BibSonomy publications for /concept/user/yish/mathematicsFri Aug 01 12:52:00 CEST 2008International Journal of Computers for Mathematical Learning2143--167Modelling Hyperbolic Space: Designing a Computational Context for Learning Non-Euclidean Geometry52000mathematics designresearch learning non-euclidean design hyperbolic geometry microworlds This paper describes and analyses the iterative design and development of a computational context for non-euclidean geometry. Drawing on three episodes from the design process, the paper discusses the epistemological implications associated with interplay between learning hyperbolic geometry and context in which that learning takes place. In particular, it explores the ways in which learners can become designers of the computational context, and the designer can become a learner. The paper concludes with a discussion of the microworld paradigm in relation to what might be called ‘advanced’ mathematics.Thu Jul 17 14:39:31 CEST 2008Proceedings of EuroPLoP 2008Guess my X and other patterns for teaching and learning mathematicsforthcomingpedagogicalpatterns patterns designpatterns design mathematics learning gmx my Thu Jun 05 13:40:11 CEST 2008Journal of Interactive MediaParticipatory design in open education: a workshop model for developing a pattern language2008openeducationalresources opensource Learning patternlanguagenetwork workshops mathgamespatterns jime08 design participatory wleformativeeassessment planetpublications educational language CaseStudies learning Mathematics my JIME patterns Architecture4Participation mythesis designpatterns PatternLanguages OER opencontent casestudies IDR Games open education methodology Technologically enhanced learning environments raise complex challenges for their designers, developers and users. Design patterns and pattern languages have recently emerged as a potential framework for addressing some of these challenges. However, the uptake of design patterns has been slow outside of the computer science community. We argue that this is largely a consequence of a weak positioning of pattern languages, as a form of delivering expert knowledge to layperson, and suggest an alternative view: the development of a pattern language as a community endeavour. In terms of open education, the workshop model can be viewed as an open production process for developing educational resources, in our case design patterns. We propose a model of pattern elicitation workshops, in which collaborative development of a pattern language provides a framework for sharing design knowledge within interdisciplinary communities. This model was iteratively developed at five international conferences. It was then postulated as a design pattern itself, encompassing a series of practices and a set of supporting tools. We believe this model could be applied in a broad range of communities concerned with the development of open digital educational resources.Fri May 30 10:54:36 CEST 2008PretoriaRevised National Curriculum Statement Grades R-9(Schools)2002revised national curriculum mythesis south sequences patterns numbers africa schools statement Fri May 30 05:56:34 CEST 2008Educational Studies in Mathematics3237-268Signs and meanings in students' emergent algebraic thinking: a semiotic analysis422000signs algebraic of to mythesis ijtme2006 CiHB representation semiotic ILE sequences-esm mathematics mathgamespatterns ijceell06 meanings sequences-ictmt7 vygotsky semiotic-cultural symbolization sequences sociocultural means social thinking approach generalization CnE07 objectification The purpose of this article, which is part of a longitudinal classroom research about students? algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and (2) to provide accounts about the students? emergent algebraic thinking. The research draws from Vygotsky?s historical-cultural school of psychology, on the one hand, and from Bakhtin and Voloshinov?s theory of discourse on the other, and is grounded in a semiotic-cultural theoretical framework in which algebraic thinking is considered as a sign-mediated cognitive praxis. Within this theoretical framework, the students? algebraic activity is investigated in the interaction of the individual?s subjectivity and the social means of semiotic objectification. An ethnographic qualitative methodology, supported by historic, epistemological research, ensured the design and interpretation of a set of teaching activities. The paper focuses on the discussion held by a small group of students of which an interpretative, situated discourse analysis is provided. The results shed some light on the students? production of (oral and written) signs and their meanings as they engage in the construction of expressions of mathematical generality and on the social nature of their emergent algebraic thinking.Fri May 30 05:56:16 CEST 2008Recife, BrazilNineteenth International Conference for the Psychology of Mathematics Education264-271Classroom Sociomathematical Norms and Intellectual Autonomy1995symbolic ijtme2006 constructivism as_conv sociomathematical mythesis sociocutural mathematics microculture sequences-esm mathgamespatterns cerme2005 ilceell06 norms sociocultural interactionism Fri May 30 05:55:37 CEST 2008Social Studies of ScienceFebruary139-68The Context of Proving362006philosophy reasoning proof situated polya mythesis mathematics discovery ijceell06 cognition ethnomethodology Discussions of mathematical problem-solving and heuristic reasoning have typically examined how proofs that are already known might be found. This approach has at least three problems: first, provers engaged in discovering proofs for themselves cannot have this perspective; second, if a proof is difficult, formulaic strategies quickly run out; third, beginning with a proof already in-hand separates reasoning about a proof from the actual circumstances in which such reasoning occurs. As an alternative approach to the study of mathematical