Calculus GHL 0.1 - Functions

Well, the first document in my Calculus GHL series is available now. We begin with some preliminary concepts, the building blocks that will allow us to do calculus. Our first section deals with functions, the basic objects we deal with in calculus. The link is down below, as well as in the portal/introduction document.

Calculus GHL 0.1 - Functions

Please post any comments or suggestions below!

Random Expectations

A friend of mine recently posed an interesting question to me. We can easily see that if we choose a random number between 0 and 1, the expected value should be one-half. Now, in this case, the only number is the same as the largest number, so that the expected value of the largest number is also one-half.

So what occurs if we choose two random numbers? Can we determine the expected value of the largest number in this case? What if there are N numbers? I work through a pretty direct interpretation for the N=2 case, then find a general approach for arbitrary N.

Random Expectations

Given the result of N/(N+1), I noticed that this is equal to 1 minus the integral from 0 to 1 of x^N. This would be a really quick solution, but I can't seem to find an interpretation of the situation which lends itself nicely to that expression. Perhaps it's a coincidence, but who knows?

Calculus GHL Series

Welcome to my first series of educational documents at the fashionablemathematician website. The forthcoming set of documents are intended to help you learn and do calculus in what I think is an unfortunately unique manner, chiefly through problem solving and direct example. As I stated in my very first mathematics post here, I believe that mathematics is best learned through problems, and many others agree with me. However, few people take this belief to such a practical extreme as I hope to here.

Traditional courses segregate in-class “lesson time” and out of class “homework time” as separate (though certainly related) periods of learning. This has a number of benefits;

• Students see concepts multiple times; when in class and while completing homework assignments.
• Students spend more time with the material (related to above)
• By introducing a period of down time, the brain has time to process material, and may possibly continue the process of learning subconsciously.

These benefits are, in my opinion, tangible, though they are not always achieved. They assume students actually go to class (something not technically required of them in many cases), and that they put requisite effort into homework to extract the intended learning value (that is, they don’t copy answers from the back of the book or their friends). The nice breakdown of time is often mangled by students as well. Rare is the student which attends each lecture in the morning, then dutifully completes the assignment later that evening, just long of enough for material to soak in, but not so long as to have forgotten much. Add a good long night’s sleep right after completion and we’re a subconscious cycle away from a perfect learning environment! More realistically, a student might attend 2 of the 3 lectures in a particular week, start on the first homework assignment Tuesday afternoon, then put off the rest until late Thursday night when he or she crams three assignments in before they’re due Friday morning. In such cases, the above benefits are largely lost, and we see tutoring offices filled with struggling students, and instructors wondering what’s going wrong when their lectures seem perfectly clear (even when they are clear).

Now, we can (and many universities probably do) invest a significant amount of time, effort, and money into getting students to adapt to the standard model (not this Standard Model) of lecture/homework, and use it to its fullest. Doing this would probably help a fair number of students learn better than they currently do, though I doubt the effects would be widespread or in individual cases, long lasting. A few reasons;

• Students do not treat school like a 9-5 job (nor should they), so perhaps increasing the rigidity of one’s schedule is not the answer.
• College responsibilities (and revelries) are much more fluid and sudden than the typical 9-5 job, so often the above model student behavior simply can’t be achieved for all courses, simultaneously.
  

Instead, what if we introduce a system which takes into account the mobile, erratic lifestyle of the target audience (these observations are increasingly true in the working world as well), and provides the same or similar benefits in a different form, one which is not dependent on the particular time which a student can spend learning. Naturally, students will still have to spend a certain amount of time to effectively learn, but it seems possible to relax some other conditions.

Some progress has already been made towards such an effort. Many schools (especially for their larger classes) put their notes online, or even better, their lectures. Pioneers, such as those at MIT OpenCourseWare, even make such materials available to the public. But this is just a temporal modification to the standard model. Students still have lectures (usually optional) and homework, just with more freedom as to when to observe the lecture. Quickly, video lectures provide both an interesting benefit and drawback; video lectures can be paused, rewound, and fast-forwarded, but offer no opportunity to ask questions.

