Thoughts On My Teaching At SCSU
Brian D. Rude, March 2003
I expect I teach math in much the same way that most other college teachers do. However there are always many ideas about teaching and learning being discussed in educational circles. Many of these ideas are not very practical, but that does not seem to prevent them from spreading, and sometimes becoming popular expectations of what a teacher should or should not do. I believe as a professional matter one should have some idea of what the new ideas are in the teaching of mathematics, but I feel no obligation to try to put into practice ideas that do not seem sensible or practical to me.
A visit to one of my classes would indicate that my main teaching method is lecture. That is basically true, but I think there is a lot more to it than that. What topics to present, what order to present them in, and how to present them, are questions that require careful thought and planning.
I do believe that a careful explanation of each topic we cover is very important, and forms by far the greatest amount of time spent in class. I do not assign group or individual projects. I make no attempt to “use a variety of teaching strategies and technologies.” I stick to what seems to work, and what seems to work, in broad terms, is a careful explanation of a topic, followed by a well chosen homework assignment, followed by giving the students feedback on that assignment, and acquiring feedback from the students. In class I encourage questions, but I do not encourage free ranging discussion. We don’t have time to get off the subject. There is a lot of material that should be covered in any math course. My job, as I see it, is to facilitate the learning of that material.
I typically start class by handing out homework that I collected the previous time and graded. Then I usually spend a few minutes answering questions about the current homework before collecting it. Then I spend the rest of the hour explaining new material. I write the homework assignment on the board at the end of the hour. When I have a good textbook to work with I simply choose problems from the book to assign. Often it happens that the text book doesn't present quite the right set of problems I believe are needed, so I work up a handout assignment. Some of these can be seen on my web site, as I will describe shortly.
I consider the daily assignment of homework crucial to student learning. I assign homework everyday, starting on day one. I don't assign anything unless I plan to grade it and get it back the next class period. I do tell students that the homework I assign is limited to what I can grade. I stress that for most students just doing the assigned work is not enough.. Math textbooks traditionally have the answers to odd numbered problems at the end of the book. Therefore I normally assign even numbered problems, but tell them to use the odd numbered problems for guidance and more practice. It would be more convenient to collect homework only once a week or so, but I have always felt that daily homework is much better for student learning. I do accept late homework, but with a 50% discount. I stress that late homework is better than no homework, because then they get feedback, as well as a few points, and something to study on for tests. I repeat this message several times during the first few weeks, for I find that a lot of students seem to assume that if they miss an assignment then they can't do anything about it. I am very lenient about waiving the late penalty for illness or family emergencies.
For the past several years I have graded 75% on tests, 15% on homework, and 10% on quizzes. Occasionally students will ask to be graded more on homework, but I feel, and explain to them, that the purpose of homework is primarily to learn the material so they can pass the tests. Learning must be demonstrated on the test.
I try to have a quiz at least once a week, but that has proven difficult to do. I probably average more like a quiz every week and a half.
I do maintain a web site, primarily as a convenience to my students. It is http://web.stcloudstate.edu/bdrude. I have a page for the assignments in each course I teach. When a student misses a class he or she can just get on the internet and get the assignment. Of course this convenience may have a disadvantage of making it even easier to skip classes, but I consider that minor compared to the benefits. My web site also contains the syllabus of each class I teach, and my office hours, again purely for the convenience of my students. I occasionally put announcements on it also. I also keep a “Miscellaneous” page on my web site, where I post links to articles that I find of interest, or educational thought that I have written up at one time or another in the past.
Math 193, Mathematical Thinking, is a freshman math course required for a number of majors on this campus, such as elementary education, child development, and nursing, and is the default required math course of a number of nonscience majors. This course is of special interest to me, for it is quite different than the typical freshman math course. I will describe the course and what I do in it in some detail, as it seems a good way to illustrate some of my ideas on teaching and learning.
Math 193 consists of a series of topics designed to give an idea of how math works. We cover eight chapters in the book, Excursions in Modern Mathematics, Fourth Edition, by Tannenbaum & Arnold. (The title seems just a bit misleading to me. It does not cover the main topics that have gone under the title “modern mathematics” in the past.) We begin with a study of Fibonacci numbers, leading through the mathematics of linear and exponential growth and a few related topics. Then we go into a chapter on symmetry in two dimensions, rigid motions, classification of symmetry of two dimensional figures, and symmetry of border patterns. Then we cover a couple of chapters on simple statistics, followed by a chapter on probability. We finish the semester with an introduction to graph theory.
Students have to pass a placement test to enroll in 193, but even so a number of students with very weak math skills are enrolled. They are supposed to be proficient at arithmetic and basic algebra. Many are not. I try to make special provision for these weak students in the topics of percents and linear equations. For several semesters now I have made the first two quizzes different from other quizzes given during the semester. Quiz 1 is a special review quiz on percents. I tell the class that we will not use class time to review percents, but that they ought to know the topic. In fact, it is assumed that they do know it. To motivate them to relearn per cents, if they have forgotten them, I tell them they can retake Quiz 1 as many times as needed to bring the score up to 10 out of 10, but they'll have to come to my office for retakes. I have about six forms of quiz 1, so a student is unlikely to get the same form twice. Each semester there are just a few students that get a score of ten the first time, but perhaps no more than 10% of the students. But over the course of the semester another 10 or 20% of the class will retake the quiz a time or two to get their grade up. I would like to tell them that a score of 10 is the only acceptable grade, but I don't really want to try to enforce that. Quiz 2 is the same arrangement for simple equations, with similar results.
