The Abstract Power of Mathematics

Yesterday, as I was walking home, I stepped upon an acorn just to hear the popping sound it makes. Stirred by a moment of pique, I picked it up to investigate what was inside. To my disgust, I saw a couple of maggot/grubby things, which made me feel bad about stepping on it, and all the other acorns upon which I have trod. This got me to wondering, how likely is it that a given acorn has a grub in it?

Since the first one I investigated had a grub in it, I got the impression that it would probably be a fairly likely thing. If only one in every hundred contained a grub, it would be terribly improbable that the first one I examined would just happen to have one. However, I wasn't interested enough to try to remember how conditional probability worked, so I just went home.

After I got home, I went to pay my rent. Walking out of the rental office I saw a maintenance van, which was apparently number 1 in their fleet. Since the van I saw was labelled one, I started wondering about the size of their fleet. If you see a van with a certain number, you can assume the fleet is at least that large, but it doesn't tell you how big it actually is. So, seeing a van with number one seems to indicate a small fleet, otherwise I would be much more likely to see a van with a large number. When I realized that this was, basically, the same problem that had occurred to me earlier, with the grubs in acorns, I decided that who was I to ignore coincidence, and decided to work out the conditional probabilities.

Most of the time people think about probability knowing, approximately, how likely outcomes are and they want to figure out what will happen. For example, if you roll a six sided die, you would be interested in knowing what number you roll. Conditional probability works in reverse, if you know you are in one of a few scenarios and you know what happens, you try to figure out what the likelihood is that you are in a specific scenario. For example, if someone rolls a six sided, eight sided, or twenty sided die and they tell you that they roll a 7, you know they didn't roll the six sided die. For more information on determining conditional probabilities see below*

For simplicity, I decided to assume one out of every so many acorns has grubs. This ignores a large number of cases, for example 2 out of every 3 have them, but we can round those to the nearest considered case. I made this assumption both because it simplifies the calculations, and because it makes the acorn problem: the first acorn I see has grubs when 1 out of n do, the same as the van problem: the van I see is number 1 out of a fleet of n total vans.

One would sort of expect lower numbers in the fleet or higher grub frequency to be the most likely scenario. But, it turns out, if you consider fleets of any number of vans to be possible, then even a fleet of one van, the most likely, turns out to be statistically impossible, or a 0 probability event, if you remember my first post on probability. This is because, although the van's number is 1 100% of the time, or 1/1, if the fleet size is one, the probability of the van's number being one over all possible scenarios turns into the infinite sum of 1/1+1/2+1/3+1/4... which diverges**

Anyway, I suppose the main point of this post is not conditional probability, or the odd things that happen when you let infinity enter things, but rather how wonderful it is that math can take two, ostensibly different problems, like vans in a fleet and grubs in an acorn, and unify them into one, underlying concept.

*Calca I: Conditional Probability

Suppose that you know something happened, say someone rolled a 7 on a die, but you don't know exactly what circumstances led to this occurring, they either rolled a die with 8 sides or a die with 10 sides. To figure out how likely a scenario is given a known outcome, one simply divides the likelihood of the known outcome in that given scenario by the sum of the likelihoods of that outcome in all possible scenario.

For example, the 7 will roll 1/8 of the time on an 8 sided die and 1/10 of the time on a 10 sided die, so the likelihood that an 8 sided die was rolled to obtain the 7 is: (1/8)/(1/8+1/10), or 5/9. This makes a certain amount of sense, the 8 sided die is more likely to produce a 7 than the 10 sided die, so if a 7 rolled, we should think the 8 sided scenario more likely than the 10 sided one.

**Calca 2: The Series 1/n

If we try to find the probability that a fleet only has one van, given that we saw the van numbered 1, out of scenarios allowing the fleet to have any number of vans, we run into a problem. The numerator should be the probability of the observed event in the specified scenario, namely seeing van number 1 in a fleet of one van, or 1/1. However, the denominator should be the sum of the probability of seeing van number one over all scenarios, which is (1/1+1/2+1/3+1/4+...).

