A New Type of Learning Community

Setting high standards is expected from all educators. Yet, I think I may have taken this to an extreme in my 2019 spring senior seminar course in algebraic combinatorics. Students walked in to class, got a copy of the syllabus along with a community agreement (sample community can be found here) and saw:

With a pile of 30 or so textbooks on the front desk I pointed out that there was only one undergraduate book on algebraic combinatorics which was published in 2013 — Algebraic Combinatorics by Richard Stanley [1]. It covered classical subjects very well, but what I wanted to do in our class was to get students to a point where they could reach open problems in the field by reading recently published research manuscripts in broad areas of algebraic combinatorics. So to solve this problem we would write our own textbook!

Imagine the students’ surprise to learn of this during the first day of the course. Not to mention the level of work such an undertaking entails, both on the students and on the professor. Fourteen students took on the challenge, and below I discuss the process of building a learning community and how through this community we reached our goal of writing a book.

Sky high expectations

“I believe I can do anything if I just try.” — Granville T. Woods

Naturally, some students felt very unprepared to take on the challenge of writing a textbook and decided to wait to take the course in a future semester. And even for those students that remained, they were not convinced that our goal was attainable. After all, it takes mathematicians sometimes decades to complete a book. So how would we be able to accomplish this within the span of a single semester course?

My motto was that we could not know whether this could be done unless we tried. So I set sky high academic standards for students, regardless of whether I or others believed that students were ready for such an academic challenge.

Contingencies and aims

The goal of the class was then clear. But what took more planning was determining the following contingencies:

• Instructional — how would I support learning activities?
• Domain — how would I help determine what students should focus on next?
• Temporal — how would I decide if and when to intervene?

With these questions in mind I set the following aims:

1. Move students progressively toward stronger understanding and, ultimately, greater independence in the learning process.
2. Create a scaffolding framework that
   allows students to meaningfully participate in and gain skill at a task that they could not
   complete unaided.
3. Build a collaborative learning community that promotes critical thinking skills better than competitive/individualistic learning environments.
4. Learn ways to deal with people that respects and highlights individual group members’ abilities and contributions.

To support the course’s goal of completing a book, I grouped students into pairs who would collaborate on writing a chapter of the book. The research topics were based on student interests, and I provided some initial manuscripts they would explore. The students’ initial job was to understand the material well enough to write about it in detail with the aim of making it accessible to students with calculus or linear algebra background. From this initial content, students then found new resources to further expand their chapter.

As is expected, writing skills would be paramount to completing our class goal of writing a book. Making sure students learned how to read research manuscripts and how to write accessible pieces of mathematics posed the major challenge. Most of our class meetings were spent in small groups working through mathematical arguments and discussing how we should be reading research articles and what constitutes a clear argument with enough detail so not to make huge jumps in logic. After all, what use is there to write another book in which we spend hours dissecting a paragraph? My mantra was “If you cannot explain it simply you don’t know it well enough.”

Logistics and assessments

Hot chocolate helps with editing

Given the goal of the class, a major part of the class assessments was based on chapter drafts. Drafts were due (approximately) every 3 weeks, they were typed in LaTeX, using the online platform overleaf. Given the small size of the class, I focused on giving extremely thorough feedback on students writing. This was the most grading I have ever done in one semester!

Belle at board teaching high school students visiting our course

Throughout the semester, students also gave short presentation describing what their chapter covered. These talks were peer-evaluated and I provided individual feedback based on what students wanted to improve. Also the talks were recorded and students were able to watch them to create a new list of things to improve on for the following presentations.

The last  piece of the course assessment were reflections. I provided students with short writing prompts from which students reflected on the experience of writing and learning math independently/collaboratively.
Reflection prompt included: How does collaboration enhance/hinder the writing and oral presentation of mathematical material? How can you leverage the strengths of each member of the group to create a better chapter and/or have a better presentation? What you think about quantitative evaluation systems like test scores and class rankings, as opposed to qualitative forms of feedback like written comments and conferences. Does one give you a better measure of the knowledge you have gained?

The reflections were key in helping me adapt the course throughout the semester. I also learned so much about the students motivation and how their confidence was growing as the semester progressed. The reflections were our way to communicate deeply and meaningfully about the purpose/value of an education and in fact one student’s reflection developed into a publication, see [2]. Reflections were a refreshing take on an assessment piece, one which I plan to include in all future courses.

Final Product

After a grueling semester we completed the book. There is of course more editing that should take place before any final access to the book is made available, and I hope to teach this course soon so that more students can contribute to it.  Below is the contents along with the authors for each chapter.

A Friendly Introduction to Topics in Algebraic Combinatorics

Chapter 1: Permutations and their peaks by Belle and Ben
Chapter 2: Combinatorics of parking functions by Alex and Maryanne
Chapter 3: Combinatorial representation theory by Franny and Anthony
Chapter 4: Combinatorics and Lucas analogues by Joanne and Francesca
Chapter 5: Chromatic polynomials by David and Katherine
Chapter 6: Combinatorial geometry by Naush and Teresa
Chapter 7: Numerical semigroups by Ben and Aesha

Parting thoughts

I have been asked if I think that more faculty should structure courses in this way. Honestly, I do not know if more people should teach in this particular way. Each institution has different student cultures. What works at a small elite liberal arts college like Williams may not work elsewhere. Moreover, this depth of active teaching is scary even for someone with ample experience. What I do know is that whatever way we structure our courses ought to help students hone their writing and public speaking skills — which will be key for all of their future careers. And we should do so by setting sky-high expectations and building a supportive community in which students can meet those standards.

Pamela and Anthony at is graduation

“This class gave me, or forced me to pick up, the confidence and ability to read a math textbook and genuinely understand the material. Above every lecture based class I’ve ever had, this experience is irreplaceable. Both because it was enjoyable and because it gave me skills that have translated into my life after school.” — Anthony Simpson

[1]. Richard P. Stanley, Algebraic combinatorics, Undergraduate Texts in Mathematics Walks, trees, tableaux, and more. Second edition of [MR3097651], Springer, Cham, 2018.

[2]. Dean, F. “Tired: A Reflection on Asceticism and the Value of Quantitative Assessment,” Journal of Humanistic Mathematics, Volume 10 Issue 1 (January 2020), pages 375-377. DOI: 10.5642/jhummath.202001.19 . Available at: https://scholarship.claremont.edu/jhm/vol10/iss1/19