How Manuscripts Can Contribute to Research on Mathematics Education: An Expansive Look at Basic Research in Our Field (2024)

Basic Research at the Origins of JRME

The expression “basic research" was an important element of the language used at the time of the founding of JRME to argue for why a new journal was needed. In their reminiscences 25 years after the founding of JRME, Johnson et al. (1994) noted that “product-oriented studies were prevalent in the mid 1960s" (p. 569) because curricula developed out of the “new math" reform required evaluation studies to document their effectiveness. Against that background, a set of articles edited by Scandura (1967) as a pilot for the new journal made a distinction between information-based research (taken as a synonym for basic research) and product-development research; Scandura noted that both kinds of research were needed because they complemented each other.

In one of the articles of that volume, Suppes (1967) wrote that “it is the ultimate objective of basic research in mathematics education to understand how students learn mathematics" (p. 1). In Suppes’s argument, “this understanding [should be used] to outline more effective ways of organizing the curriculum" (p. 1) because mere intuitions of how students learn, or even an empirical aggregation of such intuitions, constituted a shaky basis for curriculum development. Instead, Suppes said, “a major task of basic research [is] to analyze and provide a theory for the kind of learning difficulties students encounter" as they study mathematics (p. 4). Other articles in the same volume fleshed out examples of basic research done at the time. For instance, Dienes (1967) proposed the hypothesis that students’ playful engagement with multiple embodiments of mathematical ideas could support students’ abstraction of those ideas.

In their reflections, Johnson et al. (1994) noted how the midcentury mathematics education community felt “the need to link work with that of the behavioral sciences and the importance of ‘conceptual models’ and ‘theory development’" (p. 563). JRME was established in part to be a home for basic research studies that could contribute to theory development. As Becker (1970) indicated in the first volume of the journal, basic research on mathematics education was expected to expand on what had previously been done by educational psychologists, and to support the development of a theory that

would pull together what is already known about mathematics learning. In this sense we will have related the theory to the relevant research that now exists. We must, however, be prepared to face up to the cold fact that maybe not much is known about mathematics learning. The more research we can build on, the more stable the foundation of the theory. (p. 21)

We note that, for Becker, doing more research on mathematics learning would help to build a theory articulating what is known about it. Whereas Suppes had argued for basic research as a way to support the practical ends of developing curricula, Becker saw it as a foundation for more and better research.

Over the years, JRME has become a choice outlet for research that seeks to understand children’s mathematical thinking and learning, to which many publications on various aspects of the matter can attest (e.g., Carpenter et al., 1981; ; Matthews et al., 2012; Sáenz-Ludlow, 1994). Research that helps us understand how children think and learn mathematics has been and continues to be a source of valuable contributions to theory. Yet, as we have already suggested, an ecumenical approach to the gaze and focus of basic research could help us expand how we think about how the research portfolio of our field contributes to fundamental understanding and theory building. We turn next to some examples of this expansion of gaze and focus that have appeared in JRME in the 50 years since its founding.

Examples of Basic Research: Casting an Expanding Mathematical Gaze on an Expansive Set of Practices

We highlight two examples, didactique des mathématiques and ethnomathematics, that may not be immediately obvious as basic research but that fit our expansive definition. They illustrate, in particular, a gaze that sees the mathematical at various levels and that is cast on foci that extend beyond human development or human cognition.

