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Body Parts as Numerals: A Developmental Analysis of Numeration Among the Oksapmin in Papua New Guinea
Abstract
The purpose of this paper is to present an analysis of the acquisition of an indigenous body part numerational system by children who live among remote Oksapmin village populations in Papua New Guinea. The findings reported in studies 1 and 2 indicate that Oksapmin children progress from premediational to mediational phases in their use of body parts to compare and reproduce number and that this change generally occurs prior to the development of concepts of number conservation. This trend parallels findings in the United States. The findings reported in study 3 show that this general change is manifested in culturally specific ways. For instance, Oksapmin children progress from a belief that the numerical relation between any 2 body parts is determined by their physical similarity (e. g., symmetrical body parts imply numerical equivalence) to the understanding that the numerical relation between any 2 body parts is determined by their ordinal positions in an enumeration.
... Because the early stages of the development of number were personal-need driven, most early numeral systems were based on the handiest counting devices available, including body parts which serve as a recursive base for the counting system (Gordon 2004, 496;Heine 1997;Saxe 1981). The ten fingers for example provided the inspiration for the decimal system that is dominant today. ...
... Martins (2004Martins ( , 1994 cited in (Epps 2006, 265) reports that speakers of Dâw, an Amazonian language, indicate 'four' by holding the fingers of one hand separated into two blocks; for 'five' they add the thumb; for 'six' they place the second thumb against the first to make a third pair; and so on until for 'ten' all fingers are grouped into five pairs, with the thumbs together. Saxe (1981) also, reports that among the Oksapmin of Papua New Guinea parts of the body are touched sequentially to communicate number. He describes the system thus: ...
In this study, I present a constructional account of the internal grammar of Akan complex numerals. One of the main concerns in this thesis is to determine what the formation and structure of complex nominals in Akan reveal about the nature of the relation between morphology and syntax and about the architecture of the grammar. For this, we have defended a constructional view by which it is argued that the full gamut of grammatical constructions ranges from the most regular to the most idiosyncratic with varying degrees of gradation or sub-regularities in-between. If there is one “corner” of the grammar where one finds a concentration of evidence for this view, it is numeral system where some numerals have very regular formal and semantic properties whilst others have highly idiosyncratic properties. Some numerals are unmotivated mono-morphemic units whilst others are full syntactic constructions, taking the form of simple transitive clauses or double object constructions or even coordinate constructions.
... Similarly, Greenfield (1966) found that unschooled children in Senegal did not attain conservation on their own, paying more attention to the authority of the experimenter than to the quantities involved; however, children at the age of European or American children who attain conservation quickly attained the conservation concept when an intervention was carried out in which the child replaced the experimenter in manipulating the conservation materials. Saxe (1981Saxe ( , 1982 found an additional substage in Oksapmin children of Papua New Guinea due to the use of a local body-count system for digits. These are exceptions that only confirm the rule. ...
... There have been a few attempts of a more "emic" approach, notably Saxe's (1981Saxe's ( , 1982Saxe's ( , 2012b research among the Oksapmin in Papua New Guinea (PNG). He used the local body-parts counting system in assessing the number concept and locally made string bags for the conservation of length. ...
This paper reports my own perceptions of the history of research on culture and cognitive development, in the approximate period 1960 to 2000. I review in particular my own efforts to test Piaget’s theory cross-culturally, but also include other lines of research such as research inspired by Vygotsky and research on child rearing/socialization potentially linked to cognitive development. I briefly mention a research program in Bali, Indonesia, India, and Nepal on the geocentric spatial frame of reference that allowed us to disentangle a variety of eco-cultural and linguistic variables that determine the preference for this geocentric cognitive style. I also recall the “integrated framework” that I proposed in 2003 to combine the models proposed by several authors and which serves to integrate all the findings. The main conclusion is that cognitive processes are universal but that there are cultural differences in cognitive styles and pathways of development. I also discuss why the field has lost momentum, whether it is because all the questions have been answered or because new topics and research methods have evolved.
... For example Oksapmin uses an extended body part counting system where different parts of the bodies are assigned different values. However, 'kir' (= elbow), or '8' (Saxe, 1981), is not a number in the sense that it is used solely to describe value but rather the other way around: The value 8 is assigned to the elbow area of the left arm when counting. This means, although a concept of numbers is present, it is based only on the means of sequence. ...
... The third stage, indexical links, is the beginning of the transformation from iconic numerical links to indexical links, meaning that certain words that had been used as an iconic tally previously transform into words that express cardinality. For example, Oksapmin use the word 'kata' (= shoulder) to denote '10' (Saxe, 1981) in their body part counting system. If kata were to be used for a long period of time while the language develops naturally, it might eventually split off lexically from shoulder (with most likely a change in phonology or morphology as well) and become an abstract word that is solely used to express cardinality. ...
New Guinea and its surrounding islands show a myriad of different and unique aspects of people, culture and language. The frequent mixing of languages combined with relatively little value placed on language as part of one's identity creates an environment where languages change amply and quickly. There is quite a lot of attention towards the linguistic phenomena in New Guinea but, since there is so much to be studied, data on many languages is still scarce. An area that has been especially neglected is the area of numeral systems on the island. This paper provides in-depth information about many numeral systems and connects them to migration patterns, cultural practices, and cognitive development theories. A database including basic information of numeral systems found in the area was created and the languages were categorized by type and family, and were then displayed on a map. The map was then further divided into three distinct layers based on numeral system patterns found. These layers are thoroughly analyzed and attempts to explain the patterns are made. The languages included in this study are, among other factors, based on availability and completeness of data; any conclusions made within this paper are as precise as possible but are not meant to be final as new information could change or even devalue conclusions or patterns. The paper discusses phenomena such as senary numeral systems based on subitization, the stages of development in number concepts and how the systems found on New Guinea fit within these stages, and will hopefully bring us a step closer to fully understanding the concept of numbers and their development within New Guinea and its surrounding islands.