reasoning, this paper presents a detailed descriptive account of the work of finding a specific proof, including the shifting of perspectives, the wrong paths, the mistakes and the outright errors. Even the appearance of a sketched diagram or of a course of mathematical writing can suggest unanticipated possibilities for finding a proof. This material is used to illustrate the paper's central claim - that the ways that provers go about working on proofs provide the context for continuing that work and for discovering the reasoning that a particular proof is then seen to require.Fri May 30 05:55:14 CEST 2008Journal of Computer Assisted Learning2114-136Exploring the mathematics of motion through construction and collaboration222006PlanetMakingStuffTogether weblabs learning mathematics science lunarlander computer programming mathgamespatterns webreports ijtme2006 communication mythesis game constructionism modelling In this paper we give a detailed account of the design principles and construction of activities underlying a model-based approach to learning about the relationships between position, velocity and acceleration, and corresponding kinematics graphs. In these activities, students controlled the movement of objects in a programming environment, recording the motion data and plotting corresponding position-time and velocity-time graphs. They shared their findings on a specially-designed web-based collaboration system, and posted cross-site challenges to which others could react. We present learning episodes that provide evidence of students making discoveries about the relationships between different representations of motion. We conjecture that these discoveries arose from their activity in building models of motion and their participation in classroom and online community.Fri May 30 05:55:04 CEST 2008Mind, Culture, and Activity3/4171-225Studying Cognition in Flux: A Historical Treatment of Fu in the Shifting Structure of Oksapmin Mathematics122005oksapmin mathematics culture mythesis guinea papua new cognition This article extends a framework for the study of culture-cognition relations to problems of historical research and diachronic analysis. As an illustrative case, we focus on mathematics in Oksapmin communities located in a remote highland area in central New Guinea. The Oksapmin, like their neighboring Mountain-Ok groups to the West, traditionally use a 27-body-part counting system for number, and there is no evidence that Oksapmin used arithmetic in prehistory. We present a coordinated analysis of shifts in functions of a word form based on field studies completed in 1978, 1980, and 2001. These shifts are related to changing collective practices of economic exchange in which arithmetical activities are increasingly important. The word form fu has changed from its use as an intensive quantifier that means "a complete group of plenty" to one that means double a numerical value. We show how the analytic framework affords a multilevel inquiry into genetic processes of change in the Oksapmin case and argue that the approach is useful for understanding the interplay between cultural and developmental processes in cognition more generally.Fri May 30 05:54:55 CEST 2008Educational Studies in Mathematics185-92The signed algorithm and its bugs351998representation deaf arithmetic language childrean sign mythesis mathematics learning Deaf children consistently lag behind their hearing cohorts in mathematics achievement tests. It has been hypothesized that their difficulty is a consequence of their lack of covert counting strategies and reliance on memorized verbal facts. We investigated the acquisition of an alternative method to solve sums, the signed algorithm, by six profoundly deaf primary school children. Similarly to the acquisition of the written algorithm by hearing children, deaf children´s calculation errors with the signed algorithm were found to be systematic and related to the structure of the numeration system in British Sign Language. These results can be used to examine better ways of teaching arithmetic to deaf children and illustrate in a novel way the role of systems of signs in mathematical cognition.Fri May 30 05:54:46 CEST 2008New York, NYSIGCUE Outlook213-17Programming-languages as a conceptual framework for teaching mathematics41970mathematics history microworlds learning constructionism programming logo mythesis Fri May 30 05:51:51 CEST 2008New York, NYIDC '04: Proceeding of the 2004 conference on Interaction design and children141-142The child-engineering of arithmetic in ToonTalk2004constructionism weblabs mathematics programming education children toontalk mythesis KalDesignResearch learning logo Fri May 30 05:47:28 CEST 2008Educational Studies in MathematicsmaySystematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed, with no entrance and no exit. In its highest perfection it can even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible even that of mathematizing mathematics.1/23-8Why to Teach Mathematics So as to Be Useful11968education mythesis mathematics philosophy learning Fri May 30 05:47:13 CEST 2008International Journal of Mathematical Education in Science and Technology3371-384Secondary trainee teachers' understanding of convergence and continuity342003training limit mythesis number teacher sequences convergence infinity This paper explores the understanding which students who are training to be secondary mathematics teachers have of elementary concepts in mathematical analysis. Research to date has tended to concentrate on students' knowledge and understanding prior to university entrance or in the first year of undergraduate mathematics courses. The design of this research, similar in style to a viva voce, involved probing students' conceptual understanding of convergence and continuity and associated reasoning. The report tentatively concludes that even mathematically well qualified students have difficulties