Here is some talk on a TED conference about some important flaws in educational systems around the world.
What I can agree is that universities are very good at preparing students to be a future academic teachers (just look at me, it's temporary, for my PhD only, but still - I'm doin' it). It was a nice observation, anyway, I didn't think of it before. Also, most students fear doing an error or get a bad mark and it really leads to many abnormal situations. For example - during my studies we had usually some coding projects. I was well known of inventing very weird and risky topics and my friends usually were afraid of cooperating with me. They were usually choosing something safe, something on a teacher list, something well-known for them, they were good at, something that will give them a best mark. And as you see, there is a contradiction here! You should go to school to learn something new, right? And they were simply avoiding exploring new, interesting topics this way. Educational system was not rewarding learning new things, not mentioning, exploring new, innovative topics. Students just had to pick the best strategy to... survive and get a degree. That leads to many compromises.
Educational system rewards good marks and loyalty to teachers. If you have a good marks, you also have more freedom to choose your speciality and get scholarship. So you better don't tell your teacher he is wrong or try to dispute with him. It will be not good for you. Also don't try to think on your own - you will just confuse yourself during an exam and find out that solving those exam takes you too much time, since you will be "wasting" your time for thinking, instead of using automatically well-trained skills of solving well-known problems.
In my personal opinion it is all turned around. Ideally teachers should only assist students in learning, provide bibliography, expertise, hints. Students should be aware why they want to learn and improve themselves at university by asking for assistance. To motivate students there should be some obligations - student, individually, should decide after consultation with teacher about his topics he will learn (by studying, practicing, making a project etc..). After the obligation is filled, the work could be measured by several factors (how much knowledge/skills the student gained after the assignment, how creative he was, etc..). And what is also important - while making decision upon next obligation, all previous should be carefully checked, so the next one will be something new to student. At a first glance it complicates educational system a bit, but I believe with a proper management it could be quite efficient - an extension to existing computer databases could be made. There should be not much problem to manage a list of obligations/topics for every student, additionally to standard list of courses/assignments.
I really think universities should be more flexible (individual course program is not enough!) and concentrate much more on an individual teaching.
Tuesday 28 October 2008
Friday 22 August 2008
Message From The Alien God
Today, my alien fellows received a message from their God (and please don't tell me there is only one God or Universe out there!).Recently, Ken Perlin, a computer graphics icon that has my respect, was "rambling" a lot about PI (he has even written a peom that turned into the song regarding PI!). On the other hand, he criticized Carl Sagan's Contact novel for the God-in-a-circle ending. However, as my alien fellows reported to me, in their circles you can find the message from their God much, much easier and it is much more appealing !
Their world appears three dimensional to them as well. But they are defined with a different metrics. Instead of euclidean norm, they live in a metric space that is defined by a mix of euclidean and maximum norm: alien_norm(v) = euclidean(v)*god + maximum(v)*(1-god), where god is approx. 0.19. As we know, PI can be calculated as circumference of the circle divided by its doubled radius: PI = circumference/(2*radius). Now, since they are in different space, every circle in their world looks a bit like square with slightly curved sides to us (just as in the image above). If they are measuring circumference of the circle with radius equal to 1/2, they get PI = 3.7269767679... Aha! A computer scientist eye can easily spot a message here! For all unaware people out there: ASCII codes for 'H'=72, 'E'=69, 'L'=76, 'L'=76, 'O'=79. You see! Their God is much more kinder than ours. We would need to waste a lot of time to find such a message in our PI, and in their world it is just around the corner or just after the period to be more exact!
Wednesday 30 July 2008
Procedural graphics: science or art ?
Scientists credo seems to be: all hypotheses must be supported by the data. It's resonable. Typical scientific discovery recipe: gather a lot of data, use statistics, formulate "rules", falsify.
Statistics is math. Linear regression is a kind of curve fitting is a kind of approximation technique. Regularities (low-order curves) are common: planet bodies, planet orbits or pendulum oscillation. But so are irregularities: rocks, coastlines, clouds, fixational eye movement.
