Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom (original) (raw)
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American Educational Research Journal, 2003
Novel (as opposed tofadmiliar) tasks can be contextsforstudents' development of new knowledge. But managing such development is a complex activityfor a teacher. The actions that a teacher took in managing the development of the mathematical concept of area in the context of a task comparing cardstock triangles are examined. The observation is made ihat some of the teacher's actions shaped the mathematics at play in ways that seemed to counter the goals of the task. This article seeks to explain apossible rationality behind those contradictory actions. The hypothesis ispresented that in managing task completion and knowledge development, a teacher has to cope with three subjectspecfic tensions related to direction of activity, representation of mathematical objects, and elicitation of students' conceptual actions.
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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...
International Journal of Computers for Mathematical …, 2009
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.
Task Design in Mathematics Education
Margolinas, C. (Ed.). (2013). Task Design in Mathematics Education. Proceedings of ICMI Study 22 . Oxford., 2014
The context for this paper is the ordinary mathematics classroom lesson, in which the tasks set for students involve the use of computer software. The paper highlights the importance of paying attention to the intended mathematical learning of students as they work through the task, adapting their strategies as they negotiate epistemological obstacles. It draws on theoretical notions of student activity, or ‘dialectics’, within the classroom environment to suggest the sorts of activity that are likely to provoke mathematical learning, particularly highlighting the role of computers within these activities. This provides the background against which approaches to the design of mathematical tasks are recommended. The approaches focus on the intended mathematical learning of the students and the obstacles built intentionally and unintentionally into the tasks.
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.'
A student often meets a conflict between conceptual and procedural knowledge: does (s)he need to understand for being able to do, or vice versa? Hence an important research question is how pedagogical solutions affect the relation between the two knowledge types. Our theoretical analysis and practical experience evidence that desired links can be promoted when the learner has opportunities to simultaneously activate conceptual and procedural features of the topic at hand. Such activation is considered for interactive learning that utilizes an able technological tool, the ClassPad calculator. In a sequence of examples, we will show how this tool can be exploited to develop both informal and formal mathematical knowledge.
From Artifact to Instrument: Mathematics Teaching Mediated by Symbolic Calculators
2003
The evolution of calculation tools available for the learning of mathematics has been quick and profound. After the first illusions on a naturally positive integration of these tools, new theoretical approaches have emerged. They take into account individual and social processes of the mathematical instrument construction from a given artifact. In this article we show how analyzing constraints of the tool allows the understanding of its influence on the knowledge construction. We propose the concept of instrumental orchestration to design different devices which may be built in class and thus strengthen the socialized part of the instrumental genesis: instrumental orchestration is defined by objectives, configuration and exploitation modes. It acts at the same time on the artifact, on the subject, on the relationship the subject has with the artifact and on the way the subject considers this relation
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
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