January 2023<\/p>\n
Mark Saul<\/p>\n
I have lived my professional life\u2014half a century by now\u2014passing among three islands, three professional communities interested in mathematics.\u00a0 There are bridges between them, but I am always made cognizant of crossing a bridge and being \u2018somewhere else\u2019. And there are walls, some artificial and others naturally occurring.\u00a0 In this note I want to talk about both bridges and walls.<\/p>\n
I am not here discussing the contributions of each of these three communities to accomplishment or knowledge. I leave acknowledgment of these to the various journals and prize committees.\u00a0 Here I want to talk about how each community can support the other.<\/p>\n
Let\u2019s start with research mathematicians.\u00a0 In some ways, this group has been the easiest for me to work with (although I recognize that this may not be everyone\u2019s experience).\u00a0 Talking with them about mathematics is like having a conversation with a native speaker of a language not your own: even if your thoughts are not clearly expressed, they are understood, re-interpreted, and made sense of.\u00a0 Indeed, my usual interaction with a mathematician involves a content question I have, sometimes well-formulated and sometimes not, which the mathematician immediately knows the answer to.\u00a0 But it had never occurred to her or him to ask that particular question.\u00a0 We enlighten each other.<\/p>\n
Often a mathematician will initially give an explanation two or three levels above the question.\u00a0 It\u2019s comprehensible, but not \u2018digestible\u2019, until we dialogue together to make it so for students\u2014with whom the question has sometimes originated.\u00a0 That\u2019s part of the collaboration.<\/p>\n
For example, sometimes a nice heuristic proof is given that the sum of the exterior angles of a triangle is always 360 degrees.\u00a0 Simply drive a car around the triangle, keeping track of the angle through which the front wheels turn.\u00a0 These are the exterior angles of the triangle.\u00a0 Since the car ends up pointing in the same direction in which it started, the sum of these angles must be 360 degrees.<\/p>\n
But this argument cannot be complete as phrased above, because one can make the same sounds while pointing to a picture of a spherical triangle\u2014and the conclusion is not valid.\u00a0 Anyone teaching differential geometry will see what the story is here\u2014I\u2019ve asked a few times.\u00a0 But the challenge is to explain it to a tenth grader, who might have asked the question.\u00a0 In particular, the only difference in the situations that a tenth grader will understand is that Euclid\u2019s parallel postulate does not hold on a sphere.\u00a0 Where does the argument use the parallel postulate?\u00a0 It took many conversations with several mathematicians\u2014not all of them geometers\u2014to resolve this question.<\/p>\n
How would you resolve it?<\/p>\n
Researchers in mathematics education: Talking to them, in my role as a teacher, I often feel that I\u2019ve cleaned my eyeglasses.\u00a0 They have a vision of the field which cannot result solely from work in one\u2019s own classroom or lecture hall.\u00a0 For research mathematicians, they can ask questions outside the scope of the mathematics itself.\u00a0 For teachers, they find ways to transcend the intensely personal and idiosyncratic nature of teaching.\u00a0 For example, I have come to the conclusion that I cannot accurately observe my own teaching.\u00a0 I can do meta-cognition while teaching, I can analyze and process feedback from my students.\u00a0 But it is not given to me\u2014as a teacher\u2014to generalize, to say much that is useful to another teacher, or to describe what I am doing to someone without classroom experience.\u00a0 That is what education researchers do well.<\/p>\n
A signal contribution of education researchers is that they can act as a voice for the teaching profession.\u00a0 From outside the classroom, it is very difficult to understand the interactions between student and teacher.\u00a0 Education researchers, however, have developed the eyes to see what is going on in a classroom, and the language to interpret it to others.\u00a0 They can paint a wider picture of the field, one that combines the experience of many teachers.<\/p>\n
Too, they can find ways to validate teaching, checking it against external standards.\u00a0 The most obvious such standard, but in many ways the crudest, is analysis of test scores.\u00a0 But there are other ways to validate teaching: attitudinal studies, longitudinal studies, interviews with students or with professionals about their own education.\u00a0 All these methodologies are available to the researcher in mathematics education, and all can inform anyone teaching mathematics, on any level.<\/p>\n
Where do the difficulties in collaboration lie?\u00a0 Well language can be a problem.\u00a0 Sometimes it is the language of the scientific method that speaks most clearly to administrators, funders, politicians, even parents.\u00a0 But not necessarily \u00a0to the teacher.\u00a0 This creates a tension between the working classroom and the higher levels of administration and funding.\u00a0 Only skill and patience will defuse this tension.<\/p>\n
