My primary focus when presenting material in the classroom is to make it inquiry based and to convey that mathematics is a dynamic and evolving field. Students are urged to move away from memorization, which is replaced with the development of problem solving skills. A deep understanding of definitions is required, and heuristics are presented to try to make them more relatable.
This approach, based in understanding and problem solving rather than memorization, is often met with much resistance at first, but the development of such critical thinking is, in my opinion, the most important objective. In my 2016 sections of Math 383 - First Course in Differential Equations, I implemented this, in part, by focusing on word problems and “real life” problem solving. In Math 521 - Advanced Calculus I, students are almost never asked to reproduce a proof that they have seen but are instead required to learn to develop and convey their own proofs.
On the first day of a class such as Math 521, which is a proof based course, it is stressed that students need to learn to understand proofs, to construct proofs, to read proofs, and to write proofs. Traditional lectures are only of much assistance with the first of these goals. My focus goes into making lectures interactive and conversational, and these times are used to demonstrate effective problem solving strategies and proper ways to write technical solutions. This initiates, during class time, the process of getting the students to engage with problems. But even with such, the students need to do more with the material on their own, to do many exercises, etc. The methods described below are designed to encourage such.
Many students find the concept of doing research in mathematics to be completely foreign and most are aware of quite few unsolved problems. Throughout the semester, course-appropriate new mathematics, often simplified versions of my research or that of other faculty members at UNC, and open problems are described. For example, the famed Riemann Hypothesis can naturally be introduced in, e.g., Math 521 when infinite series are being discussed. I distinctly remember being introduced to Goldbach’s conjecture in an elementary number theory class; it greatly expanded my interest in the course and in mathematics as a whole.
Methods Used to Achieve Educational Goals
In recent years, I have been experimenting with a number of ideas in order to improve student performance and increase their engagement with the material. A new element or two is added every semester. Math 521 is the class that I have taught the most often, which has afforded me the most opportunity to experiment. Some of the techniques mentioned below are specific to that class. I am proud to point to my Spring 2016 evaluations in Math 521 as evidence that the culmination of these techniques has been very well received.
The methods that I have focused on recently include:
- Piazza online forums: These are becoming increasingly used on campus. Since the Spring 2015 semester, I have used piazza.com for email correspondence with students. These forums allow all students to see the reply to questions, which both provides a level of fairness as a hint to one student on a problem can now be seen by all students and a decrease in email load as the same question does not have to be answered repeatedly, which thus promotes a quicker response time. Moreover, LaTeX formatting is incorporated for clearer responses, students can reply and work together to remedy their confusions, etc.
- Warm-up problems: A lot of students tend to be in the classroom five or more minutes before the beginning of lecture. So, for several years now, I have been arriving early to the classroom to put a “warm-up” problem on the board. The students can attempt this problem collaboratively during the time before lecture starts. At times, such problems are true / false questions in order to test the students’ understanding of a definition, while sometimes the questions are an additional example of techniques developed previously. The first few minutes of lecture are then spent discussing the warm-up problem as a review of the previous lecture. These have been warmly received and help the students to recall the preceding material before seeing new topics.
More frequent assignments: In an attempt to encourage students to interact with mathematics more regularly, I typically require multiple assignments to be turned in every week. Often there is an assignment corresponding to every lecture, which is due one week after it is assigned. This is an idea that L. C. Evans at Berkeley shared with me, and while sometimes described as overwhelming in evaluations, it is more frequently reported to be beneficial, even by some of those that thought it was overwhelming. Having smaller but more regular assessments also aids with the next topic.
Large and active office hours: Using, in part, suggestions of H. Christianson to increase participation, I have focused on getting the students to consider office hours as an essential portion of my classes. Students are told that office hours are not only for those struggling in the class but are instead a time for practice, workshopping ideas, advising, etc. This forces distinct sets of office hours for separate classes, but the benefits to the students, as indicated in evaluations, appear to be tremendous. For the past two years, attendance in office hours for undergraduate courses has been so significant that I have had to reserve larger rooms in which to hold such meetings. This only further encourages participation as then I am no longer at my desk and near my work, and the impression that I am being interrupted from other things is removed. These times quickly become collaborative amongst the students, and problem solving abilities greatly increase with this practice. It also promotes one- on-one interaction with me as the instructor. A particularly notable event occurred this semester when I held a problem session for Math 521 on a Sunday evening that was three days before the exam (and for which extended office hours were going to be held on subsequent days). It was stressed that the problem session was voluntary and would only be question / answer as I had already written the exam and would not want to accidentally provide a hint to the content of the exam through my selection of problems. Nevertheless, in a class of 42 students, more than 30 attended, and we continued practicing and reviewing for 3 hours.
Glossary: This is most appropriate to Math 521 where a solid grasp of many technical definitions is essential to success. Students are told that the key to the work in this class is to first learn the definitions and then to practice a lot. A repetition of definitions is often a part of the warm up problems. To stress the importance of learning the definitions, I have begun maintaining an online glossary on Sakai for this class. Each day, after lecture, I update the glossary with any new definitions that were presented. This collects all of this essential information in one spot. Students in several cases have used the raw LaTeX file to create online flash cards, which have then been shared in the Piazza forums.
