Teaching Philosophy
Luke Oeding
When teaching mathematics, I have a very grounded goal – learn how to do the mechanical problem solving in the course – and a very ethereal goal – learn abstract mathematical thinking. Both of these outcomes will prepare them for future courses in their major and will eventually be used (at least indirectly) in their jobs. I help them accomplish these goals by setting challenges in front of them that enable them to work diligently to build their mathematical skill set. For example, I design immersive in-class exercises that help students discover concepts from examples they do, I require them to do (what they think is) a large amount of homework, and I strive to cover all the material that was promised them in the course description while still putting some of my own mathematics into the course. Allow me to highlight the following themes that shape my teaching.
Connections: How can I foster students to develop connections to their peers and to the university?
Students today have many options for higher ed, and not all of them include in-person learning. Why do students choose to sit in a classroom at my university? Some of this decision is tradition – they go where their parents went. But some of this is because students innately know that it is the people that make the education. They know that it’s likely to be the person they’re sitting next to that they’ll start a business with. The person sitting in front of them might be the one they eventually marry. The shadow of the stadium outside their classroom is the one they’ll return to year after year and continue to participate in the university family. While much of this tradition is facilitated by extracurricular activities, we should all ask ourselves if we are doing the most we can to enrich the classroom experience and set up an environment that will allow these connections to be made inside the classroom context as well. If we mathematicians are successful at this we will find happy alumni coming back after finding success in their careers and personal lives, and we hope that they will share that success with the next generation of students.
Focusing on connections led me to design a project for Abstract Algebra called #ThisIsSymmetry. The project started by asking students to take pictures of any objects on campus that had symmetry, whatever they thought that meant at the time. Throughout the semester students learned about symmetry groups, and how to codify symmetry from a mathematical perspective. The students then took this new skill and applied it to the pictures they took, and prepared descriptions of the symmetry for a broad audience. They revised these descriptions through peer review, and then 16 students each posted 5 pictures and descriptions on social media with the hashtags #ThisIsAuburn #ThisIsSymmetry #ThisIsAuburnMath. Before this project, the only media tagged with #ThisIsAuburnMath was someone complaining about their TA’s language issues. Suddenly Facebook was flooded with images and mathematical descriptions of symmetry found on Auburn’s Campus. Assuming each student has 500 friends disjoint from their classmates’ networks, then a first estimate of the impact of this project is on the order of 8,000 people reached! This concept also connected to Auburn’s fundraising campaign "Because This Is Auburn," which nurtured their mindset of staying connected to Auburn, and to Auburn Math. My hope is that these sorts of activities will enrich the learning of my students, and that they will graduate with an increased commitment to Auburn University.
I hope that my students will know the names of their classmates early in the semester. I lay the groundwork for these connections by encouraging group work, making team assignments, and taking simple steps such as introducing students to each other when they attend my office hours.
Culture: I also do my best to stay involved in the culture of the student body. I attend (and even organize) on-campus events, such as Bryan Stevenson’s book (Just Mercy) reading, I stay involved in the culture of the Auburn Family, attending the COSAM tailgate and AU Sporting events. I believe that the more I can connect with the cultural center of Auburn, the better I will be at connecting with Auburn students, and ultimately this ability to connect with my students positively affects my teaching.
Delta: I realize that students enter college at all different ability levels. What I’m most interested in is helping all my students to make the greatest possible improvement. I’m willing to work with any student who is willing to do their part and put in the effort. I try my best to help all my students to maximize their ability, both by offering my help and by directing them to campus resources.
Excellence: I’m also very interested in helping Auburn to produce the next Tim Cook. Even more, when will the next talent like George Washington Carver come out of the state of Alabama? We have excellent opportunities to find and nurture talent at Auburn, and more generally around the state of Alabama and I am constantly looking for students who have great ability. I endeavor to provide adequate challenges to these students while making them aware of additional opportunities to excel. I’ve had the opportunity to be a part of mentoring some of Auburn’s top students.
Doyon Kim was already a very talented undergraduate when he entered my Abstract Algebra course. Seeing his talent, I offered to direct an undergraduate research project for him. He came up with his own mathematical problem, worked diligently on the project, solved the problem, and ultimately his work was published in the Journal of Integer Sequences. He ultimately applied to and was accepted at some of the best graduate schools in the country and is currently having good success in a PhD program at Rutgers University.
Leann Kopp was a freshman in my Honors Calculus III course and was one of the shiest students I’ve seen. Her grades were hard to miss, however, often performing 2 standard deviations above the rest of a very talented cohort. I reached out to her and encouraged her to take more math classes (she eventually took several graduate courses and continued to out-perform even her older peers) and to do undergraduate research. She ultimately graduated with a 4.0 having won several university and college level awards and is now leading a successful career with E*TRADE Financial.
Constant improvement: I’m also constantly seeking to improve my teaching. I work hard to make sure that I’m modifying my course materials each semester, especially my exams. I want my student’s performances on exams to improve because they learned more, not because they now have a copy of last year’s exam. In addition, I’m continually developing new in-class projects that lead students to discover mathematical concepts. To be most effective these projects need to contain as of yet undiscovered concepts as easter eggs buried within. This development requires a lot of pre-emptive work on my part, but the pay-off is that students begin to understand the process of learning, and the lessons they learn from discovery tend to stick with them forever. Through constant change seeking continual improvement I keep my courses fresh and interesting each semester.
