Showing papers in "Zdm in 2004"
Journal Article•
TL;DR: GeoGebra is a new software system that integrates possibilities of both dynamic geometry and computer algebra in one tool for mathematics education.
Abstract: Dynamic geometry and computer algebra systems have highly influenced mathematics education. Unfortunately, these tools have been totally unconnected. GeoGebra is a new software system that integrates possibilities of both dynamic geometry and computer algebra in one tool for mathematics education. ZDM classification : R 20, U 70
148 citations
TL;DR: In this paper, four potential modes of interaction with diagrams in geometry are introduced, and the extent to which the latter kind of interaction may induce tensions on the work of a teacher as she manages students' mathematical work is illustrated.
Abstract: Four potential modes of interaction with diagrams in geometry are introduced. These are used to discuss how interaction with diagrams has supported the customary work of ‘doing proofs’ in American geometry classes and what interaction with diagrams might support the work of building reasoned conjectures. The extent to which the latter kind of interaction may induce tensions on the work of a teacher as she manages students’ mathematical work is illustrated.
59 citations
TL;DR: In this paper, it has been suggested that students who have been turned off by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving.
Abstract: It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.
42 citations
TL;DR: A status report on discrete mathematics in America's schools, including an overview of publications and programs that have had major impact, is provided in this paper, where the authors discuss why discrete mathematics should be introduced in the schools and the authors' efforts to advocate, facilitate, and support the adoption of discrete mathematics topics.
Abstract: This article provides a status report on discrete mathematics in America's schools, including an overview of publications and programs that have had major impact. It discusses why discrete mathematics should be introduced in the schools and the authors' efforts to advocate, facilitate, and support the adoption of discrete mathematics topics in the schools. Their perspective is that discrete mathematics should be viewed not only as a collection of new and interesting mathematical topics, but, more importantly, as a vehicle for providing teachers with a new way to think about traditional mathematical topics and new strategies for engaging their students in the study of mathematics.
39 citations
TL;DR: In this article, the authors start from the assumption that a semiotic approach might provide a fresh start in reconciling prevailing dichotomies in educational reflection and research, especially as regards the popular dichotomy of individual against social learning and constructive against receptive learning.
Abstract: The present paper starts from the assumption that a semiotic approach might provide a fresh start in reconciling prevailing dichotomies in educational reflection and research. Especially as regards the popular dichotomies of individual against social learning and constructive against receptive learning. Three exemplary sections will illustrate the salient features of a semiotic approach: sign process seen as mediating means; sign processes as creating networks of objects, signs and interpretants; and the metaphor of map and territory as a relation of sign and activity. Throughout the paper it is tried to capitalize from the tension between the semiotic approaches of Peirce and Vygotskij. Der vorliegende Beitrag grundet sich auf die Annahme, dass ein semiotischer Ansatz moglicherweise dem Versuch zu neuem Erfolg verhelfen kann, herrschende Dicho-tomien in der padagogischen und didaktischen forschung zu uberwinden. Dies betrifft besounders die beliebte Gegenuber-stellung von sozial und individuall und von konstruktiv und rezeptiv beim Lernen. In drei exemplarischen Abschnitten wird die Fruchtbarkeit einer semiotischen Perspektive demonstriert: Zeichenprozess als Mittel und Vermittlung. Zeichenprozesse als Vernetzung von Objekten, Zeichen und Interpretanten; und die Beziehung von Landkarte und Territorium als Metapher fur die Beziehung von Zeichen und Tatigkeit. Die Spannung zwischen den semiotischen Perspektiven von Peirce und Vygotskij bildet dabei einen permanenten Bezugspunkt.
27 citations
TL;DR: In this article, the authors present a method for evaluating proof instruction and some results of a video study that describes proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.
Abstract: Teaching mathematical proof is one of the most challenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are significantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.
26 citations
TL;DR: In this article, the authors report on collaborations with practitioners to examine the results of students' performances on high stakes tests as a means to strengthen practitioners' knowledge of probability and statistics and to empower their conduct of investigations on student performance.
