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.