... The body/finger counting is common in Africa especially in Rwanda and Tanzania where they form an integral part of all transaction in the market (Zaslavsky, 1999) and body parts are touched in sequence to communicate numbers. This mode of counting is prevalent among the people of Oksapmin village populations in Papua New Guinea (Saxe, 1981) as shown in Figure 2.1. The main challenge with body/finger counting is that it has to be done on a physical present human; and the process must be watched with attention. ...
... stones, cowries and seeds) to represent numbers. To represent 32,000 in a place where stones are used for counting, 32,000 (Saxe, 1981) stones will be required. Therefore, making the tally system bulky when representing large numbers. ...
... I analysed age-related shifts in unschooled Oksapmin children's use of body-part counting when they were asked to compare or reproduce groups of objects as well as when they were asked to measure string bags. In one study conducted in 1978 (Saxe, 1981), I found that although younger 305 Oksapmin children could recite body parts in the conventional sequence, they did not use reference to body parts to mediate numerical comparisons and reproductions. For example, if asked to reproduce numerically a group of stones, children would create an approximate number in their copy without the use of body parts to count each group. ...
... Con niños no escolarizados, analizamos los cambios que se producían en el uso del sistema según la edad cuando les pedíamos que comparasen o reprodujeran grupos de objetos, y también cuando les pedíamos que midieran bolsas de cuerda. En un 820 estudio realizado en 1978 (Saxe, 1981), encontré que, si bien los niños más jóvenes podían recitar las partes del cuerpo en el orden convencional, no hacían referencia a las partes del cuerpo para llevar a cabo las comparaciones y reproducciones numéricas. Por ejemplo, si se les pedía que reprodujeran numéricamente un conjunto de piedras, los niños creaban un duplicado con un 825 número aproximado de piedras sin utilizar las partes del cuerpo para contar las piedras de cada conjunto. ...
This article sketches a framework for the study and analysis of culture-cognition relations that I present in my recent book, Cultural Development of Mathematical Ideas: Papua New Guinea Studies. Taking an historical approach, I focus on the reproduction and alteration of representational forms for number and the functions that representational forms serve in collective practices in and out of school. My key argument is that, in the context of their goal-directed communicative activities in collective practices in daily life, people unwittingly reproduce and alter representational forms and the functions that these forms serve. The process leads to continuities and discontinuities in form-function relations in communities over historical time. To provide evidence for this process and its role in the cultural development of ideas, I report selected findings from research conducted in 1978, 1980 and 2001 in Oksapmin communities; I focus on historical shifts in the forms and functions of the Oksapmin body part counting system over time. I then update that body of research with a sketch of a recent 2014 follow-up study. I close with reflections on the utility of my framework for the study of culture-cognition relations in other communities and on cognitive domains other than mathematics.ResumenEste artículo esboza un modelo para el estudio y análisis de las relaciones entre cultura y cognición que he presentado en mi reciente publicación Cultural Development of Mathematical Ideas: Papua New Guinea Studies (Desarrollo cultural de las ideas matemáticas: Estudios en Papúa Nueva Guinea). Adoptando un enfoque histórico, me centro en la reproducción y alteración de las formas de representación del número y las funciones que estas formas representacionales desempeñan en las prácticas colectivas dentro y fuera de la escuela. Mi principal argumento es que, en el contexto de las actividades comunicativas dirigidas a la consecución de metas en las prácticas colectivas en la vida cotidiana, las personas involuntariamente reproducen y alteran las formas de representación y las funciones para las que esas formas se utilizan. Este proceso provoca continuidades y discontinuidades en las relaciones entre forma y función en las comunidades a lo largo del tiempo. Para mostrar evidencias de este proceso y su papel en el desarrollo cultural de las ideas, informo de hallazgos seleccionados de las investigaciones realizadas en 1978, 1980 y 2001 en comunidades oksapmin. Me centro en los desplazamientos históricos en las formas y las funciones del sistema de cómputo oksapmin basado en las partes del cuerpo. A continuación, actualizo ese cuerpo de investigación con un bosquejo de un estudio reciente de seguimiento, realizado en 2014. Finalizo el artículo con algunas reflexiones sobre la utilidad de mi modelo para el estudio de las relaciones entre cultura y cognición en otras comunidades y en dominios de conocimiento diferentes de las matemáticas.
... Traditionally, Oksapmin people use a body count system consisting of 27 body parts, as depicted in Figure 1. To count as Oksapmin do, one begins with the thumb on one hand and enumerates 27 places around the upper periphery of the body, ending on the little finger of the opposite hand (Moylan, 1982;Saxe, 1981). To indicate a particular number, one points to the appropriate body part and says the body part name. ...
... bum rip, (4) h^tdip, (5) h^th^ta, (6) dopa, (7) besa, (8) kir, (9) tow^t, (10) kata, (11) gwer, (Saxe, 1981). ...
We report an observational study of school mathematics instruction in Oksapmin, a remote Central Highlands community in Papua New Guinea. As a part of a national initiative, Papua New Guinea is engaged in implementing educational reforms that attempt to create strong links between activities in school and students' out-of-school lives. We show how reforms have mitigated some of the sharp discontinuities between activities in and out of school though at the same time they have created others. For example, the prior colonial-like system of education that did not allow use of indigenous languages in school classrooms (a discontinuity between in and out of school) has given way to elementary schools in which children are not only encouraged to use their native language in school, they are taught the indigenous mathematics of traditional life. However, in the process of incorporating the traditional mathematics into school instruction, that mathematics is altered in some fundamental ways, as teachers create explicit links to the base 10 English counting system. The transformations of out-of-school practices raise questions about the long-term impact of schooling on out-of-school mathematical practices and traditional ways.
... Therefore, children acquire learning as a culturally mediated experience (Rogoff 2003;Vygotsky 1978). To illustrate, Saxe (1981), Saxe and Esmonde (2005) reported on the mathematical system of the Oksapmin of Papua New Guinea, who used a method of counting that incorporated 27 specific parts of the body to indicate quantity. An elbow, a finger, a wrist, etc. have a specific numerical meaning. ...
