From static to dynamic mathematics: historical and representational perspectives

The nature of mathematical reference fields has substantially evolved with the advent of new types of digital technologies enabling students greater access to understanding the use and application of mathematical ideas and procedures. We analyze the evolution of symbolic thinking over time, from static notations to dynamic inscriptions in new technologies. We conclude with new perspectives on Kaput’s theory of notations and representations as mediators of constructive processes.

[1]  M. Mariotti.  Introduction to Proof: The Mediation of a Dynamic Software Environment , 2000 .

[2]  V. O. Mcbrien.  Remarks on the Foundations of Mathematics , 1958 .

[3]  Robert Kozma,et al.  Innovations in Science and Mathematics Education: Advanced Designs for Technologies of Learning. , 2000 .

[4]  J. Kaput.  Notations and Representations as Mediators of Constructive Processes , 1991 .

[5]  P. Cobb,et al.  Reflective Discourse and Collective Reflection , 1997 .

[6]  James J. Kaput,et al.  Algebra in the early grades , 2008 .

[7]  S. S. Demidov,et al.  Two Letters by N. N. Luzin to M. Ya. Vygodskiĭ , 2000, Am. Math. Mon..

[8]  J. Roschelle.  Learning by Collaborating: Convergent Conceptual Change , 1992 .

[9]  G. Goldin.  Representations and the psychology of mathematics education: Part II , 1998 .

[10]  David Williamson Shaffer,et al.  Mathematics and Virtual Culture: an Evolutionary Perspective on Technology and Mathematics Education , 1998 .

[11]  R. Goodstein,et al.  Remarks on the Foundations of Mathematics , 1957, The Mathematical Gazette.

[12]  Luc Trouche,et al.  The Didactical Challenge of Symbolic Calculators , 2005 .

[13]  C. Laborde,et al.  Approaching functions: Cabri tools as instruments of semiotic mediation , 2007 .

[14]  René Thom,et al.  Developments in Mathematical Education: Modern mathematics: does it exist? , 1973 .

[15]  Graham Flegg.  Numbers: Their History and Meaning , 1983 .

[16]  James J. Kaput,et al.  Representations, inscriptions, descriptions and learning: A kaleidoscope of windows , 1998 .

[17]  Marcelo de Carvalho Borba,et al.  Humans-with-Media and the Reorganization of Mathematical Thinking: Information and Communication Technologies, Modeling, Visualization and Experimentation , 2006 .

[18]  Patrick Morrison,et al.  The Symbolic Species: The Co-Evolution of Language and the Human Brain , 1999, BMJ.

[19]  A. Jonker.  Origins of the modern mind. Three stages in the evolution of culture and cognition , 1998 .

[20]  T. Deacon.  The Symbolic Species: The Co-evolution of Language and the Brain , 1998 .

[21]  Cristina Esteley,et al.  Humans-With-Media and the Reorganization of Mathematical Thinking. Information and Communication Technologies, Modelling, Experimentation and Visualization , 2006 .

[22]  Ricardo Nemirovsky,et al.  Body Motion and Graphing , 1998 .

[23]  Stephen Hegedus,et al.  The Effect of a SimCalc Connected Classroom on Students' Algebraic Thinking , 2005 .

[24]  Ricardo Nemirovsky,et al.  Children creating ways to represent changing situations: On the development of homogeneous spaces , 2001 .

[25]  Luc Trouche,et al.  The didactical challenge of symbolic calculators : turning a computational device into a mathematical instrument , 2005 .

[26]  L. Vygotsky.  Mind in Society: The Development of Higher Psychological Processes: Harvard University Press , 1978 .

[27]  Colette Laborde,et al.  The Hidden Role of Diagrams in Students’ Construction of Meaning in Geometry , 2005 .

[28]  Raymond Duval.  A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics , 2006 .

[29]  J. Wertsch.  Vygotsky and the Social Formation of Mind , 1985 .

[30]  Jeremy Roschelle,et al.  SIMCALC : Accelerating Students’ Engagement With the Mathematics of Change , 2000 .

[31]  Etienne Wenger,et al.  Situated Learning: Legitimate Peripheral Participation , 1991 .

[32]  Jean-Pierre Changeux,et al.  Conversations on mind, matter, and mathematics , 1995 .

[33]  L. S. Vygotskiĭ,et al.  Mind in society : the development of higher psychological processes , 1978 .

[34]  Alex Kozulin,et al.  Vygotsky's Psychology: A Biography of Ideas , 1990 .

[35]  Leslie P. Steffe,et al.  Radical Constructivism and Mathematics Education. , 1994 .

[36]  G. Salomon.  Distributed cognitions : psychological and educational considerations , 1997 .

[37]  C. Hoyles,et al.  Meaning in mathematics education , 2005 .

[38]  M. Kline.  Mathematics: The Loss of Certainty , 1982 .