And here is the discovery that influenced my interests and research topic a lot: fractal geometry. Fractals provide sort of evidence that irregular "things" can be described as simply as regular. But there is a small twist here!
For example take some long sequence of random (white noise) data. How to approximate those date ? Is there any simple recursive or any other equation that can describe this sequence or at least fit to it with a small error ?
Unfortunately, in general, it's almost impossible to find a short one. To give you more clue - white noise doesn't compress easily (just try JPG or ZIP on it).
But we can do something else with it. We change the way we think about approximation! Instead of trying to fit all the data, we can just try to reproduce their "general characteristics". Why not "approximate" white noise just by any other white noise, i.e. generated using simple LCG? If our application is audio-visual, we will not see much difference. This kind of reasoning was succesfully used in speech synthesisers or synthetic terrain/rocks generators. And taken to the extreme (applied to vast variety of signals) define what is called: procedural art.
A procedural technique researcher is trying to find methods (a set of rules, algorithms or equations) that can be used to generate very complicated signals that has the same "general characteristic" as modeled signals, while procedural artist is trying to "paint" with those methods.
I did several procedural works in my life, i.e. a movie inside weird caves and 64k intro, introducing my own procedural techniques.
More works like this can be found in so called demoscene archives, most recommended are some Inigo Quilez works. Another prominient researcher (not related to demoscene) is Dmytry Lavrov.
With a demoscene you have to be careful though: not all those little creations are fitting my procedural art definition. Some of them just use standard approximation techniques to describe regular "things" using subdivision surfaces, quantization, wavelet compression, etc... To put it straight, this creations have lower artistic value for me (but I'm not claiming they have no value at all).
Statistics is math. Linear regression is a kind of curve fitting is a kind of approximation technique. Regularities (low-order curves) are common: planet bodies, planet orbits or pendulum oscillation. But so are irregularities: rocks, coastlines, clouds, fixational eye movement.
And here is the discovery that influenced my interests and research topic a lot: fractal geometry. Fractals provide sort of evidence that irregular "things" can be described as simply as regular. But there is a small twist here!
For example take some long sequence of random (white noise) data. How to approximate those date ? Is there any simple recursive or any other equation that can describe this sequence or at least fit to it with a small error ?
Unfortunately, in general, it's almost impossible to find a short one. To give you more clue - white noise doesn't compress easily (just try JPG or ZIP on it).
But we can do something else with it. We change the way we think about approximation! Instead of trying to fit all the data, we can just try to reproduce their "general characteristics". Why not "approximate" white noise just by any other white noise, i.e. generated using simple LCG? If our application is audio-visual, we will not see much difference. This kind of reasoning was succesfully used in speech synthesisers or synthetic terrain/rocks generators. And taken to the extreme (applied to vast variety of signals) define what is called: procedural art.
A procedural technique researcher is trying to find methods (a set of rules, algorithms or equations) that can be used to generate very complicated signals that has the same "general characteristic" as modeled signals, while procedural artist is trying to "paint" with those methods.
I did several procedural works in my life, i.e. a movie inside weird caves and 64k intro, introducing my own procedural techniques.
More works like this can be found in so called demoscene archives, most recommended are some Inigo Quilez works. Another prominient researcher (not related to demoscene) is Dmytry Lavrov.
With a demoscene you have to be careful though: not all those little creations are fitting my procedural art definition. Some of them just use standard approximation techniques to describe regular "things" using subdivision surfaces, quantization, wavelet compression, etc... To put it straight, this creations have lower artistic value for me (but I'm not claiming they have no value at all).
Monday 14 July 2008
Closures and boundaries
Today, I was working on some technical report (an internal publication for the university) for my PhD. I was dealing with topology of sets in metric spaces, which is a special case of general topology, well suited just for my needs, a computer graphics field.
I had minor troubles with a definition of a set boundary, but I think I finally got it right and actually some things surprised me a bit - that's why I'm sharing it here!
