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Embodied cognition as grounding for situatedness and context in mathematics education
 EDUCATIONAL STUDIES IN MATHEMATICS
, 1999
"... In this paper we analyze, from the perspective of "Embodied Cognition", why learning and cognition are situated and contextdependent. We argue that the nature of situated learning and cognition cannot be fully understood by focusing only on social, cultural and contextual factors. One mus ..."
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In this paper we analyze, from the perspective of "Embodied Cognition", why learning and cognition are situated and contextdependent. We argue that the nature of situated learning and cognition cannot be fully understood by focusing only on social, cultural and contextual factors. One must also take into account the nonarbitrary biological and experiential constrains that shape social activity and language, and through which cognition and learning are realized in a genuine embodied process. The bodilygrounded nature of cognition provides foundations for situatedness, entails a reconceptualization of cognition and mathematics itself, and has important consequences for mathematics education. After framing some theoretical notions of embodied cognition in the perspective of modern cognitive science, we analyze a case study continuity of functions. We use conceptual metaphor theory to show how embodied cognition, while providing grounding for situatedness, gives fruitful results in analyzing the cognitive difficulties underlying the understanding of continuity.
Context as a Spurious Concept
, 1997
"... I take issue in this talk with AI formalizations of context, primarily the formalization by McCarthy and Buvac, that regard context as an undefined primitive whose formalization can be the same in many different kinds of AI tasks. In particular, any theory of context in natural language must take th ..."
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I take issue in this talk with AI formalizations of context, primarily the formalization by McCarthy and Buvac, that regard context as an undefined primitive whose formalization can be the same in many different kinds of AI tasks. In particular, any theory of context in natural language must take the special nature of natural language into account and cannot regard context simply as an undefined primitive. I show that there is no such thing as a coherent theory of context simplicitercontext pure and simpleand that context in natural language is not the same kind of thing as context in KR. In natural language, context is constructed by the speaker and the interpreter, and both have considerable discretion in so doing. Therefore, a formalization based on predefined contexts and predefined `lifting axioms' cannot account for how context is used in realworld language.
MATHEMATICAL IDEA ANALYSIS: WHAT EMBODIED COGNITIVE SCIENCE CAN SAY ABOUT THE HUMAN NATURE OF MATHEMATICS
"... This article gives a brief introduction to a new discipline called the cognitive science of mathematics (Lakoff & Núñez, 2000), that is, the empirical and multidisciplinary study of mathematics (itself) as a scientific subject matter. The theoretical background of the arguments is based on embod ..."
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This article gives a brief introduction to a new discipline called the cognitive science of mathematics (Lakoff & Núñez, 2000), that is, the empirical and multidisciplinary study of mathematics (itself) as a scientific subject matter. The theoretical background of the arguments is based on embodied cognition, and on relatively recent findings in cognitive linguistics. The article discusses Mathematical Idea Analysis—the set of techniques for studying implicit (largely unconscious) conceptual structures in mathematics. Particular attention is paid to everyday cognitive mechanisms such as image schemas and conceptual metaphors, showing how they play a fundamental role in constituting the very fabric of mathematics. The analyses, illustrated with a discussion of some issues of set and hyperset theory, show that it is (human) meaning what makes mathematics what it is: Mathematics is not transcendentally objective, but it is not arbitrary either (not the result of pure social conventions). Some implications for mathematics education are suggested. Have you ever thought why (I mean, really why) the multiplication of two negative numbers yields a positive one? Or why the empty class is a subclass of all
Do multiple representations need explanations? The role of verbal guidance and individual differences in multimedia mathematics learning
 Journal of Educational Psychology
, 2004
"... Elementary school children, some of whom were nonnative speakers of English, learned to add and subtract integers in a discoverybased multimedia game either with or without verbal guidance in English or optionally in Spanish (Groups G—verbal guidance and NoG—no verbal guidance, respectively). Grou ..."
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Elementary school children, some of whom were nonnative speakers of English, learned to add and subtract integers in a discoverybased multimedia game either with or without verbal guidance in English or optionally in Spanish (Groups G—verbal guidance and NoG—no verbal guidance, respectively). Group G members chose to listen to verbal explanations in their first language and showed larger posttest scores than Group NoG. Highcomputerexperience students in Group G outperformed the rest of the students on training session scores and a transfer test. Longer latencies to respond to practice problems affected all learning measures positively. Results support the use of verbal guidance for discoverybased multimedia games and show that multimedia games may not be equally effective for all learners. What is the role of verbal guidance in promoting mathematics learning from a discoverybased multimedia game? Are there important individual differences for which a multimedia game helps some kinds of learners more than others? These are important questions both for research and for the application of research, to improve instruction and learning outcomes in learning mathematics with multimedia programs. Our first question is concerned
Mathematics and the Biological Phenomena
"... The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, more than as a hierachically organized structure where mathematics would display the essence of phenome ..."
