Sustaining the Energy and Maintaining the Growth

I often start out each semester eager to try a few new things in the classroom, or to pay particular attention to certain aspects of my teaching.  As the semester progresses, I often find myself slipping into the pattern of past routines, less eager or able to find the time to reflect as deeply or to focus as intentionally on expanding my own skills.

Here at the University of Colorado Denver, we’re starting our fourth week of classes.  One of the classes that I’m teaching this semester is the history of mathematics.  As part of an NSF-funded grant, Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), I’m mentoring a graduate student in the use of primary sources projects in the classroom.  This is helping to sustain my intentionality with regard to my preparation as well as my choice of instructional practices. In this role, I have been pondering both how to be a good mentor as well as how to keep working to learn and grow in my own teaching throughout the entirety of the semester.

This has led me to return to some of our past blog posts that I found particularly helpful to read or write, which I want to share. Below are links to some of these past blog entries which focused directly on some aspect of classroom teaching practices, and that I want to use throughout the next few months to keep my energy level up for my teaching. I hope you can find something here to energize you as well.

The first is a link to the editorial board’s six-part series on active learning that appeared in 2015:

This was followed up by an article in the Notices of the AMS from February 2017:

This next entry by Steven Klee at Seattle University focuses on how to encourage increased student interactions during group work by having them work together at the board:

One of my all-time favorites, by Art Duval at the University of Texas at El Paso, focuses on if telling jokes and making class humorous is really beneficial to student learning, or if it unnecessarily takes away precious time that the instructor and students have together:

And, finally, a post from Allison Henrich at Seattle University, reminding us of the wonderful value of mistakes in the learning process, and sharing ideas of how to help students be comfortable with making and discussing mistakes in the classroom:

As you progress through your semester, I hope you find something in these various posts to keep you energized and growing in your own practice of teaching.

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The Joy of Mathematical Discovery

By Steven Klee, Seattle University

It persistently rises to the surface of your memory – that afternoon when you fell in love with a person or a place or a mood … when you discovered some great truth about the world, when an indelible brand was seared into your heart, which is, of course, a finite space with limited room for searing.

Arthur Phillips, Prague

It was my senior year of high school. I had spent the first half of my day taking the AIME exam. At the end of the exam, there was one problem that really intrigued me. I couldn’t stop thinking about it! It lingered in the back of my mind through lunch and gym class. When I got to my history class, I had an idea to start looking at small examples: what if there were only two houses on the street? Or three? Or four? Then I had an “a-ha” moment, which let me see a recursive pattern and ultimately led to the solution of the problem.

The joy I experienced at solving this problem was profound, and it still stands out in my mind, almost 20 years later, as a significant moment in my mathematical journey. I had had this insight that was completely new (at least it was new to me), and led me to solve a problem that was unlike anything I had ever seen before. It was exciting! It didn’t count towards my grade anywhere, but that didn’t matter. I had discovered something new, and mathematics had left an indelible brand on my heart.

My goal in this article is to examine this experience more carefully, along with the experiences of other mathematicians and scientists, to try to understand the “a-ha” moments that can be so powerful for our students. To gather data, I asked a large group of people, including high schoolers, academics, and people in industry to reflect on the following question:

Tell me about one of the first times you ever experienced joy or excitement at solving (or not solving) a math problem.  When did this happen? Do you remember the problem? What made this experience so memorable?

In what follows, I will reflect on general themes that surfaced in the responses I received in the hopes that they can help us more deeply reflect on our own teaching. I am grateful to my friends and colleagues who shared their stories. Each one was exciting and inspiring in its own way, and I regret that I was not able to include an excerpt from each of them. I would love to hear about your stories of joy and mathematical discovery in the comments section below.

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Advice for New Doctoral Advisors

By Benjamin Braun, Editor-in-Chief, University of Kentucky

It is commonly understood that graduate students need guidance and mentoring, especially as they begin the research phase of their studies with an advisor. A less-frequent topic of discussion is the guidance and mentoring that new doctoral advisors benefit from as they take on this unfamiliar responsibility. For many, if not most, mathematicians working in doctoral-granting departments, training and mentoring in how to be an effective advisor is done in ad hoc and informal ways. As a result, many new doctoral advisors work in some degree of isolation as they develop their advising styles.

In this article, I offer five suggestions for new doctoral advisors, suggestions that I believe make the advising process both more enjoyable and more effective. Knowledge of these suggestions can also be helpful for doctoral students, providing ideas of questions that might be helpful for them to ask their advisors. Continue reading

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Communicating Advanced Mathematics to Kids

By Jeremy Martin, Professor, University of Kansas

I’ve often thought that we could do a lot better job of explaining “advanced” mathematics concepts in simple language for the benefit of a wider audience. As a student, I never liked being told, “We’ll explain that to you next year.” As a teacher, I’ve always wanted be able to give real answers to students’ exploratory questions: if a Calculus I student asks me a question whose precise answer requires knowledge of manifolds and de Rham cohomology, I want to be able to distill those ideas into an answer that this student can understand. Also, I’ve always enjoyed the challenge of telling non-mathematicians about Euler’s formula, or voting theory, the Four-Color Theorem, or the game of Nim. I have experimented with trying to explain my own research in algebraic combinatorics to an intelligent layperson.

