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