This semester I was gifted with the opportunity to teach Mathematical Reasoning at the University of Tennessee, affectionately known as “The Beautility of Mathematics” (the word “beautility” being a combination of beauty and utility). The course is aimed at art, psychology, architecture, and humanities students, among others, and instructors of the course hope students not only see mathematics they did not encounter in high school, but also develop an appreciation for how the subject matter can be applied directly to their disciplines of choice.
This semester has been very important for me in developing a teaching philosophy that is flexible enough to be applicable to different courses I may teach, because it is the first semester in which I have given any deliberate consideration to this matter. Next semester I will be working with my teaching mentor to write an initial teaching statement, which will become a part of any job applications I may send out later for postdoctoral positions in academia. To some extent, my reflections may act as a starting point for that more formal document that I will write and continue to revise later on.
In the meantime, I’ve begun to identify principles that are important for the way I teach. While I think it’s always important to consider a course structure — including how to design assignments and calculate grades — the unusual circumstances presented before us with COVID-19 have illuminated an even greater importance of personally connecting with my students. Thus my initial list of ten principles for the mathematics classroom are somewhat skewed to reflect this mindset. I present them in no particular order, and as you will see, they are general enough to be applicable to different mathematics courses at the undergraduate level. Also, I try to indicate these things to my students (oftentimes explicitly) so they can at least partially understand my rationale for running my classroom in a certain way, and it invites them to talk to me if they disagree with something I am doing.
I. Modify grading/curve assignments if questions appear to be unfair or ambiguous
Teaching this class presented me with my first opportunity to write my own exams for my course, whereas in the past the exams were created by a lead instructor or course coordinator. With this power comes the responsibility I have if a question turns out to be “bad.” By that I don’t mean a question being too hard (or easy), but rather if a question has other types of issues. Sometimes a question or the directions associated with a question will be unclear or ambiguous, even to a student who understands all of the terminology used in the question. It’s hard to know in advance whether this will be the case, and I suppose the ability to realize these types of things ultimately comes from experience.
Another possibility is the inclusion of questions related to material only briefly or tangentially mentioned in class. While in some cases this is technically allowed (especially if your syllabus says something along the lines of, “you are responsible for all material covered in class and the textbook”), students often perceive this as unfair if you only spent ten minutes briefly outlining a topic, rather than spending an entire lecture period dissecting the topic in detail.
That being said, if a large portion of students individually or collectively complain about a single question, odds are there is some sort of issue. In the latter situation I like to make the problem worth a relatively small number of points, if I didn’t emphasize the related content as much in class. In other cases, I might add a small “bump” or “curve” to the exam scores. This sends a signal to the students saying, “perhaps this question wasn’t the best choice on my part.”
II. Be transparent about what’s most important and emphasize those ideas on exams
On the contrary to my previous point, inevitably you will want to indicate some topics as being more important than others. While you can explicitly say, “this is important” and underline and boldface it and whatnot, I do something else too. I like to repeatedly indicate when I am using the same idea over and over again. That idea might be a formula, or a more conceptual strategy for digesting what certain problems are asking. One of my favorite examples from the course I taught is something I call “divide and conquer:” if you wish to calculate the area or volume of a certain irregular shape, break it into pieces where each piece is a more familiar shape. Then add up all the areas or volumes separately.
Cue phrases such as “we’ve done this before” or “we will see this again in a later chapter” can be helpful to remind students of patterns interspersed throughout the course content.
III. Be open to alternative solutions to problems
I’ll start this one with an anecdote. Back in middle school, I competed in a math competition called MATHCOUNTS. My coach, the professor Po-Shen Loh at Carnegie Mellon, shared a strategy for double-checking our work while working on the problems. He said, if you finish all the problems in the allotted time, go back and try to solve as many of them as you can AGAIN, but using a different method. Ideally, you’d get the same answer both times.
This is a piece of advice I regularly give my students, except I don’t connect it to the scenario where they have spare time. Rather, I frame it as an acceptance of alternative solutions to problems, accompanied by phrases such as “that’s not what I had in mind, but it may still work.” Not every student will solve problems the way I intend, and even if they do, they might explain the idea in a different way. Being able to appreciate the thought process of students requires being able to listen to what they are saying, read their work carefully, and ask further clarifying questions.
Of course, not every seemingly creative or “different” method your students propose will actually work. If, after listening to your student’s idea, you believe there is an issue, try to point out where the error is and guide them to a different approach, perhaps the one you expected they’d use in the first place. In my classroom I usually carry this out as a dialogue, either with an individual student, a small group, or the whole class. Alternatively, this can be done while grading assignments or exams, but the interaction isn’t in real-time.
