Courses Taught
Courses Taught
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Introduction to Discrete Mathematics
Introduction to Discrete Mathematics
I produced a series of videos for my students in this course. Selected videos are available on YouTube for the general public. A sample content video is embedded here which covers the union and intersection set operations.
Most students in this course are majoring in computer science, computer engineering, or electrical engineering. There are usually a few math majors in the bunch. Regardless, each student is in this course to develop their logical rigor and argumentation skills, ability to communicate abstract and technical ideas, problem-solving skills, and their familiarity with discrete mathematical structures such as sets, relations, functions, and graphs.
One of my favorite topics in the course is on cardinalities of sets. In this case, we use functions to define what is meant by two sets having the same cardinality. We see how this definition is motivated by finite sets, and so we are introduced to a crucial mathematical process. Largely, my students will walk away from this class having a new appreciation for mathematics that comes from having their "mind-blown" with some fascinating and non-trivial mathematics.
Discrete Activity Combinatinatorial Identities.pdf"Properties of Combinations" is given once an initial lecture containing definitions and some examples of combinations has been given. The goals are to strengthen the students' understanding of the definition of combination, discover some common and useful identities regarding combinations, and gain experience with "combinatorial proofs."
Discrete Activity Intro Graphs.pdf"Introduction to Graphs" is given to introduce students to basic language and definitions used in graph theory. It culminates in the students proving the "first theorem of graph theory."
Analytic Geometry and Calculus I
Analytic Geometry and Calculus I
As I taught MA165 only at the start of the pandemic, I do not have individual content videos for the topics in the course. These are in progress. Instead, I recorded my in-class lectures for the semester. An example recording is found above.
The students in this course come from a variety of majors, but mostly I find my students are in Engineering, Mathematics, or Computer Science. While providing these students with the main methods, tools, and applications of integral and differential calculus, I find it's also important to develop "soft skills" during this course. For example, students coming in to the class typically have a difficult time producing a sequence of equalities which demonstrate a desired result or calculation. I demonstrate and share with these students how equations should be readable as a sentence. We practice "reading" these sequences of equalities and make sure what is being written is actually what is meant to be said.
Other soft-skills that I work on developing for my students: double-checking calculations as the calculations are made, working on mathematics & problem-solving in a group, and self-regulating one's own learning.
Another major learning goal is to develop the students' ability to recognize the main components of a theorem. In particular, I want my students to get better at recognizing if the hypotheses of a theorem are met or not as well as what it means in each case - does the conclusion follow? Do we know the conclusion does not follow if the hypothesis is false? Developing this skill also hones the students' attention to detail and familiarity with logical implications.
As in Discrete Mathematics, I have activity assignments for this course which are meant to be worked on in a group. Some of our class time is devoted to working on these activities which allows me a chance to address individual and specific questions from students.
MA165 Week 1.pdf"A Scenic Bike Ride: Estimating Velocity" is my first week activity. It introduces students to the concepts average rate of change and instantaneous rate of change in the context of riding a bike.
Week12NewtonsMethod.pdf"Newton's Method" is given during the chapter on applications of the derivative. I find Newton's method particularly important because it provides a gentle introduction to the method of gradient descent. While we don't do much with gradient descent in this course, many students are specializing in data science and building a solid conceptual foundation for finding mins/maxes/zeros of functions using the derivative is crucial.
Multivariate Calculus
Multivariate Calculus
I have over 60 content videos for this course. An example video is given above. You can also check out the playlist for the video content I have made for this course: YouTube Multivariate Calculus Playlist
This video, "Divergence Product Rule Example" is a popular video on my channel. In it, we compute the divergence of a vector field that is the product of a scalar field and another vector field.
Week 3 - London Tsai.pdf"Math and Art" is given at the end of the third week of the course. Students learn about quadric surfaces this week, which is a difficult topic to assess deeply. Instead of simply matching surfaces with their name, I have my students identify quadric surfaces in a couple pieces of artwork by London Tsai. Beyond this, the students develop their understanding of quadric surfaces by comparing the surfaces to each other, identifying and explaining why the artist grouped the quadric surfaces as he did in the composition, and discovering a connection between cones, hyperboloids of one sheet, and hyperboloids of two sheet in the piece titled "Algebraic Degeneration". I find this activity is helpful for students to further understand this crucial connection between geometry and analysis/calculus.