Didactique des mathématiques, originating in France in the late 1960s, was showcased in JRME by Balacheff’s (1990) article “Toward a Problématique for Research on Mathematics Teaching." Balacheff illustrated a paradigm of research on the instructional conditions under which particular mathematical ideas (in this case, the conjecture that the sum of angles in a triangle is a constant) could emerge. Rather than ask how individuals develop such notions, this approach to research took the instructional intervention as the object of study. And although this research involved the design of instructional interventions, its goal was not to evaluate curriculum. Its goals, instead, were more fundamental. They included asking epistemological questions (e.g., What tasks and properties of the milieu increase the chances for a given mathematical idea to emerge in a classroom?) and ecological questions (e.g., How do the characteristics of the institutional environment of instruction, including what its members expect themselves to be able to do, challenge or support the emergence of such mathematical ideas?). The latter type, in particular, highlighted the importance of seeing children as the specific kind of institutional beings that we call “students" when making sense of the mathematics that emerged in classrooms. Thus, in its focus, didactique sought fundamental understanding of the mathematics of classrooms while viewing those classrooms as institutionally structured, complex spaces. Its gaze, in turn, called for different ways of seeing the mathematical—for example, mathematics as the official knowledge at stake (or the knowledge designated to be taught and learned) and mathematics as the meanings entailed in the actions of students at work (e.g., theorems-in-action; see Vergnaud, 1982).

Ethnomathematics, which we are considering broadly to include the study of the mathematics “frozen" (Gerdes, 1994) in cultural artifacts, as well as the study of the mathematical characteristics of cultural practices such as those of Indigenous Peoples or of the world of work, offers a different example of research seeking fundamental understanding. JRME published, for example, the work of Carraher et al. (1987) comparing the mathematical strategies children used to solve problems when selling goods in Brazilian streets with the strategies they used when solving similar problems in school. JRME also published as a monograph Millroy’s (1992) study on the mathematical ideas of a group of South African carpenters. Other JRME publications have explored the differences between academic and everyday mathematical practices (), the mathematical practices of professionals (Hoyles et al., 2001; ), and the interaction between academic and cultural mathematical practices in classrooms (Meaney et al., 2013). In these cases as well as in others, the interest in understanding the mathematical ideas involved in cultural artifacts and practices and how they contrast or reconcile with academic mathematics recommends them as examples of basic research. This basic research expands its focus by seeing individuals and their actions in the context of different cultures. It also casts a broader gaze that recognizes cultural artifacts and practices as mathematical despite differences from academic mathematics.

Expanding Our Collective Mathematical Gaze

The examples of didactique and ethnomathematics, along with the more typical study of children’s mathematical thinking and learning, clearly do not exhaust the varieties of basic research in mathematics education. However, in their diversity, they serve to illustrate that the drive to understand practices—more than the scale of the practices—is what qualifies research in mathematics education as basic. Further, they demonstrate another quality that, in fact, also helps to distinguish mathematics education research on children’s mathematical thinking and learning (e.g., ) as distinct from psychological research done in the context of mathematics (e.g., ): The gaze of mathematics education researchers is trained on distinguishing mathematical nuances in the practices observed. This particular mathematical gaze has long been removed from the jurisdiction of the academic discipline of mathematics precisely because mathematical preparation helps but is not sufficient to enable the research practices of didactique, ethnomathematics, the study of mathematical thinking and learning, or any other form of mathematics education research. The particular gaze mathematics education researchers bring to understanding phenomena is informed by mathematics and yet distinct from the intellectual dispositions that support the practices of professional mathematicians. That gaze has been instrumental in generating a base of fundamental knowledge that is particular to mathematics education research.

Two articles in this issue further illustrate variability in what casting a mathematical gaze on practices of different scales may look like. The contribution by Weber et al. (2022) investigates how various mathematical activities support the achievement of conviction among mathematicians; those activities include two that could be described as cognitive (reading a proof and doing a calculation) as well as a third one that highlights the institutional nature of mathematical work (seeing a publication in a journal). At a different end of the spectrum, the article by Jasien and Horn (2022) strives to find the mathematics in a creative play space (Math On-A-Stick) devoid of most institutional accoutrements of the mathematical domain (either in its disciplinary or schooling sense of institutional). The authors document this mathematics of creative play as they describe a child’s sensibility for fixing the crookedness of the depiction of a heart.

