Search

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. Next, we compare mathematicians' diverse and conflicting testimony on the nature and purpose of proof. We lay especial emphasis on the diverse responses mathematicians give to the challenges that digital technologies present to older conceptions of mathematical practice. We examine the career of one much cited and anthologised paper, WP Thurston's 'On Proof and Progress in Mathematics' (1994). This paper has been multiply anthologised and cited hundreds of times in educational and philosophical argument. We contrast this paper with the views of other, equally distinguished mathematicians whose use of digital technology in mathematics paints a very different picture of mathematical practice. The interesting question is not whether mathematicians disagree--they are human so of course they do. The question is how homogenous is their mathematical practice. If there are deep differences in practice between mathematicians, then it makes little sense to use isolated quotations as indicators of how mathematics is uniformly or usually done. The paper ends with reflections on the usefulness of quotations from research mathematicians for mathematical education.

The GO-GIRL Program is a unique project, characterized by its ability to bring together pivotal elements within a youth-intervention program and foster collaboration between university and local communities in large and moderate-size urban areas through service learning. Designed to bolster the social-skill development, educational outcomes, and possible career interests of poor middle-school girls of color in math and science, the GO-GIRL Program targets those early adolescent girls at a critical point in their development and provides an intervention in the context of a supportive mentoring relationship. This commentary reflects on papers addressing GO-GIRL programs undertaken in Philadelphia, Chicago, and Bloomington, Illinois. Each paper describes the implementation of GO-GIRL in its local context and focuses on data related to the mentors' reflections about the program and its impact on them. This commentary highlights the impact of three intervention programs for mentors and mentees; key insights about the power of relationships in learning; and the value of those interventions in fostering knowledge and career interests in math and science for African American girls in poor communities.

In this commentary, I draw exclusively on the ten papers in this issue of "ZDM" focused on "Mathematical Tasks and the Student." These papers represent various cultural and theoretical stances and when reading them I was looking for common themes across some or all the papers. Based on my readings, I argue that (a) each task has the potential to promote domain-specific as well as domain-transcendent goals potentially communicating a more coherent and connected picture of mathematics, (b) tasks cannot be considered independently from their enactment and prior enactments should inform future implementations, and (c) the context of the implementation of a task (i.e., the teacher, classroom norms, and student agency) is essential and needs to be considered in the design phase.

The secondary-school mathematics curriculum is narrow in scope and technical in character; this is quite different from the nature of the discipline itself. As a result, it offers little inspiration to both students and teachers, and provides students with poor preparation for university mathematics courses and indeed for life. Over the past century, recently more than ever, there have been calls for change, for a curriculum that is true to the subject of mathematics as the creation and study of patterns and structures. While there are hopeful responses to this at the elementary level, there is almost nothing at the secondary level. Ironically, it is felt that in order to prepare students for university calculus, the secondary curriculum simply has to be what it is. This is a special case of a myth that needs to be destroyed.

This paper comments on the theoretical formulations and usage of the construct of teacher noticing in a selection of the papers in this special issue of "ZDM Mathematics Education." The analysis of how the notion of teacher noticing is used in the papers suggests that it draws attention to several interdependencies involved that have not been attended to in the past. However, the contributions in this special issue have only partially accounted for the dynamic interactions in teacher noticing, suggesting that there is potential for enriching our understanding of the complexities involved in the realm of teacher noticing. The purpose of this commentary is to stimulate the current discussion on teacher noticing by providing insights from cognitive science and the applied science of human factors, which have the potential to challenge the current understanding of noticing. In doing so, the paper sets the stage for several related constructs from these research disciplines to raise awareness of aspects that recent conceptualizations of teacher noticing may have blinded rather than enlightened.

To an unreconstructed cognitivist with experience of developing General Certificate of Secondary Education (GCSE) examinations and national tests in mathematics, the three papers which provide the core of this special issue (Morgan & Sfard; Morgan, Morgan & Tang) provided a refreshing opportunity to consider examination items from a different perspective. The integration of sociocultural/communicational and social semiotic/functional linguistic viewpoints underpinning the ESRC-funded The Evolution of the Discourse of School Mathematics (EDSM) highlighted for me the role of examination texts as purveyors of cultural meanings, communicating an authorised definition of school mathematics. The fourth paper (Alshwaikh) showed an attempt to translate this analysis to a different medium and also to a different culture and language. The fifth paper (Adler & Lerman) usefully took a step back and reminded me of the importance of who sets these cultural definitions, drawing on narratives from two contrasting countries. With little background in language and discourse the author is obviously not well placed to comment on the theoretical issues and will confine herself mainly to pragmatic and policy matters. In this brief commentary, the author shares her views on the papers chosen for the special issue, highlighting some specific data from across the three EDSM articles.

In this paper, I wish to reply to the questions posed by Gascón & Nicolás (2017) about the (possibly) prescriptive nature of the research results in didactics of mathematics: To what extent, how, under which conditions can (or must), didactics set value judgments and normative prescriptions in order to provide criteria about how to organize and manage study processes? (p. 26). Yet rather than replying, I shall suggest new questions about the deep sense of wondering about the prescriptive nature of the research results of the Western didactics of mathematics. [For Gascón & Nicolás' 2017 article, see EJ1162121.]

