College Monster: March 2007

Note: this article makes references to two distinct types of students, known as “techies” and “fuzzies”. It’s an important distinction, because the two realms of students experience quite different courses and course requirements and in general have quite different priorities with regards to education. In a quick definition, a “techie” is a student who is pursuing or plans to pursue an education in engineering or math or ‘hard sciences’ like physics, chemistry, or biology. A “fuzzy”, in contrast, is a student who is pursuing an education in the arts or ‘social sciences’, and are so named because of the “fuzzy” and ambiguous nature of their studies, in contrast to the distinct definitions and black and whiteness often encountered in “techie” fields of study.

In part 2 of this series, we’re discussing high school mathematics. Unlike my previous piece on foreign language, here’s a subject that’s near and dear to my mind – I was heavily invested into math throughout high school and even middle school, and the subject is a major part of engineering majors in general.

I’ve decided I’m going to begin each of these articles with perhaps the most illuminating question of all: why is this subject important? Yes, this is a college blog, and I’ll still be primarily writing a retrospective on my experience in high school with the hindsight of my experience so far in college, and give advice on what students should do. But in explaining that retrospective, the context of my thoughts are very specific and exclusive – despite my attempts to be as objective and comprehensive as possible, my commentary ultimately describes my situation, and stems from observations and evidence that I’ve experienced only. The best way to find your own answer, and find meaning in what I write for yourself, isn’t in all the specifics that I provide, but in viewing them through the lens of this question: what is the significance of this subject for me?

So, why is math important? For engineers and students pursuing hard sciences (or math itself), this question is easy – it’s at the core and arguably the root of their very profession. Math on its own is the most basic level of logic, and the purest – math is black and white, and everything it states and does is fact, and quantifiable. For this reason, it is the building block towards understanding the physical world around us – it is only through math that sciences could ever grow out of a mere set of observations and become serious, quantifiable, and meaningful ways to understand the universe we live in. Understanding mathematics is essential not only to understanding current science, but it’s also essential to broadening of realm of future science as well.

For that alone, there are numerous importances to understanding science, and therefore math, but we’ll cover that later in the science piece. Outside of science, however, math is simply logic. It’s not through philosophy or the rhetoric of argument that we learn logic – logic in its most pure form, before it’s tainted by perspective and language, is represented in math. It’s through mathematics that we learn the concepts of equality, inequality, representation (variables), quantification (number values themselves), and relationship (matrices & systems of equations). At higher levels we learn reciprocal actions (inverses), and the link between sums and rates (calculus), among many other things. Math is the fundamental logic, and at the end of the day, all the philosophical and rhetorical logic (not to mention scientific logic) taught in the universities, and the logic we employ in everyday life are simply abstractions and distortions of the fundamental logic represented by math.

Pragmatically, what does that mean? Understanding math at a fundamental level certainly helped me excel in understanding almost every other aspect I ever studied. But what I’m talking about here is understanding math – simply learning it, memorizing sine and cosine values or mastering formulaic methods for computing integrals doesn’t really do anything. I would argue, for example, that learning how to calculate derivatives using the old, archaic limit h->0 [(f(x+h) – f(x)) / h] method is far more important than ever learning the power rule shortcut for derivatives – though you’ll learn and use the limit method for all of a week and then never look back, and the power rule is the applicable one that scientists and engineers and mathematicians will use in all applicable instances, it is the former that actually outlines the basis behind and definition of derivation, while the latter is simply a tool used to compute and solve. It’s a moot point for ‘techies’ like me, as both lessons – the fundamental concept and the pragmatic tool – are equally important in our line of study and work. But for every ‘fuzzy’ student out there who is wondering what exactly is the point behind all this math, here is the answer: mathematics is a representation of logic at its most basic level, and an understanding of logic through math (even subconsciously, perhaps) goes a long way in making sense out of anything in any field. As noted in the example above, sure, not all math is relevant if you’re in a field that doesn’t use it. Some math is for the sake of math, or for the sake of science only (where it gets abstracted from logic, like all other subjects), but there are important logic concepts within mathematics that everyone should learn, and I’ll point these out here.