Abstract: The study reports on collaborations with practitioners to examine the results of students’ performances on high stakes tests as a means to strengthen practitioners’ knowledge of probability and statistics and to empower their conduct of investigations on student performance. Four issues are summarized: the development of their statistical reasoning, their understanding of the meaning of and relationships among the concepts of validity, reliability and fairness as applied to testing, their introduction to the history of testing and its relationship to science, society and cultural inequality, and their reports of independent inquiries. Data on performance on pre- and post-tests demonstrate growth in teacher reasoning and in their professionalism in raising important issues about testing
24 citations
TL;DR: The concept of adialectical system is developed as a general framework to describe the development of knowledge networks that mark the starting point for learning processes and semiotics is used to discuss the epistemological thesis that any cognitive access to the authors' world of objects is mediated by signs.
Abstract: A central challenge for research on how we should prepare students to manage crossing boundaries between different knowledge settings in life long learning processes is to identify those forms of knowledge that are particularly relevant here. In this paper, we develop by philosophical means the concept of adialectical system as a general framework to describe the development of knowledge networks that mark the starting point for learning processes, and we use semiotics to discuss (a) the epistemological thesis that any cognitive access to our world of objects is mediated by signs and (b)diagrammatic reasoning andabduction as those forms of practical knowledge that are crucial for the development of knowledge networks. The richness of this theoretical approach becomes evident by applying it to an example of learning in a biological research context. At the same time, we take a new look at the role of mathematical knowledge in this process.
15 citations
TL;DR: In this article, the authors present an empirical investigation of how students worked with sequences in a computer-supported environment, showing that students were able to create symbolic, numerical and graphical representations, to change between these different representations.
Abstract: Sequences are fundamental mathematical objects with a long history in mathematics. Sequences are also tools for the development of other concepts (e. g. the limit concept), as well as tools for the mathematization of real-life situations (e. g. growth processes). But, sequences are also interesting objects in themselves, with lots of surprising properties (e. g. Fibonacci sequence, sequence of prime numbers, sequences of polygonal numbers). Nowadays, new technologies provide the possibility to generate sequences, to create symbolic, numerical and graphical representations, to change between these different representations. Examples of some empirical investigation are given, how students worked with sequences in a computer-supported environment.
13 citations
TL;DR: This article introduces an investigation dealing with the question of what role the mathematical discipline “combinatorial optimization” can play in mathematics and computer science education at high school.
Abstract: This article introduces an investigation dealing with the question of what role the mathematical discipline “combinatorial optimization” can play in mathematics and computer science education at high school. Combinatorial optimization is a lively field of applied mathematics and computer science that has developed very fast through the last decades.
13 citations
TL;DR: In this paper, the authors present results of a pilot study in which primary school students allocated in two separate rooms solve mathematical problems by means of internet chatting and analyse the inscriptions emerging during the chat sessions Charles S. Peirce's semiotic approach.
Abstract: In internet-chat situations about mathematical wordproblems, the students are confronted with the fundamental issue of presenting their solving-attempts in a written or graphic form. This circumstance raises the opportunity to study the use of inscriptions as defined by Latour and Woolgar (1986). In my essay I present results of a pilot study in which primary school students allocated in two separate rooms solve mathematical problems by means of internet chatting. In order to analyse the inscriptions emerging during the chat sessions Charles S. Peirce's semiotic approach is applied, based on research methods of Interpretative Classroom Research.
TL;DR: In this article, a deeper analysis showed that the mathematics instruction in most of the East Asian countries is described as examination driven and based on memorising rules and facts, whereas the mathematics classroom in western countries aims at a meaningful and individualised learning.
Abstract: In different international studies on mathematical achievement East Asian students outperformed the students from Western countries. A deeper analysis shows that this is not restricted to routine tasks but also affects students’ performance for complex mathematical problem solving and proof tasks. This fact seems to be surprising since the mathematics instruction in most of the East Asian countries is described as examination driven and based on memorising rules and facts. In contrast, the mathematics classroom in western countries aims at a meaningful and individualised learning. In this article we discuss this “paradox” in detail for Taiwan and Germany as two typical countries from East Asia and Western Europe.