There has been a recent, and I think not too useful, trend to invoke advances in neuroscience to analyze teaching.\u00a0 I don\u2019t doubt that there are connections here.\u00a0 The trouble is that of overgeneralization.\u00a0 Many have claimed recently that neuroscience suggests, verifies, or supports particular instructional practices.\u00a0 In my view, these claims jump the gun in an intellectual land-grab.\u00a0 If they are true, the truth is on a metaphorical level: changes in the brain are parallel to changes in cognition.\u00a0 But we are quite far from peering into the brain to see if a student can solve a differential equation or has followed a geometric proof.<\/p>\n
And there is a deeper danger here.\u00a0 Certainly, it is the right time to ask questions about how physiology reflects cognition, whether or not we can answer them definitively.\u00a0 The more pernicious danger is that of ossifying the field with neo-positivist, reductionist thinking.\u00a0 There is a common perception that our knowledge is not ‘complete’ unless it is reduced to the level of physiology and ultimately of chemical interactions.\u00a0 Maybe, in time, we will be able to do this.\u00a0 But right now the ways we have to understand teaching and learning are on a vastly different level.\u00a0 I cannot see whether my students\u2019 hypothalamus glands have grown, or if their visual cortex is lighting up.\u00a0 I can only see what they do and say, and infer\u2014in completely different ways\u2014what they are thinking.\u00a0 That is part of my expertise as a teacher.<\/p>\n
That is, the danger to the teaching profession is of under-valuing pedagogical expertise.\u00a0 People with only an outsider\u2019s understanding of the educational process will dismiss the ability of the teacher in favor of the physical evidence of the CT scan.\u00a0 The profession is in enough trouble without this misunderstanding.<\/p>\n
Finally, teachers.\u00a0 I love working with teachers.\u00a0 They always have great ideas, and excellent critical faculties:\u00a0 \u201cThis won\u2019t work\u2026. That will take longer than you think\u2026.. This is good for 7th<\/sup>\u00a0 grade but not for 4th<\/sup> grade\u2026\u201d\u00a0 Their expertise in such matters is a contribution that is unique to them, and is rarely acknowledged by the other two groups I am discussing.<\/p>\n But teaching is an isolating experience.\u00a0 You are The Authority in your own classroom, and it takes some effort to surrender this authority, to struggle to understand the relation of your classroom work to the rest of the world.\u00a0 So teachers often like to do things themselves, or as part of a group of teachers, rather than relying on the energies and expertise of people in adjacent fields.<\/p>\n Also, teaching is idiosyncratic.\u00a0 Working teachers use their own personalities to interact with the students, and each teacher does it with subtle but effective variation\u2014a complicated enough situation even without taking into account the immense variation in students\u2019 personalities, or their interaction in a group. \u00a0So generalizations about what teachers should do, how they should talk, what \u2018teaching behavior\u2019 should be, quickly become weak when applied globally.<\/p>\n Lee Shulman has made famous the phrase \u201cpedagogical content knowledge\u201d, meaning the specific kinds of mathematical knowledge teachers need to be effective.\u00a0 Deborah Ball and Hyman Bass have gone far to operationalize this phrase.\u00a0 I often find that teachers lack \u201cpedagogical research knowledge\u201d, the specific ways that educational research applies to their classroom, to their students, to their own teaching styles.\u00a0 I don\u2019t think the field of mathematics education has recognized this gap.<\/p>\n But perhaps the most important misunderstanding of teaching is of its emotional content.\u00a0 At its heart, teaching is motivating: causing to learn.\u00a0 And by far the strongest motivation for any of us, and for any action\u2014stronger even than financial, I would argue\u2014is emotional motivation.\u00a0 So teachers use emotional cues to motivate, to assess, and even to get at intellectual misunderstandings. And this process is very difficult to analyze or generalize.\u00a0 A large part of the experience of teaching does not fit a scientific paradigm.\u00a0 The scalpel of the scientific method, so keen in analyzing numerical data, quickly becomes a blunt instrument when applied to the study of human interactions.<\/p>\n There are difficulties in collaboration.\u00a0 But difficulties can be transcended, with some effort.\u00a0 Why is this effort so rarely applied?\u00a0 The dark side of any collaboration, I am convinced, is insecurity.\u00a0 To work with another academic or professional, one must be willing to admit to the limits of one\u2019s knowledge and experience.\u00a0 Perhaps more difficult, one must be willing to cede intellectual \u2018territory\u2019 to another.\u00a0 An example that keeps coming up is discussion of the term \u2018variable\u2019.\u00a0 I have heard mathematicians dismiss it as meaningless, then use it (meaningfully!) in their own very next sentence.\u00a0 The problem is that it is not really a mathematical term, but a psychological one, used in the study of learning mathematics.