IV. Encourage students to at least consider studying for exams together and to form chat groups
This piece of my philosophy is in part inspired by an increased amount of student isolation caused by the COVID-19 pandemic, and luckily, the growing presence of virtual communication tools such as Zoom has made this easier. I believe that many students benefit greatly from preparing for exams in groups, because students can identify and repair holes in each others’ understanding as they work. At the same time, some additional social interaction, even if it’s behind a screen, can mitigate the feeling of isolation many people have developed during the pandemic, if only by a little bit.
Mathematical Reasoning, and many other classes, have groups built into the course through the form of regular in-class activities. These can be great starting points for group study, but remind students they can also work with other people in the class, or even students in other sections of the class, if more than one section of the course is offered. I’ve had students tell me Groupme works for this purpose, but other platforms exist to serve the same purpose.
V. Remind and encourage students to look at outside resources for inspiration
There should be no shortage of resources for practice problems, even if you, the instructor, don’t make all of them yourself. Google is a powerful tool, and for the most part students know how to use it. If they just google practice problems for a particular topic, there’s a good chance they’ll be able to find additional practice for whatever they are struggling with. I like to suggest Expii.com as a source of problems, and also expository articles and visual aids for understanding topics.
The one catch is that you, the instructor, ultimately decide what is important to include on exams, and your syllabus might not match up perfectly with what students find online. It’s best to warn them of this beforehand.
VI. Promote a growth mindset throughout the semester. If there are multiple exams ask students how they can progressively do better on each exam
Everyone has a bad day, and poor exam results happen, whatever the reason may be. If a student is feeling under the weather while taking an exam, that isn’t really their fault per se, so they should try to identify what circumstances they had control over. I like to remind students that one exam won’t “make or break them,” and they should always be looking to fine-tune their study habits and approaches for solving problems, both for the sake of their course grades and their general understanding of the material.
What if a student aces your first exam? Good for them! When this happens I like to encourage that student to help their fellow students, because trying to explain material to others forces them to assess their understanding even more. Besides that, maybe they are struggling in a different class and would benefit from shifting some energy to that class. Or, they just need to catch up on sleep. It’s OK to tell students to re-prioritize how they spend time outside of your classroom, as long as you set the precedent that classroom time should be devoted towards your class.
As a specific example, here are some comments my students provided as reflections of their midterm exam performances. I accompanied these with general responses for how they may want to resolve their issues for subsequent exams, fashioning my comments into another blogpost for a wider audience to see.
VII. Specifically for online classes: give benefit of the doubt for technology issues
Sometimes a student has trouble attending a virtual class or exam because of some technology issue that you can’t easily verify. The “my internet broke” excuse to get out of class has been beaten to death, but that doesn’t mean it won’t actually happen to students. Last spring, for instance, I had a bunch of students miss part or all of my online class because of a large storm system (which included tornadoes) wiped out electricity. If there’s no easy way to verify what happened, it’s best to just excuse assignments or attendance, provided it doesn’t happen repeatedly for the same student. Especially for one-off situations, this forgiveness can relieve stress for your students and make them feel they can trust you more.
VIII. Get to know the backgrounds of your students and provide examples of the material related to their majors; encourage them to think of their own examples.
I think this one is self-explanatory. If you can anticipate where your students might see the material you are teaching in action, there are two advantages: they are more likely to understand the material, and they are more likely to care.
In the Mathematical Reasoning course, students do group presentations where they connect course material to outside topics of their choice. This exercise forces the students to figure out why they should really care about what they’ve learned.
IX. Give your students a chance to communicate about math in writing or verbally
I think this point is especially important for math classes aimed at non-STEM majors that are very writing intensive: history, English, philosophy, you name it. They will not only understand what is “going on” better, but they can prove to you that is the case, whether this is done verbally or in writing. When students are working on group activities, I like to ask them questions to get them to explain their thought process to me, regardless of whether it’s right or wrong. Some students immediately assume they made a mistake if I start grilling them, but that’s not always the case. If they seem worried about this, I tell them I just want to make sure they can explain what they are thinking, to me and to other students.
Also, communication and writing skills are important for nearly every career path in some way.
X. Don’t be afraid to make mistakes and be willing to admit it
Another misconception students have is that their instructors are immortal gods incapable of error. This couldn’t be farther from the truth, especially because all professors were once students themselves. Even those on the higher academic pedestals can make mistakes, ranging from minor slip-ups to egregious issues.
IT HAPPENS. Live with it and move on.
It honestly takes a bit of humility on the part of the instructor to admit something has gone amiss, but your students will appreciate if you do this. This serves as a reminder to them that you, too, are human.