Partially correct constructs illuminate students’ inconsistent answers

We present a view of knowledge construction processes, focusing on partially correct constructs. Motivated by unexpected and seemingly inconsistent quantitative data based on the written reports of students working on an elementary probability task, we analyze in detail the knowledge construction processes of a representative student. We show how the nested epistemic actions model for abstraction in context facilitates following the emergence of a learner’s partially correct constructs (PaCCs). These PaCCs provide added insight into processes of knowledge construction. They are also used in order to analyze and explain students’ thinking in situations where some of the students’ answers were unexpected in light of their earlier answers or inconsistent with earlier answers. In particular, PaCCs are explanatory tools for correct answers based on (partially) faulty knowledge and for wrong answers based on largely correct knowledge.

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

[2]  Paul White,et al.  Abstraction in mathematics learning , 2007 .

[3]  M. Heuvel-Panhuizen.  Mathematics education in the Netherlands: A guided tour 1 , 2001 .

[4]  Daniel Kahneman,et al.  Availability: A heuristic for judging frequency and probability , 1973 .

[5]  E. Fischbein,et al.  The intuitive sources of probabilistic thinking in children , 1975 .

[6]  D. Schifter,et al.  A research companion to Principles and standards for school mathematics , 2003 .

[7]  Mehmet Fatih Ozmantar,et al.  A dialectical approach to the formation of mathematical abstractions , 2007 .

[8]  J. Roschelle,et al.  Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition , 1994 .

[9]  J. Piaget,et al.  The Origin of the Idea of Chance in Children , 1975 .

[10]  Jeremy Kilpatrick,et al.  Types of Generalization in Instruction: Logical and Psychological Problems in the Structuring of School Curricula , 1990 .

[11]  C. Pontecorvo,et al.  Arguing and Reasoning in Understanding Historical Topics , 1993 .

[12]  Dave Pratt,et al.  Making Sense of the Total of Two Dice , 2000 .

[13]  B. Schwarz,et al.  Abstraction in Context: Epistemic Actions , 2001 .

[14]  A. N. Leont’ev.  The Problem of Activity in Psychology , 1974 .

[15]  Dor Abrahamson,et al.  Embodied design: constructing means for constructing meaning , 2009 .

[16]  Walter Daelemans,et al.  Memory-Based Language Processing: Abstraction and generalization , 2005 .

[17]  Baruch B. Schwarz,et al.  Transformation of Knowledge through Classroom Interaction , 2009 .

[18]  Bert van Oers.  The fallacy of decontextualization , 1998 .

[19]  J. Wertsch.  The Concept of Activity in Soviet Psychology , 1981 .

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

[21]  J. Baron,et al.  An analysis of the word-superiority effect☆ , 1973 .

[22]  Per Nilsson,et al.  Different ways in which students handle chance encounters in the explorative setting of a dice game , 2007 .

[23]  M. Lecoutre.  Cognitive models and problem spaces in “purely random” situations , 1992 .

[24]  J. F. Wagner,et al.  Transfer in Pieces , 2006 .

[25]  Celia Hoyles,et al.  Windows on Mathematical Meanings , 1996 .

[26]  T. Dreyfus,et al.  Interacting parallel constructions: A solitary learner and the bifurcation diagram , 2006 .

[27]  E. Fischbein,et al.  Factors affecting probabilistic judgements in children and adolescents , 1991 .

[28]  Ron Tzur,et al.  Fine grain assessment of students’ mathematical understanding: participatory and anticipatory stagesin learning a new mathematical conception , 2007 .

[29]  Richard Noss,et al.  The Microevolution of Mathematical Knowledge: The Case of Randomness , 2002 .