27 February 2012

Swedish pedestrians receive secret message from God

In my earlier posting, titled "The Religion of Kopimism", I made the following observation:
I still haven't figured out Swedes' attitude to religion. On one hand, Sweden is supposed to be one of the least religious places on the planet. On the other hand, however, I think that a lot of Swedes behave as if they were religious. 
Strangely enough, my recent posting on "Swedish pedestrian crossing signs", and a bit of googling, has only strengthened my hunch on religiosity in Sweden.

In August 2008, an article published in the Local revealed that the following image, located at pedestrian crossings in countries all over the world, is designed to spread the word of god!

No, I am not joking. Voilà:
Prisma Teknik AB, the Swedish company behind the pedestrian signals, has now admitted that the hand is meant as a hidden symbol for God. In fact, the company says it has never made any secret of the fact.
"We want to show that there is only one way to reach God and that is up and through Jesus", CEO Jan Lund told The Local.
Though Sweden is not particularly religious, with church attendances consistently low, the country still contains some pockets of religious fervour. One of these appears to be located at Prisma Teknik, a company that supplies pedestrian signals and other technical devices to 60 markets around the world.
For a direct exhibition on the hand of god, via the site of Prisma Teknik AB, click here. (The sound is a bit annoying.) A second click, under "contact" in the English Version of Prisma Teknik shows their beliefs in full glory:
It is also thanks to the care from God that we have developed to be the successful company we are today. We want to have Jesus as a part in everything we do. Christian values and God's guidance is the foundation of Prisma Teknik. [English version of Prisma Teknik Personel page]
However, the Swedish version of the same page says:
--- --- ---   N O T H I N G   --- --- ---  [Swedish version of Prisma Teknik Personel page]
Now, clearly, this hand is the hand of a particular god. The god of the religion of the Personel of Prima Teknik. However, Prisma Teknik makes money by selling their hand-of-god boxes to Saudi Arabia. It turns out that Muslims took notice (more than Christians) that the hand depicted is not the hand of their god. And so,
[Muslims in Saudi Arabia] promptly cover up the hand [of god] when it arrives in the Arab state.
 So, next time you are at a pedestrian crossing in Uppsala, you have a choice:

Either to look slightly down and see the hand of god

Or look up and see satan herself:

Your choice.

23 February 2012

Grading exams

In many universities there is a tendency to eliminate the personal element from the grading of exams. That is, it is considered as a good thing to have an exam (or assignment) written in such a way so that even a machine can grade it. In fact, many argue that an exam should receive exactly the same grade regardless of who grades it, be it a machine or a human being.

To do this, administrators have taken all or some of the following steps:
  1. Exams have become anonymous. So the grader does not see the name of the person who submitted the exam.
  2. There are committees which decide, months before the exam is given, what the exam should be.
  3. Solutions to the exam are written and they are supposed to be model solutions, predicting the exact steps the examinee will or should take.
  4. The notion of partial credit has been established and sanctified. 
The intention of such rules may be good (you don't want to end up with a teacher who, for some special reasons, has the power to fail every single student). But this is hindered by the rules and regulations which are supposed to be uniform across university departments (e.g., the same exam rules should apply in the Department of Mathematics and in the Department of Theology), but also across universities in the same country or across countries. The latter is something that politicians in Europe decided they want to have, essentially advocating the Doctrine that all basic university degrees in a particular field should be equivalent, in the sense that a student with a BSc degree in Physics from the University of Bari Aldo Moro has the same knowledge as a student with the same degree from the University of Manchester. But politicians are politicians and we know why they can be so mistaken, willingly or not.

