CHARACTERIZING THEMATIZED DERIVATIVE SCHEMA BY THE UNDERLYING EMERGENT STRUCTURES

ABSTRACTThis paper reports on different underlying structures of the derivative schema of three undergraduate students that were considered to be at the trans level of development of the derivative schema (action–process–object–schema). The derivative schema is characterized in terms of the students’ ability to explicitly transfer the relationship between a function and its first derivative to the derivative function and the second derivative. This conscious shift of properties of derivatives is differently manifested by the students in the trans level of development of the derivative schema and can be considered evidence of the different characteristics of the thematization of derivative schema. From here we suggest that there are different underlying structures in the constructed schema due to the consciousness in which students use the relations between a function and its derivative.

[1]  Ron Tzur,et al.  Distinguishing Two Stages of Mathematics Conceptual Learning , 2004 .

[2]  G. Goldin.  A Scientific Perspective on Structured, Task-Based Interviews in Mathematics Education Research , 2000 .

[3]  Willi Dorfler.  Formation of Mathematical Objects as Decision Making , 2002 .

[4]  E. Filloy.  Piaget, J. y García, R. Psicogénesis e historia de la ciencia. México: Siglo XXI editores, 1982 , 1985 .

[5]  Ed Dubinsky,et al.  Advanced Mathematical Thinking , 2005 .

[6]  Leslie P. Steffe,et al.  Children's algorithms as schemes , 1983 .

[7]  Ed Dubinsky,et al.  Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos Analysis: Part 2 , 2005 .

[8]  Eduardo Dubinsky.  Aplicación de la perspectiva piagetiana a la educación matemática universitaria , 1996, Educación matemática.

[9]  Markus Hähkiöniemi.  Associative and Reflective Connections between the Limit of the Difference Quotient and Limiting Process. , 2006 .

[10]  Ed Dubinsky,et al.  Constructive Aspects of Reflective Abstraction in Advanced Mathematics , 1991 .

[11]  J. Berry,et al.  Promoting students’ graphical understanding of the calculus , 2003 .

[12]  Ed Dubinsky,et al.  The development of students' graphical understanding of the derivative , 1997 .

[13]  David Tall,et al.  Advanced Mathematical Thinking , 1994 .

[14]  Margaret Kinzel,et al.  Explicating a Mechanism for Conceptual Learning: Elaborating the Construct of Reflective Abstraction , 2004 .

[15]  R. Skemp.  The psychology of learning mathematics , 1979 .

[16]  J. Clement.  Analysis of Clinical Interviews: Foundations and model viability , 2000 .

[17]  David E. Meel.  Modelos y teorías de la comprensión matemática: Comparación de los modelos de Pirie y Kieren sobre el crecimiento de la comprensión matemática y la Teoría APOE , 2003 .

[18]  Leslie P. Steffe,et al.  Epistemological foundations of mathematical experience , 1991 .

[19]  Draga Vidakovic,et al.  Understanding the limit concept: Beginning with a coordinated process scheme , 1996 .

[20]  Sue Johnston-Wilder,et al.  Fundamental Constructs in Mathematics Education , 2004 .

[21]  J. Piaget,et al.  The Development of Thought: Equilibration of Cognitive Structures , 1977 .

[22]  Bernadette Baker,et al.  A Calculus Graphing Schema , 2000 .

[23]  Michelle Zandieh,et al.  The evolution of student understanding of the concept of derivative , 1997 .

[24]  M. Trigueros,et al.  Schema Thematization: A Framework and an Example , 2007 .

[25]  G. Vergnaud.  La théorie des champs conceptuels , 1989 .

[26]  Jere Confrey,et al.  A critique of the selection of “Mathematical objects” as a central metaphor for advanced mathematical thinking , 1996, Int. J. Comput. Math. Learn..

[27]  Jean Piaget,et al.  Studies in Reflecting Abstraction , 2001 .

[28]  Draga Vidakovic,et al.  Constructing a schema: The case of the chain rule? , 1997 .

[29]  Samer Habre,et al.  Students’ conceptual understanding of a function and its derivative in an experimental calculus course , 2006 .

[30]  R. Lesh,et al.  Handbook of Research Design in Mathematics and Science Education. , 2000 .

[31]  Laurel Cooley,et al.  Thematization of the Calculus Graphing Schema. , 2003 .

[32]  Guershon Harel,et al.  Advanced Mathematical Thinking , 2006 .

[33]  Guershon Harel,et al.  Advanced Mathematical-Thinking at Any Age: Its Nature and Its Development , 2005 .

[34]  G. Harel,et al.  ADVANCED MATHEMATICAL THINKING: SOME PME PERSPECTIVES , 2006 .

[35]  Patrick W Thompson,et al.  Didactic Objects and Didactic Models in Radical Constructivism , 2002 .

[36]  Gloria Sánchez-Matamoros,et al.  La comprensión de la derivada como objeto de investigación en didáctica de la matemática , 2008 .

[37]  Ed Dubinsky,et al.  Reflective Abstraction in Advanced Mathematical Thinking , 2002 .