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. The understanding came to me naturally, and I hardly noticed the process. Even first year algebra didn’t seem like a learning process, more like a set of exercises. So I had mastered a lot of mathematics (well, a lot of algorithms) before I really understood what it was I was learning.

A revelation came in ninth grade, when I was 13. Ms. Blanche Funke, a good math teacher in JHS 135, took some of us during lunch and organized us as a math team, to compete against other local Junior High Schools. Now this is work I have since spent decades doing, and I know now what could have been done. But Ms. Funke didn’t quite. Her idea was to give us advanced training in textbook algebra—not to find ways to make us think differently about that same algebra.

So she gave us the definition of an arithmetic progression, and the standard formulas. And a problem something like: “Insert 3 arithmetic means between 8 and 20.” I loved this work. Plug into one formula, get the common difference, then plug into another formula and get the three required numbers. I could see what I needed to do and took joy in starting the work.

But next to me was my friend David Dolinko, and he was busy drawing something in his notebook—some diagram of a molecule in chemistry. (Professor Dolinko has lately retired from the UCLA School of Law). I poked David. “C’mon. Let’s do this problem. It’s fun!”

David looked at me, as if annoyed at the interruption: “Oh, I did that already. Eight, eleven, fourteen, seventeen, twenty.” And went back to his drawing.

That moment changed my world. Suddenly I realized that these formulas had meaning, were trying to express something. They were expressing that the numbers were ‘equally spaced’. So David could just pick them out—the numbers were small—and didn’t have to bother with the algebra. Algebra has meaning. And if you know its meaning you can use it more effectively. Suddenly, instead of black and white, I saw the world of algebra in color.

I thought about this a long time. The colors attracted me more and more. I wasn’t just good at mathematics. I enjoyed it, and enjoyed being good at it.

Well, the next year I was still sitting next to my friend David, in the last seat, last row of a classroom in the Bronx High School of Science. We were taking geometry, the classic neo-Euclidean syllabus, taught by one Dr. Louis Cohen. He was a somewhat impersonal teacher, or so we thought, but a master of his discipline. And of teaching it. So one day he had covered (I don’t remember how) the theorem that the angles of a triangle sum to 180 degrees. The lesson had gone quickly, so he filled the time with some ‘honors’ problems: the sum of the angles of a quadrilateral, some problems with exterior angles, and so on. And to cap it off, he drew a five-pointed star on the board:

Not a regular figure, but just any one that came to hand, using the usual technique of following the diagonals of an imaginary pentagon. He then asked for the sum of the angles at the points of the star.

My hand shot up, seemingly of its own accord. “180 degrees,” I said, without quite knowing why. And to my horror, Dr. Cohen strode calmly down the aisle to my desk, with a piece of chalk in his hand, handed me the chalk, and asked me to explain to the class how I had figured this out. But I didn’t know how I had figured it out. I just saw it, with intuitive clarity. What was I going to do?

I was lucky that we sat in the back of the room. As I saw him coming towards me, I began to analyze my own thoughts. And as I walked to the front, I figured out what to say. To this day I remember my hand trembling and my voice shaking as I pointed out certain triangles, certain exterior angles, and got the angle measures all to ‘live’ in the same triangle. Dr. Cohen praised me, then gave a slicker version of the proof that must have clarified it for the other students. Of course, there are better ways even than his to prove this statement. If the reader can’t think of a nice proof offhand, take a look (for example) at https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-polygons/v/sum-of-the-exterior-angles-of-convex-polygon (accessed 6/2018). The argument can be adjusted to cover non-convex polygons.

Why is this important? Well, it is important for us to understand that the language of mathematics is a language of thought. And that thought is synonymous with intuitive thought. We sometimes get caught up in the expression of our intuitions, and fail to go back and make clear, even to ourselves, what we are talking about. This phenomenon has deep implications for teaching. How we do this, how we know it has happened, how we integrate it into the teaching of mathematics as a forma language, are all questions we must struggle with. But they are not questions that we can beg. We must somehow be sure that students can eventually understand our results on an intuitive level, whether or not we communicate with them on this level directly. Without that, we are teaching algorithms—even algorithms of proof—and not mathematics.

I invite readers to contribute their ideas to this blog about how to make mathematics accessible on an intuitive level.

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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 https://blogs.ams.org/matheducation/about-the-editors/ 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.

BRIDGES, NOT WALLS

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 PLURAL OF ANECDOTE IS DATA

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 (https://quoteinvestigator.com/2010/06/18/facts-stubborn/, 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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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:

- How do we build relationships with those around us?
- What accomplishments do we reward and recognize?
- 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.