For example, I recently coauthored a paper with the intimidating title “Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions”, which is full of extensions of eigenfunctions, fractional Schrödinger operators, Kreweras complementation, and similar jargon. I summarized it like this: “My coauthors study partial differential equations, which model things like fluid flow and heat dispersion. They draw pictures that look like tangled-up spaghetti, then try to measure the complexity of the equations by counting the holes in the tangle. Well, counting is what I do for a living, and when I saw their pictures, I was able to use what I know about counting to help them solve their problem.” Sure, that’s sweeping a whole lot of things under the rug, but really, that’s what we were doing.

And that is how I ended up editing articles about mathematics for kids.

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Learning by Teaching: Service-Learning in a Precalculus Classroom

By Ekaterina Yurasovskaya, Seattle University

Mathematics is a beautiful subject that can easily become an ivory tower. There can be a temptation for teachers and students of mathematics to shy away from the role that mathematics plays as a social force and a barrier that can put a halt to a person’s career, security, and social mobility. The mathematics education community has been studying this situation for years – for example, see this article by Rochelle Gutierrez [1]. One way to include a focus on society and its problems in a mathematics classroom is by introducing service-learning into one’s course.

Service-learning is a pedagogy that combines the course content with community service that is directly tied to the material that students are studying inside the classroom. Service-learning has traditionally belonged to the domain of social sciences such as psychology, sociology, or social work, however interest in service-learning has recently increased in STEM disciplines as well. A special issue of PRIMUS [2] was entirely dedicated to mathematical service-learning projects; an interested reader will find a wealth of helpful practical information and project descriptions there, from math fairs and tutoring to running modeling projects for community organizations. In this post, I would like to share with you my own experience with service-learning, its effect on my students’ worldview and mathematical knowledge, as well as offer some suggestions for the instructor who would wish to introduce service-learning into a math course.

Personal experience and motivation

When I first learned about service-learning four years ago, I immediately wanted to try it – and my initial motivation was practical. Precalculus students are a mathematical population at risk. Weak algebra preparation invariably hinders progress of STEM students, and severely affects performance in Calculus, a major junction in the leaky STEM pipeline. As teachers, we know that one of the best ways to learn something is to teach it ourselves: “I hear and I forget. I see and I remember. I do and I understand”. This led me to ask myself: “What if university students in my classroom had to teach algebra prerequisites to someone else? Will it help them learn and understand that material themselves?”

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On the Culture of Making Things

By Nicholas Long, Stephen F. Austin State University

In one of life’s weird coincidences, when I moved to a small town in East Texas to start my academic career at Stephen F. Austin State University (SFA) ten years ago, I didn’t know that I would be working with someone from my high school graduating class of about 150 people. Through that small quirk of life, I met a lot of the art faculty and local artists in the Nacogdoches area. I love that I get to hang out with artists and art educators. They are really cool people and they MAKE THINGS. Things that people want to look at, things that people want to discuss, and sometimes, things that people even buy.  

The idea I want us all to consider is: “How do we grow and improve the culture of making and improving things for our teaching?” When I say things for our teaching, I mean much more than just textbooks and notes for lecture; I mean software and technology that adds meaning and value for our students; I mean activities that can change the attitudes, habits, and practices of our students; I mean the many other materials and resources that will transform students and mathematics classrooms. While I will cite some examples below of projects and resources that I think are doing good things, I genuinely think we need to not compartmentalize these practices, but make them part of what we as a community do. Continue reading

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Pursuing Our Mission to Support All Students at All Ages

By Priscilla Bremser, Contributing Editor, Middlebury College

The more I teach and learn mathematics, the more I regard the subject as a powerful resource that is unfairly distributed. Clearly, I’m not alone. Search for “underrepresented” on the American Mathematical Society website and you’ll find the inclusion/exclusion blog and the Director of Education and Diversity at the AMS, for example. While it is vital to build on the work of exemplary programs at the university level, we cannot fully address the inequities in access to mathematics, and to fields that require mathematics, unless we also examine and address inequities in pre-college education.

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Students Find Their Fit in the Mathematical Community at the Marshall University REU

By Stacie Baumann, 2017 graduate of West Virginia Wesleyan College, currently a doctoral student at Auburn University, and Matthew Jones, Virginia Tech, class of 2018

Editor’s note: The editors thank Stacie and Matthew for taking the time to share their thoughts and insights with us about their REU experiences. For students who are interested in applying for an REU, lists of programs can be found here and here. To read more about student and faculty experiences with REUs, see our other articles on this topic.


