# Instructor Insights

## Instructor Interview

Below, Paige Dote describes various aspects of how she taught 18.S097 Introduction to Metric Spaces during the 2022 Independent Activities Period (IAP).

OCW: For the uninitiated, can you explain for us what metric spaces are and why they’re an interesting topic to study?

Paige Dote: Metric spaces allow us to rigorously study distance. Up to this point in math, we often want things to look perfect, using the Pythagorean theorem to understand distances in n-dimensional space. However, even in our day-to-day life this isn’t how we understand space. Consider for instance, a city with a grid system like New York. If I wanted to walk from one block to another, the distance wouldn’t be the shortest way to get there (after all, I cannot walk through walls). Rather, it would be the number of blocks I walked (times the distance of the block). So while we have a good understanding of distances in Euclidean space (as is studied in 18.100A Introduction to Analysis and 18.100P Real Analysis), the question in turn becomes “how does this generalize?” —a question this class attempts to begin to answer.

OCW: You taught this class while you yourself were a second-year undergraduate. How did you wind up in this unusual position?

Paige Dote: When taking 18.102 Introduction to Functional Analysis with Dr. Casey Rodriguez, I realized there was a small but important gap between 18.100A [real analysis on Euclidean space] and 18.100B [real analysis in more abstract settings]: metric spaces. I firmly believe there are many great reasons to take 18.100A, and felt that students should have resources to fill this gap for later courses if they wanted. I realized that this resource didn’t exist after I saw many friends struggling with missing familiarity with abstract spaces. Thus, I started advocating for this course, which gained traction with analysis faculty. I reached out to Academic Administrator Barbara Peskin to figure out the actual logistics of running a class, who put me in contact with Professor Bill Minicozzi, Associate Department Head for Mathematics, who helped me develop a syllabus. A few months down the road I put together the material to teach the class into existence!

OCW: Was this one of your first formal teaching experiences? In developing the course, how much thought did you give to your teaching technique, as opposed to the material you planned to cover?

Paige Dote: I have known since second grade that I have wanted to teach, though this is my first time being the actual teacher. Up until this point, in high school and while at MIT, I have tutored for numerous courses from trigonometry and beyond. When developing the course, I spent a lot of time fine-tuning the material pedagogically. My goal was to make it clear where this subject goes from here, motivating classes like 18.901 Introduction to Topology, 18.101 Analysis and Manifolds, 18.103 Introduction to Fourier Analysis, and 18.102 Functional Analysis. Ultimately, the material in Lebl’s chapter on metric spaces is only 30 pages long, but the power of this tool often goes unexplained, something I wanted to fix in this version of the class. I was also highly influenced by the teachers and students around me who have fostered amazing learning environments. These people influenced me to create an accessible website for students, type lecture notes, and record lectures during a peak in COVID.

OCW: You recently participated in a conference session on student participation in creating open educational resources. Can you share your vision of the role students could or should play in the educational ecosystem?

Paige Dote: I wish students took more of a part in the classroom, and I believe this can be better encouraged by professors. The most influential courses at MIT in the math department, in my opinion, are those with course material available to the public after the fact. For instance, when Professor Larry Guth, the faculty advisor for 18.S097, taught the course 18.118 Topics in Analysis: Decoupling, he asked each student to type notes for one lecture during the semester. The materials for that class are beautiful and nice to read even years later. I hope that in the long term, more resources like this can be made and made available to the public, and I believe that students can/could/should play a role in this in the classroom.

OCW: If you were to teach this course again, what would you do differently?

Paige Dote: If I were to teach this class again, I would put less emphasis on norms in Lecture 3. Norms give us a way to study distances in vector spaces, a topic covered in an introductory linear algebra course. However, linear algebra was not a prerequisite for this course, which made this portion of the lecture run a bit overtime. However, norms are a vital tool for functional analysis and greatly motivate compact sets, which is why I covered them here.

OCW: What would you like to share about teaching 18.S097 that we haven’t yet addressed?

Paige Dote: In math, there are a great number of grey areas between classes; metric spaces are a great example of this. One of the largest motivators for 18.S097 was the fact that this very important subject sometimes falls through the cracks in an undergraduate’s curriculum. This can make it harder to keep learning math in analysis and beyond. I hope we can keep identifying and creating resources for academic grey areas like this.

Also, I would like to thank the many students who helped provide feedback and guidance developing material and problem sets, including but not limited to Jacob Lerma, Omar Abdelghani, and Yuqiu Fu. I would also like to thank Professor Guth for helping develop the course and contributing interesting problems for problem sets and Professor Minicozzi for helping me navigate the logistics of creating this class.

## Curriculum Information

#### Prerequisites

Students were expected to have some experience with real analysis, such as might be gained from taking one of the various versions of 18.100 Real Analysis.

None

#### Offered

Various versions of 18.S097 Special Subject in Mathematics, each on a different topic, are offered each year during the Independent Activities Period (IAP).

## Assessment

The course was taught Pass / No Record, with grading based both on completion of assignments (three problem sets) and on participation.

## Student Information

20 students

#### Typical Student Background

Students were expected to have some prior experience with real analysis. There were students who had just finished one of the versions of 18.100 Real Analysis, as well as those who simply wanted to know more about the subject. The majority of the students were enrolled as listeners.

## How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

#### In Class (3 hours)

Met 2 times per week for 1.5 hours per session; 6 sessions total; mandatory attendance

#### Out of Class (9 hours)

Outside of class, students completed three problem sets.