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This special issue represents an attempt to build bridges between research in mathematics education and psychology. Although the disciplines differ in the way they frame specific research questions, these two fields concern themselves with many of the same problems, especially problems related to mathematical learning. Too often, however, their respective communities talk past one another, not knowing how to integrate work from the other field. In a commentary published in this journal, Dan

The articles presented in this issue contribute to our understanding of mathematical development in the domain of numerical cognition while integrating perspectives to various degrees. Some of the articles represent literature primarily from the authors’ own discipline to call for future research and application in the other field. Other articles examine interdisciplinary research relationships themselves. In “Bridging psychology and mathematics education,” Martha Alibali and Eric Knuth report on a particularly productive collaboration, which has generated several research publications and numerous insights into ways that students conceptualize mathematical equations. Additional articles—one led by educational psychologist Helena Osana and another led by mathematics educator Xenia Vamvakoussi—directly address Berch’s concerns and the obstacles he identified in relation to interdisciplinary collaborations.

In this editorial we elaborate on two epistemological obstacles to bridge building: differing perspectives on the nature of mathematics, and differing perspectives on research (which lead to different methodological choices). We draw upon findings from papers presented in this special issue to identify potentially productive ways to navigate these obstacles. We close with suggestions for future collaborations.

As noted by

Like mathematicians, mathematics educators recognize mathematics as a unique form of knowledge. However, motivation for mathematics education research comes from a desire to understand mathematics as a product of the human mind and the environments in which it develops (especially classrooms), rather than pre-existing in a Platonic world. Problematizing mathematics—what it is and how people develop it—is very much at the heart of mathematics education. Mathematics education includes several philosophies describing mathematics as a human invention (

Within mathematics education research, mathematics is often defined by a community with particular values and norms (socio-cultural perspective; e.g.,

Among mathematics education researchers who adopt a cognitive perspective, Piaget’s work in psychology has had considerable influence, especially among constructivists. In Piaget’s constructivist epistemology (

Embodied cognition espouses an epistemology of mathematics based in sensori-motor actions (

From both constructivist and embodied perspectives, mathematics is a product of “perceptuo-motor-imagined activity” (

By focusing on students’ mathematics, mathematics educators and psychologists might be better poised to benefit from one another’s research. Mathematics educators can gain a better appreciation for the utility of psychological constructs in building models of students’ mathematics, and psychologists can free themselves from the mathematics of textbooks by focusing on the mathematics that students construct. Three articles struggle with the issue of reconciling students’ mathematics and formal mathematics. The article by Matthews and Ellis provides windows into their respective disciplinary perspectives related to students’ understanding of fractions. Osana and Proulx do so for interdisciplinary research on mental arithmetic of whole number operations. Coles and Sinclair examine curricular influences on students’ mathematics and question whether there is such a thing a normal development in students’ mathematics, particularly in the context of developing number knowledge.

Research methods are more than tools; they are accepted practices in particular disciplines based on particular epistemological stances (

With this special issue we hope to share perspectives that could enable researchers from each field to accept research questions investigated in another field as worthy of investigation and consideration. The epistemological stances and common methods guide the types of questions researchers typically ask. In psychology journals, research foci are typically framed as yes or no questions of hypotheses to test, whereas in mathematics education journals the research focus may be framed as a statement of purpose. Although it may seem subtle, this difference can create barriers for readers who either claim a study in the opposite domain oversimplified the issues into dichotomous questions, or disregard a study as not having any research questions. Knowledge of the epistemological stances and goals of research in respective fields could help researchers understand these purposes and questions and the resulting methodological choices people make to accomplish research goals. This collection of articles reflects different methodological approaches across fields to show mutual respect and how collaborations provide opportunities to clarify and strengthen research questions and methodological choices.

