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Theoretical Orientations

Mirrors as Levers

Objective history is meant to function like a mirror that provides us with a reflection of the past. In contrast, effective history is meant to function like a lever that disrupts our assumptions and understandings about who we think we are. Foucault's history, with its provocative and ironic stance, conveys the message that mirrors make the best levers. (Fendler, 2010, p. 42).

In the spring semester of 2017, I took a humanities-oriented research course. In one of the early weeks, before we read about Foucault’s historiography, I asked a question “If history is the study of what has been, what is the study of what could be?”. Lynn Fendler’s response: philosophy. I share this story for two reasons. First, I share this story to show how my understanding of history has changed. Originally, I equated history writ large with objective history, with as way to look back and telling the story of what has been. Now, my own ideas about history have moved much closer to Foucault’s, particularly when considering that those stories preserved in the archives, those stories told in the annals of history, are precisely those who were included, in positions of privilege, wealth, authority, or some combination of the three (Rancière, 1994): history as an objective mirror is fraught with inequality.


What at first seems to be a limitation (inequality), however, can also be history’s advantage. If the official record, the image that we see when looking back, is the story of those included, instead of focusing on the visible, we can shift our gaze towards the invisible. Looking at the invisible—the gaps, the margins, the borders–to see what is missing, what is excluded, and what is outside the neat picture that we are presented, claiming to tell what was, can instead tell us what was excluded and ignored; this can tell us what we can work on including moving forward. It is for this reason, that I believe that mirrors (looking back) make the best levers. By looking back at what has been done in the name of mathematics education research, by looking at what has been published in three prestigious mathematics education research journals (this historian’s archive), and by mapping the references cited in those journals, I am able to see one story of what has been. Within those images of what has been, however, we can also see gaps, margins, and borders, specters of what also was, but was not deemed worthy of inclusion. These gaps indicate some (but not all) of the areas to develop moving forward.


Second, I share this story to show how chance figures into my own story; it is only because of this course that Foucault and the idea that mirrors can be levers came onto my radar. This notion of chance, that things can happen in unexpected and unpredictable ways, is another characteristic of Foucault’s history:

Foucault celebrated the role of chance in history because chance makes change easier to imagine. If we do not think of history as proceeding in some inevitable or predictable manner, then history is not so deterministic, and it is easier for us to imagine that things might be different in the future. (Fendler, 2010, p. 42).


If I am able to look back, to see a story of what has been and show that what has been done in the name of mathematics education research has shifted across time, that what has been included, and therefore excluded, has neither been fixed nor progressive—thereby arguing that what is included (and excluded) today is not the inevitable accomplishment of directed action by a cohesive collection of researchers—I might be able to restore the role of chance in the field. In fact, my goal is to show that the research foci of our field of mathematics education research are “neither discovered truths nor preordained developments, but rather the products of conglomerations of blind forces” (Prado, 1995, p. 38). This tracing of what has been, it turns out, is precisely the lever that I unpack in the next section. This lever will enable me to use “what has been done in mathematics education research” as a lever to change “what can be done in mathematics education research.” 


Bubbles and Foam

Politics is commonly viewed as the practice of power or the embodiment of collective wills and interests and the enactment of collective ideas. Now, such enactments or embodiments imply that you are taken into account as subjects sharing in a common world, making statements and not simply noise, discussing things located in a common world and not in your own fantasy. (Rancière, 2004, p. 10).


Jacques Rancière is a political philosopher and a philosopher of equality. Rancière’s notion of politics, however, is not equivalent to the practice of power mentioned in the epigraph. Instead, for Rancière, politics consists of redefining who is taken into account, who has a share in the common world, whose speech is classified as statements and not noise, and whose fantasy becomes part of a common world (2004, 2009). From this perspective, then, my goal is political since I wish to reconfigure what we can do, say, and think in the name of mathematics education research. My first step is to consider a different metaphor for the field of mathematics education research, a metaphor “to counter the view of the emergent as inevitable” (Prado, 1995, p. 38).


I have heard the field mathematics education research described as a cocktail party, a group of individuals mingling in a common space (we might call that salon mathematics education) and gathering into small groups, each having their own conversations (these conversations constitute different research foci). From this metaphor, our role as emerging scholars is to distinguish the conversations from the cacophony, to listen to the conversation, then slip into the ongoing conversation (cite exiting research). This perspective, however, limits what we can do. We cannot step into a group and begin talking about something new, we need to join the conversation that is already happening.


Instead of thinking of the field of mathematics education research as a cocktail party, I propose that we consider bubbles and foam (Sloterdijk 2011, 2016). By way of analogue, the conversation groups of the cocktail party correspond to the bubbles and the cocktail party itself corresponds to the foam. Sloterdijk, in his Spheres Trilogy comprised of Bubbles (2011), Globes (2014), and Foams (2016), offered a theorization of space and the places that people take up in space. For Sloterdijk, “humans live in spheres which give them meaning and provide them with a protective membrane” (Borch, 2010, p. 224). These spheres, the contained spaces in which people attain meaning, are bubbles and these bubbles constitute “microspherical worlds” (p. 226) each with their own rules for who is taken into account, who has a share in that micro-world, whose speech is classified as noise, and whose fantasy is valued in that common world. These bubbles correspond to the distinct research foci within the field of mathematics education research: each bubble of research has its own knowledge base, its own experts, its own expectations for research methods, types of findings, etc. The field of mathematics education research is not, however, merely an agglomeration of bubbles, it is a foam.


