The purpose of this page is to illustrate some of the various teaching artifacts I have acquired throughout the PTP experience. In the fall semester, I observed Dr. Chertock’s section of Applied Differential Equations II (MA 401) and developed a greater mastery of the material through grading homework, preparing online lecture notes, creating tests, and teaching a few classes and review sessions. Reflecting upon how the fall semester went, I realized that the students would benefit greatly from more in-depth visualization of the majority of topics covered. Moreover, since the graphs in this course are very difficult to plot by hand, I decided to use my spring teaching semester as an opportunity to experiment with incorporating technology into the classroom in various ways. As such, several of my artifacts will be related to how this implementation of technology was realized. Below is a syllabus from my course, and the remaining artifacts follow:
PTP Seminar Implementation and Lesson Plan
One example of how I used technology in the classroom was in a lesson on the Maximum Principle. My goals for the lesson were for the students to understand the meaning of the Maximum Principle and how it specifically applied to some of the famous equations we had previously been learning how to solve. Taking a cue from Dr. Bryce Lane in his PTP seminar on Motivating Students, I decided to first “romance” the students and create a positive learning environment by providing some fun historical background on Laplace and Poisson, two mathematicians whose equations we were currently studying. I then proceeded with the lesson, providing the statements of theorems, proofs, and examples (or as Dr. Lane would refer to it, the “drudgery”). Then, as an “application” of that new-found knowledge, I asked my students to use the Maximum Principle to tell me some information about the solution to a specific case of Laplace’s equation. Once they were convinced, I loaded up the mathematical software package Maple on the classroom computer, turned on the overhead display, and created a 3-dimensional plot of the solution. We were then able to visually confirm what the students had already conjectured on their own. Of course, I then proceeded to romance the students all over again by ooh-ing and aah-ing over how neat of a program Maple was, leading to many groans and head shakes. Below are some notes from the lesson plan as well as the plot we studied.
Overall, I was quite satisfied with the level of student engagement during this lesson and felt that the visualization from the plot added another layer of understanding that otherwise would have been missed. I also feel that using Dr. Lane’s “tripod” approach to teaching is an effective tool that creates a lot of balance in my lesson plans and improves the motivation of my students. In fact, whenever I now create lesson plans, I ask myself each time if I’m doing enough romancing and providing enough applications to balance and split up the drier lecture portions of the class.
Writing Assignment/Course Project
Based on the success I found with using technology in the classroom, I decided to push the idea further and required the students to create some of their own plots and animations using Maple as a project to supplement their homework assignments. Many of my students had a predisposition against Maple coming into the course based on experience with the program in previous calculus classes, but after creating some very nice 3-dimensional pictures on their own with minimal syntax problems, they warmed up to the idea. Below is a copy of one of these Maple assignments along with a sample of one student’s plots:
I feel that using these assignments to give the class more hands-on experience with using technology worked out very well for both me and the students. Once the students bought in to how “cool” the plots looked, they were highly motivated to get their syntax right and obtain the correct plots to watch their solutions vibrate and come to life. These assignments were one of the highlights of the course as far as teaching tools went, and for me they are a must-have the next time I teach this class. If anything, I would improve these assignments by using them more frequently, as my class got to the point where they were disappointed if there weren’t 3-d plots to create for a given homework. Moreover, based on the positive student response to these assignments, I have begun looking for similar ways to get my other math classes involved in using Maple to facilitate their learning.
Upon planning out the course in the fall, Dr. Chertock and I thought it would be nice to provide lecture notes to the students on a website, particularly for material that was not adequately covered in the textbook. I have included a sample of these notes below:
While my students seemed appreciative of me providing lecture notes on material not covered in the textbook, they found it somewhat awkward that lecture notes were not provided for the entire semester. One student mentioned that having lecture notes posted online for every class would make it a lot easier to get caught up on material missed due to an absence. While creating the lecture notes outside of class is quite time-consuming, I completely agree that it would be a useful tool to provide, and I hope to expand the amount of notes I am able to post in future semesters.
I recently wrote this test key to post on my class website. All of the steps have been written in full detail so that the students may use it as a review tool for future studying. Also, near the top of each question I have indicated the point distribution for achieving various steps in the solution. Note that the majority of the credit is awarded for appropriately setting up each problem, with minimal points allocated for the calculation of the correct final answer.
Throughout my teaching career, I have found that students have been very receptive to my grading policies. One of the earliest observations I made as a teacher was that students were very turned off by professors who graded their tests and quizzes with an all-or-nothing mindset. I always tell my students that while getting the correct final answer is nice, I am more concerned with their overall thought process. A minor algebraic mistake should not imply that an entire problem is counted incorrect. This is a fundamental part of my approach to teaching, and I feel that student motivation improves when they are not overly anxious about making a small mistake in their work.
