It is commonplace in the educational literature on mathematical practice to argue for a general conclusion from isolated quotations from famous mathematicians. In this paper, we supply a critique of this mode of inference. We review empirical results that show the diversity and instability of mathematicians' opinions on mathematical practice.…

This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also…

The three target articles presented in this special issue converged on an emerging theme: the importance of spatial proportional reasoning. They suggest that the ability to map between symbolic fractions (like 1/5) and nonsymbolic, spatial representations of their sizes or "magnitudes" may be especially important for building robust…

This paper is a commentary on the theoretical formulations of the five empirical papers in this special issue. All five papers use aspects of the theory of commognition as presented by Anna Sfard; however, even when the same notions (e.g., rituals or explorations) are incorporated into theoretical frameworks undergirding the research, these…

White and Gorard make important and relevant criticisms of some of the methods commonly used in social science research, but go further by criticising the logical basis for inferential statistical tests. This paper comments briefly on matters we broadly agree on with them and more fully on matters where we disagree. We agree that too little…

The editors of Mathematics Teacher appreciate the interest of readers and value the views of those who write in with comments. The editors ask that name and affiliation including email address be provided at the end of their letters. This September 2016 Reader Reflections, provides reader comments on the following articles: (1) "Innocent…

In our response to Clements and Sarama (2017), we address the 5 issues that they identify as criticisms of our Research Commentary (Kitchen & Berk, 2016). As in our original commentary, we highlight concerns we have regarding the delivery of [computer-assisted instruction] CAI programs and potential misuses of CAI, particularly at Title I…

A transcribed conversation between a mathematics educator and a disability studies educator is presented to highlight cross-disciplinary dialog as a fruitful space for recursive and reflexive engagement from which may emerge concepts to further both mathematics and disability education. The opening question, "How can I know what's possible in…

In their Research Commentary, Kitchen and Berk (2016) argue that educational technology may focus only on skills for low-income students and students of color, further limiting their opportunities to learn mathematical reasoning, and thus pose a challenge to realizing standards-based reforms. Although we share the concern about equity and about…

In 2001, Kilpatrick, Swafford, and Findell proposed a new way to look at what it means for students to be mathematically proficient. They described mathematical proficiency as comprising five intertwined strands: procedural fluency, conceptual understanding, adaptive reasoning, strategic competence, and productive disposition. The vision is that…

The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion…

n a recent article, Inglis and Alcock (2012) contended that their data challenge the claim that when mathematicians validate proofs, they initially skim a proof to grasp its main idea before reading individual parts of the proof more carefully. This result is based on the fact that when mathematicians read proofs in their study, on average their…

Humans can learn to organize many kinds of domains into categories, including real-world domains such as kinsfolk and synthetic domains such as sets of geometric figures that vary along several dimensions. Psychologists have studied many individual domains in detail, but there have been few attempts to characterize or explore the full space of…

While reading the articles assembled in this volume, one cannot help asking "Why gestures?" What's all the fuss about them? In the last few years, the fuss is, indeed, considerable, and not just here, in this special issue, but also in research on learning and teaching at large. What changed? After all, gestures have been around ever since the…

Parallels between martial arts and mathematics are explored. Misguided public perception of both disciplines, students' misconceptions, and the similarities between proofs and katas are among the striking commonalities between martial arts and mathematics. The author also reflects on what he has learned in his martial arts training, and how this…