Quotes

A well-designed quote conveys interesting and often powerful ideas and makes one think for a time disproportionately longer than the length of the quote.

I have decided to add a quotes collection on my page. I try to restrict to quotes pertaining to Academia, but I can't promise I won't divert.

Anyway, here is a quote that impressed me today:

• Most [people] discover that they have often be working in the affine plane without realizing that it could be so designated. (H.S.M. Coxeter)

Coxeter is the Grand Man of classical geometry who lived in the 20th century. Some of his books should be compulsory in middle/high school education. Alas, exactly the opposite is happening: to my total dismay I learned that Swedish schoolchildren never learn why the angle bisectors of a triangle pass through the same point. Instead, I was told, a Stockholm school made computer games a mandatory course. Not designing computer games, mind you, but playing them. No wonder that university students have no clue that there is a proof of the Pythagorean theorem. Yes, they can state it (and so can the greengrocer) but not only do they not know a proof, but--what's worse--they're not even aware that a proof is needed.

Back to the affine plane, however, even under ideal schooling circumstances, what makes the affine plane so elusive is the quick passage from Euclidean geometry to coordinate geometry (thanks Descartes!). Typically (I guess not any more), a schoolchild would learn a lot of Euclidean geometry in school but then pass on quickly to linear algebra in his/her first-year university course. The affine (and so goes for the projective) plane, responsible for a lot of elementary mathematics, would go by very quickly, if at all.

Pondering the Coxeter quote carefully is, perhaps, all is needed in order that the affine plane be re-surfaced from the stack of one's toolset.

Universities and tabloids

Have you noticed that, more and more, university web pages resemble tabloids?

The first page below is from a major university: "Best sex positions for women with bad backs" is on the front page.

The second page is from a major tabloid: "Just when we had sex, I noticed..." is on the front page.

Both are catch phrases of similar type. Their goal is to attract the customer's [sic] attention so that they click and read further, and, possibly, contribute some money. By subscribing, in the case of the tabloid, or by contributing towards the 250 thousand dollar goal, in the case of the university (top right corner of first image).

Some time ago we used to think that universities were serious institutions of higher learning and research. With some exceptions, of course, this is not the case any more. A large number of academic institutions are usurping the terms "research" and "teaching" and use them for services that have nothing to do with the original meaning of the words.

Exams: the fake certificate of knowledge

University courses are typically accompanied by exams. Exams are supposed to test the students' knowledge and are accompanied by a grade given to the student. When a student gets a good grade, they are happy they have learnt the subject. The university is also happy that it has managed to provide the knowledge to the student. The funding body of the university (e.g., the government) is also happy. Everybody is happy.

My experience, however, in several countries shows that exams are not doing what they are supposed to do. (With rare exceptions, of course). The funding bodies of universities put pressure to the universities to maximize the number of students graduating per year. Therefore, it is to the interest of the university not to have failures. Exams are designed in such a way that the students pass. In fact, they do not even test the students' knowledge. Instead, whole courses are being taught in a way that the students pass the exam.

What is absolutely incredible is that, in several countries, like Greece for instance, the student can take a resit exam as many times as they desire, until they pass! This creates an enormous pressure to the teachers to pass the students because they cannot have a growing number of students taking the same exam again and again. In many universities in the UK, students are allowed to take the exam twice per year. In Sweden, they are given three chances per year, and each exam is 5 hours long. Students realize that universities are under a lot of pressure to pass them in the exam and to give them good grades, that they now do not even care about learning the subject properly. Rather, they demand that they be taught material that is examinable. In the US, the problem is solved by grading on a curve. That is, it is predetermined that, say 20% of the students must get an A, 30% a B, and so on, and, regardless of the absolute marks, the scale is adjusted accordingly. A practical solution, actually. Many European countries do even worse things (like multiple resit exams), and often they pretend that exams are flawless testing of students' abilities. Whereas, in reality, it is often a fraud.

Students who want to learn should be aware that exams are designed so that the average student passes immediately. Therefore, those students who feel the need to learn should be aware that exams are not designed for them and should seek alternative routes. Very hard, indeed, but it's much better to tell good students the truth, rather than fictitiously boost their egos by giving them good grades, when the grade inflation is so high, that only an idiot can fail to realize what is going on.