I did not know what a Research Experience for Undergraduates (REU) was until one of my professors suggested I apply to REUs at a few different institutions. I went to a small liberal arts college with few research opportunities. I applied to a handful of different REUs and was excited when I received my acceptance from Marshall University. I was excited to spend eight weeks of my summer at a different institution and to learn what mathematical research really looked like. I was also nervous that I would be behind the other students academically. When I arrived, my nerves were calmed and the excitement continued.


If I learned anything from participating in the REU at Marshall University, it was how to be frustrated. Before the REU, I had certainly encountered a few difficult proofs in my courses, some of which I spent a couple of days thinking about. However, I had never spent an entire summer obsessed on one seemingly tiny mathematical problem. Throughout the months of June and July, I thought about mathematics at breakfast while drinking my coffee. I thought about mathematics while sitting in a classroom at Marshall. I thought about mathematics while playing cards in the evening. I went as far as to buy a 3-foot-wide whiteboard to keep beside my bed that I could grab in the middle of the night and test out propositions.

When I describe that experience to my friends outside of the mathematics community, they usually say, “That sounds awful”. But in actuality, it was the most exciting experience of my brief mathematical career. In a short span of time, I learned a great deal of exciting new mathematics that I would not have been exposed to in a normal course at my home institution. I learned a great deal more about the mathematical community in general, how research works and where I might fit in. Most importantly, I learned how exhilarating the process of figuring things out mathematically can be. Continue reading

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They Taught Me by Letting Me Wonder

By Dr. Nafeesa H. Owens, Ph.D., Program Director/PAEMST Program Lead, Presidential Awards for Excellence in Mathematics and Science Teaching, National Science Foundation*

Today we celebrate the story of Marizza Bailey, who was honored last year by the White House with one of its Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST).

Marizza Bailey (top right), with her grandmother Luz Mendizabal, her mother, and her daughter, in a newspaper clipping from 1997, when Luz was given an award by Lima, Peru, for her advancement of education in the city.

When Marizza Bailey was 12, her grandmother, Luz Mendizabal, came to live with her in California. Born in Peru, Luz put herself through graduate school to earn a doctorate degree in Mathematics Education, all while teaching full-time and raising eight children. She brought many things with her when she arrived in California: her voracious appetite for learning, her vast knowledge of mathematics, history and literature, but what Marizza appreciated most were the questions. “Why do you think that?” “What makes you say that?” “How do you know that?” A conversation with Luz was a series of questions and answers that stimulated critical thinking.

With a grandmother who was a mathematics teacher and who inspired thoughtful dialogue with her children and grandchildren, it’s no wonder that Marizza followed Luz’s footsteps and became a mathematics educator, as did Marizza’s mother before her. For this family, mathematics was not so much a career or a school subject, but a way of viewing and interpreting the world. Marizza says of her family, “They taught me by letting me wonder and allowing me to draw my own conclusions.”
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Do We Get to Work at the Board Today?

By Steven Klee, Contributing Editor, Seattle University

When I first started incorporating active learning in the classroom, I struggled with getting my students to buy into being active.  I made worksheets, put the students in groups, and excitedly set them off to discover and play with mathematical ideas.  Despite this, many students were inclined to sit silently in a group of four and work on the problems on their own.

But really, who can blame them? First, this propensity towards solitude can be explained by basic human nature: specifically, the fear of being wrong.  We don’t want to be wrong. At least, we don’t want to be wrong in front of other people.  From that perspective, working alone is safe and comfortable.  We should view our job as teachers as one of helping our students overcome this basic human inclination, as opposed to viewing it as a failure or shortcoming on their part.

Beyond this, the desire to work alone can be attributed to culture and expectations.  Many students’ formative educational years have been spent sitting silently in desks passively absorbing lectures.  If they feel this is what is expected of a math class, then it is natural for them to continue to sit silently, even if the environment is meant to be collaborative.  Of course, it is not my intention to imply that this is an issue that is entirely the students’ fault – maybe my questions weren’t sufficiently open-ended, maybe I wasn’t doing a good enough job at “selling it,” maybe the students just like working alone, maybe, maybe, maybe… The list goes on.

I tried some of my standard tricks to foster communication among the students.  I would prepare impassioned pep talks about the benefits of working with your peers. This technique flopped for obvious reasons – no one wants to listen to what they are told is good for them.  Otherwise, cigarette companies and fast food restaurants would go bankrupt and I would be much more diligent about flossing.  I’d try to lighten the mood, saying “this isn’t a library, you’re welcome to talk to one another.” I’d give a difficult problem and leave the room to get a drink of water, forcing the students to rely on one another. These strategies helped, but never served to create the classroom of my dreams – one where students discuss math problems at such a frenzied pace that time ceases to exist; one that causes passersby to wonder whether we are having a math class or developing some bizarre scientific improv comedy troupe.

Over time, I continued to reflect on my own teaching and sought advice from more experienced practitioners of active learning.  As a result, I have developed a few strategies that have been effective in my classrooms. One of the most effective strategies for me has come from eliminating those pesky desks that keep getting in the way of my students’ learning.

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