Back to the exams, however, I want to argue that *some* personal element in grading them is desirable in Mathematics, and, perhaps, in many other fields too. Here are some arguments:

  1. A student screws up in a question so badly, that it is absolutely clear that the student is in the wrong field. For example, a student who reports negative probabilities, or adds fractions by adding numerators and denominators separately, or cannot find the area of a triangle  (cases taken from personal experience) should not be encouraged by partial credit. In fact, the opposite: negative credit should be given for answers which are so wrong that do not even fall in the category of "acceptable" mistakes.
  2. On the other hand, suppose that a student is doing about average in all parts of the exam except in one question where he or she has a brilliant idea. An idea which shows that the student can think outside the box (that politicians, administrators, etc. want him or her to be in). In such a case, why not assign a mark which is a fraction larger than 1 of the intended mark? I would, and I will, whenever I can.
  3. I can also argue that anonymous marking takes away the picture a teacher builds from a student by seeing how he or she participates in the classroom. Why, is it *only* a final exam or some written homework which will determine the eventual potential of a student? What if the teacher sees that the student has an ability which cannot be measured in a written exam? 

These are some arguments based on personal experience. I am fully aware that they are, in particular the last one, quite sensitive. No rule, whatsoever, can quantify the percentage a teacher can award a student for positive impression. But then, could the problem be in the very fact that we insist on numerical values of grades?

I am also fully aware that, given the current stupidity of European laws, none of the above arguments can be considered. However, they are all very reasonable and, moreover, rational!

Let us look, as a gedanken experiment, what could happen; what should, in a rational Society, take place in order that these changes be implemented.

First and foremost is the fact that, in wanting to make degrees from different universities equal, one makes the (wrong) assumption that all teachers are equivalent. How can this be corrected? By educating the teachers or by assigning them roles roughly equivalent to their level. This is not the case, in general. Once someone has, say, tenure, as a professor of Nuclear Physics, then nobody can raise the issue that this person may, at some point, not be able to teach his subject well. The university has no means of making sure that its teachers have the skills (real skills, not degrees) they claim to have. Has anybody ever considered giving an exam to a professor?

Second thing that should happen is to openly acknowledge that not everybody can go to university or that not everybody can go to a particular field. Not all people have the same intellectual abilities, much in the same way that not all have the same physical abilities. Everybody, of course, should have the same opportunities, but the two are not equivalent. The fact that I did have the opportunity to go to training to become Olympic athlete in boxing is not equivalent to my having the skills to do so. In fact, I never had. Likewise, the statement that anybody has the same opportunity to become a theoretical physicist is not equivalent to the statement that anybody has the same skills to do so. Human brain is just an organ. Muscle is another. It is easier to accept inequities in the latter, but not the former.

Discussing these things is like asking for Democracy to become real. It won't. (Hint: a necessary condition for real Democracy is the implementation of Ostracism.) But it doesn't hurt to discuss these things. This is why we have freedom of speech, don't we?

22 February 2012

Swedish pedestrian crossing signs

Everybody knows the familiar pedestrian crossing symbol. Here is a variant of it:
 But in Uppsala, Sweden, it was decided that this symbol is sexist. Indeed, it shows a man crossing. But what about women? Are they not allowed to cross the road? Are they considered to be second-rate citizens? Surely not. So (I suppose after many deliberations, meetings with politicians, city planners, architects, engineers, the police force, statisticians, scientists and ecological farmers) Uppsala decided to replace half the signs by this one:
 Apparently, other Swedish cities have done the same. In fact, this is something that was noticed by news agencies around the world some time ago. Look, for instance, at this article from the Russian Bокруг Cвета (Around the World).

The Swedish Road Administration (Vägverket) has decided to install new signs at pedestrian crossings replacing the traditional male figure Mr Pedestrian (Herr Gårman) by Mrs Pedestrian (Fru Gårman). It is reported in The Local. In Hässleholm in southern Sweden, town officials sought to erect their own ‘Fru Gårman’ signs in the name of gender equality.
So far so good. (Stupid me I hadn't even noticed the signs. The reason could be because I cross the road from wherever I feel like and do not necessarily use zebra crossings. In fact, I didn't even know there were zebra crossings. My excuse is that they are now covered with snow and ice.) But just now it was made known to me that the above depicted signs of Mrs Pedestrian are no good. Can you guess why?