**How Do We Build Relationships With Those Around Us?**

The practice of doing mathematics is one infused with emotional complexity. In *Loving and Hating Mathematics: Challenging the Myths of Mathematical Life*, Reuben Hersh and Vera John-Steiner write:

Mathematics is an artifact created by thinking creatures of flesh and blood… Mathematicians, like all people, think socially and emotionally in the categories of their time and place and culture. In any great endeavor, such as the structuring of mathematical knowledge, we bring all of our humanity to the work… It’s a challenge for everyone to achieve balance in one’s emotional life. It’s a particularly severe challenge for those working in mathematics, where the pursuit of certainty, without a clearly identified path, can sometimes lead to despair. The mathematicians’ absorption in their special, separate world of thought is central to their accomplishments and their joy in doing mathematics. Yet all creative work requires support.

Unfortunately, our mathematical culture does not encourage us to discuss and share the emotional ups and downs of our mathematical lives. While this is also true of our society at large, at least our broader culture acknowledges the role that therapy and counseling can play to improve our lives. In mathematics, our tacit prohibition on discussing the emotional aspects of mathematics has serious consequences for our community, ranging from mental health issues, especially among graduate students, to people unnecessarily working in isolation (e.g. Dusa McDuff’s reflections here), to differences in the sense of belonging and efficacy that people feel in mathematics. As Hersh and John-Steiner point out, these consequences are particularly detrimental in mathematics, due to the nature of our work.

Whether or not individuals choose to actively share their stories with colleagues, teachers, or peers is not the point here — what matters is whether or not individuals in our community feel that they are surrounded by people who are *supportive and willing to listen without judgement*. Creating a culture of authentic inquiry in our interpersonal relationships can provide the social and emotional support that we all require to pursue the creative work of mathematics. Because most mathematicians work in a community framed within professional organizations (colleges, universities, businesses, government entities, etc.), a natural source for inspiration and guidance in this professional context is the literature on organizational culture and leadership. One accessible and relevant resource in this area is Edgar Schein’s book *Humble Inquiry: The Gentle Art of Asking Instead of Telling.* Schein describes Humble Inquiry as the type of inquiry that

maximizes my curiosity and interest in the other person and minimizes bias and preconceptions about the other person. I want to

access my ignoranceand ask for information in the least biased and threatening way. I do not want to lead the other person or put him or her into a position of having to give a socially acceptable response. I want to inquire in the way that will best discover what is really on the other person’s mind. I want others to feel that I accept them, am interested in them, and am genuinely curious about what is on their minds regarding the particular situation we find ourselves in.

Schein is careful to distinguish this form of inquiry from others, such as diagnostic inquiry, confrontational inquiry, and process-oriented inquiry. He emphasizes that “the world is becoming more technologically complex, interdependent, and culturally diverse, which makes the building of relationships more and more necessary to get things accomplished and, at the same time, more difficult.” In other words, the challenge of building authentic interpersonal relationships is not only one for mathematical culture, but for society at large. Schein also emphasizes the importance of individuals in leadership positions learning to use and model authentic inquiry.

To give a concrete idea of how this approach might be used in practice, I share below some questions that we might not ordinarily consider when we are speaking with a student, colleague, employee, or peer. However, questions such as these can powerfully change our perspective toward those around us. I am not suggesting that we routinely ask these questions in regular conversation, but rather that we have these questions in our conscious mind, that we are open to the possibility that the people we interact with have complicated and difficult lives, especially when we are having challenging conversations.

- Does the person I am talking to have access to sufficient food and housing to meet their needs and the needs of their family?
- Does the colleague or student I am talking to have personal challenges or crises I don’t know about, such as a relative, spouse, or child with a serious health issue?
- Has the person I am speaking to been a victim of abuse or assault?
- How many hours each week does this student have to work to pay for their housing and basic expenses?
- Is this student or colleague suffering from PTSD due to military or other service?
- Who is this student or colleague responsible for supporting financially?

Unfortunately, these questions reflect realities that impact many more of our students and colleagues than we might guess. Knowing the answers to these questions would not necessarily change my expectations for student learning in a course, or for job responsibilities for an employee, etc., though it might inspire me to handle situations differently or with more humility. By honestly recognizing and affirming that we are usually ignorant of important aspects in the lives of those around us, we can be more empathetic, flexible, and ethical in our treatment of and relationships with others.

