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El maestro constructivista como investigador. Cómo enseñar razones y proporciones a adolescentes
... Éste es un vehícu lo de pensamiento para que los estudiantes puedan avanzar en matemáticas. Czarnocha (1999) desarrolló un estudio al usar la técnica de Bruner (1961) y Shulman (1968), citados en Czarnocha (1999), cuya principal idea es que el alumno aprende de manera más efectiva cuando descubre el conocimiento por sí solo en vez de hacerlo por instrucción directa. El profesor actúa como agente para que el menor pueda realizar el descubrimiento. ...
... Éste es un vehícu lo de pensamiento para que los estudiantes puedan avanzar en matemáticas. Czarnocha (1999) desarrolló un estudio al usar la técnica de Bruner (1961) y Shulman (1968), citados en Czarnocha (1999), cuya principal idea es que el alumno aprende de manera más efectiva cuando descubre el conocimiento por sí solo en vez de hacerlo por instrucción directa. El profesor actúa como agente para que el menor pueda realizar el descubrimiento. ...
... This is the primary reason why we suggest the notion of schema as one of the fundamental tools for teaching-research. A simple example of such a moment of "thinking together" in the context of student-teacher interaction, which reveals students' thinking process can be found in Czarnocha, (1999). While students work with the schema of square root, teachers work with the schema of students' thinking about that subject; their interaction shows intense coordination between the two which facilitate students' final mastery of the problem. ...
A short review of the meaning and role of learning educational theories is provided through personal narrative remarks of the author, a teacher-researcher from the Bronx, NYC. The remarks based on classroom experience lead to a re-examination of educational theories in teaching-research. The remarks are coordinated with the comments of new teacher-researchers who graduated from the PDTR project and reflected upon the issue in their practice. Vygotsky's Zone of Proximal Development is discussed as an example of the theoretical framework that fits well into classroom teaching practice of concerned teachers; it is followed by an argument presenting a theoretical notion of schema as a theory particularly useful for teaching-research practice. 1 In many contributions to this book one can read about other educational theories and their use by the PDTR Project, some of which are discussed below. INTRODUCTION Bruner (1996) introduces an interesting dichotomy in the characterization of mathematical presentations: a narrative and a paradigmatic presentation, each arranging the experience in a different, irreducible, yet complementary manner. While the narrative mode pays attention to the sequence of actions in time, dwells on particular episodes for the illustration of an idea, and is highly contextual and personal, the paradigmatic mode is highly impersonal, categorical and hierarchical in its expression. The aim of the present remarks is to contribute to a discussion on how the role and need for an educational theory arise out of teaching practice of teacher-researchers. Therefore, the narrative mode of presentation is taken here as better conveying the subtleties of this experiential process, which is often difficult to grasp through the paradigmatic mode characteristic for standard research papers. Moreover, since the developmental process of teacher-researchers is not yet well understood, the narrative story-telling approach has a chance to provide glimpses into TR practice for any professional interested in the development of a teaching-research profile. Consequently, the presentation below is created by integrating the thoughts of an individual teacher-researcher, the author of this contribution, together with statements of participants of PDTR found in this volume of collected articles. 1 The remarks below are enhanced if read in conjunction with two other entries by the same author: "The Ethics of Teacher-Researchers," and "A Teaching Experiment," (Part 1), where pertinent issues concerning the learning theories are also extensively discussed.
Unit 4 presents the designs of TR investigations and teaching experiments conducted by teacher-researchers in their classrooms. It represents three type of TR activity: daily classroom TR investigations (Chapters 4.2–4.5), construction of learning trajectories through iterated classroom teaching experiments (Chapters 4.6–4.8) and two teaching experiments of opposite types (Chapter 4.9) and (Chapter 4.10).
TR/NYCity Model is based on the careful composition of ideas centred around Action Research (Lewin, 1946) with the ideas centred around the concept of the Teaching Experiment of the Vygotskian school in Russia, where it “grew out of the need to study changes occurring in mental structures under the influence of instruction” (Hunting, 1983).
This paper is part of a series of studies by the Research in Undergraduate Mathematics Education Community (RUMEC), concerning the nature and development of college students' mathematical knowledge. This project began as an attempt to explore calculus students' understanding of the chain rule and its applications. Based on the initial description of how the chain rule concept may be learned (genetic decomposition) an attempt to interpret the data using the Action-Process-Object theoretical framework is made. The insufficiency of this alone led to an extension of the Action-Process-Object-Schema epistemological framework (APOS) which includes a theory of schema development based on ideas of Piaget and Garcia. The Piagetian Triad is suggested as a mechanism for describing schema development in general, and the chain rule is used as an example. The Triad of the Intra, Inter and Trans stages of schema development provides the structure for interpreting the students' understanding of the chain rule and classifying their responses to interview questions about the chain rule. The results of this data analysis allowed for a proposed revised epistemological analysis of the chain rule. Finally, several suggestions and questions for future study are presented.
In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge and how it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions. I. ROLES OF THE RESEARCHER IN A TEACHING EXPERIMENT A distinguishing characteristic of the technique is that the researcher acts as teacher. Being a participant in interactive communication with a child is necessary because there is no intention to investigate teaching a predetermined or accepted way of operating. The current interest always lies in hypothesizing what the child might learn and finding ways and means of fostering this learning. Based on current interpretation of the child’s language and actions, the experimenter makes decisions concerning situations to create, critical questions to ask, and the types of learning to encourage. These on-the-spot decisions represent a major modus operandi in teaching experiments and the researcher has the responsibility for making them. Beyond acting as teacher, another role of the researcher is to analyze the knowledge involved in teaching. The researcher must build what Hawkins (1973) called a map and what I call a model of each child’s mathematical knowledge. Toward this end, the teaching experiment is primarily an exploratory tool, derived from Piaget’s clinical interview and aimed at investigating what might go on in children’s heads. This process involves formulating and testing hypotheses about various aspects of the child’s goal-directed mathematical activity in order to learn what the child’s mathematical
The constructivist teaching experiment is used in formulating explanations of children's mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations of theoretical constructs--that represent our understanding of children's mathematical realities. However, the models must be distinguished from what might go on in children's heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children's construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher's understanding of the children.
In the context of fractions and ratios, assessments were made of cognitive ability level, achievement, facility with general mathematical strategies, computational ability, and attitude of 435 12-year-olds. Approximately half the students were taught fraction and ratio topics by means of a concrete, process-oriented method, whereas the other half were taught in the regular way. The concrete, process-oriented approach resulted in significantly improved achievement in, and attitude toward, fractions and ratios. Also, the development of general mathematical strategies was enhanced and computational facility maintained.
Describes recent developments of the clinical interview and of the teaching experiment. Applications and extensions are discussed in the context of developing models of cognitive mechanisms in a constructivist framework. Purposes for which the clinical method has been adopted are described, and problems associated with it are discussed in relation to these purposes. A teaching experiment, conducted with 8 6–9 yr olds, that was designed to reveal constructive mechanisms that children use in establishing knowledge of numeration and addition and subtraction is reported. Connections between these 2 methodologies are examined. (60 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The study deals with qualitative features of proportional reasoning on dimensional intensive variables by eleven and thirteen year-old students. These features include types of comparison and types of strategies employed. Four proportional problems, varying in numerical and referential content were administered to 116 eleven year-olds and 137 thirteen year-olds. No significant age effect was found. The relative frequencies with which the type of comparison and various strategies were used was greatly affected by context, numerical content of the problem, and the immediately preceeding task. No type of comparison appeared to be more natural than another, as suggested in previous research.