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 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.