Natural alternatives to natural number: The case of ratio

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students’ non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students’ perceptually based intuitions about quantity and b) students’ reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

A tale of two researchers: Commonalities, complementarities, and contrasts in an examination of mental computation and relational thinking

This paper describes a research collaboration between an educational psychologist and a mathematics education researcher, namely a didacticien des mathématiques. Our joint project aimed to explore the mental computation strategies of preservice teachers in an elementary mathematics methods course and to investigate the relationship between mental computation and relational thinking. The primary objective of the paper, however, is to go beyond the data and their interpretation. We describe the commonalities, complementarities, and points of contrast that emerged between us as researchers who hail from different disciplines, but who have the same overarching interests in mathematical thinking. In particular, we untangle issues we encountered during our collaboration related to our research questions, methodologies, and epistemological stances. We detail the ways in which we navigated these issues in the context of the research and describe what we learned about our own disciplinary perspectives and each other’s. We conclude by discussing what our story offers as a means of reflecting on our individual fields and potential interactions between them.

Building and Structuring Knowledge That Could Actually Improve Instructional Practice

In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledge base (Cai et al., 2017b, 2018). This proposal would refashion our conceptions of the nature and collection of data so that teachers, researchers, and teacher-researcher partnerships could benefit from the accumulated learning of ordinarily isolated groups. Although we have discussed the general parameters for such a system in previous editorials, in this editorial, we present a potential mechanism for accumulating learning into a professional knowledge base, a mechanism that involves collaboration between multiple teacher-researcher partnerships. To illustrate our ideas, we return once again to the collaboration between fourth-grade teacher Mr. Lovemath and mathematics education researcher Ms. Research, who are mentioned in our previous editorials(Cai et al., 2017a, 2017b).

Equity Within Mathematics Education Research as a Political Act: Moving From Choice to Intentional Collective Professional Responsibility

In 2005, the NCTM Research Committee devoted its commentary to exploring how mathematics education research might contribute to a better understanding of equity in school mathematics education (Gutstein et al., 2005). In that commentary, the concept of equity included both conditions and outcomes of learning. Although multiple definitions of equity exist, the authors of that commentary expressed it this way: “The main issue for us is how mathematics education research can contribute to understanding the causes and effects of inequity, as well as the strategies that effectively reduce undesirable inequities of experience and achievement in mathematics education” (p. 94). That research commentary brought to the foreground important questions one might ask about equity in school mathematics and some of the complexities associated with doing that work. It also addressed how mathematics education researchers (MERs) could bring a “critical equity lens” (p. 95, hereafter referred to as an “equity lens”) to the research they do. Fast forward 10 years to now: Where is the mathematics education researcher (MER) community in terms of including an equity lens in mathematics education research? Gutiérrez (2010/2013) argued that a sociopolitical turn in mathematics education enables us to ask and answer harder, more complex questions that include issues of identity, agency, power, and sociocultural and political contexts of mathematics, learning, and teaching. A sociopolitical approach allows us to see the historical legacy of mathematics as a tool of oppression as well as a product of our humanity.

Disciplinary differences between cognitive psychology and mathematics education: A developmental disconnection syndrome. Reflections on 'Challenges in Mathematical Cognition' by Alcock et al. (2016)

As the participants in this collaborative exercise who are mathematics education researchers espouse a cognitive perspective, it is not surprising that there were few genuine disagreements between them and the psychologists and cognitive neuroscientists during the process of generating a consensual research agenda. In contrast, the prototypical mathematics education researcher will mostly likely find the resulting list of priority open questions to be overly restrictive in its scope of topics to be studied, highly biased toward quantitative methods, and extremely narrow in its disciplinary perspectives. It is argued here that the fundamental disconnects between the epistemological foundations, theoretical perspectives, and methodological predilections of cognitive psychologists and mainstream mathematics education researchers preclude the prospect of future productive collaborative efforts between these fields. [Commentary on: Alcock, L., Ansari, D., Batchelor, S., Bisson, M.-J., De Smedt, B., Gilmore, C., . . . Weber, K. (2016). Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41. doi:10.5964/jnc.v2i1.10]

Book Review: … And All the Children Are Above Average: A Review of The End of Ignorance: Multiplying Our Human Potential

The End of Ignorance (TEOI), laced with anecdote, is published for a lay audience, by a nonacademic press, with no real foundation in learning theory. Yet every mathematics education researcher—perhaps every educational researcher—needs to have John Mighton's book clearly within his or her sights. Mighton's curriculum approach, JUMP Math, threatens to leapfrog over the head of the mathematics education establishment, delivering mathematical learning to every child, from the profoundly learning disabled on up, at up to three or four grade levels beyond their current grade placement.