a Boundary Object?

Consider the following exchange, which took place in a meeting of a group of mathematicians, physicists, and secondary mathematics teachers. The group was jointly examining secondary calculus tasks from a recent Israeli matriculation exam, including: Let $f(x)=x+\ln (x^2-3)$. Identify the asymptotes of $f(x)$ (if there are any). Sketch the graph of $f(x)$.

The physicists immediately observed that the function behaves asymptotically like $y=x$ since the $\ln$ part is negligible compared to $x$. This observation surprised the teachers. They were very familiar with the task and to their knowledge, the function does not behave asymptotically like $y=x$ or like any other line. There was dissonance.

Is there more than one meaning of asymptotic behavior in play? Further discussion revealed that the teachers think about asymptotic behavior in terms of difference, whereas the physicists think of asymptotic behavior in terms of ratio. Specifically, the teachers interpreted the task as asking whether

$$\begin{align*} &|f(x)-x|\xrightarrow [x \to \pm \infty ]{} 0. \\ &{\text{(Here, $f(x)-x$ tends to $\infty $.)}} \end{align*}$$

In contrast, the physicists interpreted the task as asking whether

$$\begin{align*} &\left|\frac{f(x)}{x}\right|\xrightarrow [x \to \pm \infty ]{} 1. \\ & {\text{(Here, the ratio does indeed tend to 1.)}} \end{align*}$$

From this observed distinction between secondary and tertiary mathematics arose a new question:

Why is asymptotic behavior defined in the Israeli secondary mathematics curriculum in terms of difference and not in terms of ratio, particularly considering that at least in physics the ratio-based definition is more useful?

Before reading onward, we invite the reader to pause. We ask you to reflect on this question, and formulate an argument for engaging secondary mathematics students with either the ratio or difference definition of asymptotic behavior that mathematicians, physicists, and secondary mathematics teachers would all find compelling.

$$\begin{equation*} * * * \end{equation*}$$

We make two points with the illustration above. First, different stakeholders in mathematics education may hold perspectives about mathematics, teaching, and learning that are not just profoundly different but are essentially incommensurable. Sometimes this can arise when the same words are being used tacitly with different meanings 6.

Second, it may be easy or hard to resolve such incommensurabilities. Some cases are relatively simple to work through, for example by articulating the different mathematical definitions that underlie the different meanings. On the other hand, it is far more difficult to resolve more nuanced distinctions, for example the reasons why certain meanings are more useful, powerful, or salient than others within particular professional communities. These harder cases may end up hindering communication and collaboration between stakeholders, who fail to understand the source of their disagreements.

Differences in views of what is, could be, or should be in mathematics education have led to rifts between school and university communities and mathematical and educational communities—even when all hold the common desire to improve mathematics education. For example, there have been longstanding and numerous attempts to “smooth discontinuities” between school mathematics and university mathematics (e.g., 4). Yet the divide between school and university remains and may even be widening, despite efforts to find consensus 7. There is thus limited success in advancing mathematics education on the basis of consensus. The notion of “boundary object” has been at the core of research that underlines the need for alternative ways for stakeholders to work together productively even when they may never come to agreement.

We propose “boundary object” as the subject of the first What is… article in the area of mathematics education. The Notices has long published items of interest to the mathematics and mathematics education communities, from an announcement of NSF Fellowships for Secondary School Teachers of Science and Mathematics (November 1959, Vol. 6, No. 6), to Barry Cipra and Paul Zorn’s report on calculus initiatives (January 1989, Vol. 36, No. 1, pp. 49–50), to William McCallum’s article on work in education in mathematics departments (October 2003, Vol. 50, No. 9, pp. 1093–1098), to the now recurring Education section. These publications reflect the community’s investment in mathematics education, as well as the promise for those in mathematics and education to learn from each other.

