Saxe, G. B., & Esmonde, I.(in press). Studying cognition in flux: A historical treatment of fu in the shifting structure of oksapmin mathematics. Mind, Culture, & Activity. Download this article (PDF 2.5 MB)

This paper extends a framework for the study of culture-cognition relations to problems of historical research and diachronic analysis. As an illustrative case, we focus on mathematics in Oksapmin communities located in a remote highland area in central New Guinea. The Oksapmin, like their neighboring Mountain-Ok groups to the West, traditionally use a 27-body-part counting system for number, and there is no evidence that Oksapmin used arithmetic in pre-history. Based upon field studies completed in 1978, 1980, and 2001, the paper analyzes a change from a subsistence-oriented to a cash-oriented economy in which arithmetical activities are important, and the accompanying shift in functions of a word form used in mathematical activities. The word form has shifted from its use as an intensive quantifier that means “a complete group of plenty” to one that means double a numerical value. We show how the analytic framework affords a multi-level inquiry into genetic processes of change in the Oksapmin case and argue that the approach is useful for understanding the interplay between cultural and developmental processes in cognition more generally.

(Note: This paper probably best represents Geoffrey Saxe's current approach to analysis.)

Nasir, N. & Saxe, G. B. (in press). Ethnic and Academic Identities: A Cultural Practice Perspective on Emerging Tensions and Their Management in the Lives of Minority Students. Educational Researcher. Download this article (PDF <1MB)

Saxe, G. B. (2004). Practices of quantification from a sociocultural perspective. K. A. Demetriou & A. Raftopoulos (Eds), Developmental Change: Theories, models, and measurement. NY: Cambridge University Press: 241-263. Download this article (PDF <1MB)

(Note: This is an accessible paper that might be of interest to people wanting an overview of the conceptual framework that guides Geoffrey Saxe's work)

Saxe, G. B., & Esmonde, I. (2004). Making change in Oksapmin tradestores: A study of shifting practices of quantification under conditions of rapid shift towards a cash economy. South Pacific Journal of Psychology 15(1), 2-19.Download this article (PDF 1.1MB)

We report two studies about shifting practices of quantification in tradestores in Oksapmin communities (Papua New Guinea). In Study 1, we enlisted 7 local tradestore clerks to collect information about customers’ language practices of quantification, age cohort, schooling level, and cost of purchase. Analyses of 305 exchanges revealed that older cohorts tended to use indigenous practices and extensions of the indigenous language. Younger cohorts – particularly those with some schooling -- tended to use practices that involved Melanesian Pidgin. In Study 2, we analyze interviews with 9 tradestore clerks who described typical purchase transactions with customers from different age cohorts/schooling levels. Analyses of interviews revealed that elders tended to structure multi-item purchases into sequential transactions and use extensions of indigenous approaches to quantification. Schooled adults tended to purchase multiple items in a single transaction and use Pidgin quantifiers. We argue that tradestores today sustain multiple practices of quantification but also support change towards the exclusive use of Melanesian pidgin.

Presents a cultural-developmental framework for the analysis of children's mathematics in collective practices and illustrates the heuristic value of the framework through the analysis of videotaped episodes drawn from a middle-school classroom (see K. McClain, record 2002-13461-002). The framework is presented in 2 related parts. The 1st targets the children's emerging mathematical goals in collective practices, with a focus on the complex role that artifacts play in children's emerging goals. The 2nd part focuses on children's developing mathematics that takes form in their goal-directed activities: (a) microgenetic analyses concern the process whereby children structure cultural forms like artifacts to serve particular functions as they accomplish emerging mathematical goals; (b) sociogenetic analyses concern the spread or travel of mathematical forms and associated functions within a community of individuals; and (c) ontogenetic analyses concern the interplay between the forms that children use and the functions that they serve over the course of children's development. The analyses of the classroom episodes illustrates the framework as a method for furthering understanding of the interplay between social and developmental processes in children's mathematics. (PsycINFO Database Record (c) 2002 APA, all rights reserved