**What Accomplishments Do We Reward and Recognize?**

Complicating our relationships with other people is that, at least in the United States, a dominant social and cultural force is the drive to prize individual achievement over the building of relationships. This force extends throughout our society, not only in mathematics. In *Humble Inquiry*, Schein writes:

When we compare some of the artifacts and behaviors that we observe with some of the [social] values that we are told about, we find inconsistencies, which tell us that there is a deeper level to culture, one that includes what we can think of as tacit

assumptions…The most common example of this in the United States is that we claim to value teamwork and talk about it all the time, but the artifacts — our promotional systems and rewards systems — are entirely individualistic. We espouse equality of opportunity and freedom, but the artifacts — poorer education, little opportunity, and various forms of discrimination… — suggest that there are other assumptions having to do with pragmatism and “rugged individualism” that operate all the time and really determine our behavior.

How does this manifest itself in the mathematical community? As one example, publications are generally used as the currency of our realm, and it is typical that single-author publications are viewed as more valuable than publications resulting from a team of collaborators. Yet it is reasonable to ask what benefits mathematics more: having a person write a paper on their own, or having researchers build relationships and collaborative teams that are able to pool ideas and resources effectively?

Similarly, faculty are frequently given a higher evaluation for single-PI research grants than for leading a team of co-PIs on an infrastructure or education grant. Yet the NSF Education and Human Resources (EHR) and Undergraduate Education (DUE) divisions have funding available and have been actively seeking proposals in mathematics, as evidenced by Jim Lewis’s 2015 AMS Committee on Education presentation and this blog’s 2016 post by NSF program officers about the type of awards funded by EHR and DUE. Again, is this what we actually want to value in our mathematical culture? Is this what most strongly benefits the community? What is it that we collectively want to achieve, and do these recognition values reflect our common goals?

As a third example, consider two hypothetical students: a “smart” student who solves correctly every problem the instructor provides, or a student who sometimes makes errors yet is engaged, persistent, motivated, and dedicated. Which student is most likely to receive praise and support in a math course? Which is typically considered to be the most successful in mathematics? To have the most mathematical potential? To be the highest achieving? Which of these students do we typically provide with encouragement, awards, and recognition?

How will we make explicit, especially at the local level within departments and colleges, what type of collaborative activity we value, and how it will be rewarded? How will we go about recognizing and rewarding the type of activities that are needed to build supportive communities? The first step is one of the most difficult, in that we have to have clear and articulate discussions about these questions. This will almost certainly lead to serious disagreements among colleagues and peers, as mathematicians have strong beliefs about cultural issues; in many important ways, mathematical culture is quite conservative and deferential to tradition, though in my experience we rarely discuss this. Improving our habits of authentic interpersonal inquiry, which we have already seen is necessary for building better relationships, will be required of everyone involved in these types of discussions.

**What Stories Do We Tell About Mathematics and Mathematicians?**

A noteworthy quality of mathematical culture is that we frequently celebrate mathematical mythology rather than mathematical reality. For example, the stories we tell in mathematics are typically mythological, whether they are stories about “brilliant” mathematicians and their work or descriptions of “typical” career paths for mathematicians. For example, I have often overheard undergraduate and graduate students being told some variation of the story that an ordinary “successful career” in mathematics involves finishing high school, then immediately getting an undergraduate degree, then immediately completing a PhD, then obtaining a postdoc, then getting a tenure-track job, and then staying in that job until death. However, many of the mathematicians I know do not fit this simplified plotline at all. Rather, when we begin asking each other about our real stories, the stories that we usually don’t tell in public because they go against our cultural myths, we find that our realities are often much richer and more interesting than the standard narrative.

A deep commitment to the real instead of the mythological also influences our understanding of the nature and history of our discipline. We must be willing to challenge “traditional” and inherited narratives regarding the origins of mathematics, even when these narratives are strongly embedded in our culture. For example, in *The Crest of the Peacock: Non-European Roots of Mathematics*, George Ghevergese Joseph writes:

Evidence of [the contributions of Egyptian and Mesopotamian mathematicians] is not all hidden away in obscure journals or expressed in languages that tend to be ignored by many Western scholars: much is published in English in “respectable” journals and books… The reason for the neglect [of these contributions] was not that the relevant literature was inaccessible or “unrespectable” but something deeper — a serious flaw in Western attitudes to historical scholarship (one not confined to histories of mathematics or science). An excessive enthusiasm for everything Greek, arising from the belief that much that is desirable and worthy of emulation in Western civilization originated in ancient Greece, has led to a reluctance to allow other ancient civilizations any share in the historical heritage of mathematical discovery. The belief in a “Greek miracle” and the way of attributing any significant mathematical discoveries to Greek influences are part of this syndrome.