At first I was saying about boundary of the set A without mentioning about the metric space X that contains it. Usually I'm operating in R^n euclidean space, and all of my sets was subsets of R^n. But this time, I was dealing with a space constructed from a subset of R^n, specifically, a unitary box: [0..1]^n, further called I^n space, I={x|0<=x<=1}, where x is a real number. Let Y be a continuously deformed ball that is a subset of I^n . I was "intuitively" (by analogy to R^n space) thinking that the boundary of Y should be always connected (or even simply-connected). You can not imagine a ball that has disconnected boundary, right ?
As usual, it all depends on the definition on the boundary.
In topology, the boundary can be defined several ways that are equivalent for metric spaces. We can for example say, that a boundary of a set Y included by a metric space X is a set of points in X for which any open ball (with radius > 0) contains points both inside Y and outside Y.
To define an open ball in X you surely need to define a metrics of X. If it is euclidean R^n space, a ball "looks" like a ball = it is a sphere, if not, it can be almost "anything". But stick to the euclidean first. The I^n set has a boundary in R^n. It is an "empty" box. But any open ball in euclidean I^n space will not always be an open sphere - near the boundary of I^n in R^n, it will be just a part of a sphere cut by the walls of I^n box. Now, if we take any connected subset Y of I^n that touches the walls of I^n box, the set of points at the walls of I^n box will be not a part of boundary of Y! This is why we can construct Y that will have a disconnected boundary in I^n, an example provided below:
In the figure above, a boundary is denoted as a thicker line. Note that any open ball co-centered with "a" and smaller than "a" will contain only points inside Y, thus the center of "a" is not a part of the boundary of Y. We can also clearly see that the center of "b" lies on the boundary of Y (any ball around it contains point both inside and outside Y). Obviously the boundary of Y' in R^n is connected.
I had minor troubles with a definition of a set boundary, but I think I finally got it right and actually some things surprised me a bit - that's why I'm sharing it here!
At first I was saying about boundary of the set A without mentioning about the metric space X that contains it. Usually I'm operating in R^n euclidean space, and all of my sets was subsets of R^n. But this time, I was dealing with a space constructed from a subset of R^n, specifically, a unitary box: [0..1]^n, further called I^n space, I={x|0<=x<=1}, where x is a real number. Let Y be a continuously deformed ball that is a subset of I^n . I was "intuitively" (by analogy to R^n space) thinking that the boundary of Y should be always connected (or even simply-connected). You can not imagine a ball that has disconnected boundary, right ?
As usual, it all depends on the definition on the boundary.
In topology, the boundary can be defined several ways that are equivalent for metric spaces. We can for example say, that a boundary of a set Y included by a metric space X is a set of points in X for which any open ball (with radius > 0) contains points both inside Y and outside Y.
To define an open ball in X you surely need to define a metrics of X. If it is euclidean R^n space, a ball "looks" like a ball = it is a sphere, if not, it can be almost "anything". But stick to the euclidean first. The I^n set has a boundary in R^n. It is an "empty" box. But any open ball in euclidean I^n space will not always be an open sphere - near the boundary of I^n in R^n, it will be just a part of a sphere cut by the walls of I^n box. Now, if we take any connected subset Y of I^n that touches the walls of I^n box, the set of points at the walls of I^n box will be not a part of boundary of Y! This is why we can construct Y that will have a disconnected boundary in I^n, an example provided below:
In the figure above, a boundary is denoted as a thicker line. Note that any open ball co-centered with "a" and smaller than "a" will contain only points inside Y, thus the center of "a" is not a part of the boundary of Y. We can also clearly see that the center of "b" lies on the boundary of Y (any ball around it contains point both inside and outside Y). Obviously the boundary of Y' in R^n is connected.
Saturday 28 June 2008
Random nonsense
My friend is always bothering me with some silly questions
about my beliefs. He claims he is an atheist, he studies developmental and evolutionary biology. He also suspects that I believe in spaghetti and unicorns (?), whatever it means.
Let me just concentrate on the most "important" (at least to some people) question: does God exist ?
There are many opinions on God existence, a lot of them.