A little context, as always, about my educational background:

I’ve attended public schools in P-12 all my life, except for P, and so in part fortunately and in part unfortunately, I never learned mathematics through the new and alternative educational methods that are sometimes used in private schools and seemed to be ushered in some time after I finished my elementary school education (I remember sometime in the elementary school, when my younger sister asked me for help on homework, and I couldn’t understand a bit of the multiplication they were teaching because it was some completely bizarre and non-traditional alternative arithmetic lesson). In any case, I never went through that, and somehow (perhaps just from this one incident) I came away forever thinking that the next several years behind me had always learned the fundamentals of arithmetic some different way. Well if it is, I’m sorry to say I can’t comment at all on how my early education could have been affected or how you readers are affected by some alternative educational scheme. If it isn’t, then ignore this entire paragraph.

I was always a pretty good mathematics student, and I say this in the context of my above commentary on fundamentalism and pragmatism – I learned how to use all the tools and which ones to apply to which problems very well, as did most people, but beyond achieving the right output I always picked up the actual concept fairly quickly, something that, from observation over my P-12 years, not all students do even if they achieve the right grades and otherwise “do” the work right. I had algebra in the 7^th grade, and re-did algebra in the 8^th (as there wasn’t anything left), so while I entered into high school at the same level as everyone else taking geometry, I probably had a bit more refinement and experience dealing with that kind of math. Overall, my pre-college math education looked like this:

7^th grade: Algebra (independent study)
8^th grade: Algebra (same material as 7^th)
9^th grade: Geometry
10th grade: Algebra II/Pre-Trigonometry
11^th grade: Trigonometry/Pre-Calculus
12^th grade: AP Calculus AB, AP Statistics, AP Calculus BC (independent study)

I ended up scoring very highly on AP and SAT mathematics tests, and ended up at UC Berkeley under an engineering major. So far in college I’ve completed one semester of Multi-variable Calculus (3^rd semester calculus, after AB and BC) and am currently in the middle of Linear Algebra & Differential Equations.

You should thus take my context from the standpoint of a “techie”-centric student (although I have a lot of fuzzy roots, which I’ll explain when I get to the ‘social’ sciences), who went through high school on the more advanced math track, and never really struggled too much with the subject. I’ll admit that I lack any experience in the realm of students who started high school doing Algebra and took Geometry their second year, and also those who went with a 3-years-and-out approach towards math classes, although I’ll attempt to take the perspective of “what if” my education ended with Trigonometry or even Algebra II.

Perhaps it’s best to discuss mathematics with a walkthrough, starting with middle school education. The unique thing about mathematics is that it’s a strict sequence – each course is a prerequisite for the subsequent one, and once you’re on a certain track, mobility into the advanced courses is nearly impossible (which isn’t the case for English, science, or social science) – if you started out your freshman year doing Algebra, there’s not much chance to be able to take Calculus in your senior year. For that reason, your mathematics education gets decided all the way back in middle school, when you decide (or perhaps get tossed into) either pre-Algebra or Algebra as an 8^th grader.

The advice I want to give is this: every student should take Algebra as an 8^th grader, and skip straight ahead to Geometry in the 9^th grade (without any “I’m doubting myself maybe I should retake Algebra in high school?” questions), because being an entire year behind in math is a very big step back that is very hard to “make up” if you decide you want or need to be a more advanced student later, and moreover because I have very strong opinions on students taking Calculus by their senior year. However, it’s advice that I’m very reluctant to push, knowing that not every student can handle that pace (which is why this track is the “advanced” one and the standard math track starts with 8^th grade Pre-Algebra and 9^th grade Algebra) – perhaps I want to push it very badly because math to me is so relevant and all-important, and from my observations of an admittedly very-biased sample of friends, all the students I knew were completely capable of learning on this advanced track, including those that started out with 9^th grade Algebra after having taken Algebra in the 8^th grade.