... e Kédang people of Indonesia accomplish complex mensuration and numeration tasks, such as the measuring and evaluation of elephant tusks as part of the ivory trade (Barnes 1982). In various societies of Melanesia, 'body-counting' is used in place of numeral words, naming various parts of the body in sequence as a means of counting (Saxe 1981;Biersack 1982). e pebble-abacus was the central technique for performing computations in the ancient eastern Mediterranean (Lang 1957;Taisbak 1965;Schärlig 2001). ...
This Handbook explores the history of mathematics under a series of themes which raise new questions about what mathematics has been and what it has meant to practice it. It addresses questions of who creates mathematics, who uses it, and how. A broader understanding of mathematical practitioners naturally leads to a new appreciation of what counts as a historical source. Material and oral evidence isdrawn upon as well as an unusual array of textual sources. Further, the ways in which people have chosen to express themselves are as historically meaningful as the contents of the mathematics they have produced. Mathematics is not a fixed and unchanging entity. New questions, contexts, and applications all influence what counts as productive ways of thinking. Because the history of mathematics should interact constructively with other ways of studying the past, the contributors to this book come from a diverse range of intellectual backgrounds in anthropology, archaeology, art history, philosophy, and literature, as well as history of mathematics more traditionally understood. The thirty-six self-contained, multifaceted chapters, each written by a specialist, are arranged under three main headings: ‘Geographies and Cultures’, ‘Peoples and Practices’, and ‘Interactions and Interpretations’. Together they deal with the mathematics of 5000 years, but without privileging the past three centuries, and an impressive range of periods and places with many points of cross-reference between chapters. The key mathematical cultures of North America, Europe, the Middle East, India, and China are all represented here as well as areas which are not often treated in mainstream history of mathematics, such as Russia, the Balkans, Vietnam, and South America. A vital reference for graduates and researchers in mathematics, historians of science, and general historians.
... During our second plática, I wanted the prospective maestras to engage with the idea of mathematics as a cultural activity. The women discussed the Oksapmin counting system and the study done in Brazil about candy sellers (Saxe, 1981(Saxe, , 1988. The goal was for the women to expansively view mathematics as an everyday activity. ...
We use vignettes from two research projects to illustrate how we seek to engage with the messiness of becoming antiracist educators. We show how we center historias in mathematics to affirm individual experiences and create opportunities to disrupt white supremacy in math education. We find that such work is complex and nuanced, requires deep and critical engagement from researchers and prospective elementary teachers, and entails creating authentic relationships that allow for vulnerability and foster solidarity within and beyond institutional contexts.
... Bu işlem bir elin bir parmağından başlar ve vücudun üst çevresinde eşlik eden vücut parçası adlarıyla birlikte 27 yeri sayarak diğer elin küçük parmağında biter. Bu kültürde çocuğun 12 sayısı için kulağı işaret etmesi ve kulak kelimesini söylemesi gerekmektedir (Saxe, 2012;Saxe, 1981). ...
... This question made me think for a moment. Then I realized that the counting systems were as varied in Papua New Guinea as their languages, each community having a slightly different approach, some using only the right or left half of their body, some using body parts from their head down to their belly button or toes (as interpreted from Saxe, 1981, andWassman &Dasen 1994). Trade, and therefore relationships, were vital to these communities. ...
In this article, the authors argue that trans-systemic knowledge system analysis of Indigenous-to-Indigenous economics enables generative thinking toward Indigenous futures of economic freedom. The authors apply a trans-systemic lens to critically analyze persistent development philosophy that acts as a barrier to the advancement of Indigenous economic development thinking. By exploring ways in which colonial discourse entraps Indigenous nations within circular logic in service of a normative centre the need for new economic logic is apparent. Shifting to trans-systemic knowledge systems analysis to include diverse insights from Māori and other Indigenous economic philosophy, the authors show that it is not profit and financial growth that matters in and of itself. Rather, according to Indigenous definitions of wealth, economic freedom and development are constituted by value creation that aligns with Indigenous worldviews and principles. Indigenous economic knowledge centred on relationship, reciprocity and interconnectedness fosters Indigenous economic freedom.
... This question made me think for a moment. Then I realized that the counting systems were as varied in Papua New Guinea as their languages, each community having a slightly different approach, some using only the right or left half of their body, some using body parts from their head down to their belly button or toes (as interpreted from Saxe, 1981, andWassman &Dasen 1994). Trade, and therefore relationships, were vital to these communities. ...
Indigenous peoples globally are seeking new ways in which to communicate and share our worldviews. Sometimes defined as resistance research, emancipatory research, decolonising research - our research (re)presents the multiple journeys in which we live and come to know. Emerging Indigenous research methodological approaches are centring Indigenous ways of knowing, being and doing, to privilege Indigenous voices that have been suppressed through colonization. The intricate weaving of Western methodologies with Indigenous knowledges evokes agency in two emerging Indigenous researchers (from Australia and Canada) and weaves a path of reconciliation between their diverse disciplines as well as the seemingly dichotomous knowledge systems they are challenged to work within. Using metalogue, a way of authentically bringing together multiple voices through dialogue, we discuss the creative and radical Indigenous methodological approaches developed and enacted within our PhDs. The paper will provide insights to the epistemological, ontological and axiological principles that inform emerging Indigenous approaches to research.
... Therefore, children acquire learning as a culturally mediated experience (Rogoff 2003;Vygotsky 1978). To illustrate, Saxe (1981), Saxe and Esmonde (2005) reported on the mathematical system of the Oksapmin of Papua New Guinea, who used a method of counting that incorporated 27 specific parts of the body to indicate quantity. An elbow, a finger, a wrist, etc. have a specific numerical meaning. ...
... The construction of the concept of number was an important topic in Piagetian developmental psychology, and there had been previous research among the Oksapmin of the PNG highlands by Saxe (1981; summarized and expanded in Saxe 2012) who had found that the Oksapmin children developed the number concept using the local body count system in the same order of sub-stages as described by Piaget, except for an additional difficulty linked to the symmetrical body parts. Saxe, after documenting the traditional number system, had used induced situations (i.e. ...