We take the variability evident in these articles as illustrative of what an expansive mathematical gaze could include. It could, of course, examine as central elements the concepts, procedures, and theorems that mathematical research has curated over the ages, or the mathematical creations begotten by schooling (e.g., the vertical line test). But it is not limited to those. It could also examine the actions and operations that students perform in classrooms—not only to adapt cognitively to the mathematics problems they are solving but also to respond as students to the academic work they are asked to do, such as describing and responding to other students’ mathematical ideas (see Bishop et al., 2022). The mathematical gaze attends to the local tastes and dispositions that can be seen among children, workers, or members of different communities and recognizes them as germane to those of mathematicians, even if by contrast. And it looks at the institutionalized procedures that confer or deny mathematical status to things and people: Just as the notion that publication in a mathematics journal increases a mathematician’s conviction of its truth (Weber et al., 2022), other institutional activities bear on how people construct their own image of what is mathematical (Berry, 2008). For example, in schools, the notion of mathematical talent is often indicated by fast execution of arithmetic calculations, and expectations of appropriate mathematical work include individual self-sufficiency. These examples suggest that casting a mathematical gaze on education phenomena includes seeing as mathematical those institutional activities with which mathematics is associated, even if their avowed goals are not teaching, learning, or problem solving. If ethnomathematics has disposed us to train our gaze into seeing the mathematics in practices that diverge from the disciplinary norm, studies that emphasize how institutions (e.g., schools, skilled trades, and professions) make specific, normative uses of the label “mathematical" should sensitize our gaze to noticing the company mathematics keeps—and the company it excludes. This latter point suggests further reflection on what kind of understanding we could seek with basic research.

We have proposed that basic research in mathematics education needs to be seen as casting an expansive mathematical gaze to understand various mathematical practices. The word “understanding" is often associated with the orientation to describe and explain. But we ought to consider that the aim to understand may also include a critical orientation, one that inquires into what makes possible (and what limits) the work of description and explanation. Intellectual history is full of examples of how critical perspectives on ways of knowing have become part of the available tool kit of scholarship (e.g., how reductionism in science was first critiqued and then complemented by systemic, phenomenological, and emergentist perspectives).

Along those lines, the word “critical" these days often alludes to research in which the gaze of the researcher is informed with perspectives that have been silenced in, or even by, the academic mainstream. These perspectives can come from the local knowledge of Indigenous Peoples or from emerging sectors of the academy that seek to acknowledge how scientific ways of knowing have been shaped by the material conditions of the production and reproduction of the academy, including ableism, anti-historicism, capitalism, classism, colonialism, communism, isolation, linguicism, nationalism, poverty, racism, sexism, and so forth. Empirical research that informs its mathematical gaze with perspectives that acknowledge those material conditions of the production of knowledge can be quite productive in helping us to understand intact practices³ and to envision the consequences of possible practices. Furthermore, such perspectives may recommend casting our gaze on practices of mathematics education about which less is known: For example, we know little about the mathematical practices of refugee children’s play, the mathematical learning of unhoused students, or the preparation of mathematics teachers for racially segregated schools compared to what we know about similar practices in the mainstream. Critical perspectives are helping our community to gain some understanding of those practices and others as we expand the boundaries of our mathematical gaze.

Expanding the Focus of Basic Research

If basic research in mathematics education involves casting a mathematical gaze on numerous practices that happen at various scales, we suggest that the universe of potential applications of that gaze is also unlimited. After all, mathematics education practice(s), such as teaching and learning mathematics in schools, existed long before mathematics education research coalesced as an independent field (De Morgan, 1831/1898; Kidwell et al., 2008). Indeed, the first example we gave of basic research in our field, research on children’s thinking and learning of mathematical ideas, though often done by pulling children out of classrooms, was predicated on the notion that those observations may inform our understanding of how children think and learn mathematics within classrooms and schools. The variety of practices about which we seek fundamental understanding has grown as our gaze has zoomed out of intact practices of mathematics education, zoomed into the thinking and learning practices of individuals, and panned to other elements in play (e.g., teaching or textbooks; see Herbel-Eisenmann, 2007; Stiff, 1989).