The almost surefire advice is that those students who have already taken Algebra in the 8^th grade shouldn’t doubt themselves by retaking it in the 9^th. Algebra was always the big and momentous subject in middle school (maybe my perspective is exaggerated because it seemed even bigger in 7^th grade), but in the context of high school math, algebra is actually a fairly shallow subject – just variables, equations, arithmetic properties, and maybe an intro to coordinate systems, right? It’s not a core part of Geometry, and the month at the beginning of Algebra II spent reviewing over basic Algebra more or less covers and refreshes everything you learn in Algebra. I validate this speculation with my own experience (perhaps not very convincing) and observations from nearly ever 9^th grade Algebra student that the year spent in review was a waste of time (much more convincing, although I should caution against the bias of my observation samples).

As for students who have only pre-Algebra, or those 7^th graders who aren’t sure if they can handle Algebra in the 8^th grade, the answer is a lot muddier. Again, the advice I want to give is to just go for it, but in reality not every student is prepared enough to take on algebra and grasp the concept of variables at 8^th grade (although many who don’t can be, if they tried), and there’s no way for students who enter high school having only pre-Algebra to magically jump a level into Geometry. Students in this situation are essentially stuck with 9^th grade algebra, but they’ll have a choice to either stay along the basic mathematics track or work hard enough to move up a level.

Move up a level? Yes, you can; admittedly, the mathematics education track isn’t as locked in as I made it seem before. If you’re on the standard math track but want to move up a level so you can take Calculus by senior year for example, the concurrent enrollment programs offered by many community colleges is an alternative. Concurrent enrollment is a program offered by community colleges (In the Daly City area, City College of San Francisco (CCSF), Skyline College, and College of San Mateo (CSM) are probably the largest) that allow high school students to enroll into college classes and earn not only college credits but high school credits as well. Usually, high school students sign up to take a class at the community college over the summer break, and more rarely during the actual school year (with night or weekend classes). The classes are free for all high school students, but you’ll need to speak to your high school counselor and get permission to enroll into classes (you should also ask your counselor how the class would apply to high school credits).

So for students who want to move up a level, concurrent enrollment programs allow you to take Geometry during the summer after freshmen year, for example, and move straight into Algebra II for your sophomore year, assuming that you’ve already talked to your counselor and/or teacher and they’ve approved that taking the college course can fulfill your requirements for high school.

There are drawbacks, however. The first is that you’ll have to put in a lot more work – your whole summer spent going to school again, for example. It shouldn’t be a deterrent; it just means you’ll have to actually dedicate your time and effort to doing well, just as if it was any other class. The second is that, being a college class, concurrent enrollment is generally more rigorous than a typical high school class. By no means is it beyond the grasp of high school students, but it requires need more effort than a typical high school class would. Third, and perhaps the most important, is the time constraints – you would normally take a mathematics course spaced out over the entire school year – 10 months from August to June. For concurrent enrollment courses, all of that material is crammed into a 2-3 month semester – you’ll end up doing the same amount of work (1/3 of the time, but classes may go on for 3 hours instead of 1, for example), but it’ll be at a highly accelerated pace. A summer course in math is like an extended cram session – you may memorize everything and get the grade on the test at the end, but how well everything actually sticks is another matter. Depending on the kind of student you are, you may or may not learn as well as you can with a full-year course, repeatedly working on the material day-in-and-day-out; generally everyone can grasp the fundamental concepts equally and those stick, but students who just tend to memorize things well can hold onto the pragmatic tools they learn from short summer courses better. Ultimately, however, in my opinion (not in my validated commentary), a student who takes a summer of concurrent enrollment along with four years of high school math comes out better prepared and knowing more than a student who only takes four years of high school math and ends up a year behind. It should also be said that if you’re dedicated and willing enough to go to school during the summer to advance a level, you’re probably a student who would be capable of starting 8^th grade out in Algebra and starting 9^th grade out in Geometry.

