Ulrich Kortenkamp

Artifact-Centric Activity Theory—A Framework for the Analysis of the Design and Use of Virtual Manipulatives

It is a challenge to analyze the design and the use of Virtual Manipulatives due to their high complexity. As it is possible to create entirely new virtual worlds that can host objects that behave differently than any real objects, allowing for new and unprecedented actions in learning processes, we are in need of tools that enable us to focus on those aspects that are important for our analyses. In this chapter we show how ACAT, Artifact-Centric Activity Theory , can be used to analyze the design and use of a virtual manipulative place value chart.

Development of a Flexible Understanding of Place Value

In this chapter, we highlight the importance not only of an understanding of place value, but the importance of a flexible understanding. We describe the principles of our decimal place value system and the development processes of children. Embedded in artefact-centric activity theory, we present an education-oriented design of a virtual place value chart and its potential to support this development and understanding. We also present results of a qualitative study with second graders as well as results of a quantitative study with third graders that can guide further research in that area.

MAKING THE MOVE: THE NEXT VERSION OF CINDERELLA

Cinderella is a software package for interactive or dynamic geometry. Its first version was published in 1999 and was the first geometry software to be based on the theory of complex tracing3, thus avoiding mathematical inconsistencies and unmotivated discontinuities (modulo numerical errors). At the ICMS 2002 we will introduce the next major version of Cinderella and highlight its new features and new concepts.

Complexity issues in dynamic geometry

This article deals with the intrinsic complexity of tracing and reachability questions in the context of elementary geometric constructions. We consider constructions from elementary geometry as dynamic entities: while the free points of a construction perform a continuous motion the dependent points should move consistently and continuously. We focus on constructions that are entirely built up from join, meet and angular bisector operations. In particular the last operation introduces an intrinsic ambiguity: Two intersecting lines have two different angular bisectors. Under the requirement of continuity it is a fundamental algorithmic problem to resolve this ambiguity properly during motions of the free elements. After formalizing this intuitive setup we prove the following main results of this article: • It is NP-hard to trace the dependent elements in such a construction. • It is NP-hard to decide whether two instances of the same construction lie in the same component of the configuration space. • The last problem becomes PSPACE-hard if we allow one additional sidedness test which has to be satisfied during the entire motion. On the one hand the results have practical relevance for the implementations of Dynamic Geometry Systems. On the other hand the results can be interpreted as statements concerning the intrinsic complexity of analytic continuation.

Number Concepts—Processes of Internalization and Externalization by the Use of Multi-Touch Technology

In this chapter, we discuss the use of ICT, in particular of multi-touch technology, to survey and to enhance the development of children’s concepts of numbers. We will have a special focus on the processes of internalization and externalization that constitute the construction of meaning. The children externalize their concepts of numbers through touching a multi-touch screen with their fingers and thus producing tokens. The visualization of the multi-touch table contemporaneously gives feedback to the children that—in the context of the social process—leads to an internalization. Hence, the design of the user interface plays an important role. Also, the instructions given by the (nursery) teacher as well as the partners have influence on the child’s internalization and externalization. As a basis for the design and analysis of this research project, we refer to artifact-centric activity theory (ACAT). This theory helps analyze the complexity of the whole learning environment.

The importance of understanding: Model space moderates goal specificity effects

The three-space theory of problem solving predicts that the quality of a learners model and the goal specificity of a task interact on knowledge acquisition. In Experiment 1 participants used a computer simulation of a lever system to learn about torques. They either had to test hypotheses (nonspecific goal), or to produce given values for variables (specific goal). In the good- but not in the poor-model condition they saw torque depicted as an area. Results revealed the predicted interaction. A nonspecific goal only resulted in better learning when a good model of torques was provided. In Experiment 2 participants learned to manipulate the inputs of a system to control its outputs. A nonspecific goal to explore the system helped performance when compared to a specific goal to reach certain values when participants were given a good model, but not when given a poor model that suggested the wrong hypothesis space. Our findings support the three-space theory. They emphasize the importance of understanding for problem solving and stress the need to study underlying processes.

Aspects that Affect Whole Number Learning: Cultural Artefacts and Mathematical Tasks

The core of this chapter is the notion of artefact, starting from the discussion of the meaning of the word in the literature and offering a gallery of cultural artefacts from the participants’ reports and the literature. The idea of artefacts is considered in a broad sense, to include also language and texts. The use of cultural artefacts as teaching aids is addressed. A special section is devoted to the artefacts (teaching aids) from technologies (including virtual manipulatives). The issue of tasks is simply skimmed, but it is not possible to discuss about artefacts without considering the way of using artefacts with suitable tasks. Some examples of tasks are reported to elaborate about aspects that may foster learning whole number arithmetic (WNA). Artefacts and tasks appear as an inseparable pair, to be considered within a cultural and institutional context. Some future challenges are outlined concerning the issue of teacher education, in order to cope with this complex map.

Geometry Education, Including the Use of New Technologies: A Survey of Recent Research

This is a summary report of the ICME-13 survey on the theme of recent research in geometry education. Based on an analysis of the research literature published since 2008, the survey focuses on seven major research threads. These are the use of theories in geometry education research, the nature of visuospatial reasoning, the role of diagrams and gestures, the role of digital technologies, the teaching and learning of definitions, the teaching and learning of the proving process, and moves beyond traditional Euclidean approaches. Within each theme, there is commentary on promising future directions for research.

Mathematikfortbildungen mit E-Learning gestalten

Das im Beitrag dargestellte DZLM-Kurskonzept »Mathematikfortbildungen mit E-Learning gestalten« qualifiziert Fortbildnerund Fortbildnerinnen in zweierlei Hinsicht. Sie lernen zum einen, »klassische« Mathematikfortbildungen mit E-Learning-Elementen anzureichern und damit BlendedLearning-Kurse zu gestalten. Zum anderen erfahren sie, wie sie die zugrundeliegenden Prinzipien in eigenen Fortbildungen vermitteln können. Eingeleitet wird der Beitrag durch Informationen zum DZLM OnlineAngebot.

Paving the Alexanderplatz Efficiently with a Quasi-Periodic Tiling

In this paper we describe a mathematical approach to create an organic, yet efficient to create tiling for a large non-rectangular space, the Alexanderplatz in Berlin. We show how to use the refinement algorithm for Penrose tilings in order to create a polygonal tiling that consist of four different tiles and is quasi-periodic. We also derive, based on the refinement algorithm, bounds for the percentage of tiles of each type needed. Another question that is addressed is whether it is possible to describe the calculated tiling in a linear form. Otherwise, it wouldn’t be possible to use the tiling, as there must be a concise description suitable for the workers who lay out the concrete tiles.