Investigated the influence of professional development and curriculum on upper elementary students' understandings of fractions. Three groups of teachers and their students participated. Two groups implemented a fractions unit that emphasized problem solving and conceptual understanding. The Integrated Mathematics Assessment (IMA) group participated in a program designed to enhance teachers' understandings of fractions, students' thinking, and students' motivation. The Collegial Support (SUPP) group met regularly to discuss strategies for implementing the curriculum. Teachers in the 3rd group (TRAD) used textbooks and received no professional development support. Contrasts of student adjusted posttest scores revealed group differences on 2 scales. On the conceptual scale, IMA classrooms achieved greater adjusted scores than the other 2 groups, with no differences between SUPP and TRAD groups. On the computation scale, contrasts revealed no differences between IMA and TRAD, although TRAD achieved greater adjusted scores than SUPP. Findings indicate that the benefits of reform curriculum for students may depend upon integrated professional development, one form exemplified by the IMA program. Computation and problem solving items used in the mathematics assessment are appended.

Examined emergent divisions of labor in children's collective mathematical problem solving during educational game playing. 96 3rd and 4th graders were divided into 4 groups and played Treasure Hunt in pairs. The 4 groups were based on mathematics achievement and grade and included high, low and 2 mixed ability groups. 32 pairs of game players played in their classrooms twice weekly for less than 2 mo, after which posttest tasks assessed players' and nonplayers' understanding of key mathematical knowledge related to the game. Results described organization of play, their relation to solution strategies during play and individual posttest strategies. The results showed that individual goal-directed activities both sustained and were constitutive of the collective play; collective efforts to accomplish emergent problems valued in play had implications for individual goals. The authors conclude that, parallel to A. N. Leontiev's (1981) argument in his activity theory, when labor becomes divided, children often become engaged in accomplishing different goals leading to different learning outcomes. (PsycINFO Database Record (c) 2000 APA, all rights reserved)

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Saxe, G. B., & Gearhart, M., & Seltzer, M. (1999). Relations between classroom practices and student learning in the domain of fractions. Cognition and Instruction, 17, 1-24.
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Gearhart, M., G. B. Saxe, et al. (1999). Opportunities to learn fractions in elementary mathematics classrooms. Journal for Research in Mathematics Education 30(3): 286-315.

Addressed 2 questions: (a) How can we document opportunities to learn aligned with the National Council of Teachers of Mathematics (NCTM) Standards? (b) How can we support elementary teachers' efforts to provide such opportunities? The authors conducted a study of the effects of curriculum (problem solving vs skills) and professional development (subject-matter focused as part of the Integrated Mathematics Assessment Program vs collegial support) on mathematics teaching practices and students learning as assessed with a paper-and-pencil test. From analyses of videotapes and field notes, 3 scales for estimating students' opportunities-to-learn were created measuring Integrated Assessment, Conceptual Issues, and Numerics. Analyses of fractions instruction in the classrooms of 21 elementary teachers provide evidence of the technical quality of the indicators and suggest that support for teachers' knowledge may be required for a problem-solving curriculum to be beneficial. (PsycINFO Database Record (c) 2000 APA, all rights reserved)

Sketches ways of understanding cognitive development that highlight its cultural roots and ways of understanding cultural change that highlight its roots in individual development. In the account, processes of microgenesis, ontogenesis, and sociogenesis play off of one another in collective cultural practices. In microgenesis, individuals create schematizations that build on prior representational and strategic constructions (ontogenesis). In turn, these schematizations may become appropriated by others, becoming seeds for the spread of new collective forms of representation or procedures for problem solving in a community (sociogenesis). With the sociogenesis of cultural forms, individuals gain access to new forms for microgenetic schematization that become the basis for new ways of engaging in practices and the germs for subsequent ontogenetic shifts in knowledge. Such an account may not only reveal the interplay between cultural and developmental processes over the social history of traditional groups but also provide a frame for understanding the dynamics of cognitive development in collective practices closer to home.... In sketching the framework, the author draws on prior studies on arithmetic of a remote group in Papua New Guinea--the Oksapmin of the West Sepik Province. (PsycINFO Database Record (c) 2000 APA, all rights reserved)