To get a feel for how to handle this sum, let us ignore the first term, 1/1. The next term, 1/2, contributes 1/2 towards the total. The next two terms, 1/3 and 1/4, are each greater than or equal to 1/4, and so together will contribute something at least 1/2 in size. The next four terms, 1/5, 1/6, 1/7, and 1/8, are each at least 1/8, so together they will contribute something at least 1/2 in size. Continuing on this way, we notice that the next 2^n terms are always at least 1/(2^(n+1)), and so together contribute something of at least size 1/2. Thus we can think of the sum as being larger than an infinite number of 1/2's being added together. This implies that the sum cannot be a finite number, so the conditional probability of the 1 van fleet scenario is 1 divided by a limit which approaches infinity, so the probability goes to zero.

Pedagogical Problem

So, I have a relatively important decision to make before I finalize my syllabus for my Calculus I classes, and, since my friends contain a statistically significantly higher proportion of educators than the general population, I thought I would solicit some outside opinions. Accordingly, please feel free to weigh in with your thoughts in the comments, if you have some to contribute, as this is the purpose of this post. I may not follow advice received, but I always consider it.

My problem is thus: the course grade contains a portion allotted to weekly work. Homework and quizzes are the two main options. Last time I taught this course I opted for the default of quizzes, because it was the recommended option and it was the first time that I taught the course. However, despite my emphasizing the importance of working through the, non-graded, homework problems, I do not believe much of my class did, and I think their grades suffered consequentially. I could continue with this method, since their grade is, ultimately, their responsibility, however I am tired of feeling like I am failing my students, and would like to take a less scorched earth approach this year.

One option, and this is the one which I am currently favoring, is to inform them that the quiz problems will be selected from the suggested homework problems. This seems like it might provide added incentive for students to work the homework problems, as well as assist them in knowing how to study for the quizzes. On the other hand, however, it means that the quizzes will not be as representative of the exam, which is what I normally like to do with them, as on the exam I try to ask questions that emphasize what I think is important from the material.

What I had decided to do prior to thinking about it during the course organizational meeting was to simply have them submit the homework problems then grade an unannounced subset. If I am mainly trying to get them to do the homework, why introduce the middle man in the form of quizzes was my train of thought. Since there are so many homework problems, it would be incredibly time consuming for me to grade all of them, thus the unannounced subset bit. However, it occurred to me that this might still be hard for me to do efficiently if homework was messy. Thus I would have to impose some organizational requirements, which, although salutary for my students as budding mathematicians, would likely require a lot of effort on their part to implement for all the homework questions. In the end this seemed likely to result in more work for my students than simply allowing them to do the homework in whatever form makes sense to them then having them take additional quizzes.

Finally, I considered doing a hybrid of homework and quizzes. Homework most of the time, as detailed above, but four quizzes on weeks prior to exams. Ostensibly to provide a review, my actual reason for wanting these quizzes was the aforementioned goal of providing them with an example of how I write questions before they see them on the test, since I sometimes consider book problems too insipid to simply mimic.

So, if you have any thoughts on this topic, feel free to leave them with me :)

Resource Center for People with Disabilities

Time to get back to my roots and do another post about education, albeit a less sweeping one than my normal philosophical ramblings. The motivation for today's discussion is something I have to deal with as an instructor, namely, mandates from the Resource Center for People with Disabilities, or RCPD. The most common way that RCPD affects the classroom is by mandating that students receive augmented time on quizzes and exams, usually 25% or 50%.

As I was proctoring my last chapter exam for the semester, I was musing on this policy, about which I have mixed feelings. Of course, as someone who feels that education is of great value, I think it laudable that we make efforts to enable as many people as possible to partake in it. However, I think there are issues of classroom procedure, fairness, and philosophy, in order of increasing importance in my opinion.

Most trivially, it feels slightly disruptive to allow some students more time than others, even as a leveling tactic to take into account external factors. I feel my ability to insist students hand in their work if I am obviously not applying that standard to all my students. Of course, it doesn't help that I am, in general, as assertive as a wet paper bag, and that I think in an ideal world students would have all the time they desired to show their mathematical ability. However, as someone who proctors his own exams and quizzes, I do understand the practical reasons for time constraints, as I often have other things that I need to do.

In regard to fairness, there is a slightly trickier issue. First, let us consider what a grade in, just for example, a math class means. We ought not take it as an indication of a person's fitness as a human being, or even as a measure of their overall intelligence. As I understand it, ideally, the grade in a math class is meant to denote a student's level of mastery in a subject. Note that I do not say, "mastery in a subject relative to their own abilities."

Believe me, at times it breaks my heart that I do not give grades based on effort! But I believe that maintaining the integrity of our evaluation system requires that students receive grades that reflect their mathematical mastery, not necessarily the work they put into the class. While the two our quite related, in cases where a student enters a class without a mastery of the prerequisites, for example calculus students intimidated by fractions or baffled by trigonometric functions, one's effort can only do so much to remedy their foundational disadvantage. However, in the case of someone with an RCPD Individualized Education Program (IEP, I don't think this is the term used, but it is the one I remember), if I give them the same grade as someone without, I am saying that the two have an equivalent mastery of mathematics, if one is given such amount of extra time. Note, that my biggest concern is that the accommodation is not, in fact, geared toward helping their education, just their grade. While students can turn exams and quizzes into learning opportunities, it has been my impression that, for the most part, they do not do so, perhaps through lack of practice at this skill. Thus an accommodation that only affects quiz and exam times is more of an Individualized Evaluation Program than an Individualized Education Program.

Of course, for some people the problem may not be that they need twice as long to do the math as their classmates, but that they need twice as long to do it in a test setting. And here we run into the philosophical problem. If a student's problem is with the evaluation method in question, rather than actual comprehension, and Individualized Evaluation Program is an appropriate recourse. However, if someone has test anxiety, does giving them extra time to spend on the dreaded test really level the playing field. We are so enamored with the concept of standardization and uniform testing that we often try to shoehorn students to fit the model we have created for them. It isn't as though a written examination is the only method I have to gauge the level of mathematical mastery my students possess, although it is a very efficient one for handling students en masse. Hopefully, by now you get the impression that I suspect our test-driven evaluation is another symptom of the industrialization of education.

At the end of class, I am left with the suspicion that I have been fair to no one. Students without yet diagnosed individual needs are being measured against those who have obtained an advantage, and students with special needs are being evaluated in a manner which may still not adequately capture their true mathematical proficiency. This is why I prefer tutoring, it is so much better to get to work with people one-on-one and just chat about mathematics.

Probably Not

Bear with me, because we are going to start out with a little bit of easy mathematics. I promise that there is a more interesting point. Anyway, one way to think of the probability of something is how often you expect it to happen. So, if something happens 50% of the time, like a fair die rolling an even number, out of one hundred rolls you would expect about 50 of them to be even. Similarly, if you have a .5% chance of winning a raffle, you expect that you would win about once if the raffle happened 200 times, or about 5 times if it were to happen 1000 times.

Something with a 100% probability is, for practical purposes, a sure thing, and something with 0% probability is, for practical purposes, impossible. Because mathematicians are lazy, something I spent quite some time emphasizing to my class, we often call things that happen with 0% probability, 0 probability events. Since we assume that something does happen, the probabilities of all the possible different outcomes needs to add together to 100%.

As mentioned, 0 probability events are impossible for practical purposes, but with a bit of consideration one can come up with times when they would happen. Suppose you have a computer that picks a positive whole number at random out of all the positive whole numbers, with each number being equally likely. There are so many positive whole numbers that the chance of any individual number being picked is 0%, yet as soon as the computer picked one, a 0 probability event would have happened. Of course, this is impossible to implement in reality, because no computer has enough memory or processing power to choose a whole number from amongst ALL of them, but it is an example of a 0 probability event occurring, theoretically.

A similar 0 probability event is for a number, not necessarily a whole number, between 0 and 1 to be picked assuming each number is equally likely. Again, because there are so many numbers between 0 and 1, each individual number has 0 probability, but also again, this also means no computer could actually pick amongst them. So, the question becomes, can 0 probability events occur non-theoretically?