TL;DR: In this paper, an interviewstudie with zehn Schulerinnen und Schulern der Jahrgangsstufe 8 berichtet, die als qualitative Erganzung zu einer quantitativen empirischen Untersuchung with 659 Probanden durchgefuhrt wurde.
Abstract: In diesem Beitrag wird uber eine Interviewstudie mit zehn Schulerinnen und Schulern der Jahrgangsstufe 8 berichtet, die als qualitative Erganzung zu einer quantitativen empirischen Untersuchung mit 659 Probanden durchgefuhrt wurde. Die Probanden, die in der 7. und 8. Klasse an schriftlichen Tests teilgenommen hatten, wruden beim Losen geometrischer Beweisaufgaben videografiert und anschliesend befragt. Es zeigt sich, dass Schulerschwierigkeiten bei diesen Aufgaben im Wesentlichen auf das Faktenwissen, das Methodenwissen zum mathematischen Beweisen und die Entwicklung und das Verfolgen einer Beweisstrategie zuruckgefuhrt werden konnen. Wahrend schwachere Schuler in allen drei Bereichen Defizite aufweisen, liegen die Schwierigkeiten der starkeren Probanden vor allem in der Entwicklung einer Beweisstrategie.
TL;DR: Experimental mathematics is a serious branch of mathematics that starts gaining attention both in mathematics education and research as mentioned in this paper, and has been shown to be a valuable contribution to education as a whole.
Abstract: Experimental mathematics is a serious branch of mathematics that starts gaining attention both in mathematics education and research. We given examples of using experimental techniques (not only) on the classroom. At first sight it seems that introducing experiments will weaken the formal rules and the abstractness of mathematics that are considered a valuable contribution to education as a whole. By putting proof and experiment side by side we show how this can be avoided. We also highlight consequences of experimentation for educational computer software.
TL;DR: In this article, the authors present some topics from the field of discrete mathematics which might be suitable for the high school curriculum, and they choose elements from number theory and various aspects of coding theory.
Abstract: In this paper we present some topics from the field of discrete mathematics which might be suitable for the high school curriculum. These topics yield both easy to understand challenging problems and important applications of discrete mathematics. We choose elements from number theory and various aspects of coding theory. Many examples and problems are included.
TL;DR: In this article, the authors describe the way that the researchers and a group of teachers in three schools worked together to deal with issues and opportunities associated with the implementation of CAS calculators into the year 11 and 12 curriculum during 2001 and 2002 in Victoria, Australia.
Abstract: Introducing CAS calculators into the mathematics classroom has the potential to confront classroom practice more than the introduction of graphics calculators. CAS calculators are capable of performing many of the routine procedures that are taught in the secondary school curriculum and having CAS calculators available in the classroom has implications for pedagogy, interactions in the classroom and practice of routine procedures. This paper will describe the way that the researchers and a group of teachers in three schools worked together to deal with issues and opportunities associated with the implementation of CAS calculators into the year 11 and 12 curriculum during 2001 and 2002 in Victoria, Australia.
TL;DR: In this article, the authors describe an application of statics to geometrical proofs in the classroom and find that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems.
Abstract: This paper describes an application of statics to geometrical proofs in the classroom. The aim of the study was to find out whether the use of concepts and arguments from statics can help students understand and produce proofs of geometrical theorems. The two theorems studied were (1) that the medians in a triangle meet at a single point which is the centre of gravity of the triangle, and (2) the Varignon theorem, that the lines joining the midpoints of successive sides of a quadrilateral form a parallelogram. The classroom experiment showed that most students were successful in using arguments from statics in their proofs, and that they gained a better understanding of the theorems. These findings lend support to the claim that the introduction of statics helps students produce proofs and grasp their meaning.