In Scotland, grade inflation starts early on. At early elementary school. Here is a first-hand incident, happened to the son of a friend of mine in a good public school. The kids are given a multiple choice test. After taking the test, the teacher provides the students the answers and asks them to have a second go, without taking the original papers back. In other words, the teacher hints to the students that they can correct their mistakes, based on the answer sheets he gives them. My friend's son found this so hilarious, and, despite his age, he understood the fraud. In this way, the teacher makes sure that the kids in the class (well, those who get the hint that they can cheat) get very good grades. He then goes to the principal and boasts about his class's performance. The principal presents the results to the board of education. The board of education concludes that everything is very good and keeps funding the school. The students' parents are proud of their offspring's performance. Everybody is happy. And the fraud goes on.

This is how ridiculous the whole exam system has become. It is a failed currency, something which does not represent anything real. In fact, I maintain that there is often little correlation between one's grades in exams and one's abilities. Even if things were better (without several resit exams, ad infinitum, without exams that last as long as students like), writing an exam which actually tests what is supposed to test is difficult. And who has the time, or cares, to do so? And why should the teacher care? After all, no university will give such a diligent teacher an award. Awards are only given based on students' impression. If a teacher actually tries to design an exam which will test students' knowledge, then the students will be unhappy, the university will be unhappy with the teacher (who may punish him or her--real cases of this nature do exist), the government will be unhappy, and so on. The teacher who wants an easy way out, will teach the students how to pass the exam, will write exams so that students pass at the first attempt with good grades, will, as a result of this, get a teaching award, and, probably, his salary may even go up.

The story, as I have described it, is quite generic. One may ask if it applies to this or that place. My answer is that it applies much more frequently than not. So frequent and widespread is the fraud, that if one picks a country at random and a university in this country at random, one has a high chance of seeing the phenomena I described above. Notable exceptions do exist, fortunately, but they are becoming rarer and rarer.

The more widespread the fraud is, the more likely it is for a university to have thought through cover-ups, ways of  "proving" that everything works perfectly. For this reason, pedagogical bodies have been formed, which are supposed to test the quality of teaching and education. Endless bureaucrats have designed pedagogical rules which, if followed--they claim--then, undoubtedly--they claim--the education provided will be of first rate.

But the problem remains the same: no matter how many awards are given, no matter how many pedagogical controls are applied, making the exams disjoint from learning is a disgusting practice which is only paralleled by the Catholic Church's cover-up of its pedophile priests.

London Metropolitan University: the real question

We recently read in the news that London Metropolitan University has had its right to sponsor students from outside the EU revoked, and will no longer be allowed to authorise visas.

Why?

The UK Border Agency found that some students did not fulfill the residence requirements, that some did not speak proper English and some did not attend classes.

Having worked in the UK, at Heriot-Watt University, I'm all too familiar with the situation: universities (now in Sweden too) would like to attract as many non-EU students as possible, because they bring real income.

Of course, the problems identified by the UKBA may very well be significant. However, the real question, is: are the students qualified to study the field they choose? My experience from Heriot-Watt University is  that many of the students admitted there were not qualified. Obviously, they could pay the tuition, and, most likely, they did have the proper visas. Moreover, they did attend classes because they were asking us, teachers, to monitor attendance (last time I saw this happening was in high school). So, even when all formalities (visa, language, attendance) are satisfied, why is it that nobody asks the real question: do the students qualify? And when I say "qualify" I am using the verb with its proper meaning. Do they have the background (and abilities, of course) to study a particular field in a university?

Once, in Austin, TX, someone had phoned me and told me he wanted to do PhD with me. I asked him to apply. He said he could pay his way through because he had a million dollars. So what? A million, or a billion, dollars should not be a sufficient condition for getting a PhD. He was not happy with what I told him and somewhat threatened me. I told him to go elsewhere.

Obvious question should be, obviously, asked. But I don't see this happening.

The question should be asked by the admitted students themselves: "Did they admit me because I paid a hefty fee, or because I actually am able to study and have the requirements?" At the minimum, it should be asked by those students who both want to get a degree (i.e., a piece of paper) and learn. (The two, unfortunately, seem not to be entirely equivalent.)

The ex-minister of Economics and Technology of Germany, Karl Theodor Maria Nikolaus Johann Jacob Philipp Franz Joseph Sylvester Freiherr von und zu Guttenberg, paid someone to write (parts of) his PhD thesis. When that was found out, he had to resign. 