Moreover, we suggest that the expansive set of those possible foci can be better organized by reflecting on the meaning of the word “education." A fundamental duality in the usage of the word gives us a clue to how the practices of interest to research in mathematics education are expanding: First, “education" is used to identify particular settings (kindergartens, schools, colleges, museums, textbooks, the user interface of educational software, etc.) where people go to get educated. The study of classroom discourse by Bishop et al. (2022) in this issue illustrates a canonical case of how these education contexts can appeal to our mathematical gaze. When mathematics instruction occurs in institutional settings (mathematics classrooms) in which students and teachers transact, through classroom discourse, particular mathematical work in pursuit of instructional objectives (e.g., teaching and learning fractions), this instruction is evidently an important focus for mathematics education research. But in other education locations (e.g., social studies or science classrooms), practices also occur that could appeal to the mathematical gaze, even if the stakes of learning are not mathematical (e.g., when the consumption or production of social studies or science data is mediated by mathematical literacies; Heyd-Metzuyanim et al., 2021). Further, even within education settings that are dedicated to mathematics instruction, we may call on the mathematical gaze to look across the teaching and learning of particular mathematics and examine more enduring phenomena such as the construction of mathematical identities (McGee, 2015) or the components of mathematical anxiety (Ho et al., 2000).

“Education" also has a second meaning: It alludes to cultural and societal processes aimed at the reproduction of capacity, including capacities that involve doing mathematics. These processes happen in many places—not only in the formal education spaces listed earlier but also at home, at work, at play, and through the media—and, hence, they involve practices not necessarily framed as instruction. Mathematical work with fractions and percentages is involved in learning to dose drugs in nursing practice (Hoyles et al., 2001), and geometric calculations figure into solving problems of carpet-laying (Masingila, 1994). The work of mathematics researchers, as targeted by Weber et al. (2022), showcases an interesting case in which mathematics learning is present both in the activity being studied (mathematicians learning about a claim that is new to them) and in the outcome of the activity (the sanctioning of the claim as true), even if the practice being researched, mathematicians’ practice, is not officially framed as educational. Similarly, the work of teaching mathematics, beyond its instrumental roles of supporting students’ learning of mathematics or of communicating a mathematics curriculum, is another practice that appeals to the mathematical gaze: It is amenable to being described as an embodiment of tacit mathematical knowledge and rationality as well as a site of professional (even if incidental) mathematics learning (; ).

These two interpretations of mathematics education practices—the mathematically related practices of educationally sanctioned settings (including but not limited to mathematics instruction) and the learning and teaching practices that incorporate mathematics in any number of settings (including classrooms)—are obviously not disjoint. And our reflection does not discount that a centerpiece of our field’s research portfolio aims to understand students’ learning of mathematics in school classrooms. Still, we take pride in the fact that, over its half century of existence, JRME has played a role in expanding our fundamental understanding of the practices of mathematics education by publishing not only accounts of mathematical thinking and learning at school, at work, or in different cultures (e.g., Gainsburg, 2007; Meaney et al., 2013) but also explorations of the role that schooling plays in characterizing (and biasing) what society sees as mathematical (e.g., Leyva, 2021).

In all, we assert that the broad reach of the practices of interest to our field implies rather wide ecumenism in regard to what can be the focus of basic research. Our field is poised to produce basic research that casts an expansive mathematical gaze on an expanding set of mathematics education practices, and JRME is prepared to appreciate such contributions.

Basic Research in the Service of Theory Building

Basic research serves theory building not only by assisting theory verification but also by enabling theory generation. The discussion of examples in prior paragraphs suggests that whereas the theory-building orientation of basic research may be common with other domains of inquiry, the gaze used to look at a wide range of practices of mathematics education is what qualifies such basic research as specifically mathematics education research. The articles in this issue may not offer complete theories, but they do contribute incrementally to the development of theory that articulates that gaze.

