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Saturday, September 1, 2012

Surprise Quiz Paradox

The surprise quiz paradox can be stated as follows: A professor emails his students on Sunday and informs them that they will have a surprise quiz on some day throughout the week (Monday through Friday). An astute student reads this email incredulously because he perceives a logical flaw in it. He knows that the professor can't give the quiz on Friday because then the day of the quiz will have become apparent on Thursday, thereby making it unsurprising. 

Since Friday has now been eliminated as a possible day, Thursday becomes the new Friday of the week. Moreover, the same argument used to exclude Friday can now be used to exclude Thursday. This logic continues all the way to the beginning of the week, with the final conclusion reached that it would be logically impossible to give a surprise quiz. The paradox lies in that it is obvious the professor can still give the pop quiz as he pleases and surprise the students, as was his original intention.

Before we begin to look for a resolution, it would be helpful to first clarify the definition of "surprise" we are using that leads to the paradox. I define surprise in this instance as meaning roughly: "You will be unable to predict at any point throughout the week, at least a day in advance, the day on which I will give the quiz". This is the definition that leads to the core problem of Friday not qualifying as a valid surprise quiz day.

It seems, given the definition of surprise we have provided, it is indeed logically impossible to give a "surprise" quiz, and this is true. However, the paradox lies not in that the statement is logically impossible, but in that the professor can still give a surprise quiz despite all this. It is in this very inconsistency that we shall begin to find our resolution. It seems to me that the definition of surprise that is utilized in contemporary conversational language is not equivalent to the definition of surprise we have employed to establish the fallacy. It is a somewhat weaker definition, which I will now attempt to provide.

The definition of surprise that I believe the professor employs is this: "The exact day of the quiz will be unpredictable for at least the same number of days that it will be predictable". In this sense, the definition of surprise is not a binary concept and there are degrees of strength inherent within it. One individual may be more surprised than another even provided that both experienced the exact same occurrence at the exact same time.  For example, if the day of the quiz is known to someone for three days, but unknown to him for four days, the quiz will be a surprise, though not as much of a surprise as for the individual for whom the day of the quiz was known for three days, but unknown for five.

The corner case of this definition that we must account for is the case in which an individual is aware of when an event will occur for the same duration of time as he was unaware of it. I believe that, in this case, we can say that the individual is in a state that is simultaneously both surprised and unsurprised, although this state would be the theoretically lowest degree of both surprise and lack of surprise conceivable to us.

Given this, and stretching my definition to its limits, one can provide the following example: Suppose somebody tells you on Sunday that they will throw a surprise party for you within the next two weeks. If on the Sunday of the second week, someone happens to clue you in that your surprise party will be that Saturday, will it still be a surprise for you? Technically, my definition is still satisfied in this instance, since you were unaware of the exact date of the party for seven days, and will be aware of it for also seven days. However, I think most people would agree that, despite this, the surprise party will be pretty well ruined by the time it comes around. 

While this may at first appear to be a valid counterargument, it is ultimately specious. As I mentioned before, I believe that, conversationally, humans innately think in terms of degrees of surprise. If I knew of the exact time of the occurrence of a certain event for a longer time than you did, and we both knew this information for a shorter or equal amount of time than we didn't know it, I think I would be justified in saying that I am surprised, but less surprised than you, when that event finally occurs. 

Now since the individual in the surprise party example knew the exact day of the party for just as long as he didn't know it, one can say (as stated before) that, in a sense, he/she is both surprised and unsurprised. This to me still qualifies as a state of surprise, although the lowest one possible in terms of degrees of vigor. Only once the number of days of awareness exceeds the number of days of ignorance does one truly leave the realm of surprise regarding a particular event.