I'll let you think before you answer. When you're ready, scroll down to see the answer. Warning: it is hard, so I'll give you a hint: Recall my earlier posting on Ultra-Orthodox Jews' behaviour towards an Orthodox Jewish girl who was seen walking down the streets of Beit Shemesh in Israel.

I guess my hint was enough, wasn't it? So you got it right? (Please let me know if you did!) The girl depicted in the revised pedestrian crossing sign (which, I suppose, was created after many deliberations, meetings with politicians, city planners, architects, engineers, the police force, statisticians, scientists and ecological farmers) depicts a woman dressed immorally: the skirt is short; the breasts are pointy, and (come to think of it) how are we sure that the woman is not topless? Is this Saint-Tropez or what?

So then the city has decided (I presume after many deliberations, meetings with politicians, city planners, architects, engineers, the police force, statisticians, scientists and ecological farmers) to replace Mrs Immoral Pedestrian with Mrs Proper Pedestrian:

 The proof is in today's article from Upsala [sic] Nya Tidning:

Mrs Pedestrian had a very short skirt, pointy breasts and widely open legs. Now the municipality installed new signs of a more properly dressed woman. Particularly offensive were the Mrs Pedestrian signs in the Stabby and Storveta districts of Uppsala. [The matter was apparently considered so revolutionary that] Mr Christer Åkerlund of Uppsala Kommun was interviewed by foreign journalists. "It was not easy", he said, "to figure out what the road terms are in English."
So, here then is what the Beit Shemesh gentlemen should do. Rather than spit on improperly dressed women, they should (by holding the appropriate deliberations, meetings with politicians, city planners, architects, engineers, the police force, statisticians, scientists and ecological farmers) introduce the revised Swedish pedestrian crossing signs. Improper/immoral signs belong to the past.

For the sake of civility.

19 February 2012

Boycott Elsevier

Elsevier is a huge and aggressive scientific publisher. Their business practices, however, are not valued at all by academics. The reasons are:
  1. Elsevier charges very high prices for subscriptions to individual journals.
  2. Individuals cannot afford to buy them. They are offered to libraries but only in very large bundles, essentially forcing libraries to get journals they do not want.
  3. They support acts which aim at restricting free exchange of information (SOPA, PIPA, Research Works Act).
  4. They exploit the free (and most essential) services offered by referees, in order to increase their profit.
In http://thecostofknowledge.com/ we boycott Elsevier by refusing to publish in their journals, refusing to offer voluntary and unpaid services as referees and refuse to do editorial work for Elsevier.

At this day and age when we look for immediate access to papers in depositories such as http://arxiv.org/, we find it ludicrous to have greedy publishers like Elsevier who will do everything they can in order to increase their 2+ billion euro profit, including publishing journals of dubious quality, such as "Chaos, Solitons & Fractals", which was involved in a big scandal 2-3 years ago.

More details on this can be found on this statement of purpose, signed by some well-known mathematicians.

So, please visit the Cost of Knowledge web page and boycott Elsevier.

5 February 2012

This winter in Uppsala

Very pretty afternoon photographs from Ulva Kvarn, Uppsala, 9 km from my house. Compared to the rest of Europe, our neighborhood is enjoying mild weather.

4 February 2012

And now that I mentioned unlearning...

...let me also point out this posting by Eliezer Yudkowsky and this one by Ben Casnocha who mentions three things that one has to unlearn from school [the non-italicized comments are mine]:
  1. The importance of opinion: An opinion is the lowest form of human knowledge; it requires no accountability and no understanding. Schools, apparently, emphasize the role of students' opinions. "Tell me John, what is your opinion of the validity of Quantum Mechanics in the macroscopic world?" Or: "What is your opinion, Mary, about the solution of this equation?" Of course, we all have opinions, but we don't start learning by having them. Opinions are (should be) formed after a well-thought procedure.
  2. The importance of solving problems: Schools teach us to be clever, great problem solvers, something that makes us arrogant about our abilities. What schools do not emphasize is that the problems we learn how to solve have been solved by others first. And what they don't tell us is that formulating a problem is just as important as solving it. What they do to us is convince us that solving a problem is the end of any effort. And often, we become so good at problem solving of one kind that we underestimate our stupidity in solving problems of other kinds.
  3. The importance of earning the approval of others:  That grade that you get at school is presented to us as being something of such a value that even money cannot buy (or does it, sometimes?)We seek to get good grades so that we can be approved by friends, parents, the society, the employers... We seek their approval. Is the approval of others something that really proves our, say, understanding of Physics? Damn the teachers who tell us that the exam is what we should care about. Instead of trying to earn the approval of others, why don't we focus on those people who disapprove of us, people whom we cannot easily please? 
Unlearning is as important as learning (correctly).