As an example of the mythological twisting of history that occurs in mathematics courses, in the edition of Stewart’s Calculus textbook that my department uses there are no women mathematicians listed in the index except for a reference to the “witch of Maria Agnesi”, which does not discuss Agnesi’s mathematical contributions at all. This perpetuates the mythology that “no women did math” in the past. However, this does not reflect reality, as there were several prominent women mathematicians and mathematical physicists working in the 1700’s and 1800’s, such as Maria Agnesi, Laura Bassi, Emilie du Chatelet, Mary Sommerville, Sophie Germain, and Sofia Kovalevskaya, who have certainly earned a place in our standard textbooks. We need to train ourselves to reflexively identify mythological stories, and to respond to them by actively seeking the real story.

**Conclusion**

These three questions are certainly not the only ones that should be asked about the culture of mathematics, but they are all of central importance. One of the common themes inspiring these questions is that we must insist that the humanity of mathematics and mathematicians be placed on an equal footing as mathematical knowledge and discovery itself, and be recognized as equally valuable. This is certainly not a new idea, but it is one which we must continue to emphasize, speak about, and share. With this observation in mind, I will end with the following passage from Rochelle Gutiérrez’s talk *Rehumanizing Mathematics: A Vision for the Future*:

]]>If we think that mathematics is not political, not cultural, not any of these other things, then how do we remind ourselves that it is a human endeavor?… Why is it useful to me to say “Rehumanizing” instead of saying “equity”?… Rehumanizing for me… is to honor the fact that for centuries, humans… have been doing mathematics in ways that are humane. It’s not that we have to invent something new for people to be doing, we have been doing it. People see themselves as mathematical, everyone is mathematical… The “Re-” part is a way of acknowledging that there are things that have been erased, and yet people persist.

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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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!

Let’s start with ideas I explicitly share with students. Probably my biggest pet peeve with students is when they find the inverse of some function *y*=*f*(*x*) by first “swapping the *x* and *y* variables”. This is both mathematically and pedagogically unsound, as explained so completely by Frank Wilson, Scott Adamson, Trey Cox, and Alan O’Bryan in their article Inverse Functions: We’re Teaching It All Wrong. When my students make this mistake and, worse, see nothing wrong with it, I share this article with the whole class, and briefly summarize the ideas in class.

I also frequently share with students an article I wrote, One Reason Fractions (and Many Other Topics) Are Hard: Equivalence Relations Up and Down the Mathematics Curriculum. The more I look, the more I see equivalence relations throughout mathematics, causing hidden difficulties for students not just with fractions, but also with vectors, similar matrices, limits, and the difference between permutations and combinations. Beyond sharing this article with students, I keep in mind the difficulties caused when we need to work with equivalence classes as objects, so I can head off students’ confusion before it sets in.

Another article I use frequently for myself is Don’t Count Them Out – Helping Students Successfully Solve Combinatorial Tasks, by Elise Lockwood. I regularly teach Discrete Mathematics, and I now follow her advice to have students “focus on sets of outcomes” and not just the number of outcomes. I start each counting technique lesson by having students make a systematic list of all the outcomes. From the discussion that follows afterwards, some (not all) students understand and even sometimes figure out themselves the formulas that they will need to count such sets when they become too large to make an explicit list.

The article, Mathematics Professors and Mathematics Majors’ Expectations of Lectures in Advanced Mathematics, by Keith Weber made a big impression on me when it came out, and I have shared this one as well with students in proof-based courses. I probably need to review this article again, because I see myself slipping back to old habits, such as not writing down enough details of proofs, that I worked hard to reverse when I first read it.

More recently, I decided to give peer assessment a try, in an introduction to proofs course where I can’t give nearly as much individual feedback as the students need. I started with Elise Lockwood’s article Let Your Students Do Some Grading? Using Peer Assessment to Help Students Understand Key Concepts, and with the references it contains, to build out a system. In the end, it seemed that students learned more from when they assessed their peers than from the feedback they got when their peers assessed them.

Finally, a big part of my attitude these days towards students and mathematics comes from the idea of the growth mindset, that being good at mathematics (or other disciplines) is more a result of hard work than of any genetic predisposition. This idea is stated so beautifully in Ben Braun’s article The Secret Question (Are We Actually Good at Math?).

I invite you to revisit these articles, or browse the rest of our collection, to find a tasty morsel of your own from our kitchen of mathematics teaching and learning.

]]>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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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.

Ben Braun wrote a brilliant article about using open problems as homework. I reread this post at least once per year and continue to find inspiration in it. If we want students to start thinking like mathematicians, and if we want to share the joy of mathematics with them, then why not show them problems whose solution cannot simply be found in the back of a textbook? Why not push them to think deeply about a problem on their own, rewarding them for the effort they have put forth rather than for getting the “right answer”? It is unlikely that anyone will solve an unsolved problem, but it is likely that someone will become more excited about mathematics.

Related to this theme of discovery, Lara Pudwell wrote about an experimental math course she has developed, where students take ownership of problems that they explore and investigate on their own. Students engage in a journey of mathematical discovery that is typically reserved for research experiences and get to see beautiful mathematics that does not always make its way into the undergraduate curriculum.

However, as Bilbo reminds us, we also need to teach students how to keep their feet beneath them through this new experience of learning mathematics. Jess Ellis Hagman shared important lessons on working with students from marginalized groups in an active setting, and Jessica Deshler shared practical tips about promoting gender equity in the classroom on the *MAA Teaching Tidbits blog*. Art Duval’s post on kindness is one of the most beautiful pieces I’ve read recently, reminding us that teaching mathematics is as much a human endeavor as a scientific one.

To me, these posts are inspiring because they show how to incorporate mathematical adventure into the student experience, while also reminding us that the journey is difficult and the road is tough for many students. Lessons of kindness and grace, coupled with an understanding of how to balance learning styles, personality types, and issues of identity within groups are important for creating a mathematical adventure that is engaging and inviting for *all* students.

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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.

Earlier that year, two different people had forwarded to me the text of Francis Su’s powerful MAA Haimo Teaching Award Lecture, about grace in teaching. It was such an out-of-the-box way of thinking about teaching, that the most important part of teaching mathematics to students was not the mathematics itself, but the students. **Students deserve respect just because they are people, and not because of what they accomplish in the classroom.**

The following spring, at our campus’ annual teaching conference, I attended a roundtable discussion of interested faculty on the topic of Loving Kindness at the university; one of several focal points for the discussion was Francis Su’s article. Out of that roundtable grew an informal group of faculty and staff from a variety of departments who were all interested in these ideas. We began to meet for lunch monthly, initially mostly just sharing ways we had shown kindness to our students, and brainstorming how we could do more.

In the midst of hectic days, sometimes encountering colleagues who expressed less charitable views of students, it felt like an absolute oasis to join with these like-minded faculty and staff who also wanted to appreciate students for what they can do, and not what they cannot do. I was impressed by what some of my colleagues around campus were already doing in and out of their classes. One of our members told us how when students show up late to class, instead of making them feel bad, he would sincerely say, “I’m so glad you’re here.” Others ran food drives and helped homeless students, which is easier to incorporate into a sociology class than a mathematics class to be sure, but inspiring nonetheless.

By this point, I had already started to slowly experiment on my own with some things I could do to make more of a difference with students, and to embody some of what I had read in Francis Su’s article. For years I had done little bits of reaching out, by writing “Please see me” on exam papers of students who did poorly on the exam (and congratulatory messages for students who did well), but now I was determined to go beyond that. With the support and encouragement of the kindness group, I pushed myself further to make the extra effort to deliberately be kind to my students.

Around the same time, I became aware of the work by Carol Dweck (and later of that by Jo Boaler) on growth mindset, the idea that people are not born with fixed intelligence, but rather can develop skills through sustained effort. In particular, *every* student who will do the necessary work can learn mathematics. I cannot make students do this necessary work (and, as we will see, some of them have obstacles that have nothing to do with mathematics or their desire to work on it), but with my new focus on kindness I was now determined to reach out to each and every student to try to prevent them from falling through the cracks.

No single change I made was especially innovative or earthshaking, but the effect of each one was amplified by the others, and especially by my attitude. I kept in mind that my students don’t know everything already, especially about mathematics or how the university works (or else why were they there?); see the below wise cartoon. So what did I start to do differently?

**Redouble my efforts to value all student input during class:** I had already been using active learning in my classes for a long time, and so routinely incorporated student input. This often involves responding to student answers by focusing on the parts that are correct. (There is also value in highlighting mistakes.) But now I also paid attention to the effect this has on student attitude, and made sure students knew that I appreciated their response just because they were making an effort.

