January 2023

Mark Saul

I have lived my professional life—half a century by now—passing among three islands, three professional communities interested in mathematics. There are bridges between them, but I am always made cognizant of crossing a bridge and being ‘somewhere else’. And there are walls, some artificial and others naturally occurring. In this note I want to talk about both bridges and walls.

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. Here I want to talk about how each community can support the other.

Let’s start with research mathematicians. In some ways, this group has been the easiest for me to work with (although I recognize that this may not be everyone’s experience). 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. 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. But it had never occurred to her or him to ask that particular question. We enlighten each other.

Often a mathematician will initially give an explanation two or three levels above the question. It’s comprehensible, but not ‘digestible’, until we dialogue together to make it so for students—with whom the question has sometimes originated. That’s part of the collaboration.

For example, sometimes a nice heuristic proof is given that the sum of the exterior angles of a triangle is always 360 degrees. Simply drive a car around the triangle, keeping track of the angle through which the front wheels turn. These are the exterior angles of the triangle. Since the car ends up pointing in the same direction in which it started, the sum of these angles must be 360 degrees.

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—and the conclusion is not valid. Anyone teaching differential geometry will see what the story is here—I’ve asked a few times. But the challenge is to explain it to a tenth grader, who might have asked the question. In particular, the only difference in the situations that a tenth grader will understand is that Euclid’s parallel postulate does not hold on a sphere. Where does the argument use the parallel postulate? It took many conversations with several mathematicians—not all of them geometers—to resolve this question.

How would you resolve it?

Researchers in mathematics education: Talking to them, in my role as a teacher, I often feel that I’ve cleaned my eyeglasses. They have a vision of the field which cannot result solely from work in one’s own classroom or lecture hall. For research mathematicians, they can ask questions outside the scope of the mathematics itself. For teachers, they find ways to transcend the intensely personal and idiosyncratic nature of teaching. For example, I have come to the conclusion that I cannot accurately observe my own teaching. I can do meta-cognition while teaching, I can analyze and process feedback from my students. But it is not given to me—as a teacher—to generalize, to say much that is useful to another teacher, or to describe what I am doing to someone without classroom experience. That is what education researchers do well.

A signal contribution of education researchers is that they can act as a voice for the teaching profession. From outside the classroom, it is very difficult to understand the interactions between student and teacher. Education researchers, however, have developed the eyes to see what is going on in a classroom, and the language to interpret it to others. They can paint a wider picture of the field, one that combines the experience of many teachers.

Too, they can find ways to validate teaching, checking it against external standards. The most obvious such standard, but in many ways the crudest, is analysis of test scores. But there are other ways to validate teaching: attitudinal studies, longitudinal studies, interviews with students or with professionals about their own education. All these methodologies are available to the researcher in mathematics education, and all can inform anyone teaching mathematics, on any level.

Where do the difficulties in collaboration lie? Well language can be a problem. Sometimes it is the language of the scientific method that speaks most clearly to administrators, funders, politicians, even parents. But not necessarily to the teacher. This creates a tension between the working classroom and the higher levels of administration and funding. Only skill and patience will defuse this tension.

There has been a recent, and I think not too useful, trend to invoke advances in neuroscience to analyze teaching. I don’t doubt that there are connections here. The trouble is that of overgeneralization. Many have claimed recently that neuroscience suggests, verifies, or supports particular instructional practices. In my view, these claims jump the gun in an intellectual land-grab. If they are true, the truth is on a metaphorical level: changes in the brain are parallel to changes in cognition. 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.

And there is a deeper danger here. Certainly, it is the right time to ask questions about how physiology reflects cognition, whether or not we can answer them definitively. The more pernicious danger is that of ossifying the field with neo-positivist, reductionist thinking. 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. Maybe, in time, we will be able to do this. But right now the ways we have to understand teaching and learning are on a vastly different level. I cannot see whether my students’ hypothalamus glands have grown, or if their visual cortex is lighting up. I can only see what they do and say, and infer—in completely different ways—what they are thinking. That is part of my expertise as a teacher.

That is, the danger to the teaching profession is of under-valuing pedagogical expertise. People with only an outsider’s understanding of the educational process will dismiss the ability of the teacher in favor of the physical evidence of the CT scan. The profession is in enough trouble without this misunderstanding.