2 February 2012

Unlearning bad habits

One of the major obstacles in teaching at an advanced level is that we often have students with preconceived ideas. (The term "students" should be intepreted in the broader sense. It may include ourselves, for example.) Here is an example I have come across to many times, in the teaching of probability.

I ask the students to prove that if we take the points of a homogeneous Poisson process on the real line and translate each one of them by an independent random variable then we get again a Poisson process with the same rate.

For example, we may perform the translations by independent standard normal random variables.

The student often has a preconceived idea that a Poisson process is a random function of time $(N_t, t \ge 0)$ such that

  1. It starts from zero: $N_0=0$
  2. It is right-continuous, non-decreasing with values in $\{0,1,2,\ldots\}$
  3. It has stationary and independent increments.

One problem that the student faces (if he or she does not remember the definition I gave in class) is that the question above says to consider a Poisson process on the real line. So, after some thought, the student realizes that the following definition works:
Let $N'$ be an independent copy of $N$. Extend $N$ on $t \in (-\infty, 0)$ by letting $N_t := N'_{t-}$. It can then be checked that the process $(N_t, -\infty < t < \infty)$ satisfies

  1. $N_0=0$
  2. It is right-continuous, non-decreasing with integer values
  3. It has stationary and independent increments.

Rightly then, one can say that $(N_t, -\infty < t < \infty)$ is a Poisson process on the real line. Next, the student attempts to solve the problem by, say, doing this. They define the times of discontinuities of $N$ by $\cdots < S_{-1} < S_0 < S_1 < S_2 < \cdots$, agreeing, for instance, that $S_0 \le 0 < S_1$ (after all, the origin of time must be placed somewhere--and this agreement is up to us), then define $T_n := S_n +X_n$, for each integer $n$, where the $X_n$ are independent identically distributed random variables, and then try to show that the process with discontinuities at the times $T_n$ has exactly the same law as $N$.

This is an almost impossible task. The reason is that the student immediately realizes that the new points are not even ordered in the same way as their indices (something that was true for the old points). In fact, the ordering of the new points is random! Definitely, there is a first new point to the right of the origin of time, but this is not necessarily the point $T_1$. In fact, if the $X_n$ are standard normal, the first point to the right of $0$ could be the point $T_{517}$ with some probability, or the point $T_{-29}$, etc.

The student then may ask me for a hint. I try to bring in to them the idea that the above definition may be OK for some purposes, but that it has fundamental drawbacks. It is much better to first define a Poisson process as a random discrete subset of the real line such that the number of points contained in fixed (nonrandom) disjoint subsets of the real line are independent random variables. From this, it follows that the number of points in a set is a Poisson random variable with mean depending on a deterministic function of the set (this function being a nonnegative measure which, in the homogeneous case, is a multiple of the Lebesgue measure). From this definition it is not hard to show that the previous one is a theorem. Moreover, this is a definition which extends to higher dimensions and even to infinite dimensions.

But getting rid of preconceived ideas is very hard. Especially when teachers insist on a traditional way of approaching things.

One should not, actually, underestimate the fact the inertia of many teachers (as I said, the term "student" includes ourselves) to unlearn something and learn it from a different point of view. This is a major obstacle in the teaching of Mathematics.


What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant