Comparing Educational Philosophies

I have recently heard several mathematicians claim that the educational philosophies of Math Circles and the Inquiry Learning Community are essentially the same. I disagree. I will contrast the differences between these two approaches, along with two other common educational philosophies in the United States. All four approaches to math education differ significantly both in terms of the overall instructional goals and in terms of the primary methods used to achieve these goals.

In my experience, even the originators and staunchest advocates for specific philosophies incorporate the other approaches when putting their favored one into practice. Instructors should think carefully about the goals they have for a given set of students, and then choose a combination of approaches they believe most likely to meet those goals.

I would like to invite you to comment on any thoughts that you have about these lists.  A few questions that I have for readers include:

  • Which blend of these pedagogical approaches have you found congenial for specific audiences?
  • Are there other approaches that are essentially different that I did not include in this list?
  • What steps do you find helpful when transitioning students who are used to one pedagogical approach to another?
  • There is a tendency for students from high poverty schools to be exposed primarily to Traditional Math approaches. The Common Core State Standards represent an effort to slightly improve on this prevailing norm, by pushing teachers in the direction of Conceptual Math instruction. What do you think people who love math and teaching should do to improve access to high quality math education for these students?

I look forward to hearing your ideas!

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Ideas under construction: children saying what they know

Alli entered kindergarten quite skilled at mental addition and proud of her skill. Subtraction followed quickly. Near the end of her kindergarten year, Alli bounced into class and said that her father had taught her about negative numbers. To assure that I knew about them, she explained, “If you subtract 20 from 10, you get negative 10.” I asked, “And what if you subtract ten from seven?” She thought a second and chirped “Negative three.” Then she explained how to write a negative number—“Just put a minus in front”—and added “There are negative numbers and positive numbers.” And that was it. As with many conversations with 5-year-olds, this one ended as abruptly as it began. Continue reading

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Mathematical Practices

My work and that of my colleagues at Education Development Center has always put mathematical thinking—the habits of mind that are indigenous to our discipline—at the core of our work with teachers. What we’ve learned from expert teachers has led me to think more carefully about what it means to “work like a mathematician.” The attached essay details some of the things I’ve learned.

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Beyond Grades: Feedback to Stimulate Rethinking and Intellectual Growth

By Cody L. Patterson and Priya V. Prasad, Department of Mathematics, University of Texas at San Antonio

We frequently use writing assignments to encourage students to examine topics in greater depth than what we cover in class, and we emphasize to our students that writing assignments constitute one of the most important dimensions on which students’ thinking will be assessed. Yet in our early implementation of these assignments, we frequently received work that did not reflect students’ full potential for understanding the topic explored in the assignment. In these cases, because we were using a roughly linear scale to assign each submission a numerical grade, which would then become part of a student’s overall grade in the course, we faced a difficult decision.

  • We could assign low grades to work that did not meet our expectations; this would have the advantage of sending students a clear message about whether their work meets the standards of the course, but it might demotivate students or limit the potential of an otherwise competent student to earn a good grade in the course.
  • We could assign moderate-to-high grades to such work; this would lower the stakes of failure for students, but it would also require us to endorse work that does not meet a high standard.

Neither option, however, seemed to address our greatest concern: that some of our students had not explored and communicated about the topic of the assignment with the depth desired. Moreover, numerical grades allowed many students to decide that they had gained enough, grade-wise, out of the assignment, and did not need to take advantage of opportunities to revise their work. In this article, we’ll talk about our journey toward crafting and implementing a grading scheme for writing assignments that provides greater opportunity for student learning and growth. While we use writing assignments specifically in the context of content courses for preservice teachers, we believe much of our advice is adaptable to other mathematics courses. Continue reading

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Thinking Outside the Textbook

By Steven Klee, Contributing Editor, Seattle University

When two grandmasters face off in a chess tournament, they are faced with a complicated bit of game theory.  If you were in one of their positions, you would prepare for the match by studying your opponent’s games in great depth.  You would study board positions they had created, looking for weaknesses in their defenses and blunders their previous opponents (or they themselves) had made.  It would be safe for them to assume that you could have a strategy in mind to counter any of their strategies that had previously been successful.

Of course, your opponent would naturally study your body of work in the same way.  Therefore, by the time you sat down at the board, there would be a natural expectation that you know that your opponent knows that you know as much as you possibly could about them, and likewise they have the same expectation of you.

As a consequence, the natural strategy for determining who is the better player is to try to avoid these positions in the first place.  Don’t allow the board to get to a point where you have been defeated in the past.  Don’t allow the board to get to a point where you have been successful in the past because your opponent might know how to turn that position to their advantage.  Get away from what has been seen before and create a new position that truly tests the skill of each player.  There’s a term for this – chess players call it going “off book.”

To chess enthusiasts, this moment is exciting.  It’s the moment in the game when the board reflects a position that has never been recorded in a tournament.  It is an opportunity for observers to experience chess history and witness the creation of new knowledge or strategy.  Every move is new and the anticipation of what might come next is thrilling.

Why is this relevant to math education?

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On Being Imperfect

By Gizem Karaali, Pomona College

I have a secret: For the last year or so, my nine-year-old daughter and I have been trying to develop a meditation practice. This guy, Andy, who leads us daily through meditation sessions facilitated by a phone app, has become a familiar name between my daughter and me. (Even my five-year-old occasionally mentions Andy when going to bed at night; sleepy-time Andy tells us to lie on our backs and close our eyes and start by saying good night to our toes.) One day my daughter posed me a question. We had just completed our ten-minute session for the day. She was not willing to move on yet, it seemed, so I waited. She finally formulated her question and asked, very carefully, “Mom, is Andy perfect?”

This is a profound question even though it has an easy answer: “No”. If Andy is human, he is not perfect. Yet none of his flaws are really my business, because he is effectively teaching us to be better. He is consistently, with kindness, in good humor, and with no sign of condescension, telling us how we can do better.  In every session, or let me be honest, in most sessions, we learn from him.

Aren’t there always those we look up to who exemplify ideals we wish to uphold or those who represent the type of character that leaves us in awe? Reading Art Duval’s post on kindness in this very blog, listening to Francis Su’s talk on mathematics for human flourishing, digging into clear critiques of our community fearlessly dispensed by mathematicians such as Piper Harron and Izabella Laba, some of us might wistfully say: but I am not good enough. I am not as capable. I am not as kind. I am not as forgiving. I am not as insightful. I am not as brave.

Now let me rephrase that for you so as to be clear. All of the above are ways of saying the same thing: “I am not ready to be vulnerable.” All these amazing people are amazing partially because they are willing to put themselves out there, trying to live up to their own ideals. (And for some, an alternative may not even exist.) Do they ever falter? Maybe they do. It is not my story to tell. Again, like Andy, any of their possible faltering is none of my business. What is my business is what I learn from them.

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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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