Asking this question prompts some interesting observations about our world. Suppose you think of a dart board as a continuous surface, then throwing a dart at it and seeing where it sticks is a 0 probability event, as there are just so many slightly different places it could have stuck. However, if you think of the dart board as being discrete (made up of discernible blocks), even if you have to go to the atomic level, then each separate place the dart could land has some non-zero probability. Admittedly small, but more likely than impossible. So, refusing to believe that impossible events happen forces one to consider a discrete universe, which is hardly a handicap because at its basic level the universe does seem, at this point, to be discrete.

Also interestingly, consider the concept of free will. If everything in the universe is deterministic, that is what happens next is forced to happen because of what has happened previously, then what happens is the only thing that could happen and has 100% probability. Of course, this model of the universe is not conducive to a belief in free will. If multiple things are possible, but only a finitely many number at any given juncture, then free will is theoretically possible, allowing us to choose between the provided options, and each possible option will have positive probability. However, if anything is possible, as they sometimes say, then it becomes mathematically viable for possible events to have zero probability. It is not necessary, because you can add up an infinite number of positive numbers and still only get 100%, but under these conditions it becomes possible.

So, if one wants a 0 probability event to occur, one must believe that anything is possible. I find that a bit interesting. Also interesting, this thought process occurred because, on my way home from school, I came up with a joking characterization of Occam's Razor for the Middle Ages, "the explanation that requires the fewest witches must be true." Then, to be a bit more serious, I reformulated as, "the explanation that requires the fewest 0 probability events must be true." That got me to wondering if 0 probability events actually could occur, and the rest you know.

In Defense of Pi

I don't often do this, but, in honor of Pi Day, today I shall take up my words in the defense of traditional values. The traditional value in question is Pi itself. Much fuss has been made by proponents of a new circle constant, Tau, which would be equal to two times Pi. As much as it pains me to argue against the enigmatic and engaging Vi Hart, Pi is very much right.

While all concerned, except perhaps politicians and Biblical literalists, agree that Pi is an irrational value which is about 3.14, the argument is whether Pi is the value that we should consider important. Tau proponents argue that Tau is more sound pedagogically. Instead of half a circle being Pi radians, it would instead be half Tau radians. A quarter circle, a quarter Tau.

I cannot argue against this claim, it seems that high school mathematics would be made slightly simpler if we were to switch our focus from 3.14 to 6.28, however, arguing that we ought do this ignores a wealth of mathematical history. Humans have been seeking the decimal value of Pi as long as there has been civilization. The use of the Pi symbol to represent the value is over 300 years old, and was popularized by the mathematical great Euler. To cast Pi aside for slight notational clarity would be to show a remarkable disrespect for the historic foundations of mathematics.

Make no mistake, a slight notational clarity is the only true advantage of "Tau." Because "Tau" is just 2 times Pi, moving from one notation to the other is a trivial computation. If you have a value which is c*Pi, it will be (c/2)*Tau, if you have a value k*Tau, it will be 2k*Pi, easy as pie! Whatever notational clarity is gained by making the change, I would imagine more confusion would be created as we introduce a second, unnecessary, symbol for a concept that is already well represented.

Despite my nostalgia, I understand the systematic logic behind revoking Pluto's status as a planet. Despite my upbringing, I envy the computational clarity of the metric system. So please believe me when I say that I see no significant computational advantages to "Tau" over Pi!

The "Three 'R's"

"I am my thoughts. If they exist in her, Buffy contains everything that is me and she becomes me. I cease to exist. Huh." -Oz, Buffy The Vampire Slayer

The Three 'R's represent an old idea on the fundamentals of education are, Reading, wRiting, and aRithmetic. While meandering through East Lansing, I fell to pondering why I still consider mathematics education relevant in a world where computers are increasingly able to perform the calculations that students are called upon to make. While answering this question, I also hit upon why I consider my dual background in math and philosophy a natural pairing, and an overall theory of education that can be explained by the Three 'R's.