For instance, the study of intact discourse in mathematics classrooms by Bishop et al. (2022) helps us to understand and track what responsiveness to students’ mathematical ideas may look like. This contributes to developing a theory of mathematics classroom discourse centered on communication processes rather than on individual speakers. Likewise, Jasien and Horn (2022) trace the emergence of a mathematical aesthetic in the context of creative play, which contributes to developing a way of seeing the mathematical in the mundane that can be used in other contexts (particularly in play and games). Finally, the article by Weber et al. (2022) provides foundational work to warrant distinguishing mathematical proof from individual conviction, which contributes to a theorization of the roles of proof in mathematics teaching and learning.

Each article in this issue is likely to also have some immediate implications for the improvement of advocacy, policy, public communication, studying, teaching, or other practices to which we also aspire to contribute (Matthews et al., 2021). However, such implications are not all that reviewers saw as these submissions’ potential contributions to the field. In judging these articles to be worthy of publication in JRME, reviewers saw in them contributions to the field’s understanding of practices we care about. In spite of how diversely they cast a mathematical gaze and the wide variety of their foci, all three of these articles have succeeded in making a case for themselves as notable products of mathematics education research. We again urge future authors to similarly ask themselves whether and how a study helps to improve our field’s fundamental understanding of mathematics education practices as they think about articulating the contribution of manuscripts they submit to JRME. Our reviewers value when authors spell out these contributions to understanding, and the journal supports an expansive interpretation of what the focus and gaze of those contributions can be.

In Conclusion

In the foregoing, we have striven to affirm the value of basic research while expanding what we mean by basic research in mathematics education. We have done this with the goal of furnishing lines of argument that authors can use to identify how their work contributes to the field of research in mathematics education.

Of course, not all research is basic research; JRME publishes all sorts of valuable research outside of this category. One example is the applied research that at the founding of JRME was described as “product-oriented research." Fifty years later, we have more sophisticated ways of thinking about the distinction between basic and applied research (i.e., research can aim at understanding and be use-inspired; Stokes, 1997) and about the transit of knowledge from basic research to the improvement of practices (Sloane, 2008). In a future editorial, we hope to elaborate on the varieties of applied and use-inspired research in mathematics education, following on the vision we articulated in Matthews et al. (2021) for better relationships between the practice of research and the practices of advocacy, policymaking, teaching, and more.

Still, the range of practices of mathematics education that exist in the world demonstrate a variability that can and needs to be understood. Although intervening and improving those practices may be possible without fundamental understanding of them, we assert that understanding those practices—knowing how to describe and explain them and knowing, critically, how and why they exist—can help support the design of (and justify) transformative improvement efforts.

Our goals in making this point align with prior editorials’ discussions of the research infrastructure of our field (Herbst, Chazan, et al., 2021; Herbst, Crespo, et al., 2021). Beneath the material and human infrastructure that we need to build and preserve, the question of why we are doing research is central. Just as in the health sciences the study of human biology complements the study of pharmaceutical products—that is, health scientists have an interest in not only curing the sick but also in understanding the inherent nature of illness and health—mathematics education is not only about applied research.

Although the practical benefits of some basic research may be seen only at larger timescales (e.g., decades), our investment in this kind of research can help us better understand the complexities of the practices that might need improvement. This understanding can increase the public’s trust that we have warrants for our recommendations, thereby helping to justify improvement efforts as responsible and accountable. Hence, we need to cultivate our capacity to continue to pursue the growth of theory by pursuing and supporting basic research in mathematics education. This starts with researchers asking themselves and telling readers what their studies contribute to understanding. And it requires that reviewers, editors, and funders support work that casts an ecumenical, mathematical gaze aimed at understanding the many practices of mathematics education.