Geoffrey B. Saxe

Social processes in early number development

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Ethnic and Academic Identities: A Cultural Practice Perspective on Emerging Tensions and Their Management in the Lives of Minority Students:

Youth from minority groups often manage a tension between ethnic and academic identities as they are positioned and position themselves in relation to cultural practices in school and out. We argue that a framework involving three strands of analysis is necessary to understand these emerging tensions and their management in the lives of minority youth. The strands include analyses of shifts in (a) positioning that take form in face-to-face interactions, (b) positioning over developmental time, and (c) the cultural capital associated with practices themselves over the social histories of communities. We point to the importance of multimethod approaches to pursue such analyses.

Teachers’ shifting assessment practices in the context of educational reform in mathematics

Abstract This paper presents a study of primary and secondary mathematics teachers’ changing assessment practices in the context of policy, stakeholder, and personal presses for change. Using survey and interviews, we collected teachers’ reports of their uses of three forms of assessment, one linked to traditional practice (exercises), and two linked to reforms in mathematics education (open-ended problems and rubrics). Findings revealed several trajectories of change in the interplay between assessment forms and the functions that they serve. Teachers may implement new assessment forms in ways that serve ‘old’ functions; teachers may re-purpose ‘old’ assessment forms in ways that reveal students’ mathematical thinking. Our developmental framework provides a way to understand the dynamics of teacher development in relation to ongoing educational reforms.

Supporting Generative Thinking About the Integer Number Line in Elementary Mathematics

This report provides evidence of the influence of a tutorial “communication game” on fifth graders’ generative understanding of the integer number line. Students matched for classroom and pretest score were randomly assigned to a tutorial (n = 19) and control group (n = 19). The tutorial group students played a 13-problem game in which student and tutor each were required to mark the same position on a number line but could not see one anothers activities. To resolve discrepant solutions, tutor and student constructed agreements about number line principles and conventions to guide subsequent placements. Pre-/posttest contrasts showed that (a) tutorial students gained more than controls and (b) agreement use predicted gain. Analyses of micro-constructions during play revealed properties of student learning trajectories.

Studying Cognition through Time in a Classroom Community: The Interplay between “Everyday” and “Scientific Concepts”

This paper presents an analytic approach for understanding the interplay through time between “scientific” and “everyday concepts” in a mathematics classroom community. To illustrate the approach, we focus on an elementary classroom implementing an integers and fractions lesson sequence that makes use of the number line as a principal representational context. In our analysis of the communitys emerging collective practices (recurring structures of joint activity), we trace the interplay between childrens sensorimotor actions (displacing, counting, and splitting) and the mathematical definitions supported in the classroom, like definitions of unit interval or equivalent fractions. In our illustrative analysis, we find that the teacher orchestrated collective practices to support the use of actions to make sense of the formal definitions, and the use of definitions to regulate actions. Though we illustrate the analytic approach for a particular classroom community, the approach illuminates teaching-learning dynamics that transcend any particular classroom or subject matter domain.

Understanding Learning Across Lessons in Classroom Communities: A Multi-leveled Analytic Approach

This chapter presents a methodology for studying classroom communities as microcultures, with a focus on processes of teaching and learning over significant spans of time. In Sect. 11.1, we present a conceptual framework that treats classroom activity at two levels of analysis, collective and individual. Both levels are geared for understanding the reproduction and alteration of a common ground of talk and action through time. Key concerns are the emergence of collective norms and individuals’ use of representational forms to serve varied functions in classroom communicative and problem solving activity. In Sect. 11.2, we show how the conceptual framework was used to organize two related programs of empirical research. First, we present design research that led to a 19-lesson sequence on integers and fractions , which uses the number line as a central representational form. Second, we use the framework to organize an empirical analysis of a single classroom community over the 19-lesson sequence. We illustrate empirical techniques for capturing the reproduction and alteration of a common ground with shifting lesson topics. The chapter concludes with an analysis of the way the analytic approach illuminates core processes of teaching and learning and the utility of the approach for future work.