Piaget's legacy is manifest across a wide range of research endeavors in the behavioral and social sciences. In this chapter, I point to the way that Piaget's theory provides an important basis for my own work on the interplay between culture and cognitive development. I target sources of development.... In this chapter, I outline an approach for analyzing sources of conceptual development linked to cultural practices. Underlying my remarks is an argument that a systematic treatment of source requires an incorporation of both culture and individual agency in a single analytic framework. I point to the importance of integrating accounts of practice and of development in treatments of microgenesis, ontogenesis, and sociogenesis. (PsycINFO Database Record (c) 2000 APA, all rights reserved)

This chapter examines the way children's participation in cultural practices can influence their developing mathematical understandings. The author's focus is on street vending, an activity common for unschooled children in developing countries. The vendors he describes are 10-12 yr old boys who sell candy and are from poor urban areas in Brazil. The author examined the form children's mathematical goals take in the candy selling practice by conducting a series of ethnographic studies of the practice focusing on social processes that influenced the form of sellers' mathematical goals. The author also attempted to discover the characteristics of candy sellers' mathematics. To address this, interviews with individual children using practice-related mathematical problems were conducted and the understandings of sellers were contrasted with those of both urban and rural nonsellers. By contrasting sellers with nonsellers, the author was able to determine whether practice participation affected the kinds of mathematical understandings children developed.

Investigated relations between (1) student achievement in the domain of fractions and (2) the extent to which classroom practices are aligned with principles recommended by current reform frameworks (e.g., National Council of Teachers of Mathematics, 1989). Hierarchical linear model analyses were performed on classroom observation and pre- and postinstruction achievement data collected in 19 upper elementary classrooms. Alignment of classroom practices with reform principles was related to student achievement in problem solving but not in computation; furthermore, the relation differed for students who began instruction with different levels of prior knowledge. For students who started with a rudimentary understanding of fractions, the relation between measures of classroom practice and problem solving was linear. In contrast, for students without a rudimentary understanding of fractions, the relation was nonlinear. The findings demonstrate the value of reform principles as a guide for effective practice as well as the importance of a coordinated analysis of students' prior understandings and classroom practices in investigations of learning.

Reprinted from B. Scales et al (Eds.), Play and the Social Context of Development in Early Care and Education, New York, US: Teachers College Press, 1991, 143-155. (The following abstract of the original chapter appeared in record 1991-97908-009.) number development is a particularly fruitful domain for the investigation of developmental relations between culture and cognition; in focusing on aspects of the social organization of children's early number development, we gain access to a process whereby an evolving cultural construction--the number system--is communicated to children, who transform and incorporate it into the fabric of their own cognitive activities; in our research, we have been examining that process by observing how mothers teach their children (aged 2 1/2-5 yrs) to solve a counting problem... our analysis of adult-child interactions is set within a general model of cognitive development; it is our view that children's novel cognitive constructions result from dynamic interplay between their elaboration of problem-solving goals and coherent means to achieve those goals; as children identify new goals, they attempt to elaborate novel cognitive means, including conceptual structures, symbolic vehicles, and problem-solving strategies...

This chapter sketches a general framework for the study of children's learning environments informed by sociocultural perspectives of cognitive development. Central to the framework is the construct of emergent goals (EGs). EGs serve as a basis for both the analysis of children's construction of cognitive environments in practices and a conceptualization of children's learning. Guided by the framework, the authors explore questions about children's dyadic play of an educational game in their classrooms. The authors' efforts to extend the EGs framework to methods for the analysis of children's dyadic play have led to intriguing problems in coordinating analyses of what tasks the dyad is solving and the EGs that individuals are constructing and accomplishing in their joint activity. They describe 2 types of data, each of which addresses these issues. One type involves a case-by-case analysis of videotaped excerpts of joint play among 32 dyads of 3rd and 4th graders. The other involves the aggregation of case-by-case analyses through a framework-based coding scheme. In concert, these techniques provide a means of revealing general characteristics of the relations between the cognitive work that dyads accomplish as a unit and the cognitive environments that emerge for individuals.