With that aside, the only question for concurrent enrollment is whether or not a particular student is willing to put in the effort, and if he or she actually needs it. Thus begins a walkthrough of my perspective through my high school and early college mathematics experience, to hopefully reveal enough foresight for each of you to make your own decision.

Geometry

I started out my high school education with 9^th grade Geometry. It was a nice small, 20-student class, which I believe is a standardized maximum for freshmen math in California public schools. Although the majority of this is probably obvious, having 20 students means the individual questions of students can be addressed much more comprehensively, and on the flipside, although many won’t like me saying this, the low student capacity allows for classes to be much more tuned to the level of student ability – classes with more advanced students spend a lot less time bogged down by a handful of students who can’t keep up with the rest of the class, and classes with slower students aren’t hopelessly drowned out by the pace of more advanced students.

At the time, and for a long while afterwards, I detested Geometry. As I’m sure it was for many, Algebra and variables and solving for them was an absolute delight. Going into geometry, with all of its coordinates and shapes and 360 degrees (what kind of unit system is base-360??) (context: possibly I’m not a ‘visual learner’), and the doldrums of memorizing theorems and postulates and proving this and that when things were, “Just look at the thing! It’s quite obvious without a theorem, wouldn’t you agree?”, was quite an exercise in tedium and an absolute bore.

That said, in reflection geometry introduced a number of fundamental pragmatic tools – beside the basic ones like SohCahToa and various useful theorems for angles and shapes, geometry was the first in-depth application of algebra in tangible terms. You wouldn’t solve for some abstract variable, but for an angle, an edge of a shape, a perimeter. Applications in geometry is still very much a pragmatic tool – it doesn’t quite reach the level of abstraction you’ll find with vectors and n-space (fully explored in college-level linear algebra), but it’s the first stepping stone that allows you to grasp the same basic concepts in more familiar shapes and 2-dimensional-space.

The real beauty of geometry, however, and the most valuable lesson I took away, was the raw usage of logic to handle and solve problems. While, like any other field of mathematics, applying the tools to problems was part of the class, the largest part of Geometry was rather substantiating the validity of those tools. For every algebra-application “X and Y are similar shapes; find X’s dimensions given …” problem, there were two “X and Y have given properties; prove their similarity” problems. Rather than computation or tool application, Geometry was an exercise in logical analysis and conclusion.

What does Geometry mean now, as a second-semester engineering freshman? Pretty much as soon as I finished Geometry, I dropped just about every pragmatic tool save SohCahToa – throughout a science education filled with Physics, Chemistry, and Computer Science and a math education with Algebra II, Trigonometry, Calculus, and Linear Algebra, I rarely ever needed to calculate the angles of the vertices of a regular polygon, and only rarely did I ever have to do things like show congruency between sides. There were a few things, like the introduction of the Sine, Cosine, and Tangent trigonometric functions, as well as the properties of parallel and perpendicular lines, that stuck and proved to be repeatedly useful, but the entire pragmatic side of Geometry consisted simply of math-for-the-sake-of-math or math-for-the-sake-of-science tools. The extensive (and mandatory) use of logic, however, was what stuck most and prevailed as a foundation for future learning – everything that was validly useful was derived from a fundamental logical base, and understanding that logical base proved instrumental in not only knowing where any tool could be validly applied, but whether it was valid at all. As most of my later teachers would tell you, I refuted everything I was taught until I could substantiate it through my own train of logic, a refusal of the common ‘memorize, plug, and chug’ learning that lets many students pass classes but leaves them grasping for any understanding whatsoever. Although I probably hadn’t realized this point at the end of 9^th grade, the necessity for a foundation of logic for everything was went on to define my approach to education. Arguably, it was the biggest reason I ended up learning and understanding as much as I did in all my subsequent education.