TL;DR: This paper propose an approach based on the systematic exploitation of structured calculation, which builds the notion of objective mathematical proof into the curriculum for all pupils from the earliest years, and analyse the current situation in England, including explicit evidence of the extent to which current instruction fails even the best students.
Abstract: It is generally accepted that proof is central to mathematics. There is less agreement about how proof should be introduced at school level. We propose an approach—based on the systematic exploitation of structured calculation—which builds the notion of objective mathematical proof into the curriculum for all pupils from the earliest years. To underline the urgent need for such a change we analyse the current situation in England—including explicit evidence of the extent to which current instruction fails even the best students.
TL;DR: This article explored adult learners' preferences for explanations of mathematical statements in terms of kinds of reasoning and formats of presentation and concluded that familiarity and clarity influenced students' preferences more than the format or reasoning used.
Abstract: This article explores adult learners’ preferences for explanations of mathematical statements in terms of kinds of reasoning and formats of presentation. Based on data from questionnaires and interviews it is concluded thatfamiliarity andclarity influenced students’ preferences more than the format or reasoning used. A contrast between the factors influencing students’ choices and those of instructors is also reported. Implications for teaching and research are drawn from the study.
TL;DR: A portable ultrasound diagnostic device dynamically collecting diagnostic data from a patient and transmits the diagnostic data using a communication channel to a person at another location to analyze the diagnosticData.
TL;DR: In this paper, a discussion of a few topics that should be included in the syllabus for any course in discrete mathematics, independent of the audience, is presented, and several potential models for teaching the course, depending upon the interests and mathematical background of the students.
Abstract: This paper is concerned with the teaching of Discrete Mathematics to university undergraduate students. Two to three decades ago this course became a requirement for math and computer science students in most universities world wide. Today this course is taken by students in many other disciplines as well. The paper begins with a discussion of a few topics that we feel should be included in the syllabus for any course in Discrete Mathematics, independent of the audience. We then discuss several potential models for teaching the course, depending upon the interests and mathematical background of the audience. We also investigate various educational links with other components of the curriculum, consider pedagogical issues associated with the teaching of discrete mathematics, and discuss some logistical and psychological difficulties that must be overcome. A special emphasis is placed on the role of textbooks.
TL;DR: In this paper, the authors examine some characteristics of learning events of a community of mathematics educators and discuss the complexities underlying learning in such a community through (re)negotiation of practices and goals.
Abstract: The paper examines some characteristics of learning events of a community of mathematics educators. Participation in the community entailed gaining familiarity with agreed upon conventions, goals, and forms of communication. The case discussed herein is an attempt to convey the complexities underlying learning in such a community through (re)negotiation of practices and goals. The notion ofreflective discourse is borrowed to describe a group discussion involving collective reflection that constituted an occasion for meaningful learning.
TL;DR: Semiotics in Mathematics Education Research Semiotics is a lively field and a widely debated topic in mathematics education research: The PME-community had a Discussion Group on Semiotic and socio-cultural evolution of mathematical concepts during its conferences in 2003 and 2004 as discussed by the authors.
Abstract: Semiotics in Mathematics Education Research Semiotics is a lively field and a widely debated topic in mathematics education research: The PME-community had a Discussion Group on “Semiotic and socio-cultural evolution of mathematical concepts“ during its conferences in 2003 and 2004. This Discussion Group grew out of a group on “Semiotics in mathematics education” meeting during the conferences of 2001 and 2002, which had an early forerunner in PME-21 in 1997. A more local perspective may complement this worldwide activity: The association of German speaking didacticians of mathematics, the ‘Gesellschaft für Didaktik der Mathematik (GDM)’ has a working group on Semiotics since September 2000. This group normally meets twice a year to offer a forum of discussion to German speaking colleagues interested in the problems, potential and progress of semiotics in the didactics of mathematics. As a consequence of these ongoing activities, the journal “Educational Studies in Mathematics” plans to publish a special issue on Semiotics in the first half of 2005 with papers on the interaction of semiotics and epistemology, implications for teachers and students and implications for research.