My “First” Mathematical Problem and What It Means

I am inspired, by several previous blog entries, to write about my own mathematical awakening, and what I’ve learned from reflecting on it.

I went to New York City Public Schools, in the Bronx.  I always enjoyed arithmetic and mastered it easily.  I remember not knowing what ‘fractions’ were, but don’t remember learning about them, any more than I remember learning to read. Continue reading

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Greetings to all readers of the AMS Blog on Teaching and Learning Mathematics!

As editor for this blog for the coming year, I would like to invite you to continue its lively and meaningful conversation, of the quality that has been established by my predecessors.     I am most grateful to Ben Braun for setting up this useful tool for communication, and hope to continue and expand the dialogue it has afforded us.

I am equally grateful to Art Duval, Steven Klee, and Diana White for graciously agreeing to continue on this editorial board, and for Priscilla Bremser, who has retired from the board, for her service to the community.  At you will find brief biographies of each of us on the editorial board.

Meanwhile, I would like to look at two aspects of blogging that we can focus on in the coming year.


My intent in taking responsibility for this blog was to further communication in the mathematical community.  For me, communication is the most important stimulus for synergy, and lack of communication its most stubborn obstacle.

I have spent all my professional life in three distinct mathematical communities: research mathematics, mathematics education as an academic field, and classroom mathematics education.  Their interactions have always been fruitful, but also problematic.  The problems are rarely personal.  I seem to get along with most of my colleagues.  Even when we disagree, even to the extent of having words, things eventually return to a normal, collegial state. The problems arise, I think, from the institutions we live in.  Each group is rewarded for different goals and charged with different responsibilities.  And different value systems have grown up around these circumstances.

Classroom mathematics, especially on the pre-college level, is mainly the charge of our public schools, which are organized in the US by the smallest and most local units of government.  So responsibility tends to be to the community, the family, the individual student.   Teachers more and more face the problem of test preparation and accountability.  Are the students actually learning good mathematics?   Could they be learning in more efficient or more accurate ways?  The importance of these questions is—often—eclipsed by the need to demonstrate achievement by standards external to the schools in which teachers work.

Oddly, the accrual of knowledge, the collection of experiences of teachers, is the charge of a different set of institutions: our schools of education.  These are academic institutions, and people working in these schools are judged, famously, by publication.  But are their research findings having the desired effect in schools and classrooms?  Are research questions crafted to respond to the problems of teachers?  Is the mathematics being learned precise and pertinent?  These are questions that often go unasked by tenure and promotion committees in an academic environment, and sometimes also by journal editors.  In its worst cases, the dialogue spins away from the working classroom and the actual mathematics being taught.

The study of mathematics is likewise an academic discipline, and mathematicians are judge by research publications.   Mathematicians who get involved in education, who work with schools of education or public schools are sometimes seen as neglecting their duty to their own profession.  Why work on curriculum, or outreach, or teacher education, when you could publish two more research articles this year?

These three descriptions, of course, are simply slander against the very people I work with most—those who dare to cross the lines drawn by our institutions around us.  And, Dear Reader, you are more than likely to be among these renegades.

I personally would like to hear more about your successes, about how my somewhat cynical descriptions are wrong.  Perhaps most important, I would like to hear about how the problems I raise above, of institutional demands thwarting personal efforts, have been dealt with.

We need bridges, not walls.  We need doors, not fences.  How have you been building them?  What help did you get?  What obstacles did you face?


The negation of this subtitle is an old saw, whose veracity I dare to question.

It seems to me that educational research does not pay enough attention to anecdotes.  Anecdota (the more traditional plural of the word) offer two important opportunities to academic research.  The first is the formation of hypotheses.  The scientific method, the usual model for seeking knowledge, does not tell us what questions to ask or what to observe.   The wellsprings of hypotheses are unconscious: they lie in our reactions to the thoughts and actions of others, our responses to something that catches our attention in our environment.  We are not in control of our unconscious thoughts.