Examined the interplay between social and developmental processes in children's mathematics learning, with a focus on children's performance on an educational game, and the way children's interactions in play frame developmental processes involving arithmetic with base-10 blocks. 64 3rd and 4th graders were grouped in same- and mixed-grade dyads. Analyses of interactions reveal that players were frequently involved with jointly structuring arithmetical problems involving base-10 blocks. However, the arithmetical goals that members of dyads created often differed as labor became divided in their activity. Two findings of particular interest include: (1) differences in divisions of labor as a function of players' grades and grades of their opponents led to construction of different arithmetical goals; and (2) differences in goals led to different sequences in children's strategic developments, sequences that differed from the developmental trajectory in the matched controls.

Review the development of a heuristic method for the study of culture-cognition relations, focusing specifically on relations between frameworks and empirical techniques over the course of its development... culture in cross-cultural studies of moral development: early discontents with method; representing culture in practice: mathematics in the Oksapmin of Papua New Guinea; documenting emergent goals through social interactional analyses: a fine-grained study of practices; the practice-based approach applied to candy selling

Addresses a question concerning the interrelation of culture with individuals' developing mathematical understandings: how do forms of thought that have been invented, appropriated, and specialized over the course of a culture's social history come to be interwoven with individuals' developing abilities to accomplish problems in everyday life... [sketches] mainstream psychological approaches that bear on this developmental question, pointing both to insights that they have yielded, and to their shortcomings, particularly with regard to the representation of culture in the developmental process; introduce G. B. Saxe's cultural practice framework as a means of elevating culture more centrally into analyses of cognition, illustrating the framework with examples drawn from Saxe's prior work with the Oksapmin of Papua New Guinea; argue that clinical assessment interviews with children are a form of cultural practice, and show the way in which "culture" is interwoven with microgenetic and ontogenetic shifts in students' evolving approaches to the solution of an individual interview measurement task.

In their efforts to document children's understandings, researchers have often sidestepped analyses of sociocultural processes in children's mathematics; sketch a research framework in which dimensions of daily life are elevated to a central target of analysis; show that research guided by the framework is useful for informing the creation of classroom practices for children's mathematics learning, and that the research framework . . . is useful for understanding children's learning in such practices.

Saxe, G. B. (1994). "Studying cognitive development in sociocultural context: The development of a practice-based approach." Mind, Culture, & Activity 1(3): 135-157.

Sketches the development of a research framework for analyzing the interplay between culture and cognitive development in cultural practices and the methodological tensions that gave rise to the framework. The framework consists of 3 components geared for analyzing intrinsic relations between culture and cognitive development. The 1st focuses on the analysis of individuals' goals as they take form in everyday practices. The 2nd is concerned with the shifting relations between cognitive forms and cognitive functions in individuals' efforts to accomplish those goals. The 3rd focuses on the appropriation and specialization of forms structured in 1 practice to accomplish emergent goals in another. Applications and progressive refinements of the framework are discussed in analyses of practices of economic exchange in a remote group in Papua New Guinea, number play in middle and working class children in Brooklyn, New York, and candy selling in Northeastern Brazil.

Presents a framework for the study of children's learning in cultural practices and educational activities. The framework consists of 3 analytic components, each of which is grounded in a constructivist treatment of cognitive development. These components are a model for the analysis of emergent cognitive goals in practices, a model for the analysis of cognitive developments linked to emergent goals, and a model for the analysis of the interplay between cognitive developments linked to 1 practice or activity to accomplish emergent goals in another. The early history of the framework is described, as is its application to the design and analysis of a classroom practice in the US involving arithmetical problem solving in 3rd and 4th grade inner-city classrooms. The framework is also discussed in reference to A. H. Schoenfeld's (see record 1994-00430-001) standards for methodological innovations.