Can we work towards a more fundamental change? I believe we can. Here I present my attempt a new method of teaching (mathematics, though the process is likely adaptable), called Graduated Homework Lessons (GHL’s). The idea of the GHL is to provide a combinative experience of expositional learning and homework learning. One could think of it as a textbook section in which the exercises are interspersed (it is seen today, though rarely). Further, the exercises and exposition are structured to be a stand-alone learning device which requires an increasing amount of student input (hence the term graduated). This means we will typically begin with straight exposition, proceed to very detailed examples, and then slowly require the student to fill in more and more details of exercises of increasing difficulty, until the student is doing complete exercises, again of increasing difficulty. A more thorough, theoretical explanation of the idea may be forthcoming, but I think it best that we proceed by example:

Links to Available Calculus GHL’s

The documents are roughly ordered towards a "standard" calculus sequence. Until I say so, this is not complete, so there may be some holes. Starred sections are typically not required in an introductory calculus course, but are interesting and worthwhile to learn anyway.

This area will be populated with links to the GHL's as I produce them.

Chapter 0: Preliminaries

0.1 - Functions

The Mathematics of the Delegate/Superdelegate System

Although some pundits are claiming that the 2008 Democratic Primary is decided (with Barack Obama to be the nominee), in reality, the race is far from over, and very well could drag out all the way to the national convention in August. I’m a big proponent of voter equity, so this news is bittersweet to me. On one hand, many more states are getting to vote, and have their vote matter, than usually occurs (the last brokered convention occurred back in 1952). Unfortunately, the process also brings into play the entire delegate and superdelegate system, which I find nearly as unfair as the BCS system for college football rankings (the DNC delegate system gives approximately 800 to individuals, out of about 4000 total delegates, while the BCS gives a third of the vote to 6 computer algorithms). Thus, I set about finding out exactly how messed up this system is, by the numbers.

I compiled the following data from Wikipedia:

Raw Data

Data with Analysis Calculations

A note on the raw data: for ease of transcribing, I rounded the population to the nearest hundred thousand people.

Now let’s take a look at what I analyzed. My first step was to calculate the number of people represented by each “standard” delegate, each superdelegate, and each delegate in total. Of course, it is reasonable to expect some variation here, given the wide spectrum of possible population values and the number of available potential delegates. There may even be some significant variation, as occurs with electoral college vote distribution due to a minimum number of votes. Allow me to interject for a moment to point out how ridiculous it sounds for one superdelegate to represent hundreds of thousands of civilian votes. Anyway, the data, however, shows much more variation than expected:

The range for people per “standard” delegate is 40000 on the low end (Vermont), and 122798 (Texas) on the high end. We’ll see as we go along that Texas voters seem to get screwed in every category. For the superdelegates, 75000 (North Dakota) to 703846 (Florida), but since their delegates aren’t being counted, we’ll go to second place, 677143 (Texas). Poor Texas. For the overall total representation, we have 26087 (Vermont) to 103947 (Texas). This is quite staggering! When considering superdelegates, some states’ standard voters have nearly ten times less power than other states!

N.B. (note): I am ignoring whether states hold closed or open primaries for this analysis, because it would have taken a long time, and most of this is back-of the envelope style analysis. I am also aware that only a portion of the entire population is of voting age, but again, too many details.

The next thing to consider is the distribution of superdelegates. Do some states have a greater representation in this crucial 20% of delegates than their population’s proportionally purports? Why yes indeed. A quick glance at the data shows that “standard” delegates are fairly close to being proportionally allocated, and we see that they correlate well with the number of electoral college votes. Thus, if we calculate the number of delegates per superdelegate, and if the superdelegates are proportionally allocated, we should have a fairly constant value for the calculation. Of course, this does not occur:

The average number of delegates per superdelegate was 3.742, with a standard deviation of 1.714. The range was 0.500 to 7.115, with 24 of the 58 districts studied lying outside a standard deviation (more than 40%). States with notably low superdelegate representation are; Texas, Indiana, Louisiana, Kentucky, Georgia, North Carolina, Alabama, Ohio, and Florida. States with notably high superdelegate representation are; North Dakota, Rhode Island, Montana, Vermont, South Dakota, Delaware, and Wyoming.

The last thing I considered was how difficult it would be for the superdelegates to overturn the popular vote of the primary or caucus in each state. First, I tabulated the margin of victory for those states which have held elections (again using Wikipedia). Then, I determined what I called the “safe margin,” that is, the required margin to ensure that superdelegates could not overturn the popular vote on a state-by-state basis. This is given by the following formula:

formula

where T is the total number of delegates, and S is the number of superdelegates. We then compare this value to the actual margin to determine which states are “safe.” We see that there currently are 16 “safe” regions and 34 “unsafe” regions, with 7 undetermined. Interesting stuff. We’ll see how it plays out as the last few regions are decided.