It's not nice to go ahead by paying only. Money should correspond to something real too and this is a contract between the University and the student: the former is obliged to provide the latter a proper education, and the latter must have the intellectual abilities and/or skills in order to attend the course of studies proposed by the former. They should both agree on checking those abilities and skills as a vital part of the contract.

On teaching

From time to time, I write some thoughts about education issues. Not because I'm a great pedagogue, neither because I think I have the best solution to the problem. (Not even because I think that students are bad. Such cliches are just pointing towards an easy way out of the door. In fact, us, teachers are more responsible than students.)  But only because, nowadays, and in many places, "to teach" has become equivalent with "to teach recipes".

I will, from time to time, expand on the subject with my thoughts and examples from personal experiences which, trust me, I've had many, and many unpleasant ones.

There is one thing I take as an axiom (fully justified of course), from which I will never depart, and upon which I have always based and will always base my teaching. Namely:

There is no teaching without explaining what is being taught.

This is a fundamental principle, the basis of everything else. If you disagree with it, then you are essentially applying a procedure which should not be called teaching. You are simply adopting a process which should have another name. Preaching perhaps; Or something less harsh if you wish. But the fact that the word teaching has been usurped by all kinds of classroom activities, does not enlarge the meaning of the word. On the contrary, these activities only provide counterexamples (and frequently good ones) to the meaning of the word teaching.

So, if someone demands from you that, say, you teach Real Analysis by only giving the students a set of ready-made formulae, then your response should be:

"You are giving me contradicting demands: if I deliver to a classroom a set of formulae and examples on Real Analysis, without explaining to them how and why these formulae work, then you are asking me not to teach the subject. Therefore your request implies the clause `please teach Real Analysis without teaching the subject', from which an obvious contradiction arises. As it is impossible to simultaneously do something and the negation of the something, you are asking me the impossible."

Such is the only rational response to irrational demands. It all boils down to analyzing the phrases into its fundamental constituents, taking into account the aforementioned axiom. The person who is asking you to teach without explaining is asking you to do something else. And he or she has to name it. If he or she insists in calling it teaching, please insist in that he or she is violating elementary rules of logic. (If he or she claims to be a scientist but he or she fails to see the contradiction, then matters are worse: this person is violating their claim to adherence to science and is exhibiting religious traits.)

I know that this is too abstract. I will come up with examples in due time.

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?

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).

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.

Mathematics screening test for university students

I am writing this in response to my friend Joe who tends to believe that things in US education are bad. In fact, for anyone who thinks that things are bad in the particular place he or she happens to work.

This is a screening test I gave to second year university students (school of mathematical sciences of a UK  university I worked at earlier). The rationale behind a screening test is to alert the students that they should not take a further course without having basic skills acquired in earlier courses and that, if they have not learnt earlier material, they should repeat the courses before proceeding further.

The front page of the linked document contains statistics of students' responses. The remaining pages contain the questions I asked, together with sample responses. I would say that out of 60 students who took the test, there was probably one who could, perhaps, qualify as a university student. The remaining ones had no clue.

Although the document is self-explanatory, here are some of the questions, along with the most funny answers:

Q:  Given two polynomials $p(x) = \sum_{k=0}^n a_k x^k$ and $q(x) = \sum_{k=0}^m b_k x^k$, express the coefficient of the term $x^k$ of the product $r(x) = p(x)q(x)$ in terms of the coefficients $(a_k)$ and $(b_k)$.
A: $a_{k^{1/2}} b_{k^{1/2}}$.

Q: Define the concept of the derivative of a function $f : R \rightarrow R$ at a point $x$.
A: This is the distance of the point $x$ from the origin on a plain  [sic].

Q: Explain what we mean by the integral $\int_0^1 f(x) dx$ of a function $f : [0; 1]  \rightarrow R$. (The answer "area under the curve" is not acceptable.)
A: By integrating this function, we are being asked to calculate an area, and by providing definate [sic] integrals, the question asks us to provide a specific area.

Q: In how many ways can you put 5 indistinguishable balls in 7 distinctly numbered boxes and why?
A: 21/5.

Q: Expand $(a + b)^5$, where $a, b$ are real numbers.
A: $(a + b)^5 = \binom{a}{0} + \binom{a}{1} a^4 b + \binom{a}{2} \frac{a^3 b^2}{2!} + \binom{a}{3} \frac{a^2 b^3}{3!} + \binom{a}{4} \frac{ab^4}{4!} + \binom{a}{5} \frac{b^5}{5!}$.

