Teaching Philosophy

As a professional tutor, a teaching assistant, and an instructor I have refined my Socratic dialogue approach. This technique involves asking leading or guiding questions to guide students toward developing ideas themselves, rather than providing information directly. This facilitates learning and retention, but it also helps empower students; resulting in a feeling of pride and capability that they often lack in a math class. In my recitation classes students would ask questions, often with incorrect underlying premises. Instead of answering their question immediately, I would ask them questions in return or give them examples for them to consider that would not only answer the question they gave, but also led them to explicitly recognize and address the underlying misunderstanding. Some students have found this irritating, saying things like ``why don't you just answer my question, why do you ask me a question?" However, most students found this methodology extremely helpful. I regularly get comments on teacher and course reviews stating that my teaching was engaging and helpful specifically because I would talk with them, rather than talking to them, when answering their questions.

I utilized the Socratic dialogue techniques extensively and to remarkable success when teaching in a flipped classroom. I have taught both differential and integral calculus using a flipped classroom style where students watch instructional videos outside of class and then explore that content more organically in the classroom. I would often pose questions for students to work on in small groups that were designed to be just beyond the applications from the video lectures. Then I would wander between groups and ask Socratic-style questions to help them get past the various blocks that were stumping the group. After teaching my own flipped and standards-based grading calculus two class over the summer I still have students from that course coming by my office to thank me for helping them get a deeper and better understanding of calculus which they have needed in their other classes.

Starting in my second year of graduate school I began to learn programming, specifically LaTeX and the learning management system (LMS) Canvas, in an effort to begin incorporating more technology in my education materials. This included (re)writing templates for my math department and randomized problem generation which, in turn, led to my involvement in course development for new online courses and ultimately the Ximera and Xronos projects. These projects are innovative tools in online education resource building, primarily in making interactive textbooks and homework systems. I am now one of the three primary developers for Ximera and on the board of directors for Ximera.org, as well as the primary developer for the University of Florida spinoff project Xronos.

This unique circumstance has led to me using technology, not just in my classroom, but developing, implementing, and maintaining technological education tools in a range of applications across many different classrooms under many different lecturers. This has given me insight, not only into various roles technology can play, but also in how people view the role of technology in the classroom.

In most cases lecturers wish to use technology to replace older methods that suffer from difficulties of scale. Online homework systems are a great example of this. Although the quality of online homework tends to be presented better or more pleasantly, the homework systems that are still being used are essentially the same homework that was assigned from a book, just graded automatically and immediately. I view this as something of a tragedy however, as technology has vastly more potential than simply replacing what already exists in a slightly more efficient way. Instead I look for truly novel ways to use technology, not purely to assess, but also to educate, my students. For example, I have embedded Desmos interactive environments in various textbook topics and problems, providing interactive graphs to demonstrate the role of coefficients in transformations and translations of functions. This kind of real-time interactive visual feedback adds an entirely different dimension to a students education. It makes them both an active participant, as well as putting them in charge of the amount of time and effort they want to put into learning and `toying' with that concept.

The last year of my graduate program I was fortunate enough to have an opportunity to do so. Similar to how I start with adding new technological features to my classroom, instead of simply trying to take what had been used for a very long period of time, I started by thinking about how I teach myself. The biggest barrier we tend to struggle against in teaching mathematics isn't necessarily the initial teaching but rather student retention and actual understanding (instead of rote memorization). Having faced something similar in my own education I realized that the biggest influence in my retention was the depth of the contextual framework within which that information was placed. This led me to look into contextual based learning (CBT) as a course design approach. In my final year of graduate school and as a visiting assistant professor I had the opportunity to coordinate the precalculus algebra course. After looking at the previous materials I decided to develop my own course from scratch, which gave me the opportunity to put my philosophy of CBL into practice. I wrote an entire interactive textbook, including unlimited practice problems and various forms of feedback, over my time as a visiting assistant professor.

Note: You can see the interactive textbook and homework tool for my specific class HERE

My students have been surprisingly receptive to the idea of contextualizing what they are learning. Each semester I have several students come up and tell me that, for once, they really enjoyed a math class. Students also appreciate the nature of the context, which was designed to emulate real-world interactions with questions like ``your boss asks you to determine the cost of opening another store location. What data do you need to model this problem?" Students started connecting the math that we were doing, to the situations that they may honestly encounter in their life. The success, in terms of building relevant context, of CBT in my class is exemplified by one conversation I overheard between my students which started with the quote ``For the first time in my life, I know why I'm actually doing what I'm doing in a math class."

Perhaps even more striking however has been the ability for a much larger number of students to retain previously taught content as we have gone through the semester. Many students will cram for exams to try and do well, but almost immediately forget most of the content that they crammed for, which is a typical consequence of how the human brain retains information. By including some of the research in the psychology of learning into a `how to study' section at the beginning of my course, and using CBT, my students have repeatedly commented throughout the semester that they have found they retain a surprising amount of what we have covered. They typically mention this when discussing their study plans for upcoming exams and how prepared they feel.

