Digital-Concrete Materials: Revisiting Fröbel in Sketchpad Tasks

The present paper sets out to revisit Fröbel’s play theory, through the open-ended instructional materials, designed for pupils’ learning which he bequeathed to us. Many researchers have highlighted the advantages of digital or computer concrete materials including DGS manipulatives for teaching and learning. In terms of the present study, it is interesting to mention the introduction of Fröbel’s first Gift that I adapted in the DGS environment, designed to provide a play-based way of presenting/inquiring about geometric objects. The proposed DGS materials can be displayed, inquired about, and managed through properly set-up tasks, using linking visual active representations. The dynamic notions (e.g., dynamic point, segment, instrumental decoding, hybrid-dynamic objects, etc.), are taken as given and form the specific theoretical basis for the required processes. Dynamic interdependencies of tools in various sequential steps will be considered for the idea of building DGS Gifts, linked to the pupils’ level of conceptualization

Dynamic Euclidean Geometry: pseudo-Toulmin modeling transformations and instrumental learning trajectories

The present paper attempts to bridge the world of DGS technology with the world Euclid bequeathed to us in his "Elements". Competence in the DGS environment depends on the competence of the cognitive analysis as students seek to decode their ideas using the tools provided by the software. The dynamic notions (e.g., dynamic point, dynamic segment, instrumental decoding, hybrid-dynamic objects, active/ “alive” representations etc.), are taken as given and form the specific /particular theoretical basis for the constructive processes. Dynamic Euclidean constructions will be considered using pseudo-Toulmin’ diagrams. These considerations provide a theoretical basis for the idea that, in order to solve a mathematical construction problem in Dynamic Euclidean Geometry, we have to build up the interdependencies of tools in various sequential steps (based on theorems and definitions and the competence in using tools) which can be linked to the level of our conceptualization. The central idea is the following: Do the tools of Dynamic Euclidean Geometry determine a new kind of Geometry? Is Dynamic Euclidean Geometry a new kind of geometry? Does it have its own axiomatic system or its own undefined terms? In the paper, the notion of an instrumental learning path/trajectory is introduced as the interdependence/intra-dependence between dynamic tools, diagrams and mathematical objects during an instrumental decoding process. Keywords: Dynamic geometry, Euclid “Elements”, instrumental learning trajectories, Dynamic Euclidean Geometry DOI: 10.7176/JEP/12-9-09 Publication date:March 31st 202

Debunking Sustainability Excuses with Instrumentality and Expectancy Visualizations: A Physiological Perspective

This study advances the IS literature by investigating the effects of visualization on promoting sustainability knowledge and pro-environmental behaviors. Specifically, drawing on the visualization literature, we explain how the key visualization features, expectancy illustration, and interactivity affect individuals’ understanding of the impact of their behaviors on the environment, encouraging pro-environmental behaviors. Additionally, we draw on the pedagogy literature to explicate that the effects of visualization on learning outcomes and pro-environmental practices can be explained through individuals’ psychological responses in their course of interpreting the visualization. Collectively, this study presents our endeavor in understanding the roles of visualization in ecological discourse by integrating the visualization literature and sustainability research. Moreover, by unboxing individuals’ psychological processes in interpreting visualization, we offer a fresh perspective to understanding the promises and challenges of using visualization for knowledge acquisition

Externalizing the Latent Structure of Computer Games: Effect on game play, reasoning and implication for design

Computer games have initially and primarily been used for entertainment purposes. Recently, however, computer games have gained popularity in the educational and training arena. Epistemic computer games require players to think hard while entertaining them at the same time. Designing good epistemic computer games is complex and difficult. This thesis aims to further our understanding of how to design good epistemic computer games. Super Maze is a puzzle game that requires players to navigate through a maze picking up things on the way. At each junction, players can move either up, down, left or right. Four different versions of Super Maze were created. These versions differ from each other with respect to the representation of the maze and the way players interact with the maze to move through and finish it. The alternative representation to the traditional maze representation externalizes the internal structure of the maze as a tree diagram. An exploratory usability study was conducted to investigate how externalizing the internal structure of the game affects thinking and reasoning and if and how externalizing the internal structure of the game affects the gaming experienc

An alternative route to the Mandelbrot set: connecting idiosyncratic digital representations for undergraduates

Mathematics undergraduates often encounter a variety of digital representations which are more idiosyncratic than the ones they have experienced in school, and which often require the use of more sophisticated digital tools. This article analyses a collection of digital representations common to undergraduate dynamical systems courses, considers the significant ways in which the representations are interconnected and examines how they are similar or differ from those students are likely to have experienced at school. A key approach in the analysis is the identification of mathematical objects corresponding to manipulative elements of the representations that are most essential for typical exploratory tasks. As a result of the analysis, augmentations of familiar representations are proposed that address the gap between local and global perspectives, and a case is made for greater use of isoperiodic diagrams. In particular, these diagrams are proposed as a new stimulus for students to generate their own explorations of fundamental properties of the Mandelbrot set. The ideas presented are expected to inform the practice of teachers seeking to develop visually rich exploratory tasks which pre-empt some of the issues of instrumentation that mathematics undergraduates experience when introduced to new digital tools. The overarching aim is to address significant questions concerning visualisation and inscriptions in mathematics education