In the first part of this volume, I introduce a general analytic model that targets cultural practices as important contexts for study. In subsequent parts, I apply the model to single cultural practice--candy selling--as it has emerged in the lives of children living in northeastern Brazil. In candy selling, the relations between culture and cognitive development stand out in particularly clear relief and are particularly amenable to study.... In the following four chapters [Part II], my concern is to understand candy sellers' mathematical goals, which emerge as they work their trade to achieve the larger objective of economic survival.... In Part III, my concern is to extend the second analytic component--form-function shifts in cognitive development--to an analysis of candy sellers' mathematics. Guided by an analysis of the emergent mathematical goals of the practice presented in Part II, I focus now on the kinds of cognitive forms sellers use to accomplish mathematical problems and the cognitive functions these forms serve.... My concern in the following chapters [Part IV] is to extend the first two components of the research model--emergent goals (component 1) and form-function shifts (component 2)--to an analysis of the interplay between learning across the practice of candy selling and school mathematics lessons. To accomplish this extension, we need to consider first the way that emergent mathematical goals in school may be distinct from those that take form in the candy-selling practice and then consider the way cognitive forms linked to one context may be used to accomplish problems in another.

Reports results of a natural experiment that allowed comparison of children with different amounts of schooling reasoning about prices and profits of the candy that they sold as street vendors and comparison of children with equal amounts of schooling, some of whom were experienced candy sellers and others who lacked commercial experience... sellers with more schooling made more use of procedures that involved symbolic notation of arithmetic, and sellers with less schooling seemed to depend more strongly on the specific features of currency, rather than numerical symbols indicating denominations; in solving arithmetic problems, experienced sellers made more use of regrouping strategies, compared to equally schooled but commercially inexperienced students who relied on standard algorithmic procedures... findings show that the qualitative reasoning of everyday cognition and the symbolic procedures of school mathematics are blended in children's cognitive functioning.

Studied the development of topological concepts in unschooled child straw weavers (N = 69) from rural communities in northeastern Brazil, using videotaped observations of everyday teaching interactions and analysis of weavers' topological understandings, contrasting weavers of different age levels (5-15 yrs) and weavers with age-matched nonweavers. Expert weavers presented more topological information more often in demonstrations of weaving actions than in verbalizations. Weavers showed greater skill with increasing age and were more able to construct homeomorphic patterns for novel weavers than age-matched nonweavers. Despite their expertise, weavers performed poorly on tasks that required them to verbalize how to weave known patterns.

Saxe, G. B. (1989). "Transfer of learning across cultural practices." Cognition & Instruction 6(4): 325-330.

Examines presuppositions about the character of children's out-of-school mathematics and transfer of learning between settings. Discussion focuses on numerical units, procedures for manipulating units, and activity contexts. New methods need to be produced that more closely reflect children's construction of knowledge forms in everyday practices and the appropriation and specialization of these practice-linked forms to solve problems.

In 4 studies, 240 children (aged 3-12 yrs) were given tasks in which they had to make judgments about (1) the adequacy of puppets' counting activities when puppets used standard as opposed to nonstandard number words in counting and (2) the adequacy of puppets' token exchanges when the values of the same tokens varied across numerical systems. In both tasks, findings reveal developmental shifts in children's ability to distance numerical meanings from conventional symbols.

this chapter presents a conceptual analysis of studies that relate language background to mathematics achievement... extrinsic/intrinsic effects

The mathematical understandings of 23 10-12 yr-old candy sellers with little or no schooling from Brazil were compared with those of 37 nonvendors matched for age and schooling. Subjects' performances were analyzed on 3 types of mathematical problems: representation of large numerical values, arithmetical operations on currency values, and ratio comparisons. As expected, both vendors and nonvendors developed nonstandard means to represent large numerical values; most vendors, in contrast to nonvendors, developed adequate strategies to solve arithmetical and ratio problems involving large numerical values. Findings support a model of cognitive development in which children construct novel understandings as they address problems that emerge in their everyday cultural practices.