These techniques, although largely successful from my perspective, don't come without drawbacks however. The current culture of mathematics education focuses on extreme practice quantity, which isn't without merit. My techniques however take time, which must come from somewhere. Thus some of the lecture and typical course practice is sacrificed in favor of focusing on the concepts and building frameworks of learning to help students with retention and understanding. This generates two issues; a difficulty with student expectation, and a difficulty with student mechanical capability.

The difficulty with student expectation is a consequence of the fact that they have spent all of grade school `learning' that mathematics is all about memorizing formulas and doing endless computation. Despite saying repeatedly at the start of the semester that my class is different, I still get considerable push-back from students for ``making them think". A great example is a student I had in office hours say, as a direct quote, ``I do not appreciate the critical thinking skills you are trying to teach me right now." Despite trying to explain the importance of critical thinking and mathematical reasoning, these are skills that are often under-appreciated until later in life and so students will complain that they would ``rather just memorize the formulas" instead of learning why they work. This is exacerbated by the fact that I am teaching an entry level precalculus course composed almost entirely of first or second semester college freshmen.

The other difficulty is related but separate from the previous, which is the difficulty with student mechanical ability. Ultimately the goal of precalculus is to ensure that students are mechanically capable in preparation for the rigors of calculus. My approach, although seemingly better for things like retention, understanding, and real-world application of mathematical reasoning, inherently trades the training of those things for some of the demonstration of the mechanics themselves. I offset this in my course design by shifting the role of demonstrating more complex mechanics to TAs and supplemental materials such as interactive course notes and videos, after I have covered the concepts and deeper contexts of those mechanics. Despite this, students often underestimate the amount of practice that they need and they will easily conflate making mechanical mistakes and being correct; saying things like ``well, I got it right, I was just off by a negative sign." This demonstrates an underlying issue with the students' concept of what it means to learn the mechanics which I try to address explicitly. Unfortunately students often have these ideas firmly established from their previous experience in math courses.

I offset this mechanical skill loss by using the technology I've been working on to allow more mechanical practice. For instance, I have written problems that are generated using the programming language Sage, which allows for unlimited practice by students so that they can test themselves on the mechanics as often as they want, combine with dynamically created feedback with interactive tools that allow students to see the consequences of their mechanical mistakes. Since I am writing these problems as a mathematician, the randomness that is used is far more complex (and yet far more forgiving to students) than the typical randomization of coefficients used in most online homework problems.

If you would like to see some of the feedback I have gotten from students I have attached the semester-end student feedback report, generated through a questionnaire solicited by University of Florida of all students in every class at the end of every semester. Due to a change in how the reports were generated and which questions were asked, some of these forms have different (and more extensive) information than others.

SummerA MHF3202 - Sets and Logic (Introduction to Proof technique. 6 week course and first time I taught this specific course.)

Fall 2020 MAC1140 - Precalculus Algebra (This is an introductory service course, taught largely using Xronos)

Summer 2020 MAA4103 - Advanced Calculus 2 for Engineers and Physical Scientists (First time I taught this specific course.)

Summer 2020 MAA5105 - Advanced Calculus 2 for Engineers and Physical Scientists (Graduate Student Component, also first time I taught this specific course.)

Note/Disclaimer: Each of the student feedback blocks below was provided to me completely on the initiative of the student and without any kind of request, solicitation, motivation from me. Each student graciously gave permission to have their comments posted publicly, although any identifying information has been redacted. Anything I have redacted or edited for clarity will be bolded and in brackets and is done purely to protect the student's identity and/or clarify in the event of terminology specific to the course/institution that would not be obvious to outsiders.

• [Email from a student who was also an adjunct professor - regarding MAC1140 - Precalc Algebra]
  As a non-traditional student, I felt very prepared by Jason Nowell's instruction zoom videos as well as the Xronos application review. Each section was thoroughly explained and the questions throughout each section are thought provoking and helps the student learn and retain the information. As long as you follow along with the syllabus to join the zoom classes and complete the Xronos review and quizzes and practice tests, you should be able to successfully pass this class.
  

• Hello Dr. Jason,
  
  I would like to send this special email to truly express my gratitude for your helpful and insightful class. I took your MAC1140 [precalculus algebra] last semester and ended up with 99.97. I took that grade with 20% effort by simply doing all the practice exams and watching your short videos to excel in your class. However, I was doubting that whether this course can really help me to succeed in these challenging Calculus courses and I finally found that your videos build a strong math foundation for me.
  
  I just got 92 for my first MAC2233 [Business Calculus] exam with a 65 class average grade. I noticed that most contents from MAC2233 are covered by your videos and I even did not put any effort into preparing exam. Before the exam, my instructor told me that my calculator is a graphing calculator(and we are only allowed to use Texas instruments), and I should not be able to use that one. I am currently in China and it is impossible to get an authorized calculator. Finally, I did all questions by hand. I really appreciate your training [In doing math without a calculator] from last semester because you noted that math should always be done by hand, albeit Calculator is powerful, in your syllabus and asked us to take exams without the help from Calculator. Although I was trained in such ways in my country, your course incredibly wakes my computational skills.
  
  I hope you can keep such an admirable job to help more students like me to succeed in future Calculus courses. I wish you all the best and please be safe there.
  
  P.S I am really enjoying reading the book you recommended, A Mathematician's Lament by Paul Lockhart. It refreshes my understanding of Math :)