Q: Compute the (indefinite) integral $\int dx/\sqrt{x}$.
A: $-2u+C$.

The huge problem in education, around the world, is that the meaning of the verb "to learn" is frequently disassociated from the verb "to understand". This is convenient for students. It is also convenient for many teachers who do not want to bother to understand and explain. It is convenient for politicians. It is convenient for administrators. In short, it is convenient for everyone. Except that the result is the production of generations of students who get a degree in, say, mathematics, but know very little mathematics. What is worse, is that they think they know. It is more dangerous to have people who believe they know rather than people who know they do not know (and, therefore, may try to learn whenever necessary). Someone who is convinced of his/her skills will do nothing to improve them.

An example of bad mathematics textbooks for schoolchildren

A friend of mine sent me the following entry from a mathematics textbook for schoolchildren in the US around the age of 12. Here is a question asked, together with the suggested solution:

Question: Tell whether the following events are dependent or independent. If they are independent, find the probability that both events occur.
Event C: Choosing the letter F from a bag containing the alphabet.
Event D: Choosing the letter V from a bag containing the alphabet after already choosing F and not replacing the letter.
Solution: Events C and D are dependent events. Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag.

Here is the problem with the way the question is formulated. In defining event D, the writer of this has inserted the description of the sample space. Instead of clearly defining the experiment first and then the events, he/she mixed the two things together in defining the event D. The result is confusion. The correct way of stating the problem is:

We have a bag containing the 26 letters of the alphabet. We pick a letter at random, put it away, and then pick another letter at random and put it away. Define the following events:
Event C: The first letter we pick is F.
Event D: The second letter we pick is V.
Question: Determine whether the two events are independent or not.

The suggested solution is even more confusing. It says that events C and D are dependent (correct, provided you have understood what event D is, i.e., what the author wanted to say but did not say), but the explanation given is not very good: "Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag." It changes the probability of what? What does the changing of a probability have to do with the definition of dependence?

Let's see first what the correct solution is:

P(C) clearly equals 1/26.
P(D) also equals 1/26.

The reason is: From the definition of event D, as an event which has nothing to do with whatever we picked at the first pick, and because of symmetry, we can see that P(D)=1/26 as well. What I mean by this is the following: if, say, the letters were arranged in a random order on a line and defined event C to be the leftmost letter, while D the right most one, then it would have been even clearer that P(C)=P(D). This is why the two probabilities are the same.

As for the probability P(C & D) of their intersection we have 
P(C & D) = P(C) P(D | C) = (1/25) x (1/26).

The reason is this:  P(D | C) can be computed on the new sample space containing 25 letters because we are conditioning on having picked the letter F first. Since there are 25 letters remaining, we have P(D | C) = 1/25.

Since P(D & C)  ≠  P(D) P(C), it follows that C and D are dependent events.

A clearly stated problem, and a logical solution is the only way that the kids in the school can learn something which they can subsequently find useful. The kids will remember, even subconsciously, these kind of things and if they have been taught incorrectly, they will have to unlearn everything when they go to the university (provided--this is an assumption--that their university teachers know how to teach or bother to do so).

No wonder why when people ask me what I do for living and I say mathematics they invariable give the same response: "Oh, I was so bad in mathematics at school." "I understand", I reply. "but, most likely, so was your teacher".

P.S. My friend also told me the following idiotic question the kids get in their mathematics class:

Question: What do you call the shape whose area is given by the formula L⋅H?

There is no end to idiocy in this world. 

Quality vs popularity in teaching

A few months ago a Boston Globe article (4 July 2010), titled "What happened to studying?",  commented on research done by two Economics professors, Philip Babcock  (University of California Santa Barbara) and Mindy Marks (University of California Riverside), about the significant drop of number hours that university students spend studying:

The average student at a four-year college in 1961 studied about 24 hours a week. Today’s average student hits the books for just 14 hours.

In fact,

[t]he National Survey of Student Engagement found in 2009 that 62 percent of college students studied 15 hours a week or less — even as they took home primarily As and Bs on their report cards.