Over the years, I have had several opportunities of sharing field-work with anthropologist Jürg Wassmann, particularly in Papua New Guinea and in Bali, Indonesia. This is a chapter in a Festschrift in his honour. I retrace the advantages of such collaborative research between a psychologist and an anthropologist, but also its difficulties, and the sometimes delicate negotiations needed to agree on a common methodology.
... Beispiele für universelle kognitive Prozesse finden sich in verschiedenen Domänen. Dasen (2012) bezeichnet etwa die sensu-motorische Intelligenz im Sinne von Piaget und Inhelder (1972) als "Kandidaten für eine ‚starke' Universalie" (S. 56) und auch einige grundlegende mathematische Fähigkeiten können als universell angesehen werden, wie etwa das Zählen (Frank et al. 2008;Gordon 2004;Pica et al. 2004) und die Grundrechenarten (Petitto und Ginsburg 1982;Posner 1982;Saxe 1981; siehe auch Klein und Starkey 1988). Chomsky (1980) postulierte weiterhin die Existenz einer universellen Grammatik, der alle menschlichen Sprachen entsprechen. ...
Der vorliegende Beitrag beschäftigt sich mit der Frage, inwieweit Schulleistungen als universell versus kulturgebunden konzeptualisiert werden sollten. Hierfür werden theoretische Überlegungen und empirische Befunde aus zwei theoretischen Perspektiven – einer ökokulturellen und einer sozialkonstruktivistischen – überblicksartig dargestellt. Der Beitrag kommt zu dem Schluss, dass einerseits grundlegende kognitive und motivationale Strukturen, die schulisches Lernen erst ermöglichen, universell auffindbar sind und, dass auch die groben Ziele sowie Rahmenbedingungen institutionalisierter schulischer Bildung transnational ähnliche Grundzüge aufweisen. Andererseits sind Prozesse des Erwerbs spezifischer Kompetenzen und Wissensinhalte sowie die Bedingungen ihres Nachweises in Leistungsüberprüfungssituationen tief verwurzelt in dynamischen und interagierenden kulturellen Kontexten.
... Further, unlike their Western counterparts who acquire verbal numeration systems, Oksapmin children face a challenge linked to the spatial properties of their 27-body part numeration system, specifically from the fact that most body parts occur in symmetric pairs. When comparing the numerical value of symmetrical body parts, Oksapmin children therefore tended to regard them as indicating the same number (Saxe, 1981). Such challenges are not present for children in groups using only a verbal system. ...
In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively derived research agenda. Journal of Numerical Cognition, 2, 20–41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines—including anthropology, archaeology, cognitive science, history of science, linguistics, philosophy, and psychology—we argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it.
... Across cultures children and adults use their fingers to count, to communicate about numbers, and to do arithmetic. Body-based counting and arithmetic systems, all involving fingers and sometimes other body parts, emerged independently across human cultures and history, from New Guinea (Saxe, 1981) to Mayans, ancient Babylon, and Romans (Richardson, 1916), and show a wide range of variability (Bender & Beller, 2012). Given the deep relation between fingers and numbers, it is crucial to understand how and why finger counting and other finger-based interactions with numbers relate to numerical development, and how we can develop approaches in mathematics education that can harness the relevance of fingers for numerical development. ...
Even though mathematics is considered one of the most abstract domains of human cognition, recent work on embodiment of mathematics has shown that we make sense of mathematical concepts by using insights and skills acquired through bodily activity. Fingers play a significant role in many of these bodily interactions. Finger-based interactions provide the preliminary access to foundational mathematical constructs, such as one-to-one correspondence and whole-part relations in early development. In addition, children across cultures use their fingers to count and do simple arithmetic. There is also some evidence for an association between children’s ability to individuate fingers (finger gnosis) and mathematics ability. Paralleling these behavioral findings, there is accumulating evidence for overlapping neural correlates and functional associations between fingers and number processing. In this paper, we synthesize mathematics education and neurocognitive research on the relevance of fingers for early mathematics development. We delve into issues such as how the early multimodal (tactile, motor, visuospatial) experiences with fingers might be the gateway for later numerical skills, how finger gnosis, finger counting habits, and numerical abilities are associated at the behavioral and neural levels, and implications for mathematics education. We argue that, taken together, the two bodies of research can better inform how different finger skills support the development of numerical competencies, and provide a road map for future interdisciplinary research that can yield to development of diagnostic tools and interventions for preschool and primary grade classrooms.
... In his ethnographic studies on the Oksapmin, Saxe classified several strategies used to solve arithmetic problems (Saxe, 1981(Saxe, , 1982a(Saxe, , 1982b; see also Saxe & Esmonde, 2005). In one of the problems, the participants were asked to calculate 6 + 8 without using coins or other tangible objects. ...
... Arithmetic is always carried out using a number system, which has specific characteristics. In this review, we do not discuss different types of number systems, a topic which could take up the whole of the review and has been masterfully analyzed in other publications (e.g., Lancy 1983;Lean 1992;Miller and Stigler 1987;Miura et al. 1988;Owens 2001;Saxe 1981;Seron and Fayol 1994;Zaslavsky 1999). We focus on arithmetic carried out with either an oral or a written numeration system that uses place value notation, because place value notation is widely adopted around the world today. ...
This book offers a theory for the analysis of how children learn and are taught about whole numbers. Two meanings of numbers are distinguished – the analytical meaning, defined by the number system, and the representational meaning, identified by the use of numbers as conventional signs that stand for quantities. This framework makes it possible to compare different approaches to making numbers meaningful in the classroom and contrast the outcomes of these diverse aspects of teaching. The book identifies themes and trends in empirical research on the teaching and learning of whole numbers since the launch of the major journals in mathematics education research in the 1970s. It documents a shift in focus in the teaching of arithmetic from research about teaching written algorithms to teaching arithmetic in ways that result in flexible approaches to calculation. The analysis of studies on quantitative reasoning reveals classifications of problem types that are related to different cognitive demands and rates of success in both additive and multiplicative reasoning. Three different approaches to quantitative reasoning education illustrate current thinking on teaching problem solving: teaching reasoning before arithmetic, schema-based instruction, and the use of pre-designed diagrams. The book also includes a summary of contemporary approaches to the description of the knowledge of numbers and arithmetic that teachers need to be effective teachers of these aspects of mathematics in primary school. The concluding section includes a brief summary of the major themes addressed and the challenges for the future.
