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Teachers’ gestures as a means of scaffolding students’ understanding: Evidence from an early algebra lesson in
 Video Research in the Learning Sciences. Mahwah, NJ: Erlbaum
, 2007
"... During classroom instruction, teachers often attempt to scaffold students’ understanding of lesson content. But how is this scaffolding achieved? One obvious possibility is that teachers adjust the ways in which they communicate information relevant to the lesson. Surprisingly, relatively little is ..."
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During classroom instruction, teachers often attempt to scaffold students’ understanding of lesson content. But how is this scaffolding achieved? One obvious possibility is that teachers adjust the ways in which they communicate information relevant to the lesson. Surprisingly, relatively little is known about how teachers vary their communicative behavior in order to scaffold student understanding. However, video technology has greatly increased the range of behaviors that can come under rigorous study. Using video analysis techniques, we examined a teacher’s use of verbal and gestural forms of communication. In this paper, we consider the possibility that teachers use spontaneous hand and arm gestures along with their speech in an effort to scaffold students ’ understanding. Previous research has documented that teachers do indeed use gestures in classroom
VISUALIZATION OF ORDINALS
, 2007
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. ..."
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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics.
The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics
, 2012
"... The canonical history of mathematics suggests that the late 19thcentury “arithmetization ” of calculus marked a shift away from spatialdynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic con ..."
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The canonical history of mathematics suggests that the late 19thcentury “arithmetization ” of calculus marked a shift away from spatialdynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of cospeech gesture produced by mathematics graduate students while proving a theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitivehistorical case study of an incident in 19thcentury mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice.
Avoid ambiguity! (If you can)
, 2006
"... Evidence from cospeech gesture during mathematical proving ..."
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1. Are Virtual Learning Environments used to facilitate collaborative 5
, 2008
"... Selection of conference papers ..."
With the Future Behind Them: Convergent Evidence From Aymara Language and Gesture in the Crosslinguistic Comparison of Spatial Construals of Time
, 2005
"... Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego and temporalreferencepoint models. Dynamic models appear to be quasiuniversal crossculturally, as does the generalization that ..."
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Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego and temporalreferencepoint models. Dynamic models appear to be quasiuniversal crossculturally, as does the generalization that in Egoreferencepoint models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; linguistic and gestural data give strong confirmation of this unusual culturespecific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns?
With the Future Behind Them: Convergent Evidence From Aymara Language and Gesture in the Crosslinguistic Comparison of Spatial Construals of Time
, 2005
"... Cognitive research on metaphoric concepts of time has focused on differences between moving ego and moving time models, but even more basic is the contrast between ego and temporalreferencepoint models. Dynamic models appear to be quasiuniversal crossculturally, as does the generalization that ..."
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Cognitive research on metaphoric concepts of time has focused on differences between moving ego and moving time models, but even more basic is the contrast between ego and temporalreferencepoint models. Dynamic models appear to be quasiuniversal crossculturally, as does the generalization that in egoreferencepoint models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; linguistic and gestural data give strong confirmation of this unusual culturespecific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns. 1.
Numbers and Arithmetic: Neither Hardwired Nor Out There
, 2009
"... What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions fo ..."
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What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of nonmathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inferencepreserving crossdomain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws ” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. Keywords abstraction, arithmetic, conceptual metaphor, conceptual