To me, the natural product of education is not knowledge, but rather thought. In a society shaken to its foundations by the advent of the Internet, knowledge has become an increasingly available resource. On the other hand, thought, along with love, remain the cornerstones of that which is best in humanity. Thought can be further broken down into a three step process in which we must repeatedly engage to truly fulfill our calling as thinking beings. First, we obtain new knowledge, which I symbolize by Reading. Next, we synthesize the new knowledge with our existing knowledge and thought structure, aRithmatic. Finally, to complete the cycle we must make our new thoughts socially available so others to may obtain and respond to them, which is wRiting.

We can certainly specialize in one of these categories. Authors, advertisers, and my beloved educators focus on the presentation and sharing of information, all of which would fall into the wRiting category. Scientists and historians, for example, attempt to better acquire knowledge, and as such are Readers. Finally, people, such as mathematicians and philosophers, who seek to find structures in the raw data of life are practicing aRithmatic.

Although we may specialize, to fulfill our yearnings toward humanity we must complete the full cycle, although not necessarily completely alone. Thus, the mathematician need both examine previous works for background and present her or his own conclusions with at least a modicum of communicative savoir faire. Similarly, scientists need to synthesize their data, or find a mathematician to do so, then present their findings and educators must first educate themselves, then make their own separate peace with their subject material, before finally preparing it for classroom consumption.

To focus in on the middle step, the importance of aRithmatic lies in this. Somewhere between gaining knowledge and presenting our conclusions, we must add our own immeasurably valuable contribution and think for ourselves! Ideally this is the task for which we are made ready by our mathematics education. The true task of the mathematician is, once very basic ground rules are asserted, to go forth and discern what else must necessarily follow and why. Although this path has been blazed by mathematicians long dead, there is no reason that a student cannot walk it anew as they too are exposed to the shinning structure of mathematical knowledge. All too often mathematics serves the opposite purpose, and students are led to blindly memorize unmotivated methods, encouraging an arcane practice of "mathematics" which greatly resembles the showmanship of a magician, wherein things appear and disappear in somewhat predictable patterns but without respect for an underlying sense or reason.

My word choice here is not mere hyperbole, I often say to my students verbatim, "math is not magic," to emphasize that they ought know why the processes in which they are participating work. If one simply considers math to be the manipulation of numbers and symbols via memorized methods with the desire to produce some correct end value, then the questions, "why do I need to learn this?" and, "why can't I just use my calculator?" make perfect sense, furthermore, they have no good refutations. However, if one takes math to be a process by which one understands a set of rules and manipulates them to obtain logical, but not at all obvious, conclusions, then these questions make no sense, as well they shouldn't. The best computers we have are merely calculators, not true understanders.

This is why math, and more so philosophy, are so important to me. They are, perhaps even more than dancing, an essential expression of who I am and what it is to be human. Pure, magnificent, creation of new thoughts, MY thoughts, out of the read works of those who precede me, which I then write and send forth into the world that I may be known, and through being known create the possibility of fellowship and love. Thus, I implore educators to teach a comprehension, rather than calculation, based math. The math itself isn't important, it could be replaced by philosophy, law, debate, or any other subject based in the recombination of information. What is important is that we claim our birthright, the ability to create thought, without which not only may we never be human, we may never be loved.

Science Fairs

Now, I may not be the best qualified to discuss a science fair, never having participated in one, but I recently read a New York Times article on the decline of the science fair, and I wanted to respond. The whole article is a quite worthwhile read, but the line that irritated me into action was, "many science teachers say the problem is not a lack of celebration, but the Obama administration’s own education policy, which holds schools accountable for math and reading scores at the expense of the kind of creative, independent exploration that science fair projects require." It seems that viewing math and reading as somehow in opposition to creative, independent exploration is yet another example of why our education system is declining.

Setting aside, for now, the belief that reading and math are both inherently sources for creative, independent exploration, let me first argue that reading and math are cornerstones of science. While I may not have participated in any science fairs, toward the end of elementary school I did design and conduct my own experiment, with assistance from my parents of course. One convenient facet of moving so much as a child is that, if I can remember where something happened, I get a fairly accurate idea of when it occurred.

