Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a TaskTechniqueTheory Perspective (original) (raw)
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Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom
International Journal of Mathematical Education in Science and Technology, 2011
Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task-Technique-Theory (T-T-T) theoretical frame as adapted from Chevallard's Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the taskbased activity of a pair of 10th grade students-activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.
International Journal of Computers for Mathematical …, 2002
The last decade has seen the development in France of a significant body of research into the teaching and learning of mathematics in CAS environments. As part of this, French researchers have reflected on issues of 'instrumentation', and the dialectics between conceptual and technical work in mathematics. The reflection presented here is more than a personal one-it is based on the collaboration and dialogues that I have been involved in during the nineties. After a short introduction, I briefly present the main theoretical frameworks which we have used and developed in the French research: the anthropological approach in didactics initiated by Chevallard, and the theory of instrumentation developed in cognitive ergonomics. Turning to the CAS research, I show how these frameworks have allowed us to approach important issues as regards the educational use of CAS technology, focusing on the following points: the unexpected complexity of instrumental genesis, the mathematical needs of instrumentation, the status of instrumented techniques, the problems arising from their connection with paper & pencil techniques, and their institutional management. This paper is based on a plenary lecture given at the Second CAME (Computer Algebra in Mathematics Education) Symposium that took place at the Fruedenthal Institute in July 2001.
ICMI Study 25 Conference Proceedings, 2020
In this paper, I present and discuss findings from a research study aiming to investigate the activity of proving as constituted in the classroom. By drawing on Cultural-Historical Activity Theory and collaborative task design, this study explores the way the teacher is working with the students to foreground mathematical argumentation. In this paper, the collaboration between the teacher and the researcher is discussed, focusing on the design and implementation process as a vehicle to gain access to the teacher's objectives and motivations. Furthermore, the way the teacher intervened throughout the lessons is contrasted against this collaboration, so as to gain a deeper understanding regarding what drives the teacher's teaching decisions.
Conceptualizing the Learning of Algebraic Technique: Role of Tasks and Technology
2009
This article is divided into four parts. The first part presents some introductory remarks on the use of Computer Algebra System (CAS) technology in relation to the long-standing dichotomy in algebra between procedures and concepts. The second part explores the technical-conceptual interface in algebraic activity and discusses what is meant by conceptual (theoretical) understanding of algebraic technique – in other words, what it means to render conceptual the technical aspects of algebra. Examples to be touched upon include seeing through symbols, becoming aware of underlying forms, and conceptualizing the equivalence of the factored and expanded forms of algebraic expressions. The ways in which students learned to draw such conceptual aspects from their work with algebraic techniques in technology environments is the focus of the third part of the article. Research studies that have been carried out by my research group with a range of high school algebra students have found evide...
Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction
Educational Studies in Mathematics, 1993
LEVEL OF ACTING AND REFLECTING IN MATHEMATICS INSTRUCTION ~s'nlhcr. Mathematics instruction contains two conflicting demands: on the one hand, the demand for economical efficiency and for well-developed "motorways" and, on the other hand, the demand that pupils should investigate and discover for themselves and have the freedom to "pave" their own ways. It is argued that tasks with a certain richness and quality offer some steps towards a construefive handling of this dilemma. The author tries to develop some properties of powerful tasks and to sketch the structure and philosophy of one concrete system of powerful tasks for the concept of angle. The main part of this paper presents seven examples of powerful tasks: five from the system of tasks and two with regard to pupils' working with 2-D-graphic systems. The construction of powerful tasks is viewed as a valuable contribution to bringing the theory and practice of mathematics education closer together. BACKGROUND Discussing the importance of tasks in mathematics education has a long tradition. A detailed analysis of the so-called Task Didactics (Aufgabendidaktik), which is one of the marked features of Traditional Mathematics, is given by Lenn6 (1969). Traditional Mathematics was the leading stream