In this response paper to Hoyles' contribution "Transforming the mathematical practices of learners and teachers through digital technology" (EJ1200112) focuses on three points. First, more knowledge is needed on why teaching and learning practices should transform, into what will they transform, and by what or by whom will they be transformed. Second, a suggestion is made for a more specific taxonomy on the didactical functionality of digital tools in mathematics education. Third, a plea is made for a future research agenda that addresses the ways in which activities with digital tools mediate the learning of mathematics in a fruitful way. This includes the interpretation and grading of online student work through intelligent mathematical software, and the notion of embodiment, as to do justice to the bodily experiences in which mathematical experiences are rooted.

Larry Lesser is a Professor in the Department of Mathematical Sciences at The University of Texas at El Paso. He is also a UTEP Distinguished Teaching Professor whose awards include a 2016 Minnie Stevens Piper Professor Award, the 2012 International Sun Conference Scholarship of Teaching and Learning Award, a 2011 UT System Regents' Outstanding Teaching Award, and the MAA Southwestern Section's 2010 Distinguished Teaching Award.This interview took place via email from January 17-February 16, 2020.

In their commentary, "Toward a Framework for Research Linking Equitable Teaching with the Standards for Mathematical Practice," Bartell et al. (2017) provide a stepping-stone into the challenge of clarifying the interface between equity and standards setting in mathematics education by devising a framework that relates the "Common Core State Standards for Mathematics" to an explicit articulation of equitable teaching practices. In this commentary, we respond to this proposed framework and aim to clarify some key elements. Furthermore, we draw on our own positionings and scholarly interests to critique and bolster the framework by focusing on the tensions related to co-opting the Common Core for equity-oriented purposes, the framework's relationship to neoliberalism, and the role of racialized rhetoric and nondominant family and community knowledge.

Sternberg (2017) summarizes the history of identification of giftedness in the 20th century and presents a case for the shortcomings of measures such as IQ for problem-solving skills required in the 21st century. The Active Concerned Citizenship and Ethical Leadership (ACCEL) model is proposed to replace the outdated construct of IQ, particularly for the field of gifted education. In this commentary, the mathematical dimensions of ACCEL are teased out in contrast to its presence in psychometric testing. Further, what is considered relevant in mathematics for learners today is addressed in relation to the skills outlined in the ACCEL model. [For Sternberg's paper, "ACCEL: A New Model for Identifying the Gifted," see EJ1148989.]

One of the most challenging decisions made by school system administrators each year is how to assign students to teachers. This decision, usually guided by the administrator's beliefs and values, has major implications for the teacher and the student. Collins and Gan undertook a complex study to examine the impact of grouping practices on student achievement within 135 schools in the Dallas independent school district. Their study addressed three issues: (a) how schools sort students into classes, (b) the effect of these sorting practices on student performance, and (c) differences in effects for different groups of students. They looked at the "tracking effect" on high- and low-achieving students, students who had been identified for special (SPED) and gifted (GT) education, and for students with limited English proficiency (LEP). This commentary explores the implications of this study for students with gifts and talents. [For "Does Sorting Students Improve Scores? An Analysis of Class Composition." NBER Working Paper No. 18848 (2013), see ED541251.]

In this paper, I comment on the set of papers in this special issue on mathematical problem posing. I offer some observations about the papers in relation to several key issues, and I suggest some productive directions for continued research inquiry on mathematical problem posing.

While research by scholars has contributed greatly to an emerging knowledge base on Black children and mathematics, there continues to be a dire need for insightful research that de-centers longstanding accounts that have contributed to the construction of Black children as mathematically illiterate and as less than ideal learners relative to other student groups. Continued research will also help to refine the most promising theoretical and methodological approaches. In this article, the author argues for even greater attention by researchers to understand and document what it means to "learn mathematics while Black." This is not a narrow call meant to suggest that Black children are idiosyncratic in their mathematical behavior and development. This is a call with much richer aims focused on learning and identity, two centrally important considerations in children's mathematical development. Yet, while it is important to discuss the development of Black children as "children", this call echoes the claims of many scholars who have argued that it is equally important to prioritize their development as "Black" children. The author contends that it is a call for understanding mathematics learning, development, and participation among Black children within their "phenomenal Black realities," giving attention to the micro-, meso-, and macro-level forces affecting their lives and utilizing culturally sensitive research methods to account for these forces. (Contains 1 figure and 8 notes.)

This paper focuses on the kinds of notations young children make for fractional numbers. The extant literature in the area of fractional numbers acknowledges children's difficulties in conceptualizing fractional numbers. Some of the research suggests possibly delaying an introduction to conventional notations for algorithms and fractions until children have developed a better understanding of fractional numbers (e.g., Hunting, 1987; Saenz-Ludlow, 1994, 1995). Other research (e.g., Empson, 1999), however, acknowledges the interaction between conceptual understandings and the representations for those understandings. Following the latter line of thinking, this paper argues that children's notational competencies and conceptual understandings are intertwined, addressing the following research questions: (a) What kinds of notations do five and six-year-old children make for fractional numbers?; and (b) What can be learned about young children's learning of fractional numbers by analyzing the notations they make? This paper presents data from interviews carried out with twenty-four children in Kindergarten and first grade (five and six-year-old children), exploring the nuances of their notations and the meanings that they attach to fractions.