For me personally the question if "God exist" is simply ill-defined. Consider the word "exist", Webster's dictionary definition gives: "to have actual being, be". What it means? For me, not much. We just replaced a word "exist" with "be". Some people like to discuss "do we exist?" as well. They can do that, although I'm a big proponent of using the word "exist" only in a context (in fact it is a secondary definition in this dictionary). We say "between any two real numbers, at least one more real number exists". We can also ask if real numbers exist ? If so, where "they" exist ? They are part of our universe, dreams or what ? (Hint: they exist in a set of all sets) The same reasoning we can apply to God: where could God exist ? The answer is pretty obvious to me today: God exists between any two real numbers... in hell :P
about my beliefs. He claims he is an atheist, he studies developmental and evolutionary biology. He also suspects that I believe in spaghetti and unicorns (?), whatever it means.
Let me just concentrate on the most "important" (at least to some people) question: does God exist ?
There are many opinions on God existence, a lot of them.For me personally the question if "God exist" is simply ill-defined. Consider the word "exist", Webster's dictionary definition gives: "to have actual being, be". What it means? For me, not much. We just replaced a word "exist" with "be". Some people like to discuss "do we exist?" as well. They can do that, although I'm a big proponent of using the word "exist" only in a context (in fact it is a secondary definition in this dictionary). We say "between any two real numbers, at least one more real number exists". We can also ask if real numbers exist ? If so, where "they" exist ? They are part of our universe, dreams or what ? (Hint: they exist in a set of all sets) The same reasoning we can apply to God: where could God exist ? The answer is pretty obvious to me today: God exists between any two real numbers... in hell :P
Tuesday 17 June 2008
Math-fiction
I came up with a mental exercise today. I will try to predict how mathematics will look in the far future or in no time, it is a math fiction, everything is possible.
Mathematics in the XXII century
A classical mathematical notation will be replaced with a strict machine-friendly format. The concepts like sets and relations will not change, but we will construct sets and relations differently. There is a set-builder notation and it already makes a lot of troubles. A computer-friendly notation would allow automatic syntax checking, inclusion/exclusion test, or even advanced hypothesis verification. The next-generation computer-aided math package will be implemented in TeX 2.0, a new scientific publication language format that will push the cooperation between so called human inteligence to higher levels! Using new math tools we will solve most of the millenium problems with ease. We will clarify complexity theory classes, also write no more than two page long proofs for Riemann and Hodge hypothesis. In the mean time, pentagonal tiles of type 15 will be covering all of our residence floors. The new mathematical tools will be very powerful because we will proof that P is equal to NP or even we will figure out that every algorithm require not more than logarithmic to input data size steps on the so called Big-Bang machine. We will be Big-Banging parallel universes whenever sorting a column in a spreadsheet. Last but not least mathematics will hit the trenches. All those uneducated people (mostly engineers and nature "scientist") will use standarized terminology.
There was a time when "real" mathematics was done with a pencil, paper and fresh mind only, and many mathamaticians actually still believes it is the only way to do it and other tools are only unnecessary complications.
And what do you think about it ?
Mathematics in the XXII century
A classical mathematical notation will be replaced with a strict machine-friendly format. The concepts like sets and relations will not change, but we will construct sets and relations differently. There is a set-builder notation and it already makes a lot of troubles. A computer-friendly notation would allow automatic syntax checking, inclusion/exclusion test, or even advanced hypothesis verification. The next-generation computer-aided math package will be implemented in TeX 2.0, a new scientific publication language format that will push the cooperation between so called human inteligence to higher levels! Using new math tools we will solve most of the millenium problems with ease. We will clarify complexity theory classes, also write no more than two page long proofs for Riemann and Hodge hypothesis. In the mean time, pentagonal tiles of type 15 will be covering all of our residence floors. The new mathematical tools will be very powerful because we will proof that P is equal to NP or even we will figure out that every algorithm require not more than logarithmic to input data size steps on the so called Big-Bang machine. We will be Big-Banging parallel universes whenever sorting a column in a spreadsheet. Last but not least mathematics will hit the trenches. All those uneducated people (mostly engineers and nature "scientist") will use standarized terminology.
There was a time when "real" mathematics was done with a pencil, paper and fresh mind only, and many mathamaticians actually still believes it is the only way to do it and other tools are only unnecessary complications.
And what do you think about it ?
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