For some time, I’d been using index cards with each student’s name (and other information they provided) to be sure to call on students at random (and not fall victim to any conscious or unconscious biases I might have); now I used that technique more frequently, and asked for volunteers less frequently. Of course, this keeps students on their toes, but it also visibly demonstrates my belief in growth mindset and that every student can succeed. Also, I increased my awareness of, and sensitivity to, how students respond to being called on to share their ideas, including presenting homework in the front of the class. I try to keep the conversation positive, and any criticism, whether from me or from fellow students, must stay constructive and focus on the mathematics, not the person.

**Learning and using all my students’ names:** I had always learned the names of some of my students, especially the ones who participated more. But now I made it my mission to learn, and use, *every* student’s name. This was not easy for me, as I’ve never been good at remembering names or faces, which is why I’d never made the effort before. But I told myself that I have done difficult things before and, with some work, I could do this. Every semester I am not shy with how hard this is for me, as an opportunity to explicitly illustrate to students the idea of the growth mindset: Just as I believe all students can learn mathematics through dedicated work, even if it does not come easily to them, I can learn their names through dedicated work, even though it does not come easily to me.

And so I began to spend time studying our university’s student photo page for each class. I handed back all graded work individually, making sure to look each student in the face while saying their name. During the first exam of each course, I spent the entire time quizzing myself on names and faces, after asking student to make name placards, which I used to make a seating chart to help me with my self-quizzing. (I had heard this idea several years before, but had never before thought it was worth the effort it would take.) I began to use each student’s name *every* time I called on him or her, even if, at the beginning of the semester, I frequently had to start with “Remind me your name, please.”

**Sending email to students when they missed class:** I started to send email to students when they were absent from class, even though I generally do not require attendance for most of my classes. (Some learning management systems can automate this, but I find it easier, and more meaningful, to do it by hand.) In keeping with my mission of kindness, I try to phrase the message with a tone that is more helpful than derogatory: “Please let me know if you are having any problems, and if there is anything I can do to help.” About half of the students ignore (or at least do not respond to) these messages. But the others do, indeed, let me know their problems.

And while I had expected that they might have problems with mathematics and/or the class, I discovered that **many of my students have difficult lives**. They have problems outside of mathematics or school: family issues; medical issues; medical issues with family, including having to transport relatives to doctors or the hospital; mental health issues; transportation issues; and more. Intellectually, I knew this, but now I understood better. I became more impressed at how my students overcome their obstacles, and genuinely sad for the ones who did not. I could not help with most of these problems, but I could listen. And some students opened up a little more in response.

**Be (a little) more flexible about late assignments:** Knowing more about my students’ lives, and consciously working to respect their difficulties, made me more willing to bend deadlines for students with good excuses. Curiously, I have become more confident in deciding what is or isn’t a good excuse. Perhaps this is because I hear about problems with further advance notice; conversely, when I don’t, I know students had ample opportunity to let me know.

This is a good time to mention “kind” is not the same as “nice”. Being kind does not mean just giving everyone A’s, or assigning less work, or never criticizing; it does mean listening to students, respecting their lives, and responding accordingly.

When I started this journey, I made many of these changes mechanically, and had to work hard to keep kindness in the forefront of my brain. I sent absence email messages when I could, but not all the time, and had to remind myself to be sure to write things like “I’m sorry to hear your mother was sick,” because this sort of attention did not come naturally to me. But then two effects kicked in. First, the gratitude I got from some students for showing them extra attention and respect was a positive reinforcement for me to keep doing so. Then, responding with kindness became more instinctive and more comfortable. I found myself actually *feeling* sorry if a student’s mother had been sick. It became hard for me *not* to send absence email messages. My tone with students, in and out of class, grew more patient and understanding.

Kindness is certainly no panacea. Nick, the student whom I reached out to at the end of that calculus class some years ago, did not pass the course, and neither did the other two students I tried to pay more attention to that semester. But they did notice. Some students now respond to the absence emails with some variation on “Thank you for noticing. None of my other professors have ever shown this interest in their students.” As I’ve increased and intensified my kindness efforts, some students have written comments on my end-of-course evaluations that they appreciate what I do. This suggests that our attitude towards students matters to some of them as much as academic issues do.

But this doesn’t have to be an either/or situation, and kindness may even help, indirectly, with the academics. Since I have started working on kindness, it appeared to me that a few students made more of an effort in the course because of the attention I was paying to them and their issues and interests. I will continue to be kind to my students, though not because it will help them with mathematics, but because it is the right thing to do. What can you do today to show your students a little more kindness?

