General Methodologies
Outline of a typical class
Preparation: I spend time before each class to prepare lecture notes. I carefully select examples to be worked out together and create a worksheet and/or a Desmos activity for student practice. Occasionally, if many new definitions are to be introduced during the lecture, I prepare a handout with all of the relevant definitions to be distributed. I also open WeBWorK problem sets which reinforce the material in the lecture (or choose relevant textbook problems) and upload all relevant materials to the class OpenLab site.
Start of class: I begin my classes promptly, and encourage my students to be prompt themselves both verbally and occasionally via “spot checks” (details to follow). If a “spot check” is done, I begin by reviewing that homework problem. I ask the class which problems they would like for us to review together. For each student who requests a problem to be reviewed, I ask that they tell us how they began the problem and what the obstacle was they encountered. Next, I summarize what was covered in the last lecture and state the goals of the present lecture on the board.
New material: I begin by working off of an accessible example as motivation. Then I present more general definitions/theorems. We work through more examples together and ask the class “Are you ready to try some for yourself?”
On-the-fly assessment: At this point in the class I distribute a worksheet, ask students to log into their WeBWork accounts or into a prepared Desmos activity and ask the class to work individually or in groups. I circulate the room and give hints to students who need them to help them move forward. Occasionally I will pair up a student needing help with one who was able to master the material more quickly. I have students come to the board and write out the solutions to the problems. Next, we review the solutions as a class. I guide the discussion and as we move through the examples, I ask the student whose particular solution we are reviewing to verbally “walk us through, step-by-step.”
End of class: To end, I summarize the definitions/methods we learned in the present lecture and tie them to the coming lecture. I also assign WeBWorK or textbook homework problems.
Open Pedagogy and Technology
A major component of my teaching philosophy is accessible education. Below is a summary of technologies and pedagogies I use to promote an inclusive and accessible culture of learning within the CityTech community.
WeBWorK
WeBWorK is an open-source, online STEM assessment platform. Across CUNY, more than 8,500 students are using WeBWorK in the fall 2019 semester across more than 250 course sections on six different campuses. At City Tech specifically, the average year-over-year growth rate for WeBWorK usage in fall semester sections has been 50%. To date more than 20,000 City Tech students have taken WeBWorK courses at City Tech. For more details on how I use WeBWorK, see here.
OpenLab
City Tech’s OpenLab is an open-source digital platform where students, faculty, and staff can meet to learn, work, and share their ideas. It supports teaching and learning and enables connection, collaboration and community. I use it to support my teaching in the following ways:
- creating course websites to post course materials and to communicate with students (see my sites for MAT 1375, MAT 1475 and MAT 2440)
- posting a reading list to stimulate class discussion and motive writing assignments (see link) a process which I model in our Opening Gateways faculty pedagogy seminar (see link),
- encouraging all of my students to create their own OpenLab portfolios where they can showcase their important projects and classwork. This is important for prospective employers. l model this practice by keeping my own teaching portfolio on OpenLab.
Free online textbooks
Reducing financial barriers for our students plays an important role in increasing their educational access. Toward this end, along with City Tech math department Professors ElHitti, Carley, Tradler and Zhou, I authored an open source online textbook for City Tech’s developmental math students.
I’m happy to utilize freely available open textbooks wherever possible in other courses such as: MAT 0650, MAT 1375 and MAT 1475.
Desmos
Desmos is a free online graphing tool and activity builder, which millions of students and faculty around the world use for free. It can be used to generate beautiful graphs quickly, without a steep learning curve.
I use it in my classes in the following ways:
- real-time function graphing during lecture with comparisons to traditional graphing calculators,
- saved interactive graphs which I show in class to highlight concepts, and
- activities which are completed by students either in class (on iPads or tablets) or outside of class as homework
- I also used the Desmos activity built by Prof. Kate Poirier (City Tech) for purposes of Gen. Ed. assessment in my MAT 1375 in spring 2019.