Learning by Problems

Perhaps it is my personal bias as a student of mathematics, but I’ve always thought that mathematics is best learned by actually doing problems. Others agree with me, according to some 500,000 Google results (your mileage may vary), and the poster-level quote, “Math is not a Spectator Sport,” which has even spawned a book (that I haven’t read, and probably never will). Here are two compelling reasons for you to agree with me;

• Most people (read: not mathematics professors) will never have to quote definitions, theorems, or even proofs of theorems in their daily lives. To me, explicitly testing the semantics of any but the most fundamental theorems is as useless as testing chemistry students on the molar masses of elements, or testing biology students on the number of chromosomes in different organisms. These are the things everyone (even professional researchers) looks up when they go about their daily work. I’m not sure why this kind of material continues to be tested. It may be that many people do not understand that standing alone it really is useless information. It may be that people don’t realize that problems can easily be written which determine whether the appropriate definitions and theorems are known by the student. It may just be that people are lazy, and this is the easiest thing to test and grade. Regardless, in the mathematics 99.9% of people do in their lives, it comes down to problem solving.
• It is very easy (comparatively) to memorize a list of definitions and theorems, even simple algorithms for computation problems. This is because this involves a relatively lower level of thinking than problem solving. To write down the correct definition of a derivative, you only need to remember a chain of formal symbols and words. To explain the concept of a derivative you need a bit more understanding of those symbols and how they relate to other concepts, which you must also have some understanding of (we can already see how mathematics is so dangerously cumulative). To apply the definition of the derivative to a problem about physics, you must select the correct interpretation of the derivative, understand how the derivative interacts with other math and physics concepts, translate the problem into a mathematical setting, and still carry out the correct technical computations and interpret the result physically. The growth in complexity is staggering, which means that the number of distinct problems is large and unmemorizable! Further, solving any one of these problems usually requires at least a working knowledge of not just one definition or theorem, but many.

Given this, I find it incredible how little problem solving is emphasized in mathematics textbooks today (though I admit, from my experience with older textbooks, it used to be a lot worse). Now, you might crack open your favorite intro. calculus book, flip open to the section on the chain rule, and tell me; “Barry, there are 8 examples! Eight! And 72 exercises! How can this not emphasize problem solving?” I’m glad you asked.

Textbook writers love to tout how many new exercises and examples they’ve added in new editions. To be fair, it’s wonderful, we can always use more interesting, insightful, knowledge-building problems. Oh wait. I said interesting. And insightful. Forget about knowledge-building. The majority of textbook exercises are rote computations. These are, of course, necessary, as technical skills are often a prerequisite to solving real problems. Then most books will have a smaller selection of “applications” and “theoretical” questions. In these two sections, we might find a few really solid problems, ones which require an understanding of the concepts, how to apply them, and a little technical mastery. Unfortunately, the rest are either technically unwieldy, require an unnatural conceptual leap, or are of the “quote my definitions” ilk.

Now, the real tragedy is that when students, even top ones, encounter these good problems, they very often get stuck, sometimes to the point where they don’t even know how to begin the problem. I’ve seen it happen countless time as a teaching assistant and tutor for calculus students at Duke. This is what I mean by a lack of emphasis on problem solving. There is a real disconnect between the examples in the book, the computational exercises and the real problems. The first two often teach the mathematical concepts and methods well. They rarely teach student how to do mathematics well.

This is a void I’d like to fill. My philosophy, based very loosely off of the lab calculus program at Duke, is to explicitly teach problem solving skills, beginning with carefully worked and annotated examples and gradually asking students to fill in more and more of the gaps, until they can attack classes of problems on their own. To this end, this blog will house my attempts to create educational content which does it better. I’ll be starting with calculus concepts, but I don’t have any particular order I wish to pursue (this isn’t a textbook…yet). I will focus on really explaining why certain mathematical steps are taken, and the exercises will require the students to think about and explain why they’re using the methods they are. I look forward to trying out these methods and getting some feedback on what I believe are some very important pedagogical issues.