Application and Evaluation of Interaction Techniques to 3D Mathematical Structures

Mathematical cognitive tools have the ability to extend our cognitive capabilities and bring reason and understanding to complex structures and concepts. Designing mathematical cognitive tools is complicated and there is a lack of investigation conducted in this area. A mathematical cognitive tool that incorporates many forms of interaction is introduced in this thesis. It is intended to support and enhance the knowledge gained from interacting with 3 dimensional structures. A comparison study is done to investigate how these interactive features support reasoning. We compare it to a Physical Manipulative and another mathematical cognitive tool that does not incorporate multiple forms of interaction. Our findings suggest that the tool incorporating multiple forms of interaction supports learners with their tasks more successfiιlly than the minimally interactive tool and even more than the actual physical manipulative. The study also provided insight into when to use and when not to use specific interaction techniques to support different tasks

Hybrid-dynamic objects: DGS environments and conceptual transformations

A few theoretical perspectives have been taken under consideration the meaning of an object as the result of a process in mathematical thinking. Building on their work, I shall investigate the meaning of ‘object’ in a dynamic geometry environment. Using the recently introduced notions of dynamic-hybrid objects, diagrams and sections which complement our understanding of geometric processes and concepts as we perform actions in the dynamic software, I shall explain what could be considered to be a ‘procept-in-action’. Finally, a few examples will be analyzed through the lenses of hybrid and dynamic objects in terms of how I designed them. A few snapshots of the research process will be presented to reinforce the theoretical considerations. My aim is to contribute to the field of the Didactics of Mathematics using ICT in relation to students’ cognitive developmen

From Vecten’s Theorem to Gamow’s Problem: Building an Empirical Classification Model for Sequential Instructional Problems in Geometry

In the current study, I will be presenting a literature review regarding the importance of students building a problem’s representation and the role modeling a real-world problem plays in students’ progressive mathematization. I shall introduce five types of geometrical problems applying the meaning of Linking Visual Active Representations (LVARs). Concrete examples will be presented in the next sections (i.e., Euclid’s proof of the Pythagorean Theorem, Vecten’s theorem, Gamow’s problem). I shall also introduce the meanings of hybrid object and diagram, as well as the meaning of dynamic section in a dynamic geometry environment, through examples. To summarize, I created an empirical classification model of sequential instructional problems in geometry. Its contribution to our knowledge in the area of the didactics of mathematics lies in the fact that this sequence of problems is regarded as a process whereby students develop a sequentially deeper understanding and increasingly more coherent reasoning that raises their van Hiele level. Keywords: dynamic section, hybrid object, Euclid “Elements”, Pythagorean Theorem, Vecten’s Theorem, Gamow’s problem, problem-solving. DOI: 10.7176/JEP/10-5-0

A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why

In the current study the theoretical notion of instrumental learning path or trajectory is analyzed through examples based on a research synthesis. I point out the role of instrumental decoding in a static or dynamic environment, and how the competence of the participants (students –researcher/ teacher) can influence the holistic result of the learning process by creating interdependencies/intra-dependencies during the construction of instrumental learning trajectories. Instrumental trajectories are not just construction instructions, or a set of information which provides the properties of the figure as the figure is constructed. Instrumental trajectories can show us the interdependencies/intra-dependencies that exist or can be created between different tools. Dynamic Geometry allows for the creation of interdependencies and intra-dependencies between mathematical objects, diagrams and tools. In the sections that follow, I shall present three examples of instrumental learning trajectories, in which the interdependencies among the tools and meanings are analyzed. My aim is to combine and synthesize different primary qualitative research studies and make their results more generalizable. Keywords: instrumental decoding, interdependencies/intra-dependencies, instrumental learning trajectories DOI: 10.7176/IKM/11-3-02 Publication date: April 30th 2021

Player–Game Interaction and Cognitive Gameplay: A Taxonomic Framework for the Core Mechanic of Videogames

Cognitive gameplay—the cognitive dimension of a player’s experience—emerges from the interaction between a player and a game. While its design requires careful consideration, cognitive gameplay can be designed only indirectly via the design of game components. In this paper, we focus on one such component—the core mechanic—which binds a player and game together through the performance of essential interactions. Little extant research has been aimed at developing frameworks to support the design of interactions within the core mechanic with cognitive gameplay in mind. We present a taxonomic framework named INFORM (Interaction desigN For the cORe Mechanic) to address this gap. INFORM employs twelve micro-level elements that collectively give structure to any individual interaction within the core mechanic. We characterize these elements in the context of videogames, and discuss their potential influences on cognitive gameplay. We situate these elements within a broader framework that synthesizes concepts relevant to game design. INFORM is a descriptive framework, and provides a common vocabulary and a set of concepts that designers can use to think systematically about issues related to micro-level interaction design and cognitive gameplay