Thorough Sakai course sites: Courses are supplemented with thorough Sakai sites. In addition to the glossary, e.g., sites often contain notes, links, and old exams. Solutions to assignments and exams are prepared so as to model effective problem solving and communication of technical ideas. The calendar is updated with a brief summary of each lecture and a reading assignment. Upon request, access to an example of these can be granted so that it can be browsed. One can also see the requested course materials, which are arranged to model a Sakai site.
Handwritten solutions: While the typed solutions to an exam have benefits, the students are not typing their solutions during in-class assessments. So particularly in proof based courses, such as Math 521, I have begun also posting hand-written solutions to more exactly model the expectations.
In addition to the above, effort has been made to make lectures a more active time. When lecturing, the goal is to make it feel like a guided conversation where students are anticipating the next step, providing their own strategies, etc. With mixed results, I have been pausing and having students attempt problems on their own prior to going through them. I have been a member of the Finish Line Project and the Mathematics Learning Community through the Center for Faculty Excellence, and such methods are frequently discussed there. This was quite warmly received in Math 383, but in the Spring 2016 section of Math 521 completing an entire proof during such a pause was noted to be too daunting. In my Fall 2017 section of the same class, we instead did the same for smaller pieces such as writing down a definition without looking back at the notes, practicing applying a definition to a specific example, etc. This seemed to be an improvement. It kept the class actively engaged yet was manageable in the amounts of time that were available. An ongoing challenge is to further adapt these proven techniques to higher level, particularly proof based, courses.
Outside of the classroom, I have been leading a number of research projects. This direction is typically done through a series of exercises. I have, for example, started most of the students that have done reading classes with me by guiding them through a proof of the Morawetz inequality for the wave equation as an exercise. While the proof utilizes little beyond integration by parts, it illustrates a widely used technique, called a positive commutator argument, and yields an estimate that is playing an increasingly important role in modern studies of wave and dispersive partial differential equations. By posing additional follow up exercises for which they use their newfound skills, the students are pushed ever closer to modern research level mathematics.
Establishing an Equitable and Inclusive Classroom
The classroom thrives, with more active participation from each student, when every student feels included. The tone is set early in the semester with introductions. I share that I am a first generation college graduate that went to a high school that was so small that it did not offer calculus, which helps students know that I can understand some of their struggles. I mention how overwhelmed I felt during my initial day of college when my Calculus I teacher started by writing “ ε > 0 δ > 0 so that...” The effectiveness of this was noted in an “Of the Month” nomination that I received and which is included in the summary of student evaluations that is a part of these materials.
On the first day of a class, I often say, “This is an upper level course in mathematics at an elite university. It is suppose to be hard.” This helps to establish that struggling is expected, and it gives me the opportunity to assure the class that I will be there to assist and to outline in what ways. It also reinforces that those who are struggling are not alone. In order to establish a collaborative tone between the students and me, every syllabus encourages students to provide constructive feedback at any point during the semester rather than waiting until the end of the semester. Feedback on what may be done to help better prepare them for assessments is often requested.
As mentioned above, it is stressed that office hours are a collaborative time of practice and are an important piece of the course. And the online forum provides a source for ongoing communication.
To set the tone of the class, extra patience is given to questions that are addressed to the students in the first few lectures. This helps set the precedence that students are expected to respond and that the answer will not just be given to them without such. Lectures then become quite interactive as students are conditioned to provide responses to my inquiries. Questions are encouraged often, and every question is responded to. This includes the online forum. Even when grade distributions may look low, such as in Math 521, improvements are noted and effusively praised. Moreover, classroom practices, such as those discussed in the Faculty Learning Community that was a part of the Finish Line Project, are employed. These are proven to help in the retention of first generation college students and to help close achievement gaps (but without lowering the performance of the highest achieving population).
Evidence of Effectiveness
Evidence of the effectiveness of these methods can readily be found on the corresponding course evaluations where many comments reference how challenging such courses are, how much the student has learned, and how much the student enjoyed the class. If we look at the evaluation question “Overall, this instructor was an effective teacher,” the scores for the past several semesters are: 4.84 (Math 383, Section 003, Fall 2016, 43/56 responses), 4.89 (Math 383, Section 005, Fall 2016, 35/39 responses), 4.97 (Math 521, Spring 2016, 31/35 responses), 4.65 (Math 381, Spring 2016, 31/41 responses),
4.61 (Math 521, Spring 2015, 18 responses), 4.73 (Math 653, Fall 2014, 15 responses), 4.70 (Math 521, Spring 2014, 20/27 responses), 5.00 (Math 590, Fall 2013, 8/9 responses), which are all significantly above the department mean in their respective terms. A sampling of the comments on such evaluations and a further summary of the scores is included in this application as a separate document.