In this article, we aim to define a set of concepts—boundary, boundary crossing, boundary object—in a way that is accessible to mathematicians while being true to the origins and use of these concepts in mathematics education. We remark that the act of reading this article may well require boundary crossing, for definitions in education research may abide by rules that may be quite different from those used in the professional work of research mathematics. Rather than being explicit about what a mathematics educator may mean by definition, we invite the reader to consider this text without assuming a consensus on the notion of a definition. Ultimately, we aim to broaden the readers’ understandings of contemporary research in mathematics education.

Boundaries, a Myth About Cooperation, and Boundary Objects

“Boundaries” appear in geography, cellular organisms, and subsets of a topological space. They also live in the intersections of professional domains, such as mathematics, physics, education, or neuroscience.

Boundaries. When it comes to professional work, a boundary is a professional space where communities with differing social and epistemological commitments may interact or try to collaborate. Boundaries may also be seen as an opportunity for members from different communities of practice to learn from and with one another 12.

A myth about cooperation. As a research subject, boundaries arose in studies of cooperation without consensus (e.g., 9). To understand this motivation, consider an appealing—but inaccurate—model of cooperation. In this model, a group of people must work together, due to want, happenstance, or necessity. These people hold different views on how to achieve a particular goal and perhaps on how to define this goal. Through various communications, they come to consensus on what to do, how to do it, and why to do it. From that consensus, the group sets forth and succeeds.

This account is a myth in the sense that it is feasible for only a narrow set of circumstances, such as a relatively homogeneous team. This model is rarely accurate for successful long-term alliances in highly diverse fields, such as mathematics education.

Boundary crossing. There is thus a need for alternative models for cooperation between different stakeholders that function without consensus and in spite of “mixed economies of information with different values and only partially overlapping coin” 9, p. 413.

This functionality requires that different stakeholders involved in the cooperation are willing to cross boundaries, that is, to “enter onto territory in which [they] are unfamiliar and, to some significant extent therefore unqualified” 10, p. 25. In doing so, groups might need to negotiate strategies or ideas that combine thinking from different worlds 3. To explain such an alternative model of cooperation, we return to our opening illustration, featuring two different meanings of asymptotic behavior.

Boundary object. The task from the matriculation exam functioned as a boundary object: an artifact that different communities can engage with in a way that can support translation of meanings across these communities.

According to Star and colleagues, for artifacts to function as boundary objects, they need to satisfy certain properties, including the following (see 8, 9).

1.

Boundary objects need to have a structure which is common enough to be recognizable by individual members of different communities. In our case, the structure of a mathematical task for students was comfortably familiar to all sides.

2.

Boundary objects need to allow different interpretations and to be flexible enough to accommodate expectations, values, norms, and practices prevalent within different communities. In our case, there was no predetermined consensus about what is the solution of the task or even about what solving the task looks like.

The common structure and interpretive flexibility of the matriculation exam invited the different stakeholders to engage differently with the task, which led to different solutions, which in turn revealed a hidden boundary—different tacit meanings—that the participants could explore together.

However, it is often the case that boundaries are not recognized and the potential is not realized. It is not difficult to imagine the joint exploration continuing with the physicists basing their arguments on which definition is more productive in science and the teachers basing their argument on which definition is more productive pedagogically in school. In this case, we say that the different positions are not in disagreement but are incommensurable, in the sense that an argument in support of one position would be irrelevant to the other position, and vice versa.

If the “sides” had gotten entrenched in their respective areas of expertise and authority, trying to prove their own perspective “right,” there would have been no boundary crossing, and the potential of the matriculation task to inspire boundary crossing would have been lost.

Whether a boundary object affords boundary crossing depends as much on the group interactions as on the object’s properties. Once the physicists and teachers identified the two definitions and acknowledged that both are sensible and useful, they could then reflect on how and why they used these definitions (cf. 1). The physicists thought about asymptotic behavior as “zooming out” from the function, a move that conceals “local noises” and helps see “global patterns” of the phenomenon that the function represents. The teachers focused on extending secondary students’ discussion and understanding of functions and graphs. They explained that when drawing or describing graphs there is a need for questions about what the graph looks like “near the end of the page” and that the most “simple and concrete” test is whether the graph “gets closer and closer to a line.”