The new theoretical framework presented offers researchers in mathematics education novel insights into the differences between empirical studies in this domain. At the same time the description of the two meanings of numbers helps teachers distinguish between the different aims of teaching about numbers supported by diverse methods used in primary school. The framework is a valuable tool for comparing the different methods and identifying the various assumptions about teaching and learning.
... Arithmetic is always carried out using a number system, which has specific characteristics. In this review, we do not discuss different types of number systems, a topic which could take up the whole of the review and has been masterfully analyzed in other publications (e.g., Lancy 1983;Lean 1992;Miller and Stigler 1987;Miura et al. 1988;Owens 2001;Saxe 1981;Seron and Fayol 1994;Zaslavsky 1999). We focus on arithmetic carried out with either an oral or a written numeration system that uses place value notation, because place value notation is widely adopted around the world today. ...
Our aim in this topical review of the teaching and learning of number was to search for pivotal ideas and domains of research that have occupied researchers in mathematics education within the last four decades and to draw some lessons for further research.
... This hypothesis, generally referred to as Embodied Cognition, is the idea that cognition is embodied, meaning that cognition, including language, derives from the experiences in the real world that come from the body's interaction with the environment through the perceptual and motor modalities There are many different positions on what embodiment is, with respect to meaning and representation (Anderson, 2003;Wilson, 2002;Ziemke, 2001)including the view that even abstract concepts are influenced by perception-action in a dynamic world (e.g., Landy & Goldstone, 2007), perhaps via metaphors related to more concrete meanings (e.g., Matlock, 2004). Indeed, body parts have been found to often be used for this type of "grounding", that is, as a metaphor framing many abstract semantic domains, such as number, space, and emotion, in terms of body parts and physical world experience (de Leon, 1994;Saxe, 1981;Yu, 2004). There are some suggestive ideas that have been put forth that body parts may also play such a metaphoric role in our understanding of verb meaning (e.g., see hints in Richardson, Spivey, Barsalou, & MacRae, 2003), which we would not be surprised if explored and supported by future work. ...
In lieu of an abstract, here is a brief excerpt:All information enters the cognitive system through the body. Thus, it is possible that the body—and its morphology—may play a role in structurng knowledge and acquisition. This idea is particularly cogent in the case of verbs, since early learned verbs are about bodily actions and since recent advanc-es in cognitive neuroscience (Pulvermueller, 2005; James and Maouene, 2009) indicate that the neural processing of common verbs activates the brain regions responsible for the specific body parts that perform those actions. Here we provide initial evidence these body-part verb relations may also be related to the argument structures associated with specific verbs. We will conclude that in the same way that verb meaning and argument structure develop out of correlations in linguistic experiences, they may also develop out of correlations in body experiences.
... There are also cultural practices, such as the verbal count list, the recital of number words in a fixed order ("one, two, three, . . ."), and finger or other body part counting routines, that are widely practiced across many languages (11,12). These systems and practices converge toward a universal order of acquisition, starting with "one" and proceeding in line with increasing cardinality. ...
Significance
Although much research has been devoted to the acquisition of number words, relatively little is known about the acquisition of other expressions of quantity. We propose that the order of acquisition of quantifiers is related to features inherent to the meaning of each term. Four specific dimensions of the meaning and use of quantifiers are found to capture robust similarities in the order of acquisition of quantifiers in similar ways across 31 languages, representing 11 language types.
... Other work has elaborated on the significance of bodily representations at the symbolic and reflective levels of meaning. For ex-ample, while the use of fingers for counting is well documented (Gelman & Williams, 1998), Saxe's (1981Saxe's ( , 1995 research has shown cross-culturally that other bodily representations enter into counting systems. Further, earlier research by Overton and Jackson (1973) and more recently by Overton and Kovacs (200 I) has demonstrated that bodily gestures support emerging symbolic representations at least until the level of reflective meanings. ...
Chapter, discusses the importance of the concept of embodiment in the understanding of human behavior and development. The general argument is that embodiment is central to any discussion of the relation of biological systems and psychological systems or cultural systems and psychological systems. It is also argued that seriously embracing the concept of embodiment represents a move away from unproductive questions entailed in the nativism-empiricism or nature-nurture debate and toward a more productive arena of inquiry and research-the examination of questions of the nature of the relations that operate among biological systems, psychological systems, and cultural systems. There is a discussion of the role metatheory-especially relational meta-theory-plays in contexualizing the concept of embodiment. The idea that embodiment is a concept that bridges biological, cultural, and person-centered approaches to psychological inquiry is explored.
... Other work has elaborated on the significance ofbodily representations at the symbolic and reflective levels of meaning. For example, while the use of fingers for counting is well-documented (Gelman & Wil liams, 1998), Saxe's (1981Saxe's ( , 1995 research has shown cross-culturally that other bodily representations enter into counting systems. Further, earlier research by Overton and Jackson (1973) and more recently, by Dick, Overton, and Kovacs (2005) has demonstrated that bodily ges tures support emerging symbolic representations at least until the level of reflective meanings. ...
Chapter examines meta-theoretical assumptions as a ground for framing an understanding of embodiment in which the lived body and the physical are indissociable complements, and not competing alternatives forever segregated into pure forms of reality
... Other work has elaborated the significance of bodily representations at the symbolic and reflective levels of meaning. For example, while the use of fingers for counting is well documented (Gelman & Williams, 1998), Saxe's (1981Saxe's ( , 1995 research has shown cross-culturally that other bodily representations enter into counting systems. Further, earlier research by Overton and Jackson (1973) and more recently by Dick, Overton, and Kovacs (2005) has demonstrated that bodily gestures support emerging symbolic representations at least until the level of reflective meanings is attained. ...
