Abigail began by doing an ugly integral involving trigonometric substitutions. Next she pointed out that if we had happened to notice that we could write the integrand as a sum of simple things we could have avoided all the messy stuff. This served as the motivation for her presentation on the technique of partial fractions. Comments on Abigail's lecture were largely positive. Initially the only suggestions for improvements were about minor details of terminology and the order in which she presented the material. Professor Himonas then took the stage saying that while this was one of the most successful lectures he has ever attended he had a complaint about Abigail's lecture which he was saving for the end.
Professor Himonas passed out worksheets and proceeded to give a lesson on "Continuous Income Streams". The lesson was intended to illustrate some techniques for keeping students involved in the class. Himonas most frequently employed the technique of calling on a student (by name) and demanding he or she answer a question. After deriving the formula for the Future Value of a continuous income stream in a concrete situation and in collaboration with the students, he demonstrated the process of going into the general formula by replacing the concrete quantities with variables, thus demonstrating the application of one of Math 108 goals, namely :
"Understand mathematical symbols and formulas. Learn how to read and understand mathematical symbols and formulas, and to be able to express thoughts in symbols and equations. You must realize that each mathematical formula expresses a precise and clear relation between the variables involved. It is often said, that the best way for clarifying one's thoughts is to put them into an equation. Equations are not there to be memorized but to be understood. In many situations they form the bridge between mathematics and our world."
Then, he gave a problem about the Present Value of a continuous income stream as group activity which was supposed to last for about five minutes, and to be collected and graded. This gives students a little extra motivation to pay close attention to the lesson. Other ways for involving students in their learning of mathematics were discussed too.
Himonas' main argument for involving students in the way he does was that in this way the students learn how to think. He claimed that students could leave the class after Abigail's wonderful lecture without really knowing what they had been learning but that his students always leave class with the ability to do something.. Himonas said that what he hopes to teach, in a class like Math 108, is primarily the ablility to model problems (as equations), the mathematical way of thinking and only secondarily the technical skills for actually solving the equations. In fact, Himonas tells his students at the end of the semester to forget all the facts they have memorized and keep only the problem solving skills they have learned and the mathematical way of thinking.
Discussion after this addressed the issue of whether teaching techniques of integration is amenable to such a treatment (the point, after all, being to master the tools needed to solve certain problems). Himonas then argued that the point, really, is that there is always room for improvement in teaching and that involving the students in learning during class is probably the biggest improvement one can make.
(reported by Logan Axon and Alex Himonas)