My experiment involved seeding multiple little, plastic flowerpots with grass seed, then covering them with plastic 5 gallon buckets for different durations throughout the day. My hypothesis was that, since sunlight is necessary for photosynthesis, grass that was covered during less of the day would do better. More interestingly, I wanted to see how significant a small decrease in the length of the day would be and what would happen to the grass living in almost total darkness.

While I don't recall exactly what results I obtained, I do recall there being interesting differences between the pots. In order to describe these differences I used not only qualitative standards, such as the color of the grass, but also the quantitative standard of how high it grew. Any time you use a quantitative standard, there is a good chance your experiment can benefit from mathematical understanding. Most fundamentally perhaps, one can plot the data points then approximate a function to describe the relationship between sunlight and grass height. Then, if one knows math, one can translate one's knowledge about functions into further educated guesses about the behavior of grass. Also, if one is curious about the results that others have already obtained on the subject, the ability to read critically is probably going to be a crucial one to have.

That said, I happen to believe that reading and math are both fertile sources for personal creativity. Some of my most prized remnants of my grade school education are poems that I wrote, either for official assignments or personal gratification, I was a sad little emo-kid (see how I pretend that has changed...). To consider the ability to write as independent from the ability to read seems silly enough that I believe I do not need to address it, please correct me if I am mistaken. Another piece of paper that I treasure contains my verification that the power rule of differentiation works for arbitrary polynomials. This was not part of a homework assignment, but I knew the power rule and the limit definition of the derivative, and I was curious why it worked. If that isn't independent exploration, then what is?

I guess that my main point is that, not only are math and reading essential parts of scientific exploration, but personal exploration is an integral part of any education, math and reading included. Furthermore, we absolutely NEED educators at every level emphasizing this message to students. Otherwise we populate our college calculus classes with students concerned only with what is true and what the answer is, rather than why things are true and how to obtain answers for themselves. This trend is not only deleterious to our education system, but also to our society, as it seems destined to produce citizens who would rather be given the "answers" than wrestle, often futilely, with the important questions.

Mathematics, A Eulogy?

Did you know that the area of a circle is Pi times its radius squared? Did you know the area of a triangle is one half of its base times its height? Did you know for a right triangle (a triangle with one angle of 90 degrees or pi/2 radians) the sum of the squares of the two sides adjacent to the right angle is the same as the square of the side opposite the right angle, the hypotenuse (did you know that I, a mathematician, still cannot spell hypotenuse?)? More importantly, do you know why?

Earlier today I posted a Facebook blasting the mainstream method of mathematics education. This was out of various frustrations, including my frequent interactions with people in the late stages of chronic math apathy, and conversations with colleagues on the state of our students. I have colleagues, how weird is that! Much of my despair boils down to our dogmatic adherence to a system which is fairly seriously flawed. I don't claim to have THE solution, I have thoughts that I would like to see tried out, but mostly, I want to see us have the courage and vibrancy to experiment boldly with our curriculum, changing it to try to get out of this malaise.

I believe I was unduly harsh, as math is not the only subject that suffers from some deep problems endemic to our education system. One horribly harmful practice is the passing along of students who have not sufficiently mastered current materials to higher level courses. Because math is such a rich and useful subject, attempting to understand a subject without a thorough grasp of the prerequisite courses is akin to the canonical fool attempting to build a sturdy foundation upon ground made of sand. Although factoring polynomials (mostly quadratics) and negative numbers are two fine examples of things one ought master before proceeding, as they end up in fairly frequent use, by biggest concern is fractions.

I cannot understand what part of our system so scars students that they stay scared of fractions long through their math careers. If one simply wishes to work with fractions, two simple rules suffice, common denominators to add, multiply straight across. This will not convey an understanding of the nature of fractions, but it will provide you with all the "special" tools that you need to manipulate them, they won't be things of beauty, but at least they ought no longer be feared. However, as I indicate at the end of the first paragraph, the essential point of math is NOT an isolated scattering of lifeless, lonely facts, but an intricate, intertwined network of collaborating reasons why things are true.