... Other work has elaborated the significance of bodily representations at the symbolic and reflective levels of meaning. For example, while the use of fingers for counting is well documented (Gelman & Williams, 1998), Saxe's (1981Saxe's ( , 1995 research has shown cross-culturally that other bodily representations enter into counting systems. Further, earlier research by Overton and Jackson (1973) and more recently by Dick, Overton, and Kovacs (2005) has demonstrated that bodily gestures support emerging symbolic representations at least until the level of reflective meanings is attained. ...
In this chapter, we examine background ideas that ground, constrain, and sustain theories and methods in psychology in general and developmental psychology in particular. We briefly review the historical context that gave rise to particular epistemological and ontological positions in developmental psychology. Throughout the chapter, we elaborate a relational developmental systems approach that serves as the basis for integrating a variety of concepts that frequently are considered to be diametrically opposite. We demonstrate the integrative power of the broad relational metatheory—“relationism”—for a variety of central concepts used in developmental theories, including variational and transformational change; nature and nurture; biology and culture; and explanation and interpretation. We argue that instead of splitting these concepts off from each other and then additively and linearly reconnecting them, the dialectical approach of treating these concepts as parts within a synthesized whole that come in sight as points of view is a more fruitful to studying developmental phenomena. Finally, we identify the notions of person, action, and embodiment as core concepts of relational developmental systems, because these notions overcome the dichotomies that have traditionally characterized developmental theories.Keywords:developmental psychology;metatheory;relational developmental systems;change;explanation;interpretation;person;embodiment;action theory
... Saxe comenzó su trabajo transcultural en una comunidad esquimal en 1969, tratando de reconciliar sus observaciones de la vida cotidiana y la ejecución en pruebas de razonamiento moral (Saxe, en prensa). Luego, él se interesó en examinar los conceptos de número en varios escenarios culturales, lo cual contribuyó a que en la década del ochenta se diera una fase transicional en los estudios de cultura y cognición al relacionar el desarrollo de los conceptos numéricos de la gente con sus sistemas culturales y sus prácticas cotidianas (Saxe, 1981(Saxe, , 1988(Saxe, , 1991. En su trabajo posterior se ha preocupado por los análisis de la cognición in situ, tal como ésta toma forma en las prácticas culturales. ...
... Un système numérique qui a particulièrement retenu notre attention est le comptage sur les parties du corps, fréquent en Papouasie-Nouvelle-Guinée, par exemple celui des Yupno (Wassmann & Dasen, 1994) ou des Oksapmin (Saxe, 1981). Saxe (1982Saxe ( , 1999 avait déjà montré comment les Oksapmin ont adapté leur système, qui ne servait traditionnellement qu'à dénombrer des objets, pour pouvoir effectuer des additions et soustractions, ceci sous l'influence de l'introduction du système monétaire dans les années 1960. ...
... This has made the use of numbers an important tool within the society, where it is used in trade, cosmology, mathematics, divination, music, medicine, etc. Early cultures devised various means of number representation, which include body/finger counting (Zaslavsky 1973;Saxe 1981), object counting, Egyptian numerals, Babylonian numerals, Greek numerals, Chinese numerals, Roman numerals, Mayan numerals, Hindu-Arabic numerals, etc. The Hindu-Arabic numeral system, which is considered to be the greatest mathematical discovery (Bailey and Borwein 2011), is still the most commonly used symbolic representation of numbers due to its simplicity and the fact that it requires little memorisation to represent practically any number. ...
In this paper, we examine the processes underlying the Yorùbá numeral system and describe a computational system that is capable of converting cardinal numbers to their equivalent Standard Yorùbá number names. First, we studied the mathematical and linguistic basis of the Yorùbá numeral system so as to formalise its arithmetic and syntactic procedures. Next, the process involved in formulating a Context-Free Grammar (CFG) to capture the structure of the Yorùbá numeral system was highlighted. Thereafter, the model was reduced into a set of computer programs to implement the numerical to lexical conversion process. System evaluation was done by ranking the output from the software and comparing the output with the representations given by a group of Yorùbá native speakers. The result showed that the system gave correct representation for numbers and produced a recall of 100% with respect to the collected corpus. Our future study is focused on developing a text normalisation system that will produce number names for other numerical expressions such as ordinal numbers, date, time, money, ratio, etc. in Yorùbá text.
... Body parts, including the whole body, are implicit in many of the categories that are of interest to young children (e.g., proper nouns ( Tom , Sheila ), kinship terms ( mom , dad ), animal names, locations, etc.), and recurring links in agency and dynamics. They can help with metaphors and extended verb use in English (e.g., When Taylor spilled his milk on the table, he vacuumed it up with his mouth ; example from Seston, Michnick Golinkoff , Ma, & Hirsh-Pasek, 2009 ), as well as other languages (e.g., orientation in Tzotzil children: de Leon, 1994 ; counting in Oksapmin: Saxe, 1981 ). They can potentially facilitate certain situations of selection restriction, as in the snake throws the ball (for an application to texts describing scenes, see Bron, Corfu-Bratschi, & Maouene, 1989 ; for a model and its implementation, see Maouene, 1992 ). Body parts have a hierarchical structure and include parts and whole (see, for example, the importance of part–whole structure in ontologies, and in meaning diff erentiation for verbs such as run , walk , hop , etc.; Malt, Gennari, Imai, Ameel, Tsuda, & Majid, 2008 ). ...