In the first few weeks I try to stress the idea that this is a course about how math works, and that leads to the idea of mathematical processes, tools, and methods. Trying to show this broad perspective of mathematics is why we have such obscure topics as Fibonacci numbers. On day one I start out with the idea that we are going to take a very broad perspective of mathematics. The arithmetic they start learning in first grade, or the algebra they start learning in high school, are certainly good examples of mathematics, but math is much more than that. I talk about numbers being not at all needed in many areas of math. I don't know whether this is a new idea to them or not. I explain that arithmetic, algebra, and calculus, which involve numbers, form the basic structure of the math that most people are aware of, but by no means all the math that there is and that is important. I explain that very broadly speaking we can think of math as "figuring things out in a systematic way". Or we can think of math as the "science of patterns", or we can think of math as "science without objects". I tell them that they do have a good example of math without numbers in their own experience. Until I tell them, few students realize I am talking about geometry.
I spend a few minutes talking about math as a language, my view being that it is not primarily a language, but has some characteristics of a language. I talk a few minutes about math requiring exact meanings and exact notation. I also point out that in math we sometimes use the same notation to mean different things, but this is no different than the English language. We depend on context to know which meaning is meant.
Still on day one I explain inductive and deductive reasoning, and have a worksheet on that as part of homework. Good examples are hard to make up. It seems that what is an obvious example of inductive thinking to me will be reinterpreted by some student as deductive, sometimes with a very convincing explanation. I found that I get the best results when I underline the conclusion, or specifically restate the conclusion. I point out that whenever we do a problem in arithmetic or algebra we are using deductive logic, because we know the general ideas always work. But in everyday life we use a lot of inductive logic. We can learn much by looking for patterns, even though we realize those patterns may not always hold. I explain that “free exploration” is indeed a mathematical process.
Logic is the one most important mathematical tool we have. Logic is not limited to inductive and deductive, but those are the two big ones. A common pattern of mathematical thinking is free exploration leading to some informal inductive conclusions, leading to informal hypotheses, followed by identifying a few well chosen rules and deductively exploring the consequences of these few well chosen rules. I emphasize that free exploration is indeed a mathematical process, and a very important one. However it is not the end result of a mathematical effort. It is not what will ever be published. It is always a beginning mathematical process. The mathematics of lasting value comes later, the result of carefully choosing a few basic definitions and rules and exploring the results of those definitions and rules by deductive logic. I use the postulates of Euclidean geometry as an example of this in their own experience. When I take a quick poll by a show of hands I find that most members of the class have had geometry, but it is not at all clear that the essential structure of deductive logic has come through to them.
Over the past several semesters I have given greater emphasis to both imagination and intuition as mathematical tools or processes, emphasizing that, like free exploration, they might be very productive, or might not be. Also like free exploration, neither imagination nor intuition will be the end result of a mathematical effort, but can be very important in advancing toward that end result. I also tell them that trial and error is indeed a mathematical method, but one that we want to get away from when we can. “Work smarter, not harder” is a formulation of this idea that they seem to respond to.
I also address the idea of the role of practical benefit in mathematical thinking. I point out that many mathematicians almost make it a point of pride that their mathematical research has no practical benefit whatsoever. I explain that this may sound foolish at first. It would seem that mathematicians would want their work to be useful. But, I argue, this attitude has proven productive for mathematics in the long run. I use the example of complex numbers. I think most of the students have been exposed to i, connected to quadratic equations. I explain that imaginary numbers were understood to quite an extent in the 1500's, and, so far as I know were totally useless for several hundred years. However when electricity began to be understood in the 1800's, imaginary numbers were indispensable. I further argue that therefore modern things in their world, like instant messaging, could not exist without imaginary numbers, because understanding electricity requires imaginary numbers.
Over the first several weeks of the class I try to emphasize these ideas of how math works. However by necessity we must focus on trying to understand and learn the details of the topics we are studying. This requires the routine previously mentioned, a careful explanation followed by a well chosen assignment in repeated cycles of feedback. We prove very little. Most students are not well prepared to deal with proofs, and proofs can be very time consuming if done well. However I often take the opportunity to point out that what we try to understand on a common sense basis, would be subject to formal proofs in more advanced mathematics, and I try to give some idea of how these proofs would work.
Math 193 was set up several years before I arrived at SCSU. I enjoy teaching it. However I have never been sure it’s the best type of course for the students who take it. While working on my masters degree before coming to SCSU I taught college algebra. I wondered then if college algebra was the best type of math course for that group of students. Explaining how math works to students who know nothing of calculus, and never will, is a tall order. I’m certainly glad to have had the opportunity to teach something different to freshman. But I still have very little idea of what content is really best for these students.