Is the math getting more complex?

[misogynisticfeminist]

When the physics first started out, it was very much like Galileo demonstrating that all objects fall to the Earth at the same speed, something quite experimental. The math used would probably be pretty simple arithmetic. Then Newton's principia and the invention of calculus made physics all the more "difficult". And years went by when Maxwell's PDEs unified the electric and magnetic forces.

Then its followed by GR's use of Riemann geometry and string theory's very complex mathematics. Does it mean that the further we probe into nature, the more advanced our mathematical tools ought to be? And if it is so, and say, one day, these advances are essential to industry, what impact would it have on education? Would we have to start teaching calculus to 6-year olds or what?

 

[Tide]

That's an interesting question. I have two comments:

(1) The real significance of Galileo's work is that he was probably the first to "mathematize" physics and do it in a systematic way.

and

(2) Mathematics is a way of expressing or describing physical phenomena and as we progress we move onto more and more intricate phenomena we wish to describe - i.e. all the easy physics has been done already!

With proper motivation and presentation I see no reason, in principle, why sixth graders shouldn't be able to grasp calculus concepts. In Galileo's day basic algebra was regarded as advanced mathematics and we now teach many of those principles in junior high school and even in elementary school in some places.

 

[alpha_wolf]

technology may help...

EDIT: A bit more on topic than the stuff below: Yes, the deeper we go into science the more complex the math will be. This is simply due to the fact that we're trying to describe increasingly more complex systems and interactions. As for the impact on education, the problem will eventually be helped by technology - see below for an explanation. Until then, we'll just have to adapt somehow. Luckily, we're quite good at that.


--- If you're not interested in the technological aspects, skip this: ---

Sooner or later we will probably find a way to download that knowledge directly into our brains and/or to augment our mental capacities via genetic engineering or cybernetic means.

I know it sounds like far-out science fiction, and we probably are a long way from developing such technology, but we're already taking the first steps in that direction. The study of the brain's functions already has at least a few decades of work under its belt. Genetic engineering research is advancing rapidly as we speak. Cybernetic research is also under way. Experimental electrode sets have already been implanted onto the human brain as part of primitive artifical vision systems. And with nanotechnology behind the corner, our ability to perform all of these tasks will significantly improve in the not too distant future..

It's just a matter of time. The only questions are: when will we achieve that?, when will we need it?, and what would happen in the mean time?

 

[one_raven]

when will we need it?

What about "Do we need it at all?"?
Why?
Is it worth it?
Is the time and effort put forth worth it?
Do the ends justify the means?
What else falls by the wayside in the meantime as a result of that focus and effort?

We have people spending their time and effort attepting to work towards this idealized view of the merging od nature and technology that is perhaps centuries off, and people dying of starvation and ignorance every day.
Seems to me that our efforts would be put to much better immediate use with a different focus.

We need to kearn how to crawl first.

 

[alpha_wolf]

Well, yea, we migt not need it at all. But in any case we're not "wasting time and effort" working on this, since the capability will inevitably arise (or come very close to arising) from work in related areas directed toward other ends. E.g. the study of the brain will serve to cure such deseases as alzheimer's etcetera and would allow better treatment of numerous mental conditions. Genetic engineering will also help treat/cure various deseases, especially ones with a major genetic component. It may also help increase food availability, by the way (although world hunger stems more from distribution problems rather than supply defficiencies..). And so on.

 

[pmb_phy]

I'm reminded of Einstein's famous comment, i.e.

You know, once you start calculating you sh*t yourself up before you know it.

Pete

 

[misogynisticfeminist]

As for teaching calculus to 6 year old kids (i think 6-year old is a little stretching it, perhaps 9) is not really that far-out. Because I remembered I learned a lot of useless (well, not really useless) stuff around that age. I remembered that we had one topic dedicated to time ! and how to read clocks !. Not to mention one regarding money and how to count them.

So...

 

[cronxeh]

in my opinion math has pretty much always been used to simplify the most complex problems - be it integration and analytical method or be it numerical method - its always an attempt to process a lot of information quickly

 

[NeutronStar]

the invention of calculus made physics all the more "difficult".

Actually it made it much simpler, I think.

I honestly believe that the reason that most people believe that calculus (and math in general) is difficult is due to the archaic pedagogic styles of educational formalism more than anything else.

There are three really huge problem with educational formalism (in all areas, not just mathematics):

1. It is not geared toward education, instead it's geared toward competition for a degree.
2. It requires the students to learn far too many trivial superfluous topics which wastes the student's valuable time.
3. It is far too slow to in updating and improving pedagogic styles.

I'm 55 years old and I just re-took calculus I, II and III, as a refresher. It was like déjà vu from the last time I took it almost 30 years ago! They haven't changed a thing (except they use graphing calculators now). I mean, I realize that the math hasn't changed, but I would have thought that the style of pedagogy would have changed. It looks to me like its been on hold for at least 30 years!

If I could hire instructors to teach me in whatever manner I chose I would design the courses to compliment each other more directly. In other words, I'd have the math courses directly compliment the physics courses that I was talking concurrently. I'd also hire math instructors that aren't afraid to get a little concrete in their examples. So many mathematicians seem to have a phobia of concrete examples. They are so bent on abstraction that they are afraid to get specific. Personally I have always learned much better from doing specific examples first and then moving on to abstractions from there. Give me something I can apply and I'll take the abstraction from there. I promise that it won't hold me back!

On the contrary, being continuously fed nothing but abstract examples without ever being given a concrete example can indeed hold me back.

I could probably add one more thing,..

4. I personally believe that educational institutions could do much better by melding some subjects together into better-organized curricula.

I vividly remember when I was in high school I could not understand abstract algebra. To me it was just a bunch of meaningless rules that required memory. I just couldn't remember all the rules, or why they were valid.

After high school I went to a vocational school for electronics. By using Ohm's law and all the other circuit equations I quickly learned how to manipulate equations because I could see the relationships between the quantities. So I learned algebra from a phenomenological point of view. After that I was able to move on to more abstract forms of algebra because I understood the foundation of why it works.

Also, I have taught many courses on electronics over the years. I've had many students that would shiver at the word "algebra". But after showing them how quantities relate to each other using electronic circuits they were soon doing algebra with no problem. For some reason educational institutions seem to have never caught-on to the value of this phenomenological approach to teaching mathematics. I personally don't see this as limiting the abstraction at all. On the contrary I think it enhances it.

Going back to the idea of calculus. The biggest problem for most students is really the algebra. The calculus itself is relatively easy. It's the algebraic manipulations that most students have problems with. And, I believe, the reason they have such problems is because they never really learned algebra intuitively, instead they just learned it as a bunch of rules that they quickly forgot how to apply.

 

[Hurkyl]

I remember seeing lots of examples in calculus courses, many physical (particularly the related rates problems, but work and spring problems too).

I certainly agree that the difficulty many have is that they don't understand algebra at all. It seems people don't understand that mathematics has to be viewed more like a foreign language -- things you learn in earlier classes are essential to your later classes!

 

[Vadim]

but at the same time, the stuff you learn in later classes essentially forces you to understand what was happening in the previous ones.

for example, for some reason i had a real blank with logs when i took grade 12 math, but now I'm 4/5ths the way through calc I, and i have no trouble with them anymore. it gave me a different perspective on them.

i think that's where people run into trouble 90% of the time, they just need a different view at it.