While the group reached no consensus on the “right” definition for a secondary mathematics curriculum, their discussion turned to arguments about teaching and learning that drew in meaningful ways on the expertise of both sides and that could be tested empirically. The shift from dichotomous meaning (either ratio or difference) to inquiry (when, how, why, under what conditions) showed the possibility of learning from and with each other and the potential for future solutions that may differ from what either group would have come up with in isolation.

Potential Boundary Objects in Mathematics Education

Here are some potential boundary objects that a Notices reader may encounter.

Class sessions and homework in a first-year undergraduate mathematics course. Both sides (students and instructor) have a seemingly mutual goal of promoting student learning. Both interact with the course as a tool for facilitating this goal, but the two sides may hold different expectations regarding how class sessions and homework are supposed to facilitate and promote student learning and on the role of each side (e.g., 11). Moreover, because expectations are often left tacit, the student is left to infer them. The students may infer expectations that lead to counterproductive views of mathematics teaching and learning.

In this case, there may be such an imbalance of power between instructor and student that students may just accede to their perceptions of the instructor’s norms. The potential for boundary crossing would be lost.

However, if the instructor can use their authority to create opportunities to elicit discussions of expectations, there is more potential for boundary crossing. Joint explorations of these expectations can help explicate implicit assumptions about mathematics and its teaching and learning and thus aid students in the complex transition from secondary mathematics to tertiary mathematics.

Mathematical ideas discussed in a mathematics course for teachers. Both sides (teachers and instructor) seemingly have a mutual goal of supporting teachers’ professional development. Nonetheless, their understanding of why mathematics courses are required may be at odds. Consequently, many mathematics teachers find their mathematics courses irrelevant to their future teaching.

When encountering a mathematical idea such as a definition, a teacher may view the language as unnecessarily stilted. They may therefore want to resist using precision in their teaching.

Figure 1.

Prospective secondary teachers’ proposed definitions and nonexamples, produced in a mathematics course for teachers 5.

However, mathematical ideas, such as proposed definitions and nonexamples (such as those⁠¹ in Figure 1), can afford boundary crossing when teachers can see how to utilize ideas in and for teaching and when the instructor can use the ideas to create a sense of mathematics that is broad and coherent. Teachers who are invited to participate in the act of making definitions and responding to others’ definitions can come to understand why mathematical language is structured the way that it is 5. They may also come to understand more deeply the role of mathematical definition and inference in the enterprise of mathematics and mathematics teaching.

¹

Inspired by Sam Shah’s activity https://samjshah.com/2014/10/19/attacks-and-counterattacks-in-geometry/.

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Records of teaching practice for policy and research development across mathematicians and mathematics educators. Mathematician Hyman Bass recounted that in the 1990s, education researcher Deborah Ball invited various mathematicians to view videos and other artifacts of fourth grade teaching 2. Part of her purpose was to develop a theory of “mathematical knowledge for teaching” that accounted for mathematical and pedagogical entailments of teaching. The mathematicians’ purposes may have included an opportunity to be involved in teacher education, curiosity, and a desire to connect “across the aisle” to mathematics education. The joint examinations of these records of teaching practice eventually shaped educational research in teacher education as well as policy development for mathematics departments. Mathematical knowledge for teaching does not appear in guiding documents for mathematics departments prior to the 1990s. Yet, it would be unusual now for guiding documents on the mathematical education of teachers to not use mathematical knowledge for teaching as a central construct.

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Finally, a What is… article about boundary objects may be seen as a potential boundary object: We as writers, and the mathematicians reading this article, seemingly have the mutual goal of promoting mathematicians’ understanding of boundary objects and perhaps more generally of mathematics education research. We and the mathematician readers of the article intersect with this article for this purpose. However, mathematics education researchers and mathematicians may have very different expectations as to how What is… may promote understanding.

We invite readers to look for potential boundary objects in their routines and to seek out interactions that invite boundary crossing. Such interactions may involve letting go of consensus and finding complementary rather than cross purposes. With more boundaries entered, we as a mathematical community may be able to find more productive communication and collaboration towards improving mathematics education.