Were I to be given an island on which to implement my social experiments, my math curriculum would, broadly, look as follows (and mothers could keep their children...). An introduction to the positive integers, addition, and the number one. These topics are deeply connected, as you can think of one, the unit, as the starting place for all mathematics, without it there is only nothing, then addition miraculously conjures the rest of the whole numbers, as one is joined by one for two, and so forth. Here we introduce the miracle of zero, which leaves numbers unaltered Should we ever experience diminishing, as happens when one departs, we become familiar with negative numbers (not subtraction, which is a misleading myth). Now, one might find it useful to repeatedly add a number to itself, such as if students arrive on buses each carrying 32 children, 32 children after the first, 32+32 after the second, 32+32+32 after the third, and thus is multiplication formed from addition.

Now things become a little tricky, but I have faith in our citizenry to persevere. Just as zero played a special role for addition, leaving things unaltered, our unit plays the same role for multiplication. And just as the addition of -n undoes addition by n, because together they add to 0 which leaves things untouched, multiplication by 1/n undoes multiplication by n, because they multiply together to zero. Thus we run into the first fractions as reciprocals to the whole numbers, and we noticed that division too is an illusion. Consider 0*n, which we know to be n added to itself zero times, when no buses have arrived, there are no students, no matter how many are on each bus, so 0*n=0. Now suppose you are discovering this for the first time, what must you suspect about 1/0? Well, one expects multiplying by 1/0 to undo multiplication by 0, because that is the role of 1/n to undo n. So, 1/0 times 0*n should be n, no matter what n is. However, no matter what n is, 0*n always arrives at 0, so 1/0*0 needs take on more than one value to satisfy our system, since it cannot, and indeed must be 0 since anything times 0 is zero, we discover that 1/0 cannot be included in our system without breaking the patterns that we have established. Because there is no division, and n/0 is instead n*(1/0), we must conclude that, should zero occupy a denominator, we have left our reasonable system of numbers, so this is to be avoided. This has gone on long enough, but we may quickly build exponentiation from multiplication the same way we build multiplication from addition, and roots now are un-exponents, the same way negatives are un-addition and fractions are un-multiplication.

Of course, this is a bit of a mess when skated through in 2 paragraphs, rather than over the course of a year or two, but the point is that it is all interrelated. This is where the beauty of mathematics comes from, it is a mosaic, and a puzzle, and a fractal laden, dew adorned spiderweb on a cool morning. Addition and multiplication are different, not separate, and subtraction and division no longer exist, much simplifying the order of operations. Recall that these operations have the counter intuitive property that they do not commute (that is, 3-2 is not 2-3, and 2/3 is not 3/2). Since these operations no longer exist, this strange fact, an artifact of a formulaic approach to math, need no longer be considered. Now 3+(-2) and (-2)+3, as well as 2*(1/3) and (1/3)*2 can be swapped as much as is desired.

The WHY is beautiful, without it math is just an exercise in memorization and rote manipulation, two things that are fairly boring and easily done by computer. With why, math becomes a rich environment for exploring, every new piece of information not a fact, but a new piece of the puzzle which must be slotted into an awaiting rough edge. The easiest method to determine the area of a circle requires calculus, but I can prove the Pythagorean theorem on a napkin in under 5 minutes (if we are in the same place, feel free to ask me to do so, it is a beautiful proof). How amazing is that, these two facts, that we casually teach side by side in an introduction to geometry, have vastly different backgrounds and stories. Personally, I think the area of a triangle is the easiest of the three, and accordingly, I leave it as an exercise for the reader, should you be in the mood to explore. It may help to take right triangles as your point of departure, but remember that not all triangles are right triangles.

As I posted on Facebook, learning math the traditional way is akin to learning to write poetry by taking lines from the poems of the greats and sticking them next to each other in varying orders. Or, to be less absurd, it is learning about history by rote memorization of dates and events, with no thought paid to the overall correlation and causality, the beautiful, interconnected, vastness of human experience. It is learning a foreign language sentence by sentence, hardly paying heed to individual words in the phrases blindly committed to memory, let alone the system of syntax and grammar. Overall, I have to say that what is a very important question to answer, but only because you need to establish the what before you can sail off into the sunset after your elusive why's...

