In my teaching experience, I have come across three main types of students: those in lower level classes who suffer from math anxiety, those in upper level classes who have a low tolerance for anything theoretical, and math majors who want to know why things work. When teaching a class, I take the audience into account and tailor my teaching methods appropriately.
When teaching lower level math classes (typically classes that satisfy the quantification general education requirement), the biggest obstacle to learning that I've observed is the math anxiety experienced by many of my students. One of the common reasons for this anxiety is the students' belief that math ability is innate. I combat this idea by explicitly telling my students from the first day that they will have to work to do well in my class. I stress to my students that work is a good thing and failing on their first attempts is okay. In a class with three exams, I've had students come up to me after doing poorly on the first exam asking if they should drop the class. This exam takes place roughly five weeks into a 15 week semester. I encourage these students to keep going with the class, at least through the second exam. I tell them that despite doing poorly on one exam they can still succeed in the class because every failure is an opportunity to learn from their mistakes.
The math anxiety my students experience is also related to their idea that if they struggle with math, they have no chance of doing well. Again, I tell my students that there will be a struggle, but that doesn't mean they aren't cut out for math. Just as an athlete or musician must practice in order to improve, a math student must go through the repetition of homework problems in order to learn. I help my students build their math “strength” by providing many example and homework problems that they can use for practice. Though my students might not originally appreciate the repetitiveness of the examples, they are grateful for the practice by the end of the course. In my end of semester evaluations, one student noted that they sometimes felt I was repeating myself in the lectures a little too much, but then they would listen to what I was saying and realized that they missed a step. In the end, the student appreciated why I went through every step for every problem. They urged me to continue doing this in the future because the student knew it saved them from making silly mistakes and was sure it would help others.
I also work to be accessible to my students. I encourage them to ask questions, emphasizing that there are no stupid questions. If a student is confused about the material, then it needs to be addressed. I know that my efforts have been successful based on comments that I've gotten from my students. One student told me in office hours that they appreciate my willingness to answer their “silly questions” without judging them. They felt comfortable coming to me for help and as a result did very well in the class. In addition, I have received comments from several students on my end of semester evaluations saying that my accessibility decreased the math anxiety they felt.
Another challenge that I've faced in teaching lower level math courses is the fact that students struggle with the high level of abstractness of the course material. Many of the students just want to know how to solve the problem and get the correct answer. However, my goal is to show them that the material is conceptual, not just computational. In order to do this, I introduce them to a problem solving process. When I was an undergraduate student, I remember one of my professors giving us three questions to use for all of our homework problems. The questions are:
- What are we being asked to do? Identifying the question we are being asked to confront is a vital first step in problem solving and helps us filter out unnecessary information.
- How do we solve this type of problem in general? Depending on the problem, one solution technique may be better suited than another. Picking a general path to follow in a given situation helps us get started.
- How do we solve this type of problem for our particular case? We know the steps that we need to follow and can use the information from our particular problem to proceed.
When I present example problems in class, I use the three questions as a guide. If it is the first example illustrating new material, I work through the problem on the board, explaining each step as I go. However, after the first example, I ask my students to fill in the answers. I believe asking the students to work through examples (either on their own or as a class) is invaluable. From my own experience as a student I know that it is easy to watch a professor go through an example and think that you understand everything. It is only when you go to do the problem yourself that you see where your understanding is lacking. That is why I want my students to have as many opportunities to work through problems in class as possible – they can better their understanding of the problem solving process while I am still at hand to answer their questions and help them when they get stuck.
I carry this same methodology over to office hours. I enjoy office hours because it gives me an opportunity to work with my students in a one-on-one or small group environment that is not always practical in a large class setting. Instead of answering their homework questions by working through the problem myself, I turn the question back around to the student. I ask them what they have already tried or how they think they should start the problem. This description of the student's thought process allows me to better help them because I can understand where he or she went wrong and point them to the correct path. If the student says that they have no idea where to even start, I ask them if any of the problems from class or from the textbook are similar to the problem with which they are having trouble. Normally, this is enough to get them started and we continue by working through the problem together.
In the upper level math classes that I've taught, the majority of students have been engineering and other non-math majors. These students appreciate that they will be using the course material in their future careers. However, many have a low tolerance for the theoretical aspects of math. In my experience, they prefer knowing how they will use the material rather than why the methods work. These students want concrete, real-world examples. Because I have no engineering experience, I have relied on my students to provide examples. I ask them how our course material relates to things they are learning in their engineering courses or other examples they have seen. My students have told me that they enjoy contributing examples because it makes them feel that they have a hand in choosing what we study. As I teach more upper level courses, I will continue to collect examples.
Math majors and other highly committed students are not satisfied by knowing how to work through a problem, they also want to know why the method works. My main interaction with these types of students has been through a Rouge Waves REU held during the summer of 2014. During this REU, I served as a mentor to two undergraduate students whom I challenged to go beyond the calculations. Once the students understood the general background of modeling the physical system of rogue waves, I asked the students questions about how the model would change if we modified the initial problem set-up. This helped the students go from the concrete model to a more abstract idea.
Teaching is a great responsibility and the impact that one teacher can have on a student is immense. A good instructor can motivate students to work hard and give them the confidence to succeed in mathematics, whereas a poor instructor can perpetuate a student's idea that he or she lacks some innate trait for being a “math person.” As an undergraduate student, my professors helped me learn that math was something you had to work at to succeed, that it didn't matter that I sometimes struggled to understand an abstract concept. I want to inspire my students in the same way and help them succeed in mathematics as my instructors have helped me.