with some elementary concepts and that students with engineering backgrounds allow the 'dynamic' image to dominate their engagement with the concepts.Fri May 30 05:46:59 CEST 2008Educational Studies in Mathematics2309-329Tacit Models and Infinity482001intuition knowledge tacit infinity limit mathematics learning mythesis The paper analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. The influences of the respective tacit models, being generally uncontrolled consciously, may lead to erroneous interpretations, to contradictions and paradoxes. The paper deals especially with the unconscious effect of the figural-pictorial models of statements related to the infinite sets of geometrical points (on a segment, a square, or a cube) related to the concepts of function and derivative and to the spatial interpretation of time and motion in Zeno's paradoxes.Fri May 30 05:46:47 CEST 2008Educational Studies in Mathematics13-40The Intuition of Infinity101979learning mathematics limit infinity mythesis intuition Fri May 30 05:46:37 CEST 2008Educational Studies in Mathematics4371-397Humanities students and epistemological obstacles related to limits181987infinity epistemological obstacles learning mythesis limits mathematics The article presents a report on four 45 minute sessions with a group of 17 year old humanities students. These sessions were the first of a series organised with the aim of exploring the possibilities of elaborating didactical situations that would help the students overcome epistemological obstacles related to limits. Students' attitudes pertinent to the development of the notion of limit, as well as changes of these attitudes, are described and analysed.Fri May 30 05:46:27 CEST 2008International Journal of Mathematical Education in Science and Technology6887-904Projecting rate of change in the context of motion onto the context of money342003mathematics learning mythesis motion change rateofhcange modelling rate money This paper reports a study designed to probe the abstraction ability that US Algebra I students can achieve in moving the concepts rate of change and accumulation between motion and money contexts. The Algebra I students used the technologies motion detectors and Interactive Banking software during a replacement unit focused on slope, ratio, and rate of change. Clinical interviews were conducted with four students at the end of the replacement unit testing their abstraction ability across the two contexts. Results revealed that some students do not have to completely understand the relationship between rate of change and accumulation within a single context in order to be able to understand and project the concepts separately into multiple contexts. Using multiple rate of change contexts allowed the learners the opportunity to see the 'like' in the contextually unlike situation, enabling them to project these concepts into novel situations. Few research studies have examined students' understanding of rate of change outside of the motion context, and therefore no studies, to the authors' knowledge, have explored students' abstraction ability between two different rate of change contexts. This paper reports a study designed to probe the abstraction ability that students can achieve in moving between the motion and money contexts.Fri May 30 05:46:11 CEST 2008Educational Studies in Mathematics11-19Students' Understanding of Algebraic Notation: 11-15331997mathematics mythesis learning representation algebra notation Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns. Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names. The principal explanation given in the literature has been a general link to levels of cognitive development. In this paper we present evidence for specific origins of misinterpretation that have been overlooked in the literature, and which may or may not be associated with cognitive level. These origins are: intuitive assumptions and pragmatic reasoning about a new notation, analogies with familiar symbol systems, interference from new learning in mathematics, and the effects of misleading teaching materials. Recognition of these origins of misunderstanding is necessary for improving the teaching of algebra.Fri May 30 05:46:02 CEST 2008Educational Studies in Mathematics1111-133Knowledge construction and diverging thinking in elementary & advanced mathematics381999advanced procepts concepts elementary object mathematics thinking process product mythesis This paper begins by considering the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum. We base our theoretical development on fundamental cognitive activities, namely, perception of the world, action upon it and reflection on both perception and action. We see an emphasis on one or more of these activities leading not only to different kinds of mathematics, but also to a spectrum of success and failure depending on the nature of the focus in the individual activity. For instance, geometry builds from the fundamental perception of figures and their shape, supported by action and reflection to move from practical measurement to theoretical deduction and euclidean proof. Arithmetic, on the other hand, initially focuses on the action of counting and later changes focus to the use of symbols for both the process of counting and the concept of number. The evidence that we draw together from a number of studies on children’s arithmetic shows a divergence in performance. The less successful seem to focus more on perceptions of their physical activities than on the flexible use of symbol as process and concept appropriate for a conceptual development in arithmetic and algebra. Advanced mathematical thinking introduces a new feature in which concept definitions are formulated and formal concepts are constructed by deduction. We show how students cope with the transition to advanced mathematical thinking in different ways leading once more to a diverging spectrum of success.