Algebra II, Trigonometry

The next two years of math education are a bit of a blur, although perhaps it’s because it feels so long ago (a scant three years!). For the most part, Algebra II and Trigonometry/Pre-calculus taught a smattering of useful (and some not-so-useful) mathematical tools. The most important concept here was the coordinate system – Algebra II first explores this and uses it to represent equations in a graphical form. Trigonometry delves into this deeper, with more non-linear equations, and towards the end of the year (in my class at least), the presentation of alternative coordinate systems, such as polar and parametric coordinates. Understanding coordinate systems is the stepping stone to the more general topic of vectors, and within the domain of high school has applications such as dealing with vectors quantities in Physics. While it’s not necessary, understanding coordinate systems is an extremely beneficial supplement when learning vector quantities (such as any kind of Force) in Physics, a topic that surprisingly many struggle with (even at the college level, in my first semester Physics class) - as a tie-in with science, I started both Algebra II and Physics in my sophomore year and found the former to be extremely useful in understanding the latter.

For the most part, students who don’t live and die by mathematics usually don’t find the somewhat random smattering of Algebra II or Trigonometry very interesting or relevant, and it’s around this stage that many of the 3-years-and-out students elect not to go for their 4^th year of mathematics. The very next level of math, however, is what (in my experience) puts it all together in a single, shockingly relevant and simplistic system of mathematics.

Calculus

Out of any math I’ve ever taken, Calculus (taken in my senior year) proved to be the most revolutionary and eye-opening concept since Algebra. Up to this point, the majority of mathematics had simply been for the sake of math and science – learning math in order to do more higher-level math, or applying it to solve science (mostly Physics) problems. While I personally loved it and excelled, for any non-techie student, there admittedly wasn’t much appeal or immediate relevance in any of it. Calculus changes all this, with the introduction of the derivative, and later on the integral, two concepts that, for myself at least, offered a shockingly new way to think about nearly everything, from making sense of statistical data (distribution curves in particular), to understanding economic and business concepts (trend indicators, for example), to gaining an even deeper introspect into the workings of everyday physical phenomena – in short, for pretty much anything with quantifiable values, Calculus redefined it in an entirely new and elegant way. To give an example, before Calculus and the introduction of rates (derivatives) and sums (integrals), one might very well understand any individual quantity such as velocity, displacement, or time, and even understand their relations to each other (D=RT). One might also take the quantities of height, gravitational acceleration, and time and understand their relation in a state of free-fall (t = √(2h/g) ). Calculus offers a mathematical system that combines the awkward old system that used to define a new set of quantities and formulas for each different state or situation. Through Calculus, students come to learn that constant velocity and the free-fall situations mentioned above are actually just specialized forms of one single, universal equation that defines all the possible relationships between distance, velocity, acceleration, jerks, and so on, which in itself is a specialized form of the single equation which defines a general sum and any of its rates.

I’m not sure if I could quite convey this without explaining derivatives and integrals themselves, and I’m not sure anyone reading this could actually understand without understanding derivatives and integrals themselves. Suffice to say, Calculus unlocks a truly unique way to look at almost anything, specifically to look at relationships as not awkwardly defined in their own unique terms, but to boil down the definition of anything and everything into a single elegant system of simple rates and sums of other quantities. And understanding and viewing the world through the scope of this system both simplifies almost everything you already know, and makes many of the toughest new concepts easily graspable in familiar terms that you already know, by allowing you to define everything in terms of something else.

So now that all of you soon-to-be seniors know that Calculus is the most important and most applicable math ever (so don’t even consider chickening out and not taking math your senior year!), the biggest question many students face is to take AB or BC? In my senior year of high school, I did both, although not exactly in the conventional classroom setting. At my school we only offered Calculus AB as a class, although our very dedicated Calculus teacher held a weekly afterschool BC session for a few of us who essentially independent-studied the Calculus BC material. Thus, not having ever taken BC in a classroom setting over a full year (or semester), I’m not sure if my commentary is truly reflective of what most students will face, although in independent study we still covered all of the BC material and I was able to manage a 5 (out of 5) score on the BC test, for what it’s worth.