And I think this is a good thing.  The unconscious is a source of creativity, of new ideas.  So the best we can do is free ourselves, at times, from rational constraint—then later go back and examine our ideas more rationally.  But we dare not talk about this process in formal scientific investigations.  I think this blog is an excellent venue for just such conversation.  What anecdota have you found important in your life?  What have you learned from them? Can we use them as springboards for more disciplined investigation?

More formal investigation involves collection of anecdotes, or shaping of experiences into experiments, or refining the nature of the tale.  But I would argue that formal investigation begins with informal observation.  This is one sense in which data is a plural of anecdote.

Is this true even in the pristine world of mathematics?  The creation of the human mind, which may or may not deal with observation of reality?  I would argue yes.  But in fact I will not argue this.  I defer to Pólya, Poincaré, and other mathematicians who have given us glimpses into their mental workshops.  And I invite similar glimpses, or analyses of historical work, here in this blog.

Another sense in which anecdote is important is in the reification of formally achieved results.  It happens that, even when an hypothesis is the result of anecdotal observation, the process of formal investigation skews the meaning.  The need for rigor of thought, for comparison of data, can constrain the very data we are comparing.  This is the deeper meaning of the old joke about psychology, the one whose punchline is “What does it tell us about rats?”

Is this true in public policy?  After all, when we make rules for a mass of people, we must ‘act statistically,’ do the greatest good for the greatest number.  Do anecdotes have a place in this arena?  Well, yes.  Let’s get real.  And question another old saw.

“Facts are stubborn things.”  This quote has been variously attributed (, accessed 4/2018), most famously to Samuel Adams.  And I’m not sure it’s true.  In public discussion, facts can be pliable, ductile, malleable.  Even when research methods are unquestionably rigorous, the questions of which facts to adduce and how they relate to the decision being taken are themselves not data-driven.  They are matters of judgment.

I find that opinions are much more stubborn than facts.  And opinions are often based on anecdota, on cases that are personally known to the holder of the opinion, or stories—anecdotes—that ring true on an individual level.  So even in the area of public policy, if we don’t pay attention to anecdotes, to their meaning to individuals, we will not be able to act effectively.

Anecdotes about how research is used, how it plays out in the field, what effect research has on practice, can offer valuable feedback to the researcher. I invite readers to use this blog as a place to tell stories of direct experience, of the sort deemed ‘anecdotal’ in more formal academic research.








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Mathematical Culture Beyond the Classroom

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

Mathematics is the result of human curiosity and our desire to explain, predict, and explore observed and imagined phenomena. Our shared curiosity and sense of wonder is the wellspring of our mathematical culture. Yet a common sadness is felt by those who love mathematics, as we witness people’s wonder and curiosity stilled by strong cultural and social forces. As Paul Lockhart writes in A Mathematician’s Lament:

If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done… Everyone knows that something is wrong.

Many mathematicians, mathematics teachers, and mathematics fans and ambassadors share these feelings. It is natural that for many of us, our primary responses to this arise through our teaching, in an effort to help students rediscover their natural sense of mathematical joy and curiosity. However, my belief is that this situation is actually a symptom of an issue that extends beyond teaching and learning, and beyond the confines of mathematics. I believe that at its core, this issue involves our cultural responses to three questions:

  1. How do we build relationships with those around us?
  2. What accomplishments do we reward and recognize?
  3. What stories do we tell, especially about mathematics and mathematicians?

As powerful as classroom cultures and environments can and should be, I believe we must have an even grander vision for ourselves and our community. We need to find ways to change some core qualities of the culture of mathematics itself, qualities related to the three questions above. A central challenge is that these changes are generally orthogonal to cultural norms of society at large. In this article, I share some reflections on these questions, along with ideas for how we can work together to meet the challenge of improving the culture of mathematics both within and beyond the classroom. Continue reading

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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!

Continue reading

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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:

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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