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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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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.

Several people commented that their moments of inspiration came from venturing into the unknown of the mathematical landscape. Sara Billey (University of Washington) reflected on the joy of solving her first research problem:

One day after studying almost everything known about the problem, I decided to close the books, put away the previously published papers, and pull out a clean sheet of paper. I asked myself “What could I prove that was not written?” I wrote down a formula that combined a fact I knew from the literature with the problem I was trying to solve. I asked if that formula could also be true. I sat there for a while and the proof came to me. I wrote it up in my notebook, and declared that a successful day. About two weeks later, I showed this formula to another student, who got inspired to write down another, related formula. He came back a few days later and said he could prove the conjecture if a third formula was true. Well, I had the feeling I could prove the third formula by putting a bunch of things together. Sure enough, my rather intricate proof worked! It was a very exciting time, and I got to be a part of it because I forced myself to close the textbooks and ask myself a question beyond what was already written.

Matthias Beck (San Francisco State University) echoed these sentiments, writing:

I vividly remember the first original research problem I solved. I knew the literature well enough by that point that I was pretty sure that my theorem was novel, and that caused a certain sense of excitement: the thought that at this point in time nobody else had ever scribbled down what was written on my pieces of paper.

The power of making one’s mark on the mathematical landscape by discovering some fact that was previously unknown to the world is no doubt significant. The feeling of accomplishment that comes with a new research discovery has affected researchers at all levels, from undergraduate REU participants to established researchers.

On the other hand, this venture into the unknown need not be predicated on a research experience. A problem does not need to be new to the world in order for its solution to be meaningful; it just needs to be new to the student. Another respondent recalled her first memorable problem:

The problem was as follows: a pencil costs X, an eraser costs Y, and a pen costs Z. Can you buy these items in such a way that the total cost is M? The point of the solution was that X, Y, and Z were divisible by 7, but M was not. I was eleven at the time. It took me a few hours, but then it finally hit me how to solve it. I felt so excited when I finally got that “a-ha” moment.

In many cases students were moved because they had a sense of ownership of the problem and its solution. It is easy to feel a sense of ownership in research where we write papers with our names on them and other people refer to our results, but this same feeling can be fostered in the classroom. Dylan Helliwell (Seattle University) reflected on proving that the bisectors of a linear pair are perpendicular in his high school geometry class:

I couldn’t immediately put my finger on it, but this problem felt different than the others. I realized that I wasn’t solving for the measure of an angle or showing two things were congruent. I was establishing a new general fact! I was creating new mathematics! (Well, not really. Presumably the author of the textbook knew it was true, too.)

He went on to reflect more about the nature of this problem:

The statement wasn’t immediately obvious. I had to review the precise definitions and draw some examples before I believed it. Then I had to figure out the actual steps to prove it. We were using a “two-column” structure for our proofs and my proof took 31 lines! This was so much more than any of the other problems, and in the end I knew it was correct because I had proved it!

As with many research problems, this experience was significant because the student was challenged to do more than he had been asked to do before. The discovery was genuine *to him*; was new *to him*. His 31-line proof was *his proof*, and the work was meaningful because he had to think of how he could most meaningfully convey the information in those 31 lines. Tim Chartier (Davidson College) reflected on a similar experience in proving that there are infinitely many primes:

We were asked, prior to seeing the proof, to make an argument as to why we might and then why we might not have infinitely many primes. Could we run out of primes? Or, if we have some finite set of primes, is some integer large enough such that we need some new prime to form its prime factorization? Even today, I remember where I sat on campus as I pondered these thoughts. That evening, we worked on a proof of infinitely many primes in preparation for the next day’s class. In class, we developed the short proof. It was like a haiku of mathematics – elegant and focused.

This story inspires two important lessons. First, the students were not told to prove there are infinitely many primes. Instead, they were presented with the question of “are there infinitely many primes?” and asked to explore the meaning of that question. Second, the students first came up with their own proofs, and in the next class they were presented with what Erdös would call the “Book Proof” – the elegant proof that cuts to the core of mathematics. However, there was pedagogical value in this struggle against mathematics and in coming up with *a proof*, even if it was not *the proof*, because it was *their proof*. The students had ownership of the experience.