Investigated the interplay between social and developmental processes in children's numerical understandings in working- and middle-class home settings using interviews with 78 2.5- and 4.5-yr-olds of both economic classes, interviews with Subjects' mothers, and observational studies of mother-child pairs. Findings show that Subjects were regularly engaged with social activities involving number, although the nature of their numerical understandings and their numerical environments differed in that (1) younger Subjects varied from older Subjects in their numerical understandings across a variety of tasks, (2) 4-yr-olds from middle-class homes displayed greater competence on more complex numerical tasks than did their working-class peers, and (3) during mother-child interactions, mothers adjusted the goals of activities to reflect the child's ability, and children adjusted their goals to their mothers' efforts to organize the activity. Two commentaries and a reply by the authors are included.

Examined the influence of Western schooling on the development of arithmetical understandings in a total of 71 children from Grades 2, 4, and 6 and in 22 nonschooled adolescents from a nontechnological culture, the Oksapmin of Papua New Guinea. The indigenous number system of the Oksapmin consists of 27 conventionally defined points on the body. Although arithmetical activities are not a part of traditional life for the Oksapmin, in the recently introduced Western school setting, Subjects must solve arithmetical problems as a part of everyday classroom activities. In Study 1, school Subjects were observed during an arithmetic test. In Study 2, all Subjects were interviewed about solution strategies to presented problems. Results show that Subjects not only spontaneously used the indigenous system in the context of school arithmetic, but also created new forms of numerical symbolization and calculation based on that system to deal with the school arithmetical problems, a developmental process that occurred without the aid of instruction. Discussion focuses on the nature of the novel conceptual developments, factors mediating the schooling effect, and the influence of prior knowledge on learning from instruction in children from diverse ethnic backgrounds.

Provides coordinated analyses of 3 aspects of the social context of children's developing operations within a knowledge domain. Very young children in a sample of 2.5-5 yr old children who were videotaped with their mothers while engaged in a number reproduction activity often imposed the goal of producing a count of a single array on the nominal task, and their means of accomplishing the count were typically not well-developed. Older Subjects began to construe the task as having a double-array goal structure and attempted to produce numerical representations of the model using the means that they had developed to achieve single-array goals. All mothers continually adjusted the goal structure of the task during the activity itself. Analyses suggested that the goal structure of numerical activities as they occur in social interactions is an emergent phenomenon that is negotiated in the interaction itself. As children generate coherent means to achieve these socially negotiated goals, they create for themselves a system of representation that reflects achievements.

86 children, aged 4-16 yrs, from 3 remote village populations in Papua New Guinea were administered notational counting and number conservation tasks. The results replicate previous research conducted in the US that showed that children develop the use of counting to mediate the comparison and reproduction of sets prior to understanding number conservation. The significance of these findings is discussed with respect to current models of number development.

79 Subjects (20-50 yrs old) were interviewed about the addition and subtraction of coins when the coins were available to count and when they were not. Results show that Subjects solved all problems with increasingly sophisticated computational procedures. Analysis of these procedures revealed cognitive structural changes in the development of correspondence operations and functional shifts in the way in which body parts were used to effect solutions to the arithmetic problems.

Presents an analysis of the acquisition of an indigenous body part numerational system by 107 5-16 yr olds in remote Oksapmin village populations in Papua New Guinea. Findings from Studies I and II indicate that Oksapmin Subjects progressed from premediational to mediational phases in their use of body parts to compare and reproduce number and that this change generally occurred prior to the development of concepts of number conservation. This parallels findings in the US. Findings in Study III show that this general change was manifested in culturally specific ways. For instance, Oksapmin Subjects progressed from a belief that the numerical relation between any 2 body parts is determined by their physical similarity to the understanding that the relation is determined by their ordinal positions in an enumeration. (10 ref)