First of all, what is the difference between Calculus AB and BC? College calculus starts with two semesters, and high-school AB courses roughly cover the first semester of material while the BC courses cover both the first and second semester of material. For the most part, AB covers the basic concept of the derivative and integral, along with the various methods to compute the most common ones. In my BC experience (keeping in mind that I never took a BC course in the classroom setting), the BC material simply consisted of more advanced ways to solve trickier derivatives and integrals – methods like ‘integration by parts’ that, while needed to solve otherwise impossible integrals, contributed little if anything to the fundamental concepts of the derivate and integral, or any other relevant concept for that matter. Pragmatically, however, the BC material was a slightly different matter. Many of the integration methods would prove to be useful (and necessary) later on in Math. In the first math class I took in college, Multivariable Calculus (3^rd semester Calculus), complicated integrals that required knowledge of BC methods were ubiquitous (although unnecessary with regard to the actual fundamental concepts), and so the pragmatic skills learned in BC would be useful. In all subsequent math (that I’ve experienced so far – Linear Algebra & Differential Equations in my second-semester of Math), and certainly all science and any “real life” application, BC material is essentially worthless. While as a math major, abstract integrals like the integral of xcos^2(x) may be worth knowing how to calculate, the fact is that for any other field, integrals like those virtually never show up in real-world problems; outside of mathematics (and outside of 2^nd and 3^rd semester Calculus, even), I haven’t ever encountered a situation where I needed anything beyond the skills I learned in Calculus AB. Even in the rarest of rare situations where such an integral might arise, the far more efficient and common sense method is to simply use a calculator (what AB students do) or at worst look up the integral in a table (something even my post-Calculus graduate student instructors (GSIs) admit is the sensible thing to do). Methods are good to learn when they’re relevant to understanding fundamental concepts, or when situations that require them are bound to appear frequently enough to make them practical. Outside of perhaps a math major, the methods learned in BC don’t apply to any situation that anyone would encounter, either as a student or in any working profession.

That doesn’t answer the whole question, however. Useless as I think second-semester Calculus BC is, the fact is that it’s still required curriculum for many students. So the question, “should I take Calculus AB or BC in college” must be weighed against this. Fuzzy students obviously have even less use for BC material than techies, and my suggestion is that they check the requirements for the major at the college(s) they are planning to apply to – if second-semester Calculus isn’t a requirement (or part of some larger breadth requirement), don’t take it. If second-semester Calculus is required, however, I might suggest taking it in high school, depending on how confident you are about being able to pass the test. Since it’s a subject that really is irrelevant to your later fields of study, it’s much easier to pass the high school Calculus BC than it is to endure/waste an entire semester in college on Calculus, and it won’t really matter if you didn’t learn as much as you could have through an actual college course.

For techie students, my advice remains the same for similar reasons. Second-semester Calculus is almost an assured requirement, so there’s no way around it. As mentioned before, the tradeoff between a high-school BC course and a college course is that by passing the BC test you get to skip past what is otherwise a useless second-semester Calculus course in college, but will likely end up learning and retaining less (of somewhat useless material) than you would have in the college course. Although your mileage may vary, another consideration is the strict requirements that some colleges require for Calculus BC credit. At the UC Berkeley College of Engineering (which covers all engineering majors), for example, a perfect 5 out of 5 score is required to earn credit and skip second-semester Calculus – while many students from our small independent study group achieved this (note this is a biased sampling), a 5 is by no means automatic or easy, and students who struggle with Calculus may very well end up not receiving any credit for BC anyway and have a rushed/compromised AB education. Check with your college and intended major about the kind of requirements needed to receive credit for the BC test, and compare this against your proficiency as a student (if the requirement is a perfect 5, I might ask, “have I consistently been an A-student in math?”).