So Babcock and Marks  asked why.  Obvious culprits,

the allure of the Internet (Facebook!), the advent of new technologies (dude, what’s a card catalog?), and the changing demographics of college campuses

didn't seem to be the main reason for this dramatic change. Perhaps then,

students [are so] much more efficient [nowadays, as compared to 40 years ago] that more than 60 percent of [them] study less than 15 hours a week and still earn As and Bs.

But some people, like the Stanford University provost John Bravman, try offer  some cheap explanations such as

[s]tudents live very different lives
[t]hey [have] jobs while attending classes.
[they] are [not] lazy, but [simply are] too busy — busier than previous generations

 In the words of John Bravman, vice provost for undergraduate education at Stanford University,

“Much busier,” describing the “on-demand” world that students work in today. “I was a student here from ’75 to ’79. I was reasonably engaged in things. But I had so much free time compared to students today. They do so many things — it’s amazing.”

Could it be the use of computers? No, because

the greatest decline in student studying took place before computers swept through colleges: Between 1961 and 1981, study times fell from 24.4 to 16.8 hours per week (and then, ultimately, to 14).

 And, just to be sure, hours spent in studying are not the goal of education. Nobody says that a student studying twice as much as another is twice smarter, twice better or that he/she will achieve better grades or have better education. Generalizations like that are silly. But something is happening.

Even the students themselves, when asked to comment on difficulties in their education, agree on that

they simply [do] not know how to sit down and study.

A possible answer to the cause offered  by the two researchers is the following (and to me it is not suprising at all):

[There] is a breakdown in the professor-student relationship. Instead of a dynamic where a professor sets standards and students try to meet them, the more common scenario these days, they suggest, is one in which both sides hope to do as little as possible.
No one really has an incentive to make a demanding class, To make a tough assignment, you have to write it, grade it. Kids come into office hours and want help on it. If you make it too hard, they complain. Other than the sheer love for knowledge and the desire to pass it on to the next generation, there is no incentive in the system to encourage effort.

What has happened is something that everyone who takes education seriously sees, very clearly, but most people have no incentive (in fact they may be punished by the university) to talk about. It is, in other words, a common secret.

Murray Sperber, a visiting professor in the graduate school of education at the University of California Berkeley was a graduate student at Berkeley in the sixties, and was part of an upstart movement pushing for students to rate their professors. 

The idea, Sperber said, was to give students a chance to express their opinions about their classes — a noble thought, but one that has backfired, according to many professors. Course evaluations have created a sort of “nonaggression pact,” Sperber said, where professors — especially ones seeking tenure — go easy on the homework and students, in turn, give glowing course evaluations.

And, yes, this is a very likely cause of the problem. It is common knowledge that a teacher's job may rely on students' evaluations. If the lecturer teaches a demanding course then he/she knows that the evaluations will be bad and then the lecturer will suffer the consequences from the administration: "You are not a good teacher!" So, most lecturers either give up or are, simply, forced over the years to adopt a perverse system. The lecturer is forced to make the course easy, to make it popular, to, simply, work towards achieving a good evaluation from the students. Regardless of the content, if the lecturer gets good evaluations, he/she will be awarded by the administration by, say, receiving a teaching award, a pay rise, and congratulations by both the university and student parents alike. The system works because it makes most parties happy. It obeys the rules of Economy: If I try to sell a good product at the proper price, whereas everybody else is selling a similar-looking, but bad product, at half the price, then I lose, especially when customers just want the brand and not the quality. It can also be explained by the very simple fact, one we are never allowed to speak about, that some teachers, lecturers, professors, etc., find it much easier to teach the same old stuff they once learned, rather than try to educate themselves with new developments in their subject. Teaching, good teaching, is a dynamic process, one that is inseparable from research. And it requires effort. Many teachers find this a daunting task. And if they work in a place where they can, simply, be awarded for doing less, by making both their bosses and their clients happy alike, then that's what they do. The students walk away from the classroom happy, happy that they spent one hour looking at pretty pictures (and sound, and multimedia, and what have you) and happy that they don't have to study at home, but, simply, go partying for the rest of the day. So, everybody is happy.

There are universities where students' perception of how exciting the lectures are is their only goal. To wit, I once worked in a university where one of the head administrators told us:

 We are in the age of www.ratemyprofessors.com

going on to explain that the university is serious about student satisfaction and student metrics. In fact, they had prepared course evaluation forms giving students the possibility to comment on things which were absolutely hilarious. One of the questions, designed by the university administrators, was this:

Even if you never sought assistance from the lecturer, the lecturer was: (a) Very helpful, (b) Helpful, (c) Rather unhelpful, (d) Very unhelpful.