Writing Intensive and Active Learning Pedagogies
As former co-coordinator of City Tech’s Writing Across the Curriculum (WAC) program and current co-director of the Opening Gateways faculty seminar, incorporating WAC best practices and active learning pedagogies into my teaching in order to increase student learning is important to me. I have done so by
- authoring innovative projects for MAT 2440 students which weave together collaborative and individual assessement, writing, coding in Python, proof writing and logical puzzles (for more details and assignment examples see here);
- curating a class reading list of articles and books on topics related to course content (for a reading list example see here);
- leading focused in-class discussions on current events related to course content;
- implementing active learning activities utilizing high-tech and low-tech technologies and games;
- sharing my expertise on crafting effective writing assignments by accepting invitations to present on the topic to faculty in the mathematics department at the US Military Academy at West Point in September, 2014 and again in June 2018.
Interdisciplinarity
An important component of my teaching philosophy includes the idea of “range.” I believe it is imperitive to be educated broadly (in addition to having areas of specialization) in order to increase educational and career success. To support this, I
- include STEM applications and activities in my classes,
- discuss the history of mathematics wherever appropriate (see below), and
- teach lectures on quantum computation in my MAT 2440 classes and upon invitation for City Tech’s Modern Physics class Fall 2019, Spring 2019, Fall 2018, Spring 2018. See the invitations and the presentation.
Worksheets and Active-Learning Activities
A successful strategy I employ to engage students during a lesson is to distribute worksheets, have students work on activities or log into WeBWorK so they may practice concepts relevant to the current material. It is my finding that having students practice material in class is helpful to them as part of their learning process and helpful to me, as an instructor, so that I may assess the success (or lack thereof) of my presentation. I am then able to adjust my strategy and lecture style accordingly.
Since I feel so strongly about having students practice in class that, through my tenure at City Tech, I have created workbooks full of handouts and worksheets for most courses I teach. Many courses have a worksheet and handout for every single session and topic. I also have a cache of active learning and Desmos activities I utilize. For materials, see the links below:
History of Mathematics and Current Events
As I have indicated in my statement of teaching philosophy I feel it is important to incorporate some history of mathematics in my lessons. I feel it helps with student engagement and promotes well roundedness. In MAT 1175 and MAT 1275 when introducing some basic axioms of geometry I ask the students to imagine a world where one (or more) of the axioms do not hold. For instance, imagine we live on the surface of a sphere. Here parallel lines do not meet and the sum of the interior angles of a triangle will be less than 180 degrees. We then discuss how Riemann presented this non-Euclidean geometry in the mid 1800’s and how Einstein incorporated this mathematical apparatus into his theory of relativity thus changing the course of history.
When discussing the solution of quadratic equations (in any level course) I ask the students to imagine how one might (algebraically) solve a higher order polynomial equation such as a cubic or a quartic or one of nth degree equation. I then tell them the story of Évariste Galois. As a teenager he proposed that a polynomial equation of order greater than or equal to five cannot be solved by radicals. In order to solve such equations, he developed an entirely new branch of mathematics and hence became one of the founding fathers of modern algebra. (They enjoy the human aspect of the story when they hear that Galois was rejected from the Ecole Polytechnic and died in a dual over a love interest at the age of 21.)
In MAT 2440, we have intense discussions over the class reading list. We discuss issues ranging from the under-representation of women and people of color in tech, privacy in the digital age, the history of computer science and engineering, to hiring practices and policies of big tech companies. We debate ethical issues surrounding all of the above. I encourage all of my students to utilize their free subscriptions to the NYT and Wall Street Journal (through CUNY) and we discuss news articles relevant to course materials at the beginning of class.
These are just a few of the examples in which I try to impart to my students that the study of mathematics is interesting and relevant. I try to get them to see its amazing impact on our lives through its role in our quest for understanding of the universe, our development of technology and our history as a human race. I try to relate that many mathematicians were (are) exceptional human beings with interesting life stories.
Check-Ins
Another method which I employ is the “check-in.” Students are encouraged to “check-in” with me at some point between the first and second exam to review their performance. During the “check-in” I am able to give students specific advice on how to tackle any issues they are encountering with the course as well as to offer encouragement. This is one way of enabling students to take personal responsibility for their educational careers. Students who come for a “check-in” are rewarded with the option to drop their lowest exam score at the end of the semester. I have found that students who come for a “check-in” appointment are much more likely to attend office hours and seek help since the “check-in” helps “break the ice.” I have found the “check-in” appointment to be especially helpful when teaching pre-requisite level courses.
