Greetings to all readers of the AMS Blog on Teaching and Learning Mathematics!<\/p>\n
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.\u00a0\u00a0 \u00a0\u00a0I 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.<\/p>\n
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.\u00a0 At https:\/\/blogs.ams.org\/matheducation\/about-the-editors\/ you will find brief biographies of each of us on the editorial board.<\/p>\n
Meanwhile, I would like to look at two aspects of blogging that we can focus on in the coming year.<\/p>\n
BRIDGES, NOT WALLS<\/p>\n
My intent in taking responsibility for this blog was to further communication in the mathematical community.\u00a0 For me, communication is the most important stimulus for synergy, and lack of communication its most stubborn obstacle.<\/p>\n
I have spent all my professional life in three distinct mathematical communities: research mathematics, mathematics education as an academic field, and classroom mathematics education.\u00a0 Their interactions have always been fruitful, but also problematic.\u00a0 The problems are rarely personal.\u00a0 I seem to get along with most of my colleagues.\u00a0 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.\u00a0 Each group is rewarded for different goals and charged with different responsibilities.\u00a0 And different value systems have grown up around these circumstances.<\/p>\n
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.\u00a0 So responsibility tends to be to the community, the family, the individual student.\u00a0 \u00a0Teachers more and more face the problem of test preparation and accountability.\u00a0 Are the students actually learning good mathematics?\u00a0\u00a0 Could they be learning in more efficient or more accurate ways?\u00a0 The importance of these questions is\u2014often\u2014eclipsed by the need to demonstrate achievement by standards external to the schools in which teachers work.<\/p>\n
Oddly, the accrual of knowledge, the collection of experiences of teachers, is the charge of a different set of institutions: our schools of education.\u00a0 These are academic institutions, and people working in these schools are judged, famously, by publication.\u00a0 But are their research findings having the desired effect in schools and classrooms?\u00a0 Are research questions crafted to respond to the problems of teachers?\u00a0 Is the mathematics being learned precise and pertinent?\u00a0 These are questions that often go unasked by tenure and promotion committees in an academic environment, and sometimes also by journal editors.\u00a0 In its worst cases, the dialogue spins away from the working classroom and the actual mathematics being taught.<\/p>\n
The study of mathematics is likewise an academic discipline, and mathematicians are judge by research publications.\u00a0\u00a0 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.\u00a0 Why work on curriculum, or outreach, or teacher education, when you could publish two more research articles this year?<\/p>\n
These three descriptions, of course, are simply slander against the very people I work with most\u2014those who dare to cross the lines drawn by our institutions around us.\u00a0 And, Dear Reader, you are more than likely to be among these renegades.<\/p>\n
I personally would like to hear more about your successes, about how my somewhat cynical descriptions are wrong.\u00a0 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.<\/p>\n
We need bridges, not walls.\u00a0 We need doors, not fences.\u00a0 How have you been building them?\u00a0 What help did you get?\u00a0 What obstacles did you face?<\/p>\n
THE PLURAL OF ANECDOTE IS DATA<\/p>\n
The negation of this subtitle is an old saw, whose veracity I dare to question.<\/p>\n
It seems to me that educational research does not pay enough attention to anecdotes.\u00a0 Anecdota (the more traditional plural of the word) offer two important opportunities to academic research.\u00a0 The first is the formation of hypotheses.\u00a0 The scientific method, the usual model for seeking knowledge, does not tell us what questions to ask or what to observe.\u00a0\u00a0 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.\u00a0 We are not in control of our unconscious thoughts.<\/p>\n
And I think this is a good thing.\u00a0 The unconscious is a source of creativity, of new ideas.\u00a0 So the best we can do is free ourselves, at times, from rational constraint\u2014then later go back and examine our ideas more rationally. \u00a0But we dare not talk about this process in formal scientific investigations.\u00a0 I think this blog is an excellent venue for just such conversation.\u00a0 What anecdota have you found important in your life?\u00a0 What have you learned from them? Can we use them as springboards for more disciplined investigation?<\/p>\n
More formal investigation involves collection of anecdotes, or shaping of experiences into experiments, or refining the nature of the tale.\u00a0 But I would argue that formal investigation begins with informal observation.\u00a0 This is one sense in which data is a plural of anecdote.<\/p>\n
Is this true even in the pristine world of mathematics?\u00a0 The creation of the human mind, which may or may not deal with observation of reality?\u00a0 I would argue yes.\u00a0 But in fact I will not argue this.\u00a0 I defer to P\u00f3lya, Poincar\u00e9, and other mathematicians who have given us glimpses into their mental workshops.\u00a0 And I invite similar glimpses, or analyses of historical work, here in this blog.<\/p>\n
Another sense in which anecdote is important is in the reification of formally achieved results. \u00a0It happens that, even when an hypothesis is the result of anecdotal observation, the process of formal investigation skews the meaning.\u00a0 The need for rigor of thought, for comparison of data, can constrain the very data we are comparing.\u00a0 This is the deeper meaning of the old joke about psychology, the one whose punchline is \u201cWhat does it tell us about rats?\u201d<\/p>\n
Is this true in public policy?\u00a0 After all, when we make rules for a mass of people, we must \u2018act statistically,\u2019 do the greatest good for the greatest number.\u00a0 Do anecdotes have a place in this arena?\u00a0 Well, yes.\u00a0 Let\u2019s get real.\u00a0 And question another old saw.<\/p>\n