I used to change this question to:

If you sought assistance from the lecturer, the lecturer was...

Simply because the question was irrational: Why should a student who had never asked a question, or gone to the lecturer's office, be in a position to comment on the helpfulness of the lecturer? Absurd!

Another time, one of my colleagues who had a really high students' rating was one who had mental problems. So much so that he would never teach anything the students could understand but who was known to tell them what would be in the exam and even walk out of the classroom during exam and leave students alone for three hours to write their exam. When asked why he did that, he said that he had "bladder problems" and had to be in the toilet to pee. I had commented on this guy some time ago, only in passing. But he's in jail now, so students won't have to complete evaluations about him.

Now, going back to the research of Babcock and Marks, I would like to explain that I don't happen to think, as I commented above, that a lot of studying is necessarily an index of success. However, there is evidence (besides common sense) that those who are not willing to sweat over a subject won't go too far. Genius, someone said, relies on 90% hard work and 10% brilliance, or something like that.

The universities, in all parts of the world, should think about their product seriously. Clearly, the product is education and research or research and education. But if the product becomes faulty, then, however great the economic success and customer satisfaction is in the short term (the last 40 years, say), the long-run consequences will be really grave. Everybody will be crying when they will have to learn mandarin to send their kids for education in China. For the time being, Chinese still come to Europe and America. Because, "education is better". But for how much longer?

Have we also imagined that when we educate someone (ourselves, for instance) towards a certain goal then we really want to achieve that goal? For example, if I want to be a car mechanic, I'd better try to fix a car or two before getting a real job rather than pay to get a car mechanic certificate from an online college, or if I want to work towards finding a theory unifying gravity and the other 3 forces of nature (electromagnetic, weak, and strong), then I had better understand physics and mathematics well, rather than, simply, have a perfect grade on my diploma, a grade which I may possibly have achieved by filling in evaluation forms and by taking courses which are popular, taught by lecturers who use flashy multimedia

Food for thought, for teachers, students, administrators and politicians alike.

The fear of OMEGA

A few weeks ago I finished teaching (yet another time) a sort-of upper division undergraduate probability course. What I want to talk about is the beauty and fear of Ω.

As everybody knows, many undergraduate texts in probability start (pompously so) by putting the subject in its proper basis: A probability space is a triplet (Ω, F, P), where Ω is a set, F is a sigma-algebra of subsets of Ω and P is a countably additive function from F to the nonnegative real numbers such that P(Ω)=1.

And then they go on by giving the reader (only) some trite (silly) examples of probability spaces (such as the set {1,2,3,4,5,6}). After going throuh this rite, the quickly forget Ω.

Poor Ω, you seem to be condemned to death right away, from the start. We talk about you, we make you appear stupid, and then we tell the students: We shall not use this from now on.

What makes things worse is that when we speak of random variables, we immediately tell our students that we shall never write X(ω), but, simply, X. There are, of course, very good reasons for doing so, and, indeed, many times, we need not think of random variables as functions, but, simply, be able to handle probabilities associated with them.

In doing so, we immediately destroy the power of Ω, and tell the student that it's not really there. We condemn it to death. We make students fear of them. Some students graduate, they go to get a Master's, maybe a PhD later, and they reach the professorial levels, all the way having the fear of Ω. So much so, that they often miss a huge part of Probability because they are unwilling to delve into Ω and see that it is there and exists!

I am starting a campaign: Re-introduce Ω and keep it up to the surface, by giving, right from the beginning, meaningful examples where the construction (rather than the axiomatization) of Ω is used.

It took people long time to talk about Probability correctly and now what? Should we pretend we don't know what it is? And keep going on teaching the subject as if it were not understood?

No, I am NOT claiming we should teach it a la Bourbaki. No. I am just saying that, while we do speak of Probability in terms of dice, coins, coincidences, noise, etc., let us not forget that it lives on some Ω which can be used, whenever convenient.

Fractions

In his posting "a rant about fractions", Jason Rosenhouse makes some good points about the way fractions are taught in elementary and middle high school math classes. For example, he says that kids get taken points off if, when adding two fractions, they find a non-reduced result, like 10/24. Instead, teachers tell them they should have found 5/12 immediately. Read the posting for more information on silly things going on in the teaching of elementary mathematics. No wonder, says Jason, that most people end up being afraid or hating mathematics.