First, the foam metaphor is helpful since, from afar, a foam looks like a solid object. From afar, mathematics education research seems like an ontologically solid object, something that is and has been: fixed, inevitable, undeniable. Yet, from up close, we can see that the foam is comprised of many bubbles. Foams of bubbles “are fragile and protected by frail membranes, immunity maintenance is a crucial concern” (Borch, 2010, p. 232). Within the mathematics education context, these research bubbles are not fixed, they are volatile. It is necessary for those located within a particular research bubble to work towards maintaining the bubble’s boundary since bubbles are cofragile: the if one bubble pops, the neighboring bubbles will be affected (Borch, 2010). Bubbles within a foam can burst, merge, and split; new bubbles can emerge.


Second, this metaphor, together with a Foucaultian reading of history, gives us an understanding of the field of mathematics education research wherein “it is easier for us to imagine that things might be different in the future” (Fendler, 2010, p. 42), since we no longer need to change the field as a whole, nor change the conversations happening with groups of people, but rather split, burst, merge, or emerge. Revisiting Rancière, our levers need not be large, we need not burst all the bubbles, we need not completely reconfigure the foam at once: “change is the result of a thousand creeping encroachments” (Rancière, 2000, para. 8).


What Can be Seen, Said, and Thought


The distribution of the sensible refers to the implicit law governing the sensible order that parcels out places and forms of participation in a common world by first establishing the modes of perception…a system of self-evident facts of perception based on the set horizons and modalities of what is visible and audible as well as what can be said, thought, made, or done. (Rancière, 2009, p. 89.)


Rancière’s distribution [partage] of the sensible is “the system of self-evident facts of sense perception that simultaneously discloses the existence of something in common and the delimitations that define the respective parts and positions within it” (Rancière, 2009, p. 12). In other words, a partage of the sensible is a set of implicit laws that govern what we can see, say, or do as mathematics education research (the thing in common). It further indicates what is sensible: how the ways of doing, seeing, and saying fit together (parts and positions within it). Therefore, when we look at some article, book, or dissertation and decide if it is mathematics education research, if it fits within the coordinates that we use to determine if something makes sense as mathematics education research, we are operating within a particular partage of the sensible.


Each journal that publishes mathematics education research has its own aims and scope. These journal aims outline the expectations for topic, included content, acceptable theories and analyses, types of conclusions, etc. As a result, “the distribution of the sensible reveals who can have a share in what is common to the community based on what they do and on the time and space in which this activity is performed." (Rancière, 2004, p. 12). So, within the 1970s when quantitative analyses providing statistically generalizable results were dominant (see Chapter 3), there was little room or acceptance for qualitative studies: this would outline a particular partage of the sensible. Today, however, a variety of methods, foci, theories, etc. are acceptable: this outlines another partage of the sensible. Since journals often prescribe that authors connect to the ongoing conversations within their journals (bubbles), these constitute partages of sensible research. I am interested in which ideas appear in multiple partages and which appear in fewer (or are absent completely) since “the more frequently certain ideas are produced in speech and writing, the more true they seem, and the less often certain ideas appear, the less possible they seem” (Parks & Schmeichel, 2012, p. 241)


For the JRME, I present the bubbles and foam of each decade from the 1970s to the 2010s to show how the bubbles and foam change across time, showing evidence of the volatility of these foams. I also show the partage of research outlined by the research published within for the learning of mathematics (flm) and Educational Studies in Mathematics (ESM), respectively. By naming the bubbles within the JRME 2010s foam, the flm 2010s foam, and the ESM 2010s foam, I show the differences between what constitutes mathematics education research between different journals. Showing this variation likewise shows that what is mathematics education research is not fixed, there is variation in what can be mathematics education research. This furthermore shows the disagreement on what constitutes mathematics education research:

Disagreement is not the conflict between one who says white and another who says black. It is the conflict between one who says white and another who also says white but does not understand the same thing by it or does not understand that the other is saying the same thing in the name of whiteness. (Rancière, 1999, p. x.)


My goal in the end, however, is not to reach a consensual definition of mathematics education research. Establishing a singular and totalizing definition of what mathematics education research is and can be would necessarily lead to policing (Rancière, 1999), of ensuring compliance to the given definition and of maintaining the boundary to mitigate its fragility (Borch, 2010). Instead, my aim is to institute and maintain a politics of aesthetics—an aesthetics of mathematics education research—of constant refiguration of what we can see, say, think, and do in the name of mathematics education research. Taking these theories together, then, is the reason for the title of this dissertation: Perturbations of the Sensible. Perturbation of the Sensible as in a change to coordinates that figure what is sensible as mathematics education research, in what we can see, say, think, and do in the name of mathematics education research.


In Visual Complexity (2011), Manuel Lima outlines five purposes of cartography: (1) in the spirit of documenting, to map something new; (2) in the spirit of clarity, to make the network more understandable in its representation; (3) in the spirit of revelation, to identify “a hidden pattern in or explicit new insight into the system” (p. 81); (4) in the spirit of abstraction, to illustrate some feature beyond the data of the network itself, and (5) in the spirit of expansion, to set the stage for more exploration.


In this way, the present these maps aim to:

(1) document the citation relationships between those articles published in the JRME (1970-2019), ESM (2010s), and flm (2010s);

(2) provide clarity regarding the patterns, quantity, and nature of citations across time and journals;

(3) reveal the densely-connected bubbles of research within the data which correspond to research foci of the field;

(4) provide an abstraction of these bubbles into foams of the field which present different perspectives on what constitutes the field of mathematics education research across time and journal contexts; and

(5) generate interactive citation maps to set the stage for more exploration.

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