Many people who found inspiration in proving a theorem or solving a hard problem echoed a sentiment of joy in the realization that there was more to mathematics than rote calculation. José Samper (University of Miami) said

The first problem I remember enjoying was during a math competition in 8^{th}grade. I remember it well: There are 100 people on an island, some always lie, the rest always tell the truth, the islanders all know who lies and who tells the truth. A reporter comes to the island, lines everybody up, and asks the N-th person if there are at least N liars. Everybody answers “yes.” How many liars are there?

This problem made me realize that math could be more than a bunch of dull computations.

I was surprised to learn that several people had deep learning experiences as a result of rote computation. Rachel Chasier (University of Puget Sound) recalled learning her times tables:

After computing the multiples of 9 by hand, I quickly devised my own algorithm: to compute 9*N, put N-1 in the tens place and 9-(N-1) in the ones place so that the digits sum to 9. I tried explaining this to my friend, but it only made them more confused. This was one of the first times I realized I was thinking about math differently than other people and that I had a mathematical mind.

Similarly, Luke Wolcott (Edifecs Software) recalled

In early elementary school we learned about long division, and this set off a competition with me and a friend to divide the biggest numbers we could manage. I remember the passion with which I filled a 8.5 x 11 sheet, the long way, with a really big number, then drew the division bar over it and to the left, and came up with a (shorter) number to put on a piece of paper to its left. I remember the joy I felt when I realized that a list of the first nine multiples of the divisor would be very helpful, and reduced this enormous long division problem into repeated comparison and subtraction.

And finally, Lucas Van Meter (University of Washington) added

When I was in 8^{th}grade I wrote down all the squares and took their differences. I was surprised to find they were all odd. Then I took the differences of the differences and was amazed to find they were all two. Then I decided to do the same thing with cubes and finally found the differences of the differences of the differences were all equal to six. What makes this memory stick is that it was one of the first times I made a mathematical discovery on my own with no outside intervention. It felt like a personal discovery of my own.

I was surprised by these three reflections because we tend to hear that students dislike mathematics because it seems like a bunch of rote, boring computations, while these stories all seemed to stem from that rote computation. But perhaps this shouldn’t be so surprising. The important takeaway seems to be that the inspiration stemmed from discovering something new as a result of playing with all the mathematics they had at their disposal.

Finally, a number of people recalled the feeling of being struck by the simple elegance of a solution to a problem they had failed to solve. Jonathan Ke (Kamiak High School) recalled:

One day, my dad showed me a book filled with mathematical puzzles and questions, one of which was to add up the numbers from 1 to 100. I found a calculator, plugged in as many numbers as I could, got bored, and gave up. Then my dad showed me a video of how Gauss found the sum. I was amazed at the trickery he used and the mathematical explanation of why it worked. I realized that math is far more than just bunch of formulas that I choose to plug-and-chug and get an answer. It is far more complicated and beautiful.

Similarly, a colleague who works in industry wrote:

It was actually a simple problem if you knew trigonometry, but at that time I didn’t (I was in seventh grade). The obvious way to find the angles didn’t work, and I had no clue how to solve it despite a lot of effort. It turned out to be a proof by picture – just a picture – but once the teacher drew it, it was like a bright light at noon right after a pitch black midnight. The discovery was meaningful because it was the result of suddenly and deeply understanding something that you couldn’t understand before…there was joy in transitioning from being hopelessly clueless to knowing. It was one of the first times I saw that you could understand something, but first you had to make it more complicated.

In these stories, we see that failing to solve a problem can also lead to a meaningful experience. Again, the important aspect of these stories seems to be that the students had time to play with the problems first. They devoted considerable efforts to solving the problem, which led to a deeper appreciation of why the ultimate solution was so elegant.

What should these stories mean to us as teachers? On the one hand, many of them contain ideas that are prevalent in leading teaching philosophies:

- We should work to make it clear that mathematics is more than a set of arbitrary rules that govern mindless computations.
- We should create an environment in which students are encouraged to explore and share their ideas, ranging from observations about multiplication tables to new ideas about unsolved problems.
- Students need time to explore and struggle with ideas on their own before they see an elegant and perfectly rigorous solution to a problem.

So how do we do this? Some of these issues have been addressed in this blog and in other places, while others present ongoing issues to be overcome:

- How do we create a grading system in which students can be rewarded for working on a difficult problem as opposed to getting the “right answer”?
- How do we empower students to view themselves as creative problem solvers as opposed to human calculators?
- How do we assign interesting, substantive problems whose solutions cannot be found through a simple Google search?
- In many instances, people were inspired by mathematical explorations of their own design, not by problems that had been assigned to them. How can we foster this type of exploration in the classroom?

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