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

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

DGS Cui-Rods: Reinventing Mathematical Concepts

Curriculum materials are instructional materials produced to be used for teaching and learning. Successful teaching requires new materials and innovative approaches. In the present study, I shall present DGS Cui-rods an instructional material for effective teaching and learning of mathematical concepts. I created them in the Geometer’s Sketchpad environment. Dynamic geometry environments allow students to discover and reinvent a wide range of mathematical concepts and their relationships. During problem-solving situations, students are able to construct meanings. They are led to create their personal representations of mathematical concepts and transform them. The design of activities in the learning environment as a part of the instruction thus has a crucial role to play. In the next sections, I shall describe how learning through DGS Cui-Ros affects students’ cognitive structure’s transformations and consequently their cognitive growth

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

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

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

A Linking Visual Active Representation DHLP for student’s cognitive development

In the next sections I shall describe a ‘dynamic’ hypothetical learning path (DHLP) for the learning of the concept of parallelogram in geometry, which helped the students of the experimental team to raise their van Hiele levels. The design of the DHLP started with a ‘thought experiment’ with which I imagined a learning path for the understanding of the parallelograms, trying simultaneously to predict the reactions of students. I shall also describe the aims I had posed, as well as the points of the research process in which I changed the route of the path in order to introduce a new tool, due to students’ cognitive conflicts or other obstacles which occurred. Using examples, I will describe the research process and (a) the design and redesign of the DHLP through linking visual active representations and (b) the students’ competence in the mental or verbal decoding of these representations and in using the tools that affect their development of the thinking levels. Finally, I shall extend the conceptual framework of Linking Visual Active Representations to introduce what arises from the research process

Students Learning Progression through Instrumental Decoding of Mathematical Ideas

The current study aims to focus on mathematical tasks for students’ mathematical literacy and problem solving literacy. Excerpts are presented from dynamic hypothetical learning paths [DHLP]s and students’ learning progression. The excerpts center around activities aimed to develop the students’ geometrical thinking through the development of their ability to solve real-world problems. The students cooperated in class or worked individually to represent the images using their static or dynamic means and tools (e.g. compass and ruler, a computing environment, interactive boards, dynamic geometry software). My further aim was the students to utilize transformation processes for representations by instrumentally decoding their ideas on static and dynamic objects. An important role for the students’ cognitive development was the design of propositions and theorems (e.g the Pythagorean Theorem), through Linking Visual Active Representations (LVAR). Especially for the latter option an essential role has played the dynamic geometry software, Geometer’s Sketchpad. Furthermore, the paper provides examples that contain rich mathematical material; therefore, student’s mathematical modeling through instrumental decoding of mathematical ideas is the means of reinforcing students’ conceptual knowledge

An ‘Alive’ DGS Tool for Students’ Cognitive Development

Abstract— A basic goal of the current study, which is an excerpt from a larger study, is to analyse students’ interactions in the context of their working on transformations of tools, and specifically of custom tools in a microworld, the Geometer’s Sketchpad. Custom tools are encapsulated objects created in a DGS environment. The construction of a custom tool and its subsequent implementation in a pair of students are the focus of this study. Custom tools can serve as structural units of knowledge, as conceptual objects and hence as ‘schemes’. Moreover, they can become an ‘alive’ active tool for students’ cognitive development. The paper will include the following parts: (a) how students learn in a constructivist framework; (b) a description of the van Hiele model, and especially the meanings of ‘symbol and signal character’; (c) how a DGS environment functions as an ‘alive’ microworld; (d) the role of artifacts-[custom] tools as instruments-[custom] tools; (e) the research methodology of the current study (f) a detailed description of the experimental process (g) discussion and conclusion