Faculty Mentor Feedback
Throughout both the observation and teaching semesters of the PTP program, Dr. Chertock observed my teaching and provided her comments:
Dr. Chertock’s comments and evaluations over the past two semesters have for the most part been very positive, and it has reaffirmed my confidence level in my teaching ability for an upper-level math major course. While she did not provide any areas to work on in her above observation form, through many conversations with her over the past year she has given me several tips and pointers on small corrections I could make during my teaching. For example, during one observation when I was covering the heat equation, she caught me describing the movement of heat along an insulated rod as “dispersing” rather than the correct term, “diffusing.” Up to that point, I had been using the words interchangeably. Dr. Chertock explained the context in which each word should be used, and I now know the appropriate setting for each. Relying on her expertise to provide these types of suggestions has been a welcome experience and has improved my attention to detail when lesson planning.
Peer Mentor Feedback
Also, another PTP fellow, Helen Melito, observed my class on a day where I was incorporating technology into the lesson. I specifically wanted her to gauge students’ reactions to the use of technology and provide feedback on how effective of a learning tool it was. Her observation report is included below:
Helen’s comments provided some great insight into my level of teaching effectiveness for the class. I learned that my use of Maple in supplementing the lecture was well-received by my students, as well as my ability to ask a lot of questions to help keep them engaged. However, there were also some surprising observations made by Helen that I would not have noticed without this experience. First of all, she noted that there were a few periods throughout the class where I fell into a habit of talking to the board rather than facing the class. While some of my students requested that I say what I write (see my student comments below), this can lead into a bad habit of always talking to the board if you’re not self-aware of it. I appreciated Helen pointing this out and immediately corrected it. Helen also noted that a couple of my students were off-task throughout portions of the lecture, and with the classroom map she had created, she was able to tell me exactly which students they were. I made a note of these students and was able to then keep a closer eye on them for the remainder of the semester. Overall, I very much enjoyed this peer mentoring experience as a way to provide a different window into my classroom and improve my teaching.
In addition to feedback from my faculty mentor and other PTP peers, I also conducted a mid-semester evaluation to gather student comments on how they felt the class was going so far. I asked them to both point out areas they enjoyed and areas that could be improved. Below are some of their comments:
“I like that you post test solutions and homework solutions on the course website. I think that it would benefit most students if you gave a list of additional problems for us to practice before each test.”
“This is probably the most challenging math course I’ve taken, but I enjoy it.”
“Use of Maple in-class and in homeworks is awesome and very helpful for making the classical equations feel real. I really like that you spend so much time deriving the formulas we use; it really enhances our understanding of the methods we employ.”
“It’s helpful to me when you say what you are writing on the board as you are writing it so I don’t have to keep looking up and down to copy it.”
“Shorter and more frequent homeworks would be more enjoyable.”
“I appreciate your teaching style. I don’t always remember things from past classes and you thoroughly explain problems that clarify the material for me.”
I was happy to get an overall positive response from my students in the mid-semester evaluation (especially on the use of technology in the course), but they also provided some great feedback on easy ways to improve the class. One common request was to make the homework assignments shorter and more frequent. In the early parts of the semester, I spaced out the homework assignments a bit too much, and the amount of content in each one overwhelmed some of the students. This was a quick fix and one that I was happy to make to adjust to the specific needs of the class. Another interesting comment above was the one about saying what I’m writing to facilitate note-taking. While this was viewed as a positive among students, it sometimes leads into the habit of talking to the board for extended periods of time, as noted by my peer mentor above. Therefore, while I continued to say what I wrote, I became much more conscious of breaking away from the board, facing the class, and providing an alternate explanation of the material.
To contrast these comments with student responses from previous semesters, please refer to my feedback page.
Overall, I am very satisfied with my experience teaching this course. Upon teaching the class again, I would certainly continue to require computer plots and animations in the homework assignments and use these tools throughout course lectures. I felt that the use of technology provided nice breaks in the lecture and led to a great deal of student motivation. I would also continue providing online lecture notes and providing a course website with homework/test solutions, as these features are universally viewed in a positive light by students. I still would like to find other hands-on ways to get the students involved during lectures, perhaps bringing in a vibrating drum-head or some other physical application of the equations we study. I would also like to experiment with group work during the class as another way to stimulate student activity and break out of the traditional lecture format. I will look forward to the next opportunity I have to teach this course, but I am also excited to borrow many of the ideas that were developed over this past year and use them in all of the classes I teach!