Mathematics for All?

Every few years, I teach a first-year seminar called “Mathematics for All.” The course description begins:

What kinds of mathematical knowledge are necessary for full participation in contemporary democratic society? How well, and how fairly, do our schools educate students in quantitative skills and reasoning? By what measures might we judge success?

To put it another way, what would an equitable mathematics education system look like? In this post, I reflect on some articles published on this blog that support our efforts to move toward fairness.

A good place to start is in our own classrooms. Once we acknowledge the disproportionate distribution of access to mathematics experienced by our own students, we can make use of Six Ways Mathematics Instructors Can Support Diversity and Inclusion, by Natalie LF Hobson. One of the six ways is to “[e]ncourage your students to embrace a growth mindset,” which Cody L. Patterson explores in Theory into Practice: Growth Mindset and Assessment.

My seminar includes a service-learning project. As Ekaterina Yurasovskaya demonstrates in Learning by Teaching: Service-Learning in a Precalculus Classroom, such a project, while challenging on several levels, can benefit both the community being served and the students. If my own experience is any guide, the instructor can also gain some unanticipated lessons about mathematics learning in the early grades.

Attending to equity and inclusion is hard work. When I need to take a step back for an energy recharge, I go straight to contributions from Ben Braun, our founding Editor-in-Chief. His Aspirations and Ideals, Struggles and Realities is rich with inspirational ideas. I’ve assigned The Secret Question (Are We Actually Good at Math?) to my own students. It means a lot to them, and the resulting conversations are deep and illuminating.

Let’s not forget about the struggles our own colleagues may continue to face as they work within the flawed systems that Ben describes so well. A useful reading in this regard is Student Evaluations Ratings of Teaching: What Every Instructor Should Know, by Jacqueline Dewar. The author points out that “‘ratings’ denote data that need interpretation,” and gives useful guidelines for interpretation. While not focusing exclusively on the question of bias, the article does cite sources on that topic, including this study published in 2016.

Moving on to other aspects of our professional lives, Viviane Pons describes An Inclusive Maths Conference: ECCO 2016 . Having been to dozens of conferences, many of them quite worthwhile, I was fascinated by the intentional design details that made this one special, and wish I’d been there to experience it!

A simple Announcement of a Statement from the American Mathematical Society’s Board of Trustees reminds us that we can work toward the greater good within our professional societies.

While I’ve had plenty of my own “secret question” moments in a lifetime of learning mathematics, I recognize the benefits of mathematical habits of mind to me as an individual and as a citizen of the world. Those benefits should be available to everyone. We can all work toward that end, and I hope you’ve found some ideas here on how you might help.

 

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Eating Our Own Cooking: What I’ve Actually Used or Shared

By Art Duval, Contributing Editor, University of Texas at El Paso

A popular saying in business (or so I’ve read) is to “eat your own cooking”: Use the products your own company makes.  I suppose there are several motivations to do this: to demonstrate faith in your own work; to be your own quality control team; to make your product visible; etc.  What does that have to do with teaching and learning mathematics?

The best part about being on the editorial board for this blog continues to be the privilege of working with a talented group of editors and with all sorts of creative authors, who collectively have an incredible variety of important things to say.  (F. Scott Fitzgerald: “You don’t write because you want to say something, you write because you have something to say.”)  As a result, I sometimes feel like I am drowning in interesting ideas, with not nearly enough time to try them all.   Today I would like to tell you about the articles we’ve published here that contain ideas I’ve tried myself and/or shared with students and colleagues.  In other words, to answer the question “What have I eaten of our own cooking?”  Bon appétit!

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From the Editors: Changes for the Editorial Board

By Benjamin Braun, Editor-in-Chief

I want to thank all of our readers, subscribers, and contributors — we appreciate your feedback and ideas through your writing, social media comments, and in-person conversations at mathematical meetings and events. We will continue to strive to provide high-quality articles on a broad range of topics related to post-secondary mathematics, and we welcome your feedback and suggestions. In this post I share two upcoming changes for our editorial board.

First, I will step down as Editor-in-Chief at the end of May 2018. I am thrilled to announce that Mark Saul will serve as the next Editor-in-Chief for On Teaching and Learning Mathematics starting on June 1, 2018. Mark has extensive experience in mathematics education at the K-12 and postsecondary level, both in the classroom and through outreach programs. He also has substantial editorial experience, including editorial service to the Notices of the AMS, Quantum, and The Mathematics Teacher.

Second, following four years of service as a founding Contributing Editor for our blog, Priscilla Bremser will step down from the editorial board in May 2018. Priscilla has made many excellent contributions to our blog, and I deeply appreciate her dedication, insight, and passion for improving the teaching of mathematics. I look forward to hearing more from Priscilla in the future as a contributing author!

 

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What Do Hobbits Know About Mathematics?

Sam: If I take one more step, I’ll be the farthest away from home I’ve ever been.

Frodo: Come on, Sam. Remember what Bilbo used to say: ‘It’s a dangerous business, Frodo, going out your door. You step onto the road, and if you don’t keep your feet, there’s no knowing where you might be swept off to.’

For many students, it is scary to be pushed to think differently about mathematics or to participate in a different type of classroom environment (for example, a flipped classroom, IBL classroom, active learning classroom, etc.).  These new experiences create a certain level of discomfort in adapting to new styles and expectations, which makes it easy to pine for the comfortable ways that math has “always been taught.”  Of course, this emotional response can be just as strong for teachers as it can be for students.

In the end, we want our students to gain a deeper understanding of mathematics.  It can be easy to think we need to take every student on a grand adventure like the Hobbits in The Lord of the Rings, to show them how to battle (mathematical) orcs or dragons, and to bring them to a crowning achievement of casting the one ring (perhaps with unity) into the fires of (mathematical) Mount Doom.  But maybe that isn’t what the students need, especially at the beginning of their college careers.  Maybe they just need us to encourage them to go one step further in their mathematical journey than what they had previously thought was possible.  In this post, I would like to highlight a few of my favorite articles that have centered on the theme of creating dynamic and supportive learning environments where students can get swept away in mathematical exploration and play.

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Kindness in the Mathematics Classroom

by Art Duval, Contributing Editor, University of Texas at El Paso

Several years ago, I was teaching a calculus course which included three students who were especially struggling with the material, in spite of regularly attending class. I have a distinct memory of one day, about two-thirds of the way through the semester, when one of these three students, “Nick” (a pseudonym), was the last to leave the classroom, and I thought, “I could do something.” I stopped Nick on his way out the door so we could talk about how he was doing.

I usually have about 50-100 students in all my classes combined, and it had been easy for me to fall into the passive habit of thinking, “I can’t watch out for all of them, and so they have to contact me if they are having problems.” I had always strongly encouraged students to visit me during my office hours, or to email or even call me at home, and I was always very happy to help students who did ask for help. Until then, though, it was their job to reach out to me, instead of the other way around. But not that day, when I stopped Nick on his way out of class. What led me to that point? And what did I do with that impulse afterwards? In a word: Kindness. Continue reading

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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:

https://blogs.ams.org/matheducation/category/active-learning-in-mathematics-series-2015/

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

http://www.ams.org/publications/journals/notices/201702/rnoti-p124.pdf

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:

https://blogs.ams.org/matheducation/2017/09/18/do-we-get-to-work-at-the-board-today/

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:

https://blogs.ams.org/matheducation/2015/07/10/dont-make-em-laugh/

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:

https://blogs.ams.org/matheducation/2017/05/01/i-am-so-glad-you-made-that-mistake/

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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