Merit Based Salaries for Teachers and The Importance of Control Groups

I feel like my blog has sort of become me complaining about my life. This is not what I want it to be, mostly because I don't think that is either interesting or thought provoking. Last post may be excused because it provided an excellent example of Sartre's existential dilemma, but my complaints today are more mundane, as such, they shall be relegated to the end of this post.

With the US falling behind in educational achievement, our public school system has come under heavy scrutiny. Even more so as the documentary, Waiting for "Superman" hitting the pop culture radar. One of the controversial "fixes" proposed is tying teachers' salaries or bonuses to evaluations of their merit. I, as a semi-educator, view this suggestion with a little trepidation, not that I have any illusion that it will affect me one way or another.

My biggest problem with merit based pay is that teachers are not responsible for the quality of their students, for the most part. Perhaps my perspective on this is skewed by coming from the field of mathematics, but much of my students' success or failure is predetermined by their level of preparation when they enter my classroom. Unlike the private sector, I cannot weed out people who's performance I become responsible for if they have no business being in my classroom, except by failing them. This leads to me often trying to cobble a rudimentary working understanding of integral calculus for some of my students on top of a mathematical background where they still fear and distrust fractions and do not actually understand exponentiation. I think there are some Biblical verses condemning what I do (Matthew 9:16,17), but the task of reteaching almost the whole of mathematics to my students is daunting.

Granted, if one makes certain assumptions about the uniformity of mathematical backgrounds from class to class, then comparing educators in the same school teaching the same subject might yield some results as to the comparative efficacy of those educators. However, comparing educators who have students prepared by one school as opposed to another seems bound to confuse the individual educator's merit with that of the system that has prepared their students.

Finally, education is not a passive activity like watching the television. In order for students to get more out of it, they must put more into it. Mathematics may have come easily to me, but I was (almost) always actively trying to understand what was going on. When you understand the background, each new layer of complexity becomes relatively simple to understand, if you try. For example, on a recent exam, my students were asked to set up a partial fraction decomposition problem. Not recognizing that (x^2-4) factors was, at a glance, strongly correlated with not knowing to put Ax+B in the numerator over an irreducible quadratic factor. There is no reason that these concepts should be linked, one is a rather subtle question relying on mathematical skills they should already have, namely factoring, and the other is a simple fact about partial fraction decomposition that they ought to have learned in my class, yet, the two mistakes were almost always made simultaneously.

Granted, it is a lot easier for me to take college students to task for not investing in their education than it would be to criticize cute little elementary schoolers. My point, however, is not that we should blame the student's for their own academic successes and failures, but rather that we need to create the proper support for students to encourage them to invest in their own education from an early age. This is not something teachers can be responsible for alone, I believe, as I have said before, that our educational apathy is a cultural institution, and that students are often receiving deeply negative messages in the home and in pop culture about the value of education.

Ok, that is my two cents on that. As mentioned, I have some other thoughts on education kicking around, which may make it into a post. In regards to my own life, as mentioned we recently had a test, so grading that took up a bunch of time. Taking even more time was the Group Theory assignment due Monday, that our class just turned in today. Between those two tasks, I have been rather stressed recently, but they are now over, so I can focus on my other problems. Prominently featured among those are the fact that my laptop bricked Monday, so I have to decide what to do about that. Additionally, while sprinting to catch the bus this afternoon I caught my foot on an uneven spot on the sidewalk, causing me to slam right-side first into the sidewalk/floor of the bus. To catalog my injuries, I have a couple stubbed toes on my right foot, from catching on the sidewalk, a skinned right knee from the sidewalk, a tight pain in the middle of my right ribs, where my chest collided at a frightful speed with the edge of the bus entrance, and a skinned right elbow from the bus floor. Aside from a slight tenderness in my left elbow, which didn't bleed at all, my left side is in fine shape. This occasionally leads me to exclaim in surprise and appreciation at how much pain the left side of me is NOT feeling, which illustrates the importance of a control group!