We have seen how, in the paradox, the clause that begins the chain of reasoning that ultimately leads to the surprise quiz becoming a logical fallacy is: "A surprise quiz cannot be given on Friday". The logical definition that we gave at the beginning of this article most definitely satisfies this clause. Let us now, however, try it out with the alternative (and in my belief, more common and natural) definition that I have just provided. Suppose the professor indeed plans the quiz for Friday. On Sunday (the day the students first receive the email), the students are unable to predict the day of the quiz. The same goes for Monday, Tuesday, and Wednesday. Only on Thursday does it become readily apparent that the quiz will be on Friday. This means that for four days the students were unaware of what day the quiz will be on, as opposed to the two days during which they were.

Thus, the definition that I have provided is satisfied, and it is now possible to give a surprise quiz on Friday, thereby not creating the domino effect that leads to the paradox. Suppose that, for robustness, instead of a five day week, we have a two day week, with Monday and Friday being the only days. Then on Sunday, the students are unable to predict whether the quiz will be on Monday or Friday. On Monday, they can safely say the quiz will be on Friday. The definition I provided remains satisfied, as the students were both unable and able to predict the quiz for exactly one day. 

I anticipate that the main point of contention with this argument will be my supposedly commonplace definition of "surprise". Clearly none of us think about the definition I have provided when we use the actual word. I argue that, although this may be the case, subconsciously we more often than not still employ some version of my definition. To illustrate this point further, allow me to provide a simple thought experiment: Suppose that a doctor tells you that within the next ten years, you will die unexpectedly at any minute. Suppose further that you are 100 percent certain that he is correct. Finally, suppose that your death does not occur for the next nine years, 364 days, 23 hours, and 59 minutes. Then and only then will you be sure that your death must occur in the following (the 60th) minute.

According to the initial definition I provided for the justification of the paradox, and given this thought experiment, your death ceases to be a surprise following the end of the 59th minute. However, I believe that most people, myself included, would still attribute to the moment of your death a "surprising" quality, given the vast amount of time that you were unaware of the exact minute of your impending doom (as well as the equally vast amount of time you were anticipating it), relative to the tiny amount of time that you were. Although this is an extreme example, I believe it adequately demonstrates that there are indeed gradations to the state of surprise. 

Thus, I believe the paradox lies mainly within a discrepancy between the logical definition of the word "surprise", and the conversational definition (like, for example, with the word "or"). It is my strong belief that the concepts invoked here can be applied generally to all permutations of the surprise quiz paradox.

Saturday, June 23, 2012

Colors and the Speed of Light: A Thought Experiment

Suppose that we are moving at the speed of red light through a vacuum. Suppose further that, directly in front of us, there is an object that is also red.

Let us first refresh ourselves on how color is actually perceived by the human eye. White light is comprised of equal parts of all the other colors (or types of light) in the visible spectrum, all which have different wavelengths, frequencies, and energy levels. When this light strikes an object, part of it is absorbed by the object, and part of it is reflected, deflected, and diffracted back in various ways. These lights are the ones that we interpret as colors. So, in terms of our red object, it is absorbing all the types of light that aren't red. The red light photon bounces off the object and into our eye, where it interacts with photopsins. The color that we thus see depends on the energy level of the photon that enters our pupil (in this case 1.77 eV, relating to red light), as this determines the strength of the electrical signal sent to the brain.

Now, since the color that we register is dependent on the energy of the photons that breach our eye, and since the energy level of light is calculated via multiplying Planck's constant by the frequency of the wave that accompanied those photons, the color that we detect is also ultimately related to the frequency of the light.

Since we are moving at the speed of red light, towards an object that is emitting red light, the frequency with which the red light waves coming from that object hit our eye should theoretically be doubled (assuming the object is aligned with our sight in a perfect 180 degree angle). Doubling frequency should then double the energy with which the photon interacts with our photopsins, doubling the strength of the electrical signal to the brain. Since double the energy of red light is 3.54 eV, what we actually see in front of us is not a red object, but a violet one (since 3.54 eV corresponds to violet light).

Some further interesting propositions can be gathered from all this. For example, if we were moving at the speed of red light and were instead approaching an object that was emitting infrared light (which we cannot see), it is possible that we would perceive that object as violet as well. On the other end of the spectrum, if we were still moving at the speed of red light and approaching say, a blue object, the object would now relatively be giving off ultraviolet light, which we also cannot see. Thus, the invisible becomes visible, and vice-versa.