Prior work on argument structure development has shown connections between abstract verb meaning and argument structure; neuroimaging and behavioral studies have shown connections between verb meaning and body effectors. Here we examine the contingencies between verbs, their most likely body region pairing, and argument structure. We ask whether the verbs used in six common syntactic frames are specifically linked to one of three main regions of the body: head, arm, leg. The speech of 20-month-olds (N = 67), 28-month-olds (N = 27), and their mothers (N = 54) (CHILDES: MacWhinney, 2000) was examined. for the use of early-learned verbs (MCDI: Fenson, Dale, Reznick, Bates, Thal, & Pethick, 1994). In total, 89 verb types in 3321 utterances were
coded for their associations with the h e a d, a r m , and l e g body regions (associations taken from Maouene, Hidaka, & Smith, 2008). Significant non-random relations are found both overall and for each age group in analyses using multiple chi-square tests of independence and goodness-of-fit. These results are discussed in terms of their relevance for both argument structure development and embodied cognition, as evidence
supporting a developmental path that has not been previously examined, in which the infant can use early and concrete perception-action information to learn later abstract syntactic achievements.
... This means cultures with external representational systems for numbers, alphabets and so on (Menary, 2007;Dehaene, 2009). Although cultures without writing systems can achieve a degree of mathematical sophistication by using a limited repertoire of linguistic signs or, like the Oksapmin of Papua New Guinea, using body parts to represent quantities (Saxe, 1981), they cannot represent or compute mathematical functions that go beyond addition, subtraction and basic arithmetic. They can, of course, learn to do so. ...
Expertise is extended by becoming immersed in cultural practices. We look at an example of mathematical expertise in which immersion in cognitive practices results in the transformation of expert performance.
... The skills inherent to these cultural practices might, at first blush, be viewed as aconceptual and, as such, hardly bearing on mathematical reasoning and learning. Yet as recent theoretical and empirical work, including our own, suggests, our shared biology implies that even the most abstract of mathematical concepts may first be embodied, then verbally articulated, and finally reified in conventional semiotic forms (Núñez, et al., 1999;Saxe, 1981). Such issues are more than academic. ...
Learning scientists are only beginning to appreciate the potential of synergy between two concurrent developments—theory of embodied cognition and technology of embodied interaction. We characterize and evaluate this prospective synergy from a sociocultural perspective. First we analyze learning in explicitly embodied cultural practices (e.g., surfing), then analogize to the implicitly embodied practice of mathematics. We next contextualize this analogy via interpreting data collected in a design-based research study, in which twenty-two 9-to-11-year-olds developed notions of proportionality through participating in guided problem-solving activities in an embodied-interaction space. In both surfing and mathematics, we argue, learners develop "embodied artifacts," i.e. body-based and modular rehearsed actions. Embodied artifacts lend individuals entry into disciplinary competence via participation in action, refinement of operations, and integration into activity structures. Furthermore, embodied artifacts may become "conceptual performances," wherein performance not merely augments, but stands for and constitutes understanding.
... Other work has elaborated on the significance ofbodily representations at the symbolic and reflective levels of meaning. For example, while the use of fingers for counting is well-documented (Gelman & Wil liams, 1998), Saxe's (1981Saxe's ( , 1995 research has shown cross-culturally that other bodily representations enter into counting systems. Further, earlier research by Overton and Jackson (1973) and more recently, by Dick, Overton, and Kovacs (2005) has demonstrated that bodily ges tures support emerging symbolic representations at least until the level of reflective meanings. ...
Number systems constitute one of the major domains in which language is seen as a source of variation in cognition. The notion that the features of a language's numeral system index cognitive complexity has been pervasive in anthropological linguistics since the nineteenth century. In particular, languages with small numerical vocabularies have attracted enormous interest, but other features where linguistic relativity is invoked include systemic irregularity and the presence of multiple parallel numeral systems (numeral classifiers and object-specific counting). Quite independently, the comparison of graphic numerical notations has imputed cognitive advantages, such as the idea that the Roman numerals limited mathematical progress. These discussions of linguistic relativity have been interwoven with issues of social complexity; in place of a pure language-thought relationship, the discussion has been framed through a triad of language, cognition, and social structure. To evaluate whether and how lexical numerals and numerical notations affect numerical cognition, an activity-based explanatory model is proposed in which materiality, discourse, and practice mutually constitute knowledge systems. This allows us to move past the idea that language structure has direct cognitive effects, without denying that language is relevant to numerical cognition.
this work includes a series of aspects of the
study of numerals, from different perspectives and with different
applications. It is particularly relevant the application
of diachronic analysis to the establishment of a date for the
Libro de Alexandre. some of those aspects were already
treated by the author in different publications; however, they
have not received a joint treatment so far.
Key words: Alexandre, Arabic, Chinese, Japanese, Latin,
Malay, numeral, loanword, Nahuatl, Quechua, Romance,
Spanish, Tagalog, typology.
Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role.
Humans and other species have biologically endowed abilities for discriminating quantities. A widely accepted view sees such abilities as an evolved capacity specific for number and arithmetic. This view, however, is based on an implicit teleological rationale, builds on inaccurate conceptions of biological evolution, downplays human data from non-industrialized cultures, overinterprets results from trained animals, and is enabled by loose terminology that facilitates teleological argumentation. A distinction between quantical (e.g., quantity discrimination) and numerical (exact, symbolic) cognition is needed: quantical cognition provides biologically evolved preconditions for numerical cognition but it does not scale up to number and arithmetic, which require cultural mediation. The argument has implications for debates about the origins of other special capacities - geometry, music, art, and language.
ABSTRACT
The purpose of this study was to investigate selected predictors (i.e. participants’
background, math prior knowledge, motivation, learning strategies, teaching strategies
and practices, and instructional materials) on math performance (final grade, successful,
not successful and dropouts) in an elementary algebra course of a developmental
mathematics program at a community college. Two data sets were collected from two
semesters (Fall 2003 and Spring 2004). Four main considerations adopted from a
tetrahedral model provided a framework for analyzing the data. These included (1)
learner variables (2) teacher variables (3) material variables and (4) criterial measures.
Results showed positive and statistically significant relationships between
intrinsic goal orientation, self efficacy for learning and performance, effort management
strategies and math final grade and a negative relationship with teaching strategies and
practices. The logistic regression results reproduced the correlation results with
statistically significant likelihood ratios of success, not successful, and dropouts
associated with class level, pre-algebraic skills, self-efficacy for learning and
performance, effort management strategies and the general teaching strategies and
practices in the final model. The Final Logistic Regression Model explained successful,
not successful and dropouts with an average accuracy of 63%. The model would likely be
improved with additional information about the teachers, because some of them had a
statistically significant association with successful students. However, the statistically
significant associations found between successful math performance and teaching
strategies and practices were negative.