How will your BC-proficiency play out in post-second-semester-Calculus math? I’ve currently completed a single semester of third-semester Calculus (multivariable) and am halfway through another semester of Linear Algebra and Differential Equations, so my experience is fairly limited, but so far my BC education has either been monumental or trivial, depending on your priorities (I tend to view it as trivial). Having taken Calculus BC as a weekly independent study-like class, my retention of the material wasn’t nearly what it could have been with a full-on AP class, much less a dedicated semester-long college course, and this partially resulted in myself having pragmatic issues finishing out problems when I arrived at third semester Calculus (Multivariable). I had no hindrances in learning and comprehending all the fundamental concepts of multi-dimensional integrals and vector derivatives, however, as I had learned all that I needed from Calculus AB, and simply applied that to n-dimensional space. The result was satisfaction in having learned and understood all the fundamental concepts, coupled with a somewhat less-than-satisfactory grade because I wasn’t able to compute out all the problems, partially because of a lack of second-semester Calculus (BC) skills. Keep in mind that my situation is perhaps much more exaggerated than the typical high school student – I learned Calculus through a one hour-a-week afterschool class, while students who take actual BC courses get to have a full 5-days-a-week class, and thus will probably be far more prepared and retain far more material than I ever did.

Statistics

Usually the alternative senior-year math course for those who elect not to take Calculus, statistics is unique not only because it’s usually the only math course that doesn’t follow in the prerequisite chain of Algebra-Geometry-Algebra II-Trigonometry-Calculus (meaning you can jump into it anytime instead of following a strict order), but also because it’s a very self-contained subject – unlike other math, where you learn abstract math techniques and wait for a chance to apply them elsewhere in say science, statistics teaches you both the techniques and the myriad of statistical applications. I took the Statistics AP course concurrently with Calculus in my senior year, so my educational experience with statistics is likely more comprehensive than the non-AP Statistics classes that many schools offer or also offer.

Most students will get a taste of statistics in their Trigonometry class – when I took this in junior year we covered some basic concepts like probability, permutations, expected values, and normal distributions. Having taken Trigonometry my junior year, I think I was much more familiar with many of the initial concepts in Statistics, and this helped a lot to dive right in to the core subject matter – since I was already familiar with and understood the basic idea, I didn’t need to spend as much time figuring out what certain things meant (like the idea behind an expected value or how to interpret distributions), and could dive right in to learning how to use and apply them, which is what statistics is all about. Especially since statistics doesn’t really make use of any other mathematical concepts beyond basic algebra (exponents and logarithms are very important however), anyone who’s taken or is taking Algebra II should be able to handle statistics just fine (I think concurrent Algebra II might be a requirement as well – I’m not too familiar with this anymore).

I thoroughly enjoyed Statistics, and I might even venture to say that Statistics is one of the most appealing Maths for students who don’t like Math. The reason is that, unlike other subjects that are heavily weighted in abstractness that start with techniques and create problems for them, leaving a lot of students disinterested and wondering “When am I ever going to use this?”, statistics begins with the problems themselves and then finds techniques that allow students to solve them. While in any other math class a student might learn the Pythagorean Theorem and now have Sally mark a point and walk x meters down the river and measure an angle θ so she can find how wide the river is, this kind of example lacks immediate relevancy – people don’t encountering measuring the width of rivers in everyday life, and even if they do, they’d just take a measure straight across the river using a tape measure or some other device, not some cockamamie Pythagorean Theorem method! In statistics, we begin with very real-life problems – error probabilities and allowances with manufacturing facilities, or distributions for characteristics like a population’s height, and only then do we break out and learn about the tools like standard deviation and binomial distributions to solve them. I think I’m starting to talk too much from an educator’s perspective instead of a student’s, so in a nutshell, Statistics is very interesting, relevant, and great fun, even if you’re someone who’s been scared off by all other kinds of math in the past.