All references to color energy were taken from Wikipedia.

Saturday, June 16, 2012

On Performance Based Evaluation and Meritocracy

The American school system, and probably most of our workforce, prides itself on being a meritocracy, or a system in which our rewards are proportional to our performance on examinations. This doesn't seem too suspicious when placed in this light, but meritocracy has its dark side as well.

One's performance is often considered to be a function of one's ability and effort. However, this assumption is not always accurate. When we base the rewards reaped in society on one's results on examinations, we open the door to gamesmanship. What do I mean by this? The purpose of education no longer becomes to learn, but to maximize one's score on a test. Why does this occur? Because in a meritocracy, nobody cares about the process that was taken to get to a certain score. All that matters is the score, the whole score, and nothing but the score. Thus, students gain a very narrow minded view of the system, in which they do not study out of interest, but out of the desire to get the highest grade possible on an exam. I mentioned some of these points in a previous article titled "Studying the Tests".

What's so bad about this anyway? That depends on how useful the knowledge on the exams actually is for the real world. Since the students in a meritocracy will spend most of their time learning the tests, it would make sense to design those tests around useful skills. If the test is simply designed to measure intellect or some other abstract concept, then the existence of a meritocracy may actually be harmful for society.

The above argument is often countered by the belief that, even if an exam tests skills that are largely pointless to society, a high score demonstrates a strong work ethic, assuming that the tests are designed in such a way that they can be studied. It is also argued that there is nothing wrong about testing for intellect, since a higher intellect demonstrates a higher capacity for learning and mentally adapting to various challenges. This makes those with higher intellects ideal employees, as they can quickly pick up any necessary skills missed in school once they reach the job.

To a certain degree, both these points are correct. It does often take much studying, and thus work ethic, to do really well on exam, but it also means that individuals with a higher baseline level of talent can skate by the exams with minimal effort. Since these individuals were gifted with their talent, they have not demonstrated any capacity for hard work by doing well on the evaluation. The same goes for the point about testing for brain power as a good means of finding successful employees. Those with higher intellect will not demonstrate any work ethic by doing well on exams based largely on intellect. Does a brilliant but lazy person really make for a better employee than a duller but highly productive one? I think not. Both bring something different and useful to the workforce, and both should be given a chance to prove themselves.

So, how can we better keep this situation from arising? The key is to make sure that examinations and other performance evaluations do not lean too far to the side of testing fluid intellect, or the intellect that we are largely born with. Performance evaluations should base themselves just as much on crystallized intelligence (or learned facts) as they do on talent. For example, in a typical college classroom, attendance is either not required, or counts for very little at all. It is even more surprising when professors begin to wonder why students aren’t going to class. In a meritocracy, there is simply no point in coming to class if the student feels he can do just as well on the exams without showing up. Rectifying this is simple: Make attendance mandatory and give it more weight during the calculation of the final grade. There are few clearer signs of work ethic than consistently attending lecture.

There are certainly many other examples that could be generated, but the basic concept remains the same. To avoid spawning an unproductive workforce, we must make sure that we emphasize effort and crystallized intelligence in meritocratic examinations and evaluations. We must make sure that we emphasize that effort matters as much for success as talent does.

Wednesday, June 13, 2012

On the Theory Behind Workplace Web Filtering

Many modernly equipped firms in today's world, especially large and old-fashioned ones, follow a policy of website filtering, by which the employee is restricted from browsing to websites deemed to be either unsafe or not conducive to a productive work environment. Thus, popular social media and entertainment websites such as YouTube and Facebook are blocked, as well as more standard email websites, such as Gmail and Yahoo Mail.

The logic behind the policy is simple enough to speculate about. A firm believes that by limiting the access to these websites, an employee will begin to consider that, since he can't procrastinate in an enjoyable matter, he may as well perform his work duties. In essence, the idea is to make the work duty the most entertaining thing the employee could do on the job.