Information on the counting systems of 12 East Strickland and Bosavi languages is collated. In seven cases the body‐part tally system is symmetrical, with cycle lengths varying from 27 to 35. In four cases, the tally system is asymmetrical or truncated and in one case detailed information is not available. Methods of counting beyond one cycle have been described for all but one of the Bosavi languages but not for any of the East Strickland languages. An additional 2‐cycle or 2, 5‐cycle system is indicated for several East Strickland languages but not for any Bosavi language. Comparison with the
counting systems of languages beyond the Strickland‐Bosavi region – especially with Ok languages to the northwest and Huli to the northeast – suggests a process in which the terminology of body‐part tally systems is progressively disembedded from bodily commitment such that counting words assume the status of cardinal numbers and, thereby, facilitate expressions of the commensurability of difference.
This chapter analyses the emergence of ethnomathematics as a field of research. It starts with the work of some isolated forerunners like Wilder and Raum, and moves to D’Ambrosio’s ethnomathematical research program, and the simultaneous gestation of other concepts, like indigenous, socio-, informal, spontaneous, oral, hidden, implicit, and people's mathematics. It compares various conceptualisations and paradigms of ethnomathematics. The influence of Freire's ideas on a series of scholars working in the field of ethnomathematics is stressed . The second part of the chapter presents an overview of ethnomathematical literature, continent by continent. The third and final part discusses some ofthe basic assumptions associated with the use of ideas from ethnomathematics in education. Some complementary and partially overlapping trends in educational experimentation are considered from an ethnomathematical perspective.
The present study is an investigation of the interplay between social and developmental processes in children's numerical understandings in working- and middle-class home settings. Methods included interviews with 78 middle- and working-class 2½- and 4½-year-olds to assess their numerical understandings, interviews with the mothers about their children's everyday number activities, and observational studies of the mother-and-child pairs in interaction during prototypical number activities. Our results provide evidence that the children in the study were regularly engaged with social activities involving number, though the nature of children's numerical understandings and their numerical environments differed in the following ways. (1) Younger children differed from older children in their numerical understandings across a variety of tasks that varied in their goal structure complexity: recitation of counting words, production of cardinal values for single arrays, numerical comparisons and reproductions, and arithmetic transformations. 4-year-olds from middle-class homes displayed greater competence on tasks with more complex numerical goals than did their working-class peers. (2) At home, variation in the complexity of children's everyday number activities paralleled our findings of age and social class differences in children's numerical understandings. (3) During mother-child teaching interactions, mothers adjusted the goal structure of a given activity to reflect their children's abilities to structure numerical goals, and children adjusted their goals to their mothers' efforts to organize the activity. In a minority of contexts, working-class mothers simplified the goal structure of the activity to a greater extent than did middle-class mothers. Overall, there were few differences between the middle- and working-class dyads in the complexity of numerical goals elaborated during interactions for children of equivalent age and ability. These results support a model in which children's numerical environments are understood to be negotiated in their everyday activities-a negotiation that leads children's achievements to be linked at once to their own understandings and to the sociocultural context of their development.
The authors describe an emerging paradigm of educational research that pairs theories of embodied learning with a class of immersive technologies referred to as mixed reality (MR). MR environments merge the digital with the physical, where, for example, students can use their bodies to simulate an orbit around a virtual planet. Recent research supports the idea that body activity can be an important catalyst for generating learning, and new technologies are being developed that use natural human physicality and gesture as input. However, existing research on embodied learning technologies has been disparate, driven largely by specific technical innovations and constraints, and often lacking a clear focus on establishing their efficacy in educational contexts. On the basis of the unique characteristics of these technologies and on their own experiences conducting research in this area, the authors put forth six precepts for embodied learning technology researchers that pertain to the rationale, design, and execution of empirical studies.
Habilitationsschrift Fachbereich Psychologie und Sportwissenschaften der Johann Wolfgang Goethe -Universität Frankfurt am Main 2002 Mack, W. (2002). Die Wahrnehmung kleiner Anzahlen und die Entwicklung des Zahlenverständnisses beim Kleinkind. Unveröffentl. Habilitationsschrift, FB Psychologie und Sportwissenschaften, Goethe-Universität Frankfurt a. M.: Frankfurt a. M.
In many cultures, one of the earliest representations of number to be learned is a finger-counting system. Although most children stop using their fingers to count as they grow more confident with number, traces of this system can still be seen in adulthood. For example, an individual's finger-counting habits appear to affect the ways in which numbers are implicitly associated with certain areas of space, as inferred from the spatial–numerical association of response codes (SNARC) effect. In this study, we questioned the finger-counting habits of 98 participants who make explicit, idiosyncratic associations between number and space, known as number-space synaesthesia. Unexpectedly, neither handedness nor finger-counting direction (left-to-right or right-to-left) was associated with the relative positions of 1 and 10 in an individual's number-space synaesthesia. This lack of association between finger-counting styles and number-space synaesthesia layout may result from habitual use of synaesthetic space rather than fingers when learning to count; we offer some testable hypotheses that could assess whether this is the case.
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.
The authors have declared no conflicts of interest for this article.
© 2014 John Wiley & Sons, Ltd.
This paper presents new data on children's acquisition of counting skills. Three aspects of counting were studied: the formation of the cardinality rule that the last number named during counting denotes the number of objects in an array, the mastery of the counting procedure or the coordination of ordered number names and objects counted, and the growth of the knowledge that x + 1 is greater than x. A model was outlined which posits the hierarchic integration of six number skills to account for the growth of the knowledge that x + 1 is greater than x and the development of number conservation. The six skills are: the cardinality rule, the counting procedure, acquisition of more x's, judgments of relative numerosity, pattern recognition of small numbers, and one-to-one correspondences.