Math is a creative process, where true learning happens in the middle of not knowing but still trying. It is an active process that happens through a mix of observing others, trial and error, and peer instruction. My classroom is an environment in which students engage in these steps, so they develop creative solutions and critical thinking within both theoretical and applied mathematics. I do this by identifying the support each student needs, including group work in lecture, and incorporating real world applications of theory.
The most common response I hear when introducing my field of study is “I could never do that” or “I was never good at math.” While math may come more easily to some people, I have seen time and again that even weak math students can become proficient through reasonable effort and with the right guidance. It is my job to provide that specialized guidance so they can benefit from a knowledge of mathematical concepts. Even if they won’t ever use calculus, the patterns of thinking and reasoning developed through studying math are transferable to any discipline.
During my time at Duke University, I had the opportunity to work with students who came into the class discouraged. At Duke, two pathways are provided for calculus instruction, one of which includes time for pre-calculus review. The first course of this more foundational pathway is often filled with students who have little confidence in their math skills. When I taught Laboratory Calculus and Functions I, I found that some effort and preparation on my part, as well as theirs, helped students quickly and effectively overcome their initial apprehension. I implemented a pre-course assessment to identify areas of weakness and then incorporated those topics into the following lectures, worked one-on-one with students, and connected them with the myriad tutoring resources that Duke offers. These efforts reassured students that they did indeed have the background knowledge needed for the course and would be able to succeed. Throughout the semester I monitored each individual’s progress through short, low-risk assessments such as end of class quizzes or homework assignments to identify misconceptions and gaps in comprehension early on. Mistakes common throughout the class were discussed in the next lecture and struggling individuals were encouraged to come to office hours.
As I implemented these strategies in the first class I taught, I found that I was focusing too much on the very basics during these short in-class quizzes. This successfully identified the students who struggled the most, but failed to prepare more proficient students for exams. I learned that students need to be exposed to a variety of difficulty levels and recieve feedback on them to be pushed past only learning the basics. Since then, I strive to ensure that the first time I learn about a struggling student is never after a large exam. These steps ensure that every student has the support and direction they need to succeed - if they put in the effort.
I have found the traditional lecture to be a useful medium for presenting students with examples of basic techniques and tools, but without additional guidance, many students give up when they don’t know how to approach a problem right away or when they get a wrong answer on the first attempt. To familiarize students with the patience needed to succeed in math, I encourage students to try different approaches to a problem before guiding them to the most expedient approach. I do this by incorporating group discovery time into lectures when I know students already have the capacity to derive or intuit the formula or method in question. I then refine and correct the discoveries they made. This approach enables students to explore mathematical concepts independently, resulting in them remembering the lesson better and developing their ability to make their own deductions - a critical skill in higher level classes, graduate level classes, and other subjects.
When teaching the derivative rule for exponential functions during Laboratory Calculus and Functions I, I split the class into small groups and had each compute estimates for the limit definition, then find the ratio of the derivative to the function for two different bases. This was a chance to reinforce the idea of a limit as well as explore what the derivative could be. As the class compiled their results on the chalkboard, they noticed that the ratio stayed the same independent of the point the derivative was taken at, but that different exponential functions had different ratios of f' to f. For the rest of lecture, I then could derive the actual formula and help them understand why this ratio appeared. This approach helped them to remember the derivative rule more easily as well as introduced them to a way of exploring derivatives, as in higher mathematics sometimes the derivative of a function cannot be directly derived but must first be intuited and then proven.
My classroom balances the necessary constraint of building a theoretical foundation with the need to prepare mathematics students for how the theory applies to the real world. As a calculus instructor at Duke, I accompanied lectures with lab sessions where students worked with data sets to apply the topics they were learning. For example, students learned how to understand the meaning of mathematical concepts through working together to apply a linear model to data describing cancer mortality rates as a function of radiation exposure. Students discovered that slope described the change in cancer mortality as exposure increased and that the y-intercept was the baseline cancer mortality rate. Thus, an abstract mathematical model became a meaningful tool that they used to suggest a course of action to reduce cancer mortality rates. These lab sessions give the students agency and utility where the math may otherwise feel static and abstract.
In my experience, a concise explanation of how a concept can connect to a job or product someone uses can spark interest and motivation in most students. In this way, a student’s burgeoning motivation for mathematical comprehension is tied to their existing interests in another subject. My emphasis in applied and computational math during my undergraduate studies along with my graduate work in computational biology and bioinformatics has prepared me with a multitude of experiences about how math connects with life and technology which I will apply in the classroom. I pull from these regularly when discussing math principles with my students.
Near the end of the semester for Duke’s Laboratory Calculus II, students use their knowledge of integration, sequences, and series to compute Fourier series. These problems are hard, often requiring three complex integrals for one homework problem. However, during my experience teaching this class, I have seen students attacking these problems with renewed enthusiasm after hearing just a few examples of how Fourier series are used in the real world – including acoustics, signal processing, image processing, and audio processing. Knowledge of real world applications has the power to motivate and inspire students as well as reinforce mathematical theory. I strive to include real world applications in each of the topics covered in class.
Most of a student’s learning happens outside the classroom during their own time to practice, study, and reflect; however, the time inside the classroom is critical to building the pattern of persistence, collaboration, and a love for how math can describe and shape the world. I continually learn from my own experiences and from other colleagues about how to engage students inside and outside the classroom. I strive to share my love for mathematics and to change students’ perspective that math is a strict set of algorithms into the idea that math is alive and has space for creativity through helping students of all backgrounds engage in mathematical discovery and connect the abstract to the real world.