However, the logic against such a policy is just as simple, and perhaps even more convincing. Who is to say that, if an employee can't procrastinate a certain way, he won't simply substitute it with another? Also, the reasoning that work will automatically become the most fun thing to do at work after such a policy becomes a bit more suspect when we examine it closely. An employee still has the options of communicating with other employees, taking a walk, getting a coffee, etc., once his favorite websites are taken away. In my opinion, there is no shortage of the number of things that an employee would be able to come up with in order to avoid his duties, if he was really so inclined.

Another factor that I believe deserves consideration is the mental fatigue aspect of being expected to work continuously for many hours a day. It is possible that, at least for some people, the break associated with browsing to certain sites for small periods of time is not only a desired, but a necessary break. Here is an example of such a situation: Suppose an employee's job is to type as many pages as possible per day. An employee taking no breaks may waste no time doing other things, but his average number of pages per hour may begin to decline as a function of time because of mental fatigue. Suppose there is another employee that takes periodic breaks on entertainment websites. While this employee does lose all the time that he spends on those websites, it is feasible that, following one of these breaks, the worker's mind is refreshed and his average number of pages per hour increases substantially (if temporarily). This example could also be easily adapted to include quality of work, for which one may see similar results as with quantity. Thus, it is also conceivable that, at the end of the day, the total number of pages completed by the procrastinating worker could be greater than that by the diligent worker.

Another interpretation of the mental refreshment aspect is to think of it also as a possible relaxant. An individual may be worried about a certain email he is expected to receive or send out, or some other quick business he must attend to on a restricted website. The stress that comes from this anxiety will obviously be a hindrance to one's productivity. By simply allowing this individual to quickly take care of this business, he will be able to stop thinking about it and get back to work with a renewed sense of vigor.

One more point I'd like to consider is the force of habit. Before finding employment, most people have spent countless years developing the habits that allow them to be the most comfortable and to work the most efficiently that they can. Some people listen to music, some people keep a chat window open, some people check the news. From the very fact that these habits have remained with this person to this day, and from the fact that this person was able to secure the job in the first place, there is the direct implication that the person's working habits are of sufficient quality for him to be employed at that particular workplace. Why then, would it make sense to throw off a worker's habits by restricting certain websites? Even if it is assumed that the employee can quickly adapt to new work habits, there will be a learning curve of variable length during which productivity will naturally be stunted.

I’d like to make it amply clear that I am not advocating procrastination in the workplace. In fact, I am advocating quite the opposite, as I hypothesize that disabling workplace web filtering will ultimately improve employee performance.

Wednesday, June 6, 2012

Improving Professorial Quality

Anyone who has spent a considerable amount of time as a student in academia has no doubt had the displeasure of taking a class with a sub-par professor. What this article seeks to propose is a methodology by which universities can systematically improve the quality of their professors and help minimize the probability of this happening to anyone else.

Before we begin, it would be helpful to examine why this problem of poor professors arises even in the top universities. One reason, I believe, is a lack of incentive, most often as a result of the granting of tenure. Tenure is essentially a lifetime appointment until retirement, with dismissal only possible in the case of "due cause". The interpretation of this "due cause" is naturally very variable, as most such legal terms are, but it is probably safe to say that it would take quite a bit for a tenured professor to be forcibly removed from the job.

The fear of under-performance resulting from the privileged position of tenure is hardly a new topic of discussion, though it remains a very powerful argument. How can one expect a faculty member to retain a desire for providing quality teaching when his job doesn't depend on it, except in the most egregious cases? This problem is exacerbated even further when the professor is particularly famous, making his status even more unquestionable. It is only to be expected that a professor's yearning to teach will eventually fall victim to entropy under such an environment. For example, many students can attest to the fact that some professors use the same slides, homeworks, tests, etc. for literally decades, sometimes not even bothering to change the date at the top of the handouts. While this is not a problem in and of itself, and can actually be logical when the teaching program in question is successful and pleases the students, more often than not it is the terrible and ineffective programs that get recycled for years and years.

