The typical undergraduate enters college with an unfortunate perception of mathematics, viewing it as a field with a fixed set of rules handed down from on high in the days of yore. Many students come to math classes expecting to receive magic formulas that will instantly bring them to the answer. It is my challenge and my joy as a math teacher to redirect my students' attention away from the final destination of an answer to the journey of reasoning that lies behind the solution.
I feel strongly about helping my students learn to reason mathematically because their mathematical journey will ultimately lead them to the unknown. My engineering students hope to build new structures, new cars, and new technology. Many of the dynamics will be the same as in earlier technology, but each advance brings a new set of differential equations to describe the development. I can only cover a finite number of types of equations in class, but there are infinitely many questions waiting to be asked. The students who can reason for themselves are prepared to adapt to the new challenges and overcome any mathematical obstacles in their path.
It takes time and effort to shift students' thinking; I cannot simply tell them to go forth and understand. By getting my students actively engaged with the material, I can lead them to understanding. To this end, I promote self-study, class discussions, one-on-one discussions, and group cooperation.
I think the first steps toward understanding are best made outside of the classroom. My students know what problems will be assigned in advance, so I urge them to attempt the questions before coming to class. Some initially fear that struggling with unfamiliar problems is not an effective use of their time, but I challenge them to persevere, as those who make this extra effort are better prepared to face future variations on these problems.
Since many of my students have thought about the material before class, we are able to have lively discussions. I strongly support the old saying, “mathematics is not a spectator sport.” By actively participating in class, my students ensure that they are on solid mathematical footing, because they have to be able to quickly make logical deductions. Promoting participation means far more to me than simply checking comprehension: as much as possible, I want my students to be engaged in the process of discovery. I want my students to view me not as an oracle but as a guide on their quest; I ask my class “what if” questions to lead them toward the next theorem or result we need. When a student volunteers a formula from a different course, I ask the entire class to justify this formula before we proceed.
Keeping students actively involved in class keeps them coming back. Many mathematics classes have poor attendance because the students are bored - not inspired to think. Usually, students who skip class find themselves overwhelmed by the material, and their grades suffer. One of my classes in Autumn 2007 had every single student present the day I distributed course evaluations. Their final exams were a delight to grade, because their performance reflected their attendance and enthusiasm.
I encourage my students to continue their journey together even outside of the classroom. After they turn in their homework, I post my solutions online. I suggest they form study groups and discuss my approach, how it differs from theirs, and how their individual solutions differ. Much insight can be gained by contrasting alternative methods. I tell my students that even though I am their instructor, I still learn from seeing their work, so I know they can learn from each other.
At the end of the quarter, I ask my students to consider sharing their work with the entire class. Specifically, I pick out a number of problems and ask them to volunteer to write up one solution each. The response has been impressive; in Autumn 2007, my students gave me a total of 48 pages of solutions. Many no doubt sign up out of enlightened self-interest, because they want a large pool of sample problems. The creation of this resource is one of my goals as well, but I like this exercise because it motivates the students to think carefully about problems and write a clear presentation of their solutions.
Even when self-study, group-study, and class discussions are not sufficient, I strive to make sure no student falls by the wayside. I encourage my students to visit me during my office hours or talk to me before or after class. I am certainly happy when a student just wants to thank me for my presentation on series solutions to differential equations, but I urge them to let me know if my pace or my emphasis needs adjustment. Some students come to me with questions that they know are deep: one asked, “What do sinh and cosh mean?” That student and I discussed important qualities of functions in general, as well as the specific properties of hyperbolic trigonometric functions. Other students are more specific and just want to know where they made a mistake in a particular problem. I point out their mistakes, but then we discuss ways of catching errors before they become an issue.
I encourage many different paths toward understanding the course material, because I recognize that no two of my students are identical; each student learns in a way that is uniquely his or her own. Not everyone will be a math superstar when they leave my class, but I want everyone to be much closer to true understanding than when they started.