Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task

Conocimiento de matemáticas especializado de los estudiantes para maestro de primaria en relación al razonamiento proporcional

“…En la interpretación del número racional se consideran cinco subconstructos: razón, operador, parte-todo, medida y cociente (BEHR et al, 1997). El subconstructo razón es una comparación multiplicativa entre dos cantidades (PITTA-PANTAZI; CHRISTOU, 2011) (Tarea 2, Tabla 1).…”

Theoretical Discussion on Mathematical Knowledge for Teaching from Constructivists' Perspective

“…The first group based their study from children's mathematics whereas the second group started from semantic analysis of expert mathematicians. For instance, in fractions study of students' learning, Steffe and his colleagues (e.g., Steffe & Olive, 2010) conducted teaching experiment with children to understand 'mathematics of children' that comprise 'mathematics for children', whereas the Rational Numbers Project (Behr et al, 1997) has analyzed mathematical knowledge for fractions (as a system of rational numbers) in terms of the strategies students use to solve tasks that are representative of a conceptual domain. The Rational Numbers Project based their study of rational numbers on Kieren's (1976) work which breaks the rational number into subconstructs-part-whole, quotient, ratio number, operator, and measure and have suggested that a complete understanding of rational number requires an understanding of each of those subconstructs separately and also an understanding of the relationships among the subconstructs.…”

“…Research methods under the second perspective investigate teachers' mathematical knowledge are diverse and different from the first group as the principle of knowledge in pieces reflects on this group's studies of teacher knowledge; whereas teaching experiment (Steffe & Thompson, 2000) was the only method they used, the latter group used a variety of methods. Some researchers (Behr et al, 1997;Post et al, 1991) provided teachers with tasks fairly representing the range of knowledge and reasoning in specific conceptual domains, and inferred knowledge elements through survey items and interviews by examining teachers' strategies in solving non-routine tasks; others (Gutstein & Mack, 1998;Izsák, 2008;Izsák et al, 2008;Lehrer & Franke, 1992) have concentrated on particular cognitive structuremathematical knowledge for teaching -among knowledge, goals, and beliefs, and tried to infer knowledge elements in further grain size in the context of classroom teaching or tutoring. Even though their research methods and data were different one another, their ultimate goal seems to provide more detailed knowledge elements for teacher knowledge.…”

“…Studies under this category examined teacher knowledge in fine-grained size and tried to understand teachers' thinking from their practice or problem solving strategies (e.g., Behr, Khoury, Harel, Post & Lesh, 1997;Izsák, 2008;Izsák, Tillema & Tunç-Pekkan, 2008;Lehrer & Franke, 1992;Post, Harel, Behr & Lesh, 1991). The epistemological stance, as constructivists, under the latter domain has been elaborated by Smith, diSessa, 3 Simon (1995) provided a framework to analyze teaching in context, Hypothetical Learning Trajectories (HLTs), which consist of the learning goal that defines the direction, the learning activities, the hypothetical learning process, and the teachers' hypotheses of students' knowing and understanding that will evolve as the actual learning trajectory as they work with their children.…”

“…Whereas little study did explicitly deal with teacher knowledge under the first group, there are some studies that tried to examine teachers' deeper unit structure under the second group; Behr et al (1997) interviewed 30 preservice elementary teachers to explore strategies on working with tasks focused on one of the rational number subconstructs, operator. They provided conceptual unit analysis based on the conceptual units structure (Behr, Harel, Post & Lesh, 1994) and decomposed the operator subconstruct into another subconstructs: duplicator and partition-reducer, stretcher and shrinker, and multiplier and divisor.…”