The solution I propose to this phenomenon is quite simple, and focuses on exploiting the same factor of incentive that falls prey to indifference. Most schools already have in place a ratings system by which students can evaluate the performance of their professors (and if they don't, they should). Instead of penalizing poor performance, why does a university not monetarily encourage strong performance? For example, a professor would be granted salary bonuses from the university corresponding to the average rating that he has received from his students. The beauty of this method is that more famous professors can still receive the higher base salary that they likely feel they deserve, nor is the capacity for the professors to do research lessened, since the awards associated with that would remain intact as well. Tenure is also not threatened since good performance is rewarded, but poor performance is not penalized.

One obvious problem to this solution is the lengths that some would go to in order to capitalize on the system. Some professors may believe that the quickest and most reliable way to improve their student ratings would be to simply to give A's to everyone in the class. Although, in my opinion, the connection between grade received and the ratings given to the professor remains tentative (since the ratings are anonymous), I will admit the theory is feasible. Although some grade inflation will probably occur, I believe the normal checks that are in place (or should be in place) to make sure a suspicious amount of A's aren't being doled out should provide enough of a counter-thrust to this snag. Another problem with this system is that its implementation would almost certainly cause tuition to increase, as the potential for very large expenditures due to an over-achieving staff is not necessarily unlikely.

A second reason for the existence of incompetent professors is for the self-evident reason that they simply don't know how to teach. Too often is the mistake made by universities that eminence in one's field implies some sort of natural predisposition for teaching. This makes no more sense than holding the belief that a talented athlete would necessarily make a good coach. A professor may have the most brilliant things in the world to say, but they are worthless to the students if he cannot express himself properly. Furthermore, an often neglected part of being able to teach is having the ability to design reasonable and helpful assignments, tests, and classes in general. I can't count the number of times I have taken exams that are difficult solely for the reason that the test is poorly constructed, or have done assignments that only tangentially have any relationship to the format of those exams.

How could we alleviate this situation? To some degree, the incentive structure I proposed above would help because it would prod the professor to improve his own teaching. However, it is possible that a professor could simply have no idea how to become a better teacher on his own. To combat this, the school could provide some basic courses in education upon the professors introduction to the university. To maintain quality control, refresher courses could be given at certain intervals after that, and officials specializing in teaching inspection could be assigned to sit in at any lecture at random (like is already done in some high schools, although often not randomly). Obviously, these structures are not free and could, as stated before, definitely have the potential to increase tuition. However, all that being said, I see few better investments for an institution than increasing the excellence of its professors. Better professors lead to a better school reputation and better educated students, who will thus later have more capacity to contribute to the university and continue this cycle.

One more topic I'd like to discuss are teaching assistants, or TAs. These are essentially either aspiring professors, or more often simply graduate students, who are required to work for the professor for a certain amount of hours in exchange for the research stipend and facilities that they receive from the university. They are often in charge of grading homeworks/tests, and teaching recitations. Moreover, if there is anyone who can be expected to care less about what a student thinks about his teaching more than a tenured professor, it is most certainly a teaching assistant.

My problems with teaching assistants are multiple. Firstly, since the spotlight on their teaching ability is even less bright (arguably) than that which is on the professors, and since most of them are so busy focusing on their research, the quality of teaching is understandably expected to be rather low. My bigger issue, however, is that teaching assistants are often in charge of almost all the grading in a class. Why is this an issue? In my experience, most teaching assistants do not even bother to attend the professor's lectures. Thus, although by no means guaranteed, one can expect some degree of disconnect between what the professor teaches and how the TA grades. For example, if I attend every single lecture and listen attentively to the professor, and feel it in my best interest to write something on the exam that the professor mentioned in class, it is not unlikely that a TA could disagree with it and dock me some points. What have I to prove to him that the professor did in fact say it? Nothing really, whereas if the professor had graded the test, or if the TA had attended the lecture, the probability of me not losing points would have increased. 