Here is something else, from my recent university experience. I taught for a while in a department of Statistics & Actuarial Mathematics (where some very funny things are happening in eduation, both for students and teachers alike) in the UK. In Spring 2009 I was asked to solve some exercises for an elementary probability class, for first or second year students. The students had taken calculus before. We had to compute a certain integral (related to a density function) and I asked the students to do this by themselves. I asked a student to tell me her answer and she, correctly, responded b-2b/3.

"So," I continued, "this is equal to what?"
"I don't know", replies the student (who had done the integra correctly), "I'm not good with fractions."
"What about the rest of the class?", I asked the handful of students who were present. 

Apparently nobody knew how to subtract two thirds from 1.

So, remembering my elementrary school days, I turned back to the blackboard, and drew a pie:

(Actual picture from the lecture)

"So, if I cut a pie into three pieces and take out two, how many pieces remain?, I ask.
"One", replies the student.
"Very good", I enourage.
"Oh, that was easy", says another, "even my daughter could have done this".

When Jason, correctly, expressed his frustration with the teaching of fractions, he referred to elementary and middle school education. I repeat that the example above is taken, from personal experience, from university education.

There seemed to be something very, very, wrong in this system. This is why I left.

Interview with Gromov

In the March 2010 issue of the Notices of the American Mathematical Society, there is a very good interview with Mikhail Gromov, prominent mathematician and the recipient of the 2009 Abel Prize of the Norwegian Academy of Science and Letters. I would like to quote just a couple of little excerpts from this interview, leaving aside the majority of it for future comments which will appear on this blog.

As you know, in the UK, in some
of the universities, there are faculties of homeopathy
that are supported by the government. They
are tremendously successful in terms of numbers
of students. And anybody can learn that nonsense.
It is very unfortunate.

Alas, this is true. However, it is easy to spot fake university programmes, like those offering courses on homeopathy. What is harder is to spot university programmes that hide themselves under the guise of science. Indeed, imagine, for instance, a university offering a course on the Mathematics of Homeopathy, say, luring students who aspire to learn applied mathematics. Such a hoax would be harder to spot.

A second excerpt, from the same interview, goes as follows. (It is a reply to the question on how education should change to get better adapted to very different minds.)

Look at the number of
people like Abel who were born two hundred
years ago. Now there are no more Abels. On the
other hand, the number of educated people has
grown tremendously. It means that they have not
been educated properly because where are those
people like Abel? It means that they have been
destroyed. The education destroys these potential
geniuses—we do not have them! This means that
education does not serve this particular function.
The crucial point is that you have to treat everybody
in a different way. That is not happening
today. We don’t have more great people now than
we had one hundred, two hundred, or five hundred
years ago, starting from the Renaissance, in spite
of a much larger population. This is probably due
to education. This is maybe not the most serious
problem with education.

What he says is absolutely true: Despite the fact that the population has grown by almost an order of magnitude in 200 years, despite the fact that the majority of people are able to (and do) enter a "higher education establishment", the number of thinkers like Abel has not grown in proportion to the world population. Education, says Gromov, is responsible for this. Indeed, many universities nowadays offer vocational training instead of university education. There is nothing wrong with vocational training: we need car mechanics, accountants, paramedics, fire-fighters, etc. What is wrong is to substitute university education with vocational training. And what is much worse is to claim that we are not doing so; that we are still offering "high quality" education, whereas, in reality, we are training people to do a specific job. This is were the problem lies. And this is why Gromov, in the first excerpt above, gave the homeopathy example as an example of what is happening in universities nowadays.

What is mathematics (education) for?

Underwood Dudley, in a critical article in the latest Notices of the AMS, states it accurately:

What mathematics education is for is not for
jobs. It is to teach the race to reason. It does not,
heaven knows, always succeed, but it is the best
method that we have. It is not the only road to
the goal, but there is none better. Furthermore,
it is worth teaching. Were I given to hyperbole I
would say that mathematics is the most glorious
creation of the human intellect, but I am not given
to hyperbole so I will not say that. However, when I
am before a bar of judgment, heavenly or otherwise,
and asked to justify my life, I will draw myself up
proudly and say, “I was one of the stewards of
mathematics, and it came to no harm in my care.”
I will not say, “I helped people get jobs.”