Consistent with methodological approaches typically employed in psychology—what

Alibali and Knuth, as well as Osana and Proulx, explicitly discuss their respective methodological approaches and conflicts within their respective cross-disciplinary collaborations. Both offer summaries and insights about how researchers from respective disciplines consider reliability and validity in their methodological choices. Through the process of reviewing manuscripts that appear here and others that were reviewed, we encourage realistic expectations when choosing to adopt a new methodology or theoretical perspective, because there will be a learning curve. Experimental researchers with strong research may find a new appreciation for the difficulties of qualitative approaches they attempt to adapt. Similarly, qualitative researchers who try to quantify their results may find they need to test assumptions, statistically control variables, and cautiously interpret their results in light of the seemingly objective statistical output (

What does it mean to build disciplinary bridges? This special issue offers windows into a variety of ways to forge and define cross-disciplinary bridges to strengthen our ability as researchers to more fully solve problems of mathematical learning. The articles in the special issue serve as examples of ways we might apply research from one field to the other. Additionally, some of the articles reflect long-standing cross-disciplinary partnerships.

As Alcock et al. noted, “cross-citations are comparatively rare” (2016, p. 22). Similarly, a key criticism from reviewers for this special issue was that authors from a single field seeking to bridge to another field claimed no or little literature existed in the other field about the topic under investigation. We view this critique as a fundamental obstacle to building bridges and identify a few contributing factors: a) knowing where and how to search for literature in the other field; b) bridging disciplinary jargon or terms used to find the constructs of interest; and c) appropriately interpreting the literature from another field. Consequently, we offer suggestions gleaned through the process of creating this special issue in order to help researchers overcome each of these obstacles.

In another article that sought to encourage communication between psychological research fields and educational research in mathematics,

Terms used in each field may have dissimilar meanings, and each field may be unaware of search terms to describe the same or similar underlying construct. Even though Google Scholar and other means are available to all researchers, the search results may not be as productive without having built relationships with critical colleagues in other fields to learn each other’s terms. For example, in this journal (begun by an editorial board from psychology and child development) “number” is used as a proxy for mathematics, writ large. In mathematics education research, “number” carries a more limited meaning referring to the domain of number and operations and generally excluding other domains, such as algebra, statistics and probability, and geometry. To search within a cognitive psychology journal it may help to use “math” to narrow the search, whereas in mathematics education journals researchers might specify a very particular topic such as “integers.”

If one seeks to build bridges without an interdisciplinary collaborator, seeking a critical colleague as a conversation partner or informal reviewer would be valuable to ensure that we appropriately use theories and interpret results from the other field correctly. We suggest this in response to reviewer comments on multiple manuscripts.

Alibali and Knuth suggest that researchers of numerical cognition attend conferences and serve as reviewers in the other discipline. During the development of this special issue, the journal established a partnership with the Mathematical Cognition and Learning Society (

When researchers speak of theory, they naturally take for granted that they will be understood at the level they intend to be understood. Yet, because these various levels of meaning are simultaneously at play, what researchers take for granted may not be shared by others. They sometimes find themselves talking about a theory at one level, while their colleagues are thinking about it at quite another level. At best, this leads to muddled communication. At worst, researchers are left wondering how intelligent people could be so obtuse as to misunderstand what they mean. (Flinders & Mills, 1993, p. xiv as cited in

The issue described in the quote occurs even within disciplines, but is further compounded when we seek to understand and apply theories across disciplinary boundaries. When graduate students in education learn how to conduct research, they typically learn about qualitative and qualitative research paradigms and the epistemological perspectives that lead to such methodological choices. Most often, their research experiences on projects and the articles they read reinforce the qualitative paradigm. In contrast, graduate students in psychology may not have a course that introduces them to qualitative research and related epistemologies. Instead they read quantitative studies with experimental designs and work on research projects that implement these methods. Alibali and Knuth’s paper describes the issue this might raise for communication.

Whereas research in psychology often explicitly refers to mechanisms, mathematics education research rarely does and often uses theoretical frameworks in implicit ways. Related obstacles that arose during the production of this special issue lead us to suggest that questions for further discussion could comprise an entire special issue in itself. What do mathematics education researchers consider a theory? What do cognitive psychologists consider a theory, and how does it relate to a mechanism?

In producing this special issue, we appreciate the support of John Towse, editor of the

The authors have no funding to report.

The authors have declared that no competing interests exist.