of mathematics education in Germany (and in a similar way in Austria) until the middle of this century, and then was progressively displaced by New Mathematics (which in turn is being pushed back more and more). Task Didactics is characterized by a partition of the mathematical subject-matter into specific areas (e.g., fractions, percentages, triangles, quadrilaterals). Each area is determined by a special type of task which is systematically treated, progressing from simple to more complex tasks (combination of simple tasks). Cross-connections (e.g., regarding fundamental ideas or structures) are not worked out in detail. In general, the teacher teaches theories and methods and the pupils have to apply them by solving tasks. How much has this situation changed? Recent empirical research studies, like those of Bromme (1986) or Clark and Yinger (1987), show that even nowadays mathematics teachers plan and organize their instruction on a large scale with the help of tasks. Research with regard to tasks takes different directions. There are many contributions to general considerations about tasks, for example: Wittmann (1984) views teaching units as the integrating core of mathematics education, incorporating mathematical, pedagogical, psychological, and practical aspects in a natural
The Journal of Mathematical Behavior, 2001
The question of problem-solving activities in didactic institutions is critical in mathematics education for two important reasons. It is a main factor of learning according to Piaget, and it is a means for students to try to align their behaviors to expected institutional references. Mathematical reasoning during problem solving in didactic institutions is studied in the present work as a complex system of interfering constraints. Results tend to show that this system may be understood as being ruled by ternary interactions between three poles: the student, the teacher, and the knowledge itself. Simultaneously, theoretical and pragmatic considerations are focused on problem solving in mathematics: the specific epistemological difficulties of each domain of knowledge to be studied, the computational asymmetry between mathematical concepts and procedures, and the influence of implicit teacher expectations through students' decoding of local ''didactic contracts.'
International Journal of Computers for Mathematical Learning, 2006
This paper addresses the dialectical relation between theoretical thinking and technique, as they coemerge in a combined computer algebra and paper-and-pencil environment. The theoretical framework in this ongoing study consists of the instrumental approach to learning mathematics with technology and Chevallard's anthropological theory. The main aim is to unravel the subtle intertwining of the students' theoretical thinking and the techniques they use in both media, within the process of instrumental genesis. Two grade 10 teaching experiments are described, the first one on equivalence, equality and equation, and the second one on generalizing and proving within factoring. Even though the two topics are quite different, findings indicate the importance of the co-emergence of theory and technique in both cases. Some further extensions of the theoretical framework are suggested, focusing on the relation between paper-and-pencil techniques and computer algebra techniques, on the issue of language and discourse in the learning process, and on the role of the teacher. KEY WORDS. Algebra, computer algebra, instrumentation, technique in algebra, technology, theoretical thinking in algebra Activity 1 Part I Part II Part III Equivalence of Expressions Comparing expressions by numerical evaluation Comparing expressions by algebraic manipulation Testing for equivalence by re-expressing the form of an expression-using the Expand command Tools CAS P&P CAS
UNDERSTANDINGS REVEALED BY ENGINEERING STUDENTS' ACTIONS IN A CAS TASK
Understandings revealed by engineering students' actions in a CAS task, 2018
Understanding is a continuous pursuit of making sense of lived experiences. We considered how a mathematical modeling task could invite varied levels of understanding. When engineering students learn in a computer algebra system environment, they often jump straight into programming activities and struggle to make sense of computerised outputs. This paper aimed to identify actions that can mediate broader levels of understandings. We used the Pirie-Kieren model of growth in mathematical understanding to identify the actions of engineering students r egistered at a South African university. Thirteen participants voluntary collaborated to work on a modeling task, which was deductively analysed using content analysis. The findings revealed an interdependence between paper-and-pen, computerised and reflective actions in the development of understandings. Scaffolded and folding-back actions can assist students to oscillate back and forth towards more cogent understandings. Specific arrangement and sequence of subtasks can assist students to reconcile new understandings with past understandings. Well-designed modelling tasks can facilitate unconventional levels of understanding when students learn with a computer algebra system.