Homework
The Math Moleskine: Working consistently with the course material in the form of exercises is one of the best ways to learn mathematics. I expect that students will spend 2-3hours outside of class for each hour in class solving problems from the textbook per week. I ask each student to purchase a Moleskine notebook (or something similar), separate from their class notebook. They use the Moleskine to record all their work for textbook assignments. After completing an assignment they take pictures of the pages with AdobeScan and turn in a pdf. I also instruct them to highlight in red any questions they had or problems they couldn’t solve. I can then grade and provide feedback on the assignments without having to keep their work from them.
Process: I encourage working together on homework; however each student must be able to solve the problems on their own and must turn in their own work. They may consult as many outside sources (electronic or print) as they like as long as they cite their sources. Since back of the book answers are often provided, and there are many online resources, solutions manuals, etc., I spend a lot of effort explaining to students that learning is about the process, not the output. When students trust the process of practicing doing mathematics, confronting challenging problems and coming up with their own solutions (instead of copying someone else’s) they are building their capacity to do mathematics. In recent years, especially during the pandemic, students have begun to trust me more on this point, and their improved outcomes have been a testament to this.
In a college math class, the expectations are quite different than other disciplines and what students may have experienced in high school. The biggest difference is that they’re expected to learn by doing. Their learning is directly proportional to the amount of work they put in outside of class. I encourage my students that the biggest thing they can do to help themselves learn is to figure out what it means to “do” homework.
Superficial learning occurs when a student sees a homework problem and immediately goes to watch someone work that problem (like on Chegg or using a bad tutor) and then copy what they saw. Going straight to the answer short-circuits the learning process, and this is not “doing” the homework. Even worse, since later problems build on previous parts of the text, a bad work process early causes even bigger issues later.
Significant learning happens by starting with a problem one doesn’t know how to do, going back to the text to refresh one’s memory on the tools provided, re-working examples (asking how they got from this step to that?), thinking critically about the problem, and finding a way to combine the prior tools and techniques to solve the problem. Checking one’s understanding of the problem and its solution can be done, perhaps, by explaining to someone else how to do the problem. Learning is further solidified by writing (in complete sentences) how to solve the problem. Mastery of the concept is evidenced when the student is able to solve problems they’ve never seen before. Expert level achievement is when one can read an application and figure out what the problem is in the first place, and then use prior skills to solve that problem. My goal is to help students to develop good habits of absorbing information, struggling to solve challenging problems, overcoming small challenges, and becoming proficient with the topics in the course.
Terms like “engaged and active learning” or “group learning” are prevalent in the education landscape. Rather than ignore these evidenced-based practices, I have worked to embrace them. Some keys for me to carry this off successfully without compromising content and achievement standards are the following:
Micro-doses: Students come to my class to hear my expert opinion on the topic. However, attention spans being what they are, I break up lectures with short vignettes on how the math concept is used in application, and I pepper my lectures with short “solve this right now!” pauses to allow students to participate in the lecture.
Macro-doses: Once every 2-3 weeks I include guided activities that students will do during class in groups, while I float around the class to help each group overcome challenges. These activities have multiple levels of questions ranging from simple “do this calculation” to discover the hidden mathematical gem, and sometimes include a social engineering problem as well - for example, in one classroom we have hexagonal desks which make dividing work evenly by 6 a challenge. Here is a sample activity for an early linear algebra class. The prompt is the following: There is a finite number of 2 by 2 matrices with 0-1entries. Find them all and compute a formula for the n-th power of each. Students learn division of labor, pattern discovery, experiment, mathematical induction, computational efficiency, and some bright students even find the Fibonacci sequence and one of the most efficient ways to compute a given term.
Alignment: Realizing that there will be different achievement levels in my classes, I align my assessment to test each kind of achievement (prerequisite, novice, intermediate, advanced), and I broadcast this to my students early in the syllabus and often throughout the class. I also make them well-aware of my expectations for each grade before they take the test so that they know in advance what success on an exam will be. For instance, if I make 160 points available on an exam, but I expect that earning 80 of them will be a B and 100 will be an A, I tell them this first. This way they can leave the exam with an accurate feeling on their achievement rather than spending a week thinking they failed and that only by the grace of the curve did they pass. This psychological shift allows my students to maintain their motivation throughout the semester. I also am careful to align the examinations and the classroom activity. Particularly with in-class activities, since students have spent an entire class period discovering a concept, and additional time outside of class writing up their work, I design exam questions that are parallel (but not identical) to the work they did. In this cycle of challenge-achievement-testing students can see their own progress and can be very satisfied with their outcomes.
Side challenges: New teaching methods present unique challenges to students we might not have expected, such as the problem of the “lonely-only,” i.e. that student who looks at the classroom and doesn’t see anyone who is like them. Lonely-only students tend to struggle to find a group to work in, and often this is a blockade to accessing the full set of resources that might be available to their peers. I try my best to watch for these social cues and do some social engineering to mix the groups, introduce students to each other and try to make connections that might not be made without my encouragement. I expect my students to participate in activities during class (electronically, in writing, and in person). To help with this I design activities that are interesting, challenging and have multiple entry and exit points to encourage maximum engagement from every single student.