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The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, more than as a hierachically organized structure where mathematics would display the essence of phenomena. The concept of "unity" in the biological phenomenon is then discussed. In the second part, a foundational issue in mathematics is revisited, following recent perspectives in the physiology of action. The relevance of the historical formation of mathematical concepts is also emphasized. Part I: Reflections on Mathematics in Biology Introduction: hierarchies of disciplines. When hearing biologists about working methods in their discipline, one may often appreciate traces of the emotions of a scientific experience of great 1 In Proceedings of the International Symposium on Foundations in Mathematics and Biology: Problems, Prospects, Interactions, Invited lecture, Pontifical Lateran Univer...
Mathematical Intuition vs. Mathematical Monsters
, 1998
"... Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of ..."
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Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of intuition. We examine several famous geometrical, topological and settheoretical examples of such monsters in order to see to what extent, if at all, intuition is undermined in its everyday roles.
The Cognitive Foundations of Mathematics: The Role of Conceptual Metaphor Handbook of Mathematical Cognition New York: Psychology Press
"... analyze the biological foundations of human cognition. A crucial component of their arguments is a simple but profound aphorism: Everything said is said by someone. It follows from this that any concept, idea, belief, definition, drawing, poem, or piece of music, has to be produced by a living human ..."
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analyze the biological foundations of human cognition. A crucial component of their arguments is a simple but profound aphorism: Everything said is said by someone. It follows from this that any concept, idea, belief, definition, drawing, poem, or piece of music, has to be produced by a living human being, constrained by the peculiarities of his or her body and brain. The entailment is straightforward: without living human bodies with brains, there are no ideas — and that includes mathematical ideas. This chapter deals with the structure of mathematical ideas themselves, and with how their inferential organization is provided by everyday human cognitive mechanisms such as conceptual metaphor. The Cognitive Study of Ideas and their Inferential Organization The approach to Mathematical Cognition we take in this chapter is relatively new, and it differs in important ways from (but is complementary to) the ones taken by many of the authors in this Handbook. In order to avoid potential misunderstandings regarding the subject matter and goals of our piece, we believe that it is important to clarify these differences right upfront. The differences reside mainly on three fundamental aspects:
Why It is Important to Build Robots Capable of Doing Science
 University Cognitive Studies
, 2002
"... Science, like any other cognitive activity, is grounded in the sensorimotor interaction of our bodies with the environment. Human embodiment thus constrains the class of scientific concepts and theories which are accessible to us. The paper explores the possibility of doing science with artificial c ..."
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Science, like any other cognitive activity, is grounded in the sensorimotor interaction of our bodies with the environment. Human embodiment thus constrains the class of scientific concepts and theories which are accessible to us. The paper explores the possibility of doing science with artificial cognitive agents, in the framework of an interactivistconstructivist cognitive model of science. Intelligent robots, by virtue of having different sensorimotor capabilities, may overcome the fundamental limitations of human science and provide important technological innovations. Mathematics and nanophysics are prime candidates for being studied by artificial scientists.
Where There’s a Model, There’s a Metaphor: Metaphorical Thinking in Students ’ Understanding of a Mathematical Model
"... The central question addressed in this article concerns the ways in which applied problem situations provide distinctive conditions to support the production of meaning and the understanding of mathematical topics. In theoretical terms, a first approach is rooted in C. S. Peirce’s perspective on sem ..."
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The central question addressed in this article concerns the ways in which applied problem situations provide distinctive conditions to support the production of meaning and the understanding of mathematical topics. In theoretical terms, a first approach is rooted in C. S. Peirce’s perspective on semiotic mediation, and it represents a standpoint from which the notion of interpretation is taken as essential to learning. A second route explores metaphorical thinking and undertakes the position according to which human understanding is metaphorical in its own nature. The connection between the two perspectives becomes a fundamental issue, and it entails the conception of some hybrid constructs. Finally, the analysis of empirical data suggests that the activity on applied situations, as it fosters metaphorical thinking, offer students ’ reasoning a double anchoring (or a duplication of references) for mathematical concepts. There is a substantial agreement on the belief that introducing mathematics applications and modeling in school curricula promotes mathematical topics. The listed advantages of such activities include the oftencalled psychological argument, according to which mathematical modeling and applications have a positive influence on mathematics learning to the extent that they provide meaning to mathemat
1 On the Meanings of Multiplication for Different Sets of Numbers in Context of Geometrization: Descartes’ Multiplication, Mathematical Workspace and Semiotic
"... In this article, I briefly present an experimental lesson – from Descartes ’ multiplication to the geometric meaning of the product of complex numbers – connecting multiplication and some of its geometric meanings. I will also present some examples of our experiments conducted in French high schools ..."
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In this article, I briefly present an experimental lesson – from Descartes ’ multiplication to the geometric meaning of the product of complex numbers – connecting multiplication and some of its geometric meanings. I will also present some examples of our experiments conducted in French high schools and some elements of the analysis process to identify the possibilities of connecting multiplication and some of its geometric meanings through the production of interactions between different registers of representation and a semiotic mediation within a Mathematical Work Space (MWS). « For every question and every phenomenon, in order to follow the normal psychological route of scientific thought, we must go from an image to a geometric figure, and then from a geometric figure to an abstract form. » (Translation, Bachelard, 1938, p. 10.) Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics TeachingResearch Journal OnLine, it is distributed for noncommercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MTRJoL. MTRJoL is published jointly by the Bronx Colleges of the City University of New York.