debate as to whether maths skill is a prerequisite for being a good
programmer, but more in the way som eof the arguments have gone. Therre has
been considerable debate as to whether there is a link/relationship between
maths and programming and whether you need maths to program in some domains. I
think these threads missed the point. I don't believe anyone has argued there
isn't a relationship between maths and programming and I think its pretty
obvious that in some problem domains, you will need advanced skill in the maths
associated with that domain. I also think maths can't hurt in making someone a
better programmer, but I'm not convinced it is required. It has been argued
that the sort of logical thinking geneally associated with maths is important
for programming. However, I find that a fairly arrogant assumption because it
implies only mathematicians have the skills/discipline for that type of
thinking. This simply isn't true - there are many many areas of study, work and
life that require those sorts of skills and which are done by people who have
little maths skill. it is even possible that the type of person who is good at
maths may be that way because they already have the skills before they start
the math. Maybe all that doing maths does is help you to refine those skills
you already have. If this is the case, then that same person would likely be
good at programming without doing any maths at all. maybe maths will make them
even better or maybe it won't have any noticeable affect.

I did notice a number of posters make reference to historical change and
changes in the sort of knowledge you require. I think this is very relevant. In
fact, I've long thought that in the long term, programming will become more
trivial (and probably more boreing) as the process becomes more abstract. Early
programmers had to put a lot of effort into making things small, tight and
efficient because storage was expensive, CPUs werre slow and you were often
working 'close to the metal'. You actually implemented the algorithms to sort
your data or the hashing and collision resolution functions for that hash table
or the low level numerical functions you needed. These days, you are more
likely to use the built in library functions and leave optimisation to the
compiler (hands up those who remember going into the assembly code to unwind
loops by hand to make them faster or use less memory).

Tim


--
tcross (at) rapttech dot com dot au
.

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