How to resolve this? I understand that professors cannot feasibly grade all the material that they assign, as well as have the time left to perform their research to a respectable degree (or at least I hope this is the case). This, after all, is part of the reason for the existence of teaching assistants in the first place. However, since TAs are also often subject to student ratings, I see no reason why they also cannot be part of the system that I described before, except instead of bonuses, the TA would receive an increased stipend for his performance as a recitation leader. Also, since TAs are not tenured and are sometimes professors in waiting, I see no real reason why there should not be a punishment or warning mechanism in place for TAs with especially bad ratings. They are, after all, potential educators of the future, and it would behoove the university to set some kind of basic standard to help the TA refine his art for years to come.


The professorship is an honorable position, but it is not so honorable as to excuse a gross neglect of duty. Professors are paid to be as much teachers as researchers, and the ultimate nature of the products of the educational system rests in their hands. I do not see a more expedient way of sharply enhancing this educational system than to enhance the efficacy of its distributors and molders.

Monday, May 28, 2012

Buying Textbooks

If you're reading this right now and you're a student, you're most likely getting ripped off when buying your textbooks. This isn't an affront to your intelligence, but simply the conclusion of my belief that the average student buys the majority of his textbooks from the campus bookstore, which more often than not charges exorbitant prices. I understand that universities need to make money too, and that I have no right to judge their pricing decisions, nor will this article do anything to hurt their revenues. It is also true that what I have thus far written will most likely shock no current students, as the steepness of campus book markups is well-known throughout academia. 

So why is it that the majority of students (I have no facts to support that it is indeed the majority; only a strong suspicion) still insist on these inefficient purchasing habits. The most likely answer is also the most logical one: To avoid getting ripped off for textbooks, a certain amount of base knowledge and research is necessary. This is not to say that it is inherently any more difficult than walking over to the college store, but the initial acquisition of know-how that gets one to the point where they can use that know-how with minimal effort can have a high personal cost. This phenomenon is akin to say, changing a bike tire tube. There is nothing impressively complex in the process, but many people still see a professional about it because the cost required to learn how to change the tire proficiently can be uncomfortably high.

Getting back on point, I would like to lessen this cost for the readers by providing a few tips and websites that should make them much thriftier textbook purchasers. First off, I have found that websites which aggregate the costs of a few online bookstore websites and present a list of the lowest prices to be the most effective. A good example of this is CampusBooks. The site will present you with rental, used, international, new, and digital versions of the book you are looking for. Rentals are often the cheapest of the five options (Chegg is the biggest player in the textbook rental market), but remember that you cannot later sell these books back on the market to recoup some of your losses (or even possibly turn a small profit).

Possibly the next cheapest choice would be the digital version, otherwise known as an e-Book. Keep in mind that although they are often inexpensive, they also cannot easily be later sold on a major open online marketplace (more on these later on). There are also more obvious limitations, such as the need to have either a high-end phone, tablet, or computer to view them. Although in this day and age this shouldn't constitute an enormous problem, it can occasionally turn into a nuisance.

Used books are the next best option. Be suspicious of unusually great values found in this section (as well as the new section), as these are most likely international editions. International editions are produced abroad, usually using cheaper materials and shoddier craftsmanship, and while this is usually not a problem, it can become one when the page numbers and the order of practice questions are askew. Nothing stings quite as much as unwittingly doing questions 15 through 20 when your professor requested 1 through 5. There is no easy way to tell which internationals are winners and which are lemons, but I prefer to err on the side of caution and just pay the extra dollars for a regular used edition.

After the semester ends, you have a few options as far as trying to recover some of the cash you dished out. You can obviously sell the books to a fellow student at your university; many schools provide some type of system that helps you connect with potential buyers (you can, by the way, obviously use these same systems to purchase a book as well). Alternatively, you can attempt to sell that book back to your bookstore, although I question the quality of the deal you would receive there. Finally, you can attempt to sell the book through online marketplaces, such as Amazon or Ebay. This takes some initial experience as well to do correctly, but once you get into a groove with it, I feel the best values can usually be found through this method.

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