How is statistics important? Well, unlike the other mathematics courses, statistics isn’t really on the prerequisite list for anything, which is a shame because it’s probably the mathematical subject with the broadest and most immediate appeal and applicability to students studying in any field. Take a moment to think about the activities and tasks you’ll perform in your intended field of study, both academically and professionally. If you’re doing any kind of engineering or hard science, it goes without saying that all of data you collect from tests and experiments need statistical analysis to interpret them into any kind of useful information, and any product one might design or build would need to fit within certain specification and error parameters. If you’re studying business or economics, everything from market indicators to company finances and economic trends compose a major if not integral part of running any business, and the accuracy and significance of any of that information relies heavily not only on good statistical analysis, but good statistical practice in obtaining that data in the first place. The same goes for any of the ‘social sciences’ – anthropology or sociology or psychology deal very much with individuals and more “fuzzy” observations and notes, but at the end of the day, valid and meaningful information and conclusions can only be derived from collecting large amounts of such fuzzy data and interpreting them statistically. Even for the everyday person, regardless of college education or profession, statistics are encountered on a day-to-day basis, with information received from media outlets (television, newspapers, government or research reports) or when using statistical information to make decisions and policies. Thought it may not be readily apparent at first glance, statistics very much run the world, and while someone else may end up as the bean counter, it’s very important to know whether that person is counting the beans correctly and what all those beans mean in the first place.

I would recommend statistics almost universally to any student. In my experience at least, the curriculum isn’t nearly as rigorous as most of the other math courses, and it’s surprisingly approachable even by those who don’t have particularly strong math backgrounds. And at the end of the day, it’s a subject that almost anyone will find useful, both from fundamental and pragmatic standpoints.

In Summary

So I’ve just hit the 12^th page as I’m typing this out in Word, and I figure I better write a nice conclusion to sum this all up, particularly for all of you who haven’t the time to sift through the entire preceding portion of this article.

Math, from the simplest of arithmetic, to algebra and calculus and branching off into statistics, is the first, most fundamental step to understanding the world. Admittedly, a lot of high school mathematics gets muddled along with burdensome pragmatic methods which aren’t always useful and relevant, something necessitated by the generalized and mixed situation of high school education – high school mathematics is very much geared towards techies that will enter into math or scientific or engineering fields, and while it has very important lessons for fuzzies as well, much of this comes buried under things that many students won’t care about.

From both a techie and fuzzy standpoint, the most important thing I’ve learned throughout high school and now in retrospect is that mathematics is perhaps the most important step in furthering the capability to understand almost every other subject taught in school. This is very apparent when students first learn Algebra or Calculus, which first introduce the broad, overarching, important-to-everyone concepts, but a bit less so in all of the in-between classes that simply build on and abstract those ideas. For that reason it’s a good idea for any student to advance their math education as far as possible, particularly for the sake of reaching the Calculus plateau in senior year. The in-between classes, while still important from a pragmatic standpoint, don’t matter as much, and for that reason I wouldn’t get discouraged if they seemed boring or even if I was struggling a bit.

The biggest lesson that I’ve admittedly stumbled upon after the past two months of writing this (it’s been a busy on-and-off piece) is a general one, however, that really applies to all education, at every level. Education teaches a lot of things, but most all of it can be broken down into fundamental concepts and pragmatic applications. In every field of study there are fundamental concepts, and these are usually relevant to everyone, no matter what they’re studying or doing in their lives. Coincidentally, it’s these concepts that are most often the easiest part of a subject – there isn’t much work required, just critical thought in order to grasp the subject. In most fields, after these fundamental concepts come many pragmatic exercises and applications (although these in themselves may be prerequisites to understanding the next level of fundamental concept). For the most part, these aren’t relevant unless they’re a part of your study or working field, or cover an application that has broad and/or everyday significance. At the end of the day, it’s the pragmatic applications that make a career (partly why education gets more specialized and application-intensive once you start college), and the fundamental concepts that enrich a person and make them into better learners. Rather than grades or test scores, these are the most successful priorities one can possibly have throughout their high-school education.

Labels: algebra, calculus, education, geometry, high school, mathematics, statistics, trigonometry

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Thursday, March 08, 2007

Reflections on my high school education in the field of Mathematics