Finally, teachers. I love working with teachers. They always have great ideas, and excellent critical faculties: “This won’t work…. That will take longer than you think….. This is good for 7^{th} grade but not for 4^{th} grade…” 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.

But teaching is an isolating experience. 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. 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.

Also, teaching is idiosyncratic. Working teachers use their own personalities to interact with the students, and each teacher does it with subtle but effective variation—a complicated enough situation even without taking into account the immense variation in students’ personalities, or their interaction in a group. So generalizations about what teachers should do, how they should talk, what ‘teaching behavior’ should be, quickly become weak when applied globally.

Lee Shulman has made famous the phrase “pedagogical content knowledge”, meaning the specific kinds of mathematical knowledge teachers need to be effective. Deborah Ball and Hyman Bass have gone far to operationalize this phrase. I often find that teachers lack “pedagogical research knowledge”, the specific ways that educational research applies to their classroom, to their students, to their own teaching styles. I don’t think the field of mathematics education has recognized this gap.

But perhaps the most important misunderstanding of teaching is of its emotional content. At its heart, teaching is motivating: causing to learn. And by far the strongest motivation for any of us, and for any action—stronger even than financial, I would argue—is emotional motivation. 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. A large part of the experience of teaching does not fit a scientific paradigm. 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.

There are difficulties in collaboration. But difficulties can be transcended, with some effort. Why is this effort so rarely applied? The dark side of any collaboration, I am convinced, is insecurity. To work with another academic or professional, one must be willing to admit to the limits of one’s knowledge and experience. Perhaps more difficult, one must be willing to cede intellectual ‘territory’ to another. An example that keeps coming up is discussion of the term ‘variable’. I have heard mathematicians dismiss it as meaningless, then use it (meaningfully!) in their own very next sentence. The problem is that it is not really a mathematical term, but a psychological one, used in the study of learning mathematics. That there is a concept there is pretty much clear from the ubiquity of the term, but what that concept is needs significant exploration—by people with various backgrounds and expertise. 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.

Who’s the villain of this piece? I think no one person or group. Difficulties I’ve perceived in collaboration are not the result of negativity, but of struggle. And yet there are villains, of a sort. The villains are our institutions, that have needs that transcend our own, influence our identities and agendas, and keep us siloed.

Institutions shape our relationships. 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. (I learned this working at NSF.) The typical education grant, say for materials development, is in the millions of dollars. The typical mathematics research grant is in the hundreds, or even just tens, of thousands, an order of magnitude lower. More money means more institutional clout, so education specialists acquired more influence in academia. Mathematicians understandably resented this, and this resentment goes far to account for the rancor that accompanied any purely intellectual dialogue during that period.

Institutions shape research agendas. One reason that education research tends to be skewed towards sociological (rather than anthropological) methods is that the former can be accomplished more quickly. So the young faculty member seeking tenure—or the established scholar seeking influence–can rack up publications more quickly. A close study of teacher-student interactions, on the other hand, is slow and painstaking, and often requires the development of new tools. Too often, educational research does not answer teachers’ most burning questions.

And it’s not just academia. Funding agencies have their own needs. For example, longitudinal studies (even data-intensive studies), which give us deeper results about the effects of our teaching, take years to accomplish. 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.

There’s an old joke about a person who’s dropped his key at night near a park bench. A helpful passerby asks: “Where did you drop your key?”

The reply: “Over there, at the other end of the bench, where it’s dark.”

“Then why are you looking at this end of the bench?”

“Why should I look at the other end? It’s too dark. I’ll never find anything.” 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.

Institutions shape our teachers. 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. 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. The usual label for the latter is ‘professional development’. 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.

Institutions. We can’t live with them and we can’t live without them. But we are constrained to live within them.

Having read over this column several times, I am concerned that it be taken as a series of complaints. That’s not my intention here. Rather, it is to celebrate the productive and enjoyable times I have had collaborating with teachers, with mathematicians, and with educational researchers. Please keep asking those questions. Keep offering your expertise to the rest of us, and keep on admitting to the limits of your own.

I thank all of you who have been helping us to build bridges.

ACKNOWLEDGMENTS

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.