A pdf of this statement is also available.
Teaching Is a Learning Experience
Teaching is as much a learning experience for me as it for my students, possibly more so. I constantly find that I have to adjust how the material is presented as well as how it is tested. As instructor of record, I have taught 2 sections of Laboratory Calculus I, while I served as a Graduate Student Teaching Assistant (GTA) for 2 sections each of Laboratory Calculus II and of Linear Algebra. All of the calculus classes, containing at most 25 students, are typically for freshmen who have not yet declared a particular major or plan to pursue one not requiring further courses in mathematics. The challenge with these classes is finding how best to present the material with the minimal level of abstraction, determining exactly where students have trouble with the concepts.
Learning from Assessments
Although tests, homework quizzes, and lab reports are the primary means for graded work in these courses, these methods are not sufficient for assessing student learning. The class moves quickly, so learning what concepts are difficult for the students after grading assignments benefits more the students of the following year than the current ones. Nevertheless, common pitfalls are addressed in a timely manner. For instance, after grading every test, I write comments in my exam file as to which questions worked well and which were unapproachable, which helped me determine that breaking up more difficult questions into parts would be more approachable for the students. (See sample questions 2 and 3.) These changes were made between the 2nd and 3rd test the same semester. The troublesome concepts or common mistakes are also addressed in class after each test as well as when later material depends on such clarification.
Learning from Colleagues
In one of my 8:45 am classes, I felt there was a disconnect between myself at the board and the students, as hardly anyone was asking questions and most looked confused. I started asking other instructors about ways they kept the class engaging the material over the course of the lecture as well as evaluating how well the students digested concepts on a more immediate basis. I have started using 90-second quizzes such as Write down everything you know about concave functions.
to check understanding of definitions or property tests. After observing one of the postdocs teach, I have also tried incorporating short multiple choice exercises (sample question 4) into the lecture to allow the students to immediately grapple with new material. Each student has a set of flashcards color-coded A
through D
to vote on an answer, so that after I see discrepancies from the true answer, I can ask the students to talk to their neighbor about why they chose a particular answer to help them practice explaining their arguments. They then vote again and usually reach the correct consensus; a student may then explain how one can find the right answer, and I can follow up with any details that still seem unclear. Meeting with the postdoc later also clarified different styles for these questions. I am still looking for other ways to help the students learn and develop precision in their articulation of the concepts on a more immediate basis.
Learning from the Students
Working with students individually during office hours also helps improve my explanations of concepts as well as determine common places of confusion. They help me reconsider whether the intuition I have been using is helpful for them as well as what part of the problem should be emphasized next time. Sometimes, the concept the question addresses is different from the subconcept the student needs to approach the problem. For example, I have found asking the students to check the units when they translate a mathematical modeling problem to the corresponding differential equation helps them arrive at the correct equation much more often, based on classwork and test responses. (See sample question 1.) I enjoy when ideas become clear for the students, especially when they are happy the concepts are now clear too. Nevertheless, some students tell me when my explanations are confusing, whether in office hours or through the course evaluations at the end of the semester, so I still need to develop several other ways to view each concept in class. The students have helped me in this regard; for example, in sample question 3, part (d), one student asked whether one could appeal to L’Hospital’s rule instead of returning to definitions. She was indeed correct, leading to a more direct solution than I had anticipated. Because the office hours for each of the instructors of Lab Calculus I are open to all sections, when students from other sections came to mine, I found this simpler solution to be less confusing for the students.
Future Directions
As a mathematician, I often make or must find connections between different fields, particularly from dynamical systems, algebraic topology, and probability. I find it fascinating when any of these fields sheds light on the others. I should like to help students at the appropriate level discover the joy of these connections in a topics or seminar course, especially to see how applications can benefit from pure
math ideas, especially in biology. In any case, small guided projects may help the students in this discovery process; the lab component of the calculus courses worked this way. The students would work through a series of related questions to help them investigate a concept that we would see in more detail in a following class or had just seen in a previous class. When I took classes, I particularly enjoyed the cases where the concept was discovered in the homework first, as it gave the opportunity to connect the results with previous knowledge and integrate it with the new knowledge we would see later in class. I should like to investigate incorporating such approaches in my future classes as well as determine for which students they are most helpful. I should also like to incorporate interactive demos, such as the Geogebra software, during class or in homework to help the students develop intuition faster about the concepts, such as the slope of secant lines approaching that of the tangent line, especially for the visual or tactile learners. I have found such software helpful even in my own research, and it should help supplement note taking during class.
Motivation
One of the contrasts from taking undergraduate classes to teaching them was the different levels of motivation students bring to class. Learning for the sake of fulfilling a requirement is very different from learning because you trust or believe it to be useful for your future work. I did not realize this distinction initially as the Linear Algebra sections for whom I prepared lectures consisted of engineers, who knew this material was supposed to be useful to them in some way. When I later started teaching calculus, I slowly realized the because it’s useful
argument no longer applied. I still thought finding the appropriate motivation to learn was the responsibility of the student, though such a task would be difficult if the students were not used to doing so, and increasingly difficult if they find themselves struggling. As instructor, I thought I must help the students who were willing to keep trying as well as show that starting to try again
does not have to be difficult.
When I started asking other colleagues as mentioned earlier, I thought the main cause for a lack of motivation in class was a perceived lack of enthusiasm for the subject on my part. To address the latter, one of the other instructors offered to switch classes with me for a day; however, we learned it did not make a significant difference. When other graduate students observed my lectures, they thought I was enthusiastic, but my voice could be louder. When I asked colleagues about ways to engage the material, I had assumed that better presentation would be the only way to help the class. However, after the semester was over, I asked one of the senior teaching faculty what he did for motivating the students. He emphasized to be sure to get to know the students as people, not just know their names; it should be clear to them you care about them as individuals and not base their worth on their performance in the classroom, even subconsciously. Having been concerned primarily with the material, I realize this aspect of teaching was not given sufficient attention on my part, and I plan to focus on it and my speaking volume in my future classes. I look forward to learning still other ways I can improve my teaching in the future, as it is an ongoing process.
Michael P. Casey 