\u00a0 That there is a concept there is pretty much clear from the ubiquity of the term, but what that concept is needs significant exploration\u2014by people with various backgrounds and expertise.\u00a0 Since the term is used so often in mathematical contexts, it is hard for the mathematician to let go of it, and acknowledge the need for analysis on the level of pedagogy.<\/p>\n Who\u2019s the villain of this piece?\u00a0 I think no one person or group.\u00a0 Difficulties I\u2019ve perceived in collaboration are not the result of negativity, but of struggle.\u00a0 And yet there are villains, of a sort.\u00a0 The villains are our institutions, that have needs that transcend our own, influence our identities and agendas, and keep us siloed.<\/p>\n Institutions shape our relationships.\u00a0 For example, one of the least discussed aspects of the late and very much unlamented Math Wars of the last century is their political and economic roots.\u00a0 (I learned this working at NSF.)\u00a0 The typical education grant, say for materials development, is in the millions of dollars.\u00a0 The typical mathematics research grant is in the hundreds, or even just tens, of thousands, an order of magnitude lower.\u00a0 More money means more institutional clout, so education specialists acquired more influence in academia.\u00a0 Mathematicians understandably resented this, and this resentment goes far to account for the rancor that accompanied any purely intellectual dialogue during that period.<\/p>\n Institutions shape research agendas.\u00a0 One reason that education research tends to be skewed towards sociological (rather than anthropological) methods is that the former can be accomplished more quickly.\u00a0 So the young faculty member seeking tenure\u2014or the established scholar seeking influence–can rack up publications more quickly.\u00a0 A close study of teacher-student interactions, on the other hand, is slow and painstaking, and often requires the development of new tools.\u00a0 Too often, educational research does not answer teachers\u2019 most burning questions.<\/p>\n And it\u2019s not just academia.\u00a0 Funding agencies have their own needs.\u00a0 For example, longitudinal studies (even data-intensive studies), which give us deeper results about the effects of our teaching, take years to accomplish.\u00a0 And since the funding horizon for most research is at most five years, researchers undertaking such studies are taking a chance on their ability to continue them.<\/p>\n There\u2019s an old joke about a person who\u2019s dropped his key at night near a park bench.\u00a0 A helpful passerby asks: \u201cWhere did you drop your key?\u201d<\/p>\n The reply: \u201cOver there, at the other end of the bench, where it\u2019s dark.\u201d<\/p>\n \u201cThen why are you looking at this end of the bench?\u201d<\/p>\n \u201cWhy should I look at the other end?\u00a0 It\u2019s too dark.\u00a0 I\u2019ll never find anything.\u201d\u00a0 Sometimes researchers look for the key in areas that are easier to investigate, rather than addressing the questions that are more likely to hold the key.<\/p>\n Institutions shape our teachers.\u00a0 Teachers are (typically) employed by a school district, which has its own institutional needs: to look good, to attract students, to prove to their constituency that they are doing the job of education efficiently.\u00a0 And so, for example, districts generally under-value both the contributions of teachers to the larger profession and the need for teachers to communicate with each other and the outside world.\u00a0 The usual label for the latter is \u2018professional development\u2019.\u00a0 I like to apply this phrase in two senses: the development of the professionalism of individual teachers, and the development of teaching as a profession, and not just an occupation.<\/p>\n Institutions.\u00a0 We can\u2019t live with them and we can\u2019t live without them.\u00a0 But we are constrained to live within them.<\/p>\n Having read over this column several times, I am concerned that it be taken as a series of complaints.\u00a0 That\u2019s not my intention here.\u00a0 Rather, it is to celebrate the productive and enjoyable times I have had collaborating with teachers, with mathematicians, and with educational researchers.\u00a0 Please keep asking those questions.\u00a0 Keep offering your expertise to the rest of us, and keep on admitting to the limits of your own.<\/p>\n I thank all of you who have been helping us to build bridges.<\/p>\n ACKNOWLEDGMENTS<\/p>\n I would like to thank the following people, who have contributed to the writing of this column and to my thoughts on the subject over the years: Richard Askey, Deborah Ball, Hyman Bass. Douglas Clements, Ed Dubinsky, I. M. Gelfand, Karen King, Judy Roitman, Hung-Hsi Wu.<\/p>\n <\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" January 2023 Mark Saul I have lived my professional life\u2014half a century by now\u2014passing among three islands, three professional communities interested in mathematics.\u00a0 There are bridges between them, but I am always made cognizant of crossing a bridge and being \u2018somewhere else\u2019. 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