My teaching philosophy centers on two values: curiosity and empathy. I believe that my growth as an educator depends on my willingness to refine my teaching practices, materials, and assessments as I learn from both experience and research. If I want our students to be curious about mathematics, I ought to be curious about how they learn, what they struggle with, what the assessments I give actually reveal, and what assumptions I bring to my work. At the same time, pedagogical expertise is meaningless without genuinely caring that our students are learning and feel supported both in and out of the classroom.
As an Assistant Teaching Professor, I have strived to make each of my classes feel like a supportive learning community for our students. To do so, I have drawn on both experience and research. Below, I highlight some of the specific tools and techniques I have been working on.
The classroom is a learning community
When I first began teaching as a graduate student of mathematics in 2008, I acted on instincts rooted in my experiences as a student and the methods and attitudes of the teachers I admired. I identified a common thread in these teachers: the ability to model sustained inquiry and to foster a sense of community in the classroom - i.e. the classroom is a learning community where each member has a role to play and mistakes are to be celebrated as learning opportunities. This principle continues to inform my teaching.
My approach to fostering a sense of community begins with a view of the teacher not as a vessel for content, but as a facilitator of learning. In practice, this means that I ask a lot of my students in the classroom and, in turn, provide a lot of feedback on in-class work and assessments. I ask students to contribute ideas for how to solve problems or prove theorems, and then explore their ideas to see where they went right and where they went wrong. I often have students work together in small groups on challenging exercises and concepts to open up a dialogue. When doing so, one of my greatest challenges is resisting the urge to give hints or answers too soon when walking around the room to check on each group’s progress. I find that when I am able to resist that urge, my students often end up thinking more deeply about a concept or problem and learning from their mistakes and from one another.
In lecturing, I attempt to model the types of behavior I hope to see in my students. First and foremost, I try to maintain a level of enthusiasm and curiosity about the mathematics I teach. Even after years of teaching calculus, I am always looking for new insights, and I relish the opportunity to make discoveries on the spot in front of the class. Secondly, I often interweave stories from my undergraduate career of times when I struggled with certain concepts or problem types in the hope of offering insights into how one overcomes such difficulties. Thirdly, I emphasize the importance of clarity in mathematics and the role of precision in our notation and exposition by writing clearly, without skipping steps, and with proper notation - I have included some materials that I use to support these ideas in this dossier.
Technology allows for the community to stay active outside the four walls of the classroom. To this end, I put a great deal of effort into maintaining a website with lecture notes, video lectures, and extra resources for my students. To provide students with more opportunities to practice the skills we develop in class, I use online homework systems such as WebWork.
The role of research in teaching
After two years as an instructor at Drexel University, I began my work on a doctorate in education. This formal education and the research I have done since completing my degree have had a profound impact on my teaching.
Teaching with worked examples
My dissertation research was on the efficacy of a worked-examples-based framework for introductory mathematical proof-writing. In my research, I found that when students are new to proof-writing, they benefit more from studying worked examples of proofs (that provide explanations for each step and an overall framework for working through a proof) than they do from attempting their own proofs. After having an opportunity to study proofs that are worked out entirely, they attempt to fill in the missing pieces to partially completed proofs. Over time, this type of scaffolding can slowly be removed, placing more of the responsibility on the student until, eventually, they are problem-solving/proof-writing on their own.
In my calculus classes, I have attempted to apply a similar philosophy. When introducing a new unit, I will often give students several examples to study and discuss before they attempt to solve any problems on their own. Then, I give them problems that are very similar to the worked examples with some of the steps worked out for them. Gradually, they work their way to exercises, which differ significantly from the original examples, without any scaffolding. Finally, I prompt them to reflect on the differences and similarities between examples. While I continue to improve upon this process, I have found it to be very successful thus far, especially when students are working in groups on challenging problem-solving activities.
Asking right questions
Not long ago, I co-authored a paper on teacher questioning with colleagues from Rutgers University, Montclair State University, and Temple University in which we examined the questioning habits of eleven instructors of upper level mathematics courses. In this study we coded each question asked during a recorded lecture according to the type of contribution it elicited from students. My work on this project raised my own self-awareness about the types of questions I ask in the classroom. What I found through my own reflection is that, like most of the subjects in our study, many of the questions I ask of the classroom are either checks for understanding (e.g. does everyone understand?'') or simple calculations (what is the sine of pi over four?”).
Since conducting this study, I have put considerable effort into asking more substantive and conceptually demanding questions of my students - e.g. “how can we apply the idea of the Riemann sum to find the volume of a solid of revolution?” This type of conceptually demanding question places more responsibility on the students to think through what they have learned and to contribute mathematical content to the class.
I have also become much more mindful of the amount of time I wait after posing a question. Our study found that many instructors wait only a couple seconds after posing a question. This does not leave students with much time to think through a concept or problem. Though the silence can be uncomfortable at first, waiting just five or ten seconds longer increases the likelihood of student participation. Over I time, this approach turns a lecture into a conversation.
Assessment
During 2020 and 2021, I participated in the Faculty Fellows program offered through the Center for the Advancement of STEM Teaching and Learning Excellence (CASTLE), here at Drexel. The goal of this program was to bring together faculty from different STEM departments to discuss best practices for remote teaching and learning (in light of the pandemic lockdown).
For my project, I experimented with ways of introducing more writing into online assessments. Both the NCTM standards and the Common Core standards highlight the importance of incorporating writing in math curricula. The NCTM includes communication as one of its five process standards. To meet this standard, students must be able to organize their mathematical thinking and communicate it clearly to peers and teachers. The Common Core’s standards for mathematical practice include the ability to construct viable arguments, to explain mathematical problem-solving to oneself and others, and to communicate mathematical ideas clearly.
I found that, by including more writing, exams can become a learning opportunity not just for students, but for me as well. Their written explanations shed far more light on what they do and don’t understand than calculations alone, and this in turn allows me to craft better lesson plans and materials.
Conclusion
Math is a subject that challenges many students, not just intellectually but also emotionally. To support our students along both of these dimensions, I treat the classroom as a learning community and leverage experience and education to continually refine my teaching practices. This, for me, is where the joys of teaching lie: connecting with students and learning more about how to be an effective educator every day.