Math chalkboards and mandalas...

There is a quiet, almost ceremonial moment at the end of a mathematics lecture that often goes unnoticed. After an hour of careful construction—definitions laid down, computations layered, structures revealed—the chalkboard stands filled to its edges with meaning. Every line is deliberate. Every symbol carries weight. And yet, in a matter of seconds, it all disappears beneath the sweep of an eraser.

What remains is not the board, but the understanding.

Your analogy to the Tibetan mandala captures something profound about this act. In the creation of a mandala, monks spend hours or days placing colored grains of sand with extraordinary precision, forming intricate geometric patterns that symbolize the universe. The work is patient, disciplined, and deeply meaningful. Yet, upon completion, the mandala is not preserved. It is dismantled—its grains brushed away, often returned to a river—embodying the principle of impermanence.

So too with the chalkboard.

In presenting a mathematical theory, the dense writing, the multiple perspectives, the unfolding of structure, these form a kind of intellectual mandala. Each method illuminates the same underlying reality from a different angle, much like the symmetries within a mandala reflect a deeper order. The mathematical chalkboard, filled to capacity, becomes a temporary universe of thought.

And then it is erased.

This erasure is not loss. It is, in a sense, completion. The purpose of the work was never the chalk marks themselves, just as the mandala was never meant to endure as an object. Both are vehicles—means through which understanding, insight, or contemplation arises. Once that purpose is fulfilled, attachment to the physical form becomes unnecessary, even contrary to the spirit of the practice.

There is also humility in this act. Mathematics, for all its precision and permanence as a discipline, is practiced in moments that are fleeting. The lecture ends, the board is cleared, and the space is made ready for new ideas. What persists is carried internally: in memory, in intuition, in the quiet restructuring of thought that follows a deep encounter with a problem.

In this way, the mathematician at the chalkboard and the monk creating a mandala share a common gesture. Both engage in acts of careful creation that culminate not in preservation, but in release. And in that release, there is a recognition: the true substance of their work lies not in what is seen, but in what is understood.

My retirement

I decided to retire at the end of this academic year, 2025-2026. After 26 years as an ONU professor (Full Professor since 2011, including being a Mathematics and Statistics Department Chair 2014-2018), one year at Stanford as an EPGY Research Associate, 3 years at Washington State as an Assistant Professor, in addition to the 4 years of being a Mathematics PhD student (including serving as a Teaching Assistant) at Penn State 1992-1996. Since 2003, I was the Senior Capstone research advisor for 37 Mathematics students at ONU, several of them continuing and finalizing their graduate degrees after leaving ONU, others becoming successful Teachers or Engineers. Thank you, all!

on a busy day - one of my math chalkboards

 

A short classroom exercise on partially ordered sets

 

Memories - a beautiful campus and an abstract - 27 years ago

I vividly remember that talk. 

People were great! Added bonus: the campus and the nature were amazing. 

I could recover my abstract from https://scholarworks.umt.edu/mathcolloquia/7/

 

So... about 27 years years ago:

 

Presentation Date 10-23-1997 

Abstract A classical problem considered by Davenport asks for the number of occurrences of a certain pattern of quadratic residues and nonresidues among the set of integers modulo a prime p. We will take this problem as a starting point of an itinerary revealing the interplay between quasirandom structures over finite fields, definable subsets of finite fields, character sums and coding theory. 

Additional Details Thursday, October 23, 1997 4:10 p.m. in MA 109 Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Are these the same proof?

In his blog, Timothy Gowers presents two proofs of the fact that the product of k consecutive positive integers is a multiple of k! and asks  

"whether there is some sophisticated perspective from which it is possible to regard them as sort of the same". 

There are many interesting answers from the readers, among them being Terence Tao.

Math Education: "more than good grades and test scores..."

An interesting article at The Conversation:

Real equity in math education is about more than good grades and test scores - by Megan Staples, Associate Professor of Mathematics Education, University of Connecticut

"With more than three decades of experience as a researcher, math teacher and teacher educator, I advocate for expanding what equity means in mathematics education. I believe policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power – and true equity."

Read the full article at  

https://theconversation.com/real-equity-in-math-education-is-about-more-than-good-grades-and-test-scores-215518  

The poetry of recursion

After teaching what I believe to be the highest-numbered MATH class in the catalogue (MATH 4671 Probability and Graphs - a.k.a. Random Graphs - a quite difficult class), next year 2024-2025 I would like to take at step towards the other end of the spectrum, and teach Mathematics for Liberal Arts (MATH 1201). 

It will be a challenge. 

I want to get into the mindset of a poet, and I want to make said poet feel a friendly spirit in the abstract realm. That would be difficult indeed.

For that I need a guiding theme, a guiding spirit - and I found it: the poetry of recursion. 

Many thanks and gratitude to my students!

At the end of every semester, I find myself amazed, impressed, and grateful seeing amazingly uplifting comments from my students. This gives me the power to move on. Some excerpts from evaluations for my Discrete Mathematics class - Fall 2023:

"I think this course is a great introduction to higher level math, its related language, terminology, as well as higher-level working logic. I enjoyed this class as it dove into many of the mysterious, hidden patterns that I have seen in math for several years. It is interesting to learn all of these things and finally learn the "secrets" behind math."

 

"The materials in discrete math reminded me a lot of the materials in the digital logic class I took the semester prior. I enjoy all things math and I also enjoyed that digital logic class as well. It was nice to be able to build upon that knowledge and become more familiar with the theory behind topics such as k-maps, modules, and decryption. Decryption and it's multiple theories was one topic that I think interested me the most."