Coordinating Numeric and Linear Units: Elementary Students’ Strategies for Locating Whole Numbers on the Number Line

Two investigations of fifth graders’ strategies for locating whole numbers on number lines revealed patterns in students’ coordination of numeric and linear units. In Study 1, we investigated the effects of context on students’ placements of three numbers on an open number line. For one group (n = 24), the line was presented in a thematic context as a “race course,” and, for a second group (n = 24), the line was presented as a conventional number line. Most students in both groups placed consecutive whole numbers at appropriate linear distances, but the thematic context group was more likely to place nonconsecutive whole numbers at appropriate linear distances. In Study 2 (n = 24), students placed numbers on lines marked with two numbers. Most students placed a third number appropriately when the marked numbers were consecutive whole numbers, but not when the labeled numbers were nonconsecutive whole numbers. The findings reveal fifth graders’ conceptual difficulties in coordinating numeric and linear units on the number line and a thematic context that can support this coordination.

Cultural context of cognitive development.

UNLABELLED The cognitive problems that children formulate and solve in their daily lives necessarily take form in a cultural context. We review and illustrate two dominant approaches to study relations between cultural context and cognitive development, and we point to the limitations and affordances of each. Using a dichotomous approach, scholars employ a methodology that sharply differentiates cognition from cultural context, treating elements of cultural context as independent variables and elements of cognition as dependent variables. The approach often leads to propositions about transcultural features of context that influence the cognitive development of individuals. In contrast, using an intrinsic relations approach, researchers create units of analysis that capture relations between cognition and cultural context, investigating their mutual grounding in daily activities. We also review a small but important body of research that extends these approaches to diachronic analysis. This research seeks to understand shifting relations between cultural context and cognitive development over historical time. WIREs Cogn Sci 2014, 5:447-461. doi: 10.1002/wcs.1300 For further resources related to this article, please visit the WIREs website. CONFLICT OF INTEREST The authors have declared no conflicts of interest for this article.

RESPONSE: Genetic Method and Empirical Techniques: Reply to Hatano and Sfard's Commentaries on Cognition in Flux

Some time ago, Geoffrey Saxe (1994) published an article in Mind, Culture, and Activity that invoked a distinction between a method of inquiry and empirical techniques. Techniques are approaches to data gathering like observations, interviews, and surveys, as well as research designs. They are a tool kit for empirical work. Deeply linked to conceptual frames and units of analysis, a method of inquiry is something bigger. A method, like the genetic method outlined in our article, positions the investigator to pursue particular leads while leaving others to remain unexplored, if not unnoticed. It supports an investigator’s efforts to parse nature at its joints, given theoretical assumptions about the character of knowledge and knowledge change. Our method of inquiry targets forms and functions used in collective practices, with a focus on continuities and discontinuities in processes of micro-, onto-, and sociogenetic change. In the case at hand, we set our sights on the lexical expression fu, using it as a microcosm to unravel some complex issues in culture–cognition relations. Sfard (this issue) and Hatano’s (this issue) commentaries spark a host of interesting issues or tensions between our techniques and our genetic method, and in our reply, we comment on three: (a) our genetic method and techniques to document naturally occurring conversation, (b) cross-cohort design techniques and what they afford in relation to our method, and (c) epistemological groundings of our method and techniques. In each case, we make efforts to clarify issues and in so doing point to productive tensions between methods and techniques in fieldwork.

Approaches to Reduction in Treatments of Culture-Cognition Relations: Affordances and Limitations

Gauvain and Munroe take on a provocative question when they ask: how is cultural change related to the cognitive development of individuals? To address the question, they report an ambitious project that contrasts the cognitive development of people from two small-scale traditional societies (in Kenya and Nepal) with those from two industrial societies (in American Samoa and Belize) [Gauvain & Munroe, 2009]. The findings produced are consistent with other studies that have investigated similar questions in less targeted ways: individuals from more industrialized communities perform more successfully on IQ-like measures than individuals from small-scale traditional groups. Further, across communities children’s performances on the IQ-like measures showed the expected correlations with variables associated with industrialization (like the adoption of radios in homes, the use of why? questions with children, and the reduction of open-fire cooking). The conclusion they draw is that, as small-scale traditional communities shift to industrial societies, individual members advance in cognitive development, and the authors point to their correlational analyses as ways of understanding mechanisms.