Discusses stages in child development in which children acquire the use of number operations and number vocabulary and are eventually able to combine the 2 effectively. The cases of 2 brain-lesioned adults demonstrate that the use of operations and words are not necessarily linked in cognitive functioning. However, there may be important interactive relations between number words and number operations, such that they influence each other during the course of development. This interaction also means that children who have difficulties in acquiring quantitative language or who are delayed in elaborating numerical operations will also face difficulties with the other construct. Remediation should encourage the child to play upon interactive relations between number symbols and number operations so that the development of one might support the development of the other. (9 ref)

On the basis of Piaget's theory (1970, 1972), it was hypothesized that developmental changes in operational reasoning would be necessary for children to understand the determinate and indeterminate age relations implied by birth-order names within and across sex. To test this formulation, 29 Ponam islanders (8-23 yrs old) were interviewed. Findings support the hypothesis. (7 ref)

42 children 2, 4, and 6 yrs of age were required to count 2, 3, 7, or 8 objects in 2 conditions--one in which they could not use pointing gestures and the other in which they were encouraged to do so. With increasing age, Subjects' counting accuracy improved across both conditions; however, the 4-yr-olds' performance was significantly better with gestures. It is argued that the functional relation between the recitation of number names and Subjects' pointing gestures in counting changes over the course of development. (7 ref)

Analyzed cognitive developmental changes in the use of a convention for measurement among the Oksapmin, a recently contacted cultural group who live in a remote section of New Guinea. The Oksapmin measurement system consists of conventionally defined points on the arm and is used in practical activities involving the measurement of string bags, a common cultural artifact. The Subjects consisted of 103 individuals, including unschooled children (mean age 8.4 yrs), unschooled adults (mean aged 29.5 yrs), Grade 2 children (mean age 10.9 yrs), and Grade 6 adolescents (mean age 13.9 yrs), who were administered 3 types of tasks assessing their ability to use their conventional system to produce comparisons of lengths. It was found that the Oksapmin develop an understanding of the necessity of equivalent units despite the fact that units of measurement are not equivalent in the Oksapmin system since individuals' arms vary in length.

A battery of Piagetian tasks was administered to 2 9-yr-old boys with normal measures of IQ, language ability, and reading but who were unable to acquire elementary numerical skills and manifested other Gerstmann syndrome-related cognitive deficits. Results show neither Subject had advanced to the Piagetian stage of concrete operations, indicating that while Subjects' inability to acquire numerical skills was due to a delay in the stage transition, the acquisition of reading skills was not dependent on the stages of operational development. (10 ref)

Examined the differences between 45 5-, 7-, and 9-yr-olds' ability to estimate their accuracy for large set sizes on tasks of 3 levels of counting difficulty. ANOVAs revealed that younger Subjects tended to believe that they counted accurately, independent of their actual accuracy. With increasing age, Subjects' estimates of their counting accuracy increasingly corresponded to their actual counting accuracy. Findings support the view that children increasingly monitor the relation between their counting activity and the numerical products of their counting over the course of development. (8 ref)

Saxe, G. B. (1979). "Developmental relations between notational counting and number conservation." Child Development 50(1): 180-187.

Examined the developmental relationship between a child's use of counting as a notational symbol system to extract, compare, and reproduce numerical information, and the development of number conservation. In Study 1, 66 4-6 yr olds were administered notational-counting and number-conservation tasks. Analysis of Subjects' profiles across tasks indicated that children develop quantitative counting strategies (but do not necessarily count accurately) before they develop number-conservation concepts. In Study 2, the generality of this sequence was tested. 44 7-9 yr old "learning-disabled" Subjects who reportedly were developing atypical counting skills were administered notational-counting and number-conservation tasks. All Subjects who conserved number also used quantitative counting strategies, although some frequently counted arrays inaccurately. The significance of these findings is discussed with respect to existing models of counting/number-conservation relations, and an alternative formulation is suggested based on the new findings. (10 ref) (PsycINFO Database Record (c) 2000 APA, all rights reserved)