This is easily one of my favorite courses to teach as students have already built up a nice mathematical vocabulary and have developed their communication skills to the point where more focus can be spent on the concepts, rather than on developing these "soft-skills". Many students are engineering majors, but I always find a couple math majors as well. Since many students are interested in applications, several of the activities I developed involve applying a concept learned in class to a real-world problem. You can see such an example below of applying the concept of curvature to a civil engineering problem.
One of the major skills I want my students to develop in this course is the ability to ask questions and explore in order to understand a concept on a deeper level. One way in which I do this is by asking questions that have the students identify, correct, and fix a mistake in a "non-solution" that I present to them as "this is how several students answered this question in the past."
I also want my students to be able to make connections not only between mathematical topics, but also to make connections between the course topics and their own major/profession. One way I do this is through a comprehensive oral exam which counts as 20% of the final exam grade.
The oral exam instructions as well as a selected activity assignment are displayed below.
You can access all my worksheets, videos, learning objectives, and a weekly plan for this course here: Multivariate Calculus (V. F22)
Oral Exam Instructions and Question List.docxWeek 5 - Designing a Highway.pdfAs I mentioned above, many of my students in this course are engineers. This activity, "Designing a Highway", puts students in the perspective of a civil engineer who is designing an easement curve for entering/exiting a freeway. Students apply what they have been learning about curvature to this problem. A satisfying end-result can be found which gives students a sense of satisfaction having successfully applied mathematics to a real-world scenario.
Graph Theory
Graph Theory
Check out the playlist for the video content I have made for this course:
Two popular videos are just below.
In this graduate level (575) graph theory course, I guide both advanced undergraduates and graduate mathematics students to understand some fundamental concepts in Graph Theory. The students typically come in to the class with some experience with mathematical proofs, a solid understanding of discrete mathematical structures, and a well-developed logical foundation. One major goal of the course is to prepare students for research in graph theory. The course project on an unsolved problem in graph theory is assigned to meet this end.
Topics covered in the course include:
Fundamental concepts - order and size, degree, paths and distance, cycles, complete graphs, and the adjacency matrix
Trees - characterizations, (minimum) spanning trees, Cayley's Tree Formula, Kruskal and Prim's algorithms, and the Matrix Tree Theorem
Historical Problems and Open Questions - Bridges of Konigsberg, Hamilton's Icosian Game, the Four Color Theorem, the Reconstruction Conjecture, and the Graceful Tree Conjecture.
Hamiltonian and Eulerian graphs
The automorphism group of a graph
Graph coloring
Ramsey theory
First Week Activity.pdfInstructions and Topics List.pdfMathematical Modeling
Mathematical Modeling
As a capstone course, students leave the course having created an original work of mathematical modeling.
We encounter discrete and continuous dynamical systems, statistical models, cryptography, machine learning, and network models.
We develop our understanding of mathematical tools and theories such as gradient descent, neural networks, optimization, graph theory, differential equations, linear algebra, and number theory.
And we use tools like MATLAB, Python (e.g., Sympy, Numpy, NetworkX libraries), and Excel to create our own mathematical models.
During our unit on Network Science, we use Gephi to visualize network data. The network shown here is the Western States Power Grid of the United States. Vertices are colored by modularity class and vertices are sized based on their betweenness centrality.
Data found in:
[D.J. Watts and S.H. Strogatz, Nature 393, 440-442 (1988)]
Activity for Week 3 - Hanging Chain.pdfI wanted to be as hands-off as possible with the activities in this class. This way, students can gain confidence with the modeling process. "Hanging Chain" takes the material we developed on curve fitting and differential equations and has the student apply it to determine a more accurate model of a hanging chain.
Activity for Week 12 - Zacharys Karate Club.pdf"Zachary's Karate Club" is a classic introductory lesson to communities in network science. Taking the key points of the narrative, I have created an activity which enables students to get hands-on experience importing network data, running algorithms, and interpreting the results. While there are some other programs which can be used for this activity, I recommend Gephi as it has an easy learning curve (once we get installed) but is powerful enough to gather the data needed.
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