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Supporting group cognition in an online math community: A cognitive tool for smallgroup referencing in text chat
 Journal of Educational Computing Research (JECR) special
"... cognitive tool for smallgroup referencing in text chat ..."
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cognitive tool for smallgroup referencing in text chat
How are words stored in memory?: Beyond phones and phonemes
, 2007
"... A series of arguments is presented showing that words are not stored in memory in a way that resembles the abstract, phonological code used by alphabetical orthographies or by linguistic analysis. Words are stored in a very concrete, detailed auditory code that includes nonlinguistic information inc ..."
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Cited by 14 (4 self)
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A series of arguments is presented showing that words are not stored in memory in a way that resembles the abstract, phonological code used by alphabetical orthographies or by linguistic analysis. Words are stored in a very concrete, detailed auditory code that includes nonlinguistic information including speaker’s voice properties and other details. Thus, memory for language resembles an exemplar memory and abstract descriptions (using letterlike units and speakerinvariant features) are probably computed on the fly whenever needed. One consequence of this hypothesis is that the study of phonology should be the study of generalizations across the speech of a community and that such a description will employ units (segments, syllable types, prosodic patterns, etc.) that are not necessarily employed as units in speakers’ memory for language. That is, the psychological units of language are not useful for description of linguistic generalizations and linguistic generalizations across a community are not useful for storing the language for speaker use.
Keeping meaning in proportion: The multiplication table as a case of pedagogical bridging tools. Unpublished doctoral dissertation
, 2004
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Is the Brain a Quantum Computer?
"... We argue that computation via quantum mechanical processes is irrelevant to explaining how brains produce thought, contrary to the ongoing speculations of many theorists. First, quantum effects do not have the temporal properties required for neural information processing. Second, there are substant ..."
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We argue that computation via quantum mechanical processes is irrelevant to explaining how brains produce thought, contrary to the ongoing speculations of many theorists. First, quantum effects do not have the temporal properties required for neural information processing. Second, there are substantial physical obstacles to any organic instantiation of quantum computation. Third, there is no psychological evidence that such mental phenomena as consciousness and mathematical thinking require explanation via quantum theory. We conclude that understanding brain function is unlikely to require quantum computation or similar mechanisms.
Formal notations are diagrams: Evidence from a production task
"... Although a general sense of the magnitude, quantity, or numerosity of objects is common in both untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but wellspecified situations—to behave formally, that is—are skills unique to people t ..."
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Although a general sense of the magnitude, quantity, or numerosity of objects is common in both untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but wellspecified situations—to behave formally, that is—are skills unique to people trained in symbolic notations. These symbolic notations typically employ complex, hierarchically embedded structures, which all extant analyses assume are constructed by concatenative, rulebased processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notations—in short, that formal notations are a kind of diagram. We examined selfgenerated productions in the domains of handwritten arithmetic expressions and typewritten statements in a formal logic. In both tasks, we found substantial evidence for spatial representational schemes even in these highly symbolic domains. It is clear that mathematical equations written in modern notation are, in general, visual forms and that they share some properties with diagrammatic or imagistic displays. Equations and mathematical expressions are often set off from the main text, use nonstandard characters and shapes, and deviate substantially from linear symbol placement. Furthermore, evidence indicates that at least some mathematical processing is sensitive to the particular visual form of its presentation notation (Cambell, 1999; McNeil & Alibali, 2004, 2005). Despite these facts, notational mathematical representation is typically considered sentential and is placed in opposition to diagrammatic representations in fields as diverse as education
EMBODIED SPATIAL ARTICULATION: A GESTURE PERSPECTIVE ON STUDENT NEGOTIATION BETWEEN KINESTHETIC SCHEMAS AND EPISTEMIC FORMS IN LEARNING MATHEMATICS
"... Two parallel strands in mathematicseducation research—one that delineates students ’ embodied schemas supporting their mathematical cognition and the other that focuses on the mediation of cultural knowledge through mathematical tools—could converge through examining reciprocities between schemas a ..."
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Two parallel strands in mathematicseducation research—one that delineates students ’ embodied schemas supporting their mathematical cognition and the other that focuses on the mediation of cultural knowledge through mathematical tools—could converge through examining reciprocities between schemas and tools. Using a gesturebased methodology that attends to students ’ hand movements as they communicate their understanding, data examples from design research in two domains illustrate students ’ spontaneous spatial articulation of embodied cognition. Such embodied spatial articulation could be essential for deep understanding of content, because in performing these articulations, students may be negotiating between their dynamic imagebased intuitive understanding of a concept and the static formal mathematical formats of representing the concept. Implications for mathematics education are drawn. The growing body of literature on ‘situated cognition ’ and ‘cognition in context ’ (e.g., Lave & Wenger, 1991; Hutchins & Palen, 1998) is informing research in mathematics education. In particular, we are challenged to think of mathematical cognition not as “abstract ” inthehead processes devoid of concrete grounding, but as phenomenologically, intrinsically, and necessarily dwelling in student interactions with objects in their environment (Heidegger, 1962;
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
A glimpse at the metaphysics of Bongard problems
, 2000
"... Bongard problems present an outstanding challenge to artificial intelligence. They consist of visual pattern understanding problems on which the task of the pattern perceiver is to find an abstract aspect of distinction between two classes of figures. This paper examines the philosophical question o ..."
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Bongard problems present an outstanding challenge to artificial intelligence. They consist of visual pattern understanding problems on which the task of the pattern perceiver is to find an abstract aspect of distinction between two classes of figures. This paper examines the philosophical question of whether objects in Bongard problems can be ascribed an a priori, metaphysical, existence  the ontological question of whether objects, and their boundaries, come predefined, independently of any understanding or context. This is an essential issue, because it determines whether a priori symbolic representations can be of use for solving Bongard problems. The resulting conclusion of this analysis is that in the case of Bongard problems there can be no units ascribed an a priori existence  and thus the objects dealt with in any specific problem must be found by solution methods (rather than given to them). This view ultimately leads to the emerging alternatives to the philosophical doc...
Embodied semiotic activities and their role in the construction of mathematical meaning of motion graphs
, 2008
"... Abstract This paper examines the relation between bodily actions, artifactmediated activities, and semiotic processes that students experience while producing and interpreting graphs of twodimensional motion in the plane. We designed a technologybased setting that enabled students to engage in e ..."
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Abstract This paper examines the relation between bodily actions, artifactmediated activities, and semiotic processes that students experience while producing and interpreting graphs of twodimensional motion in the plane. We designed a technologybased setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students ’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.
Mathematics as Reference System of Life: preliminary observations
, 2009
"... We forward hypothesis that all what we refer to as mathematics are cognitive aspects of life, moreover, we have right to refer to mathematics as reference system of life. Mathematics and cognition are not distinguishable between themselves because what we call mathematics refer to the functionality ..."
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Cited by 5 (3 self)
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We forward hypothesis that all what we refer to as mathematics are cognitive aspects of life, moreover, we have right to refer to mathematics as reference system of life. Mathematics and cognition are not distinguishable between themselves because what we call mathematics refer to the functionality by means of what (or via what) we are created by nature, or by God, be it question of our religious persuasion. Thus, according to this hypothesis, mathematics turns out to be considrable more as primary in many points as before, when we attributed to mathematics role of sort of descriptor of nature. When we are going to say that mathematics is reference system of life, we mean that today's mathematics is only some starting state of what might be referred to as mathematics as subject / object of reality.