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

Stavroula Patsiomitou

doi:10.7176/ikm/11-3-02

Abstract

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 30 th 2021

Highlights

Simon (1995, p. 136) introduced the idea of Mathematics Teaching Cycle and created a diagram in order to represent the way that a learning trajectory is an ongoing modification of three components: “(a) the learning goal that defines the direction, (b) the learning activities and (c) the hypothetical learning process”
The analysis of the didactic actions led me in 2014, to the development of the Didactic cycle of Mathematics (e.g., Patsiomitou, 2014, p. 35) adapting the diagram created by Simon (1995, p.136) so that I took into account the use of technology in the mathematics didactic cycle, the modelling of activities with linking visually active representations (LVAR) within the software, and the evaluation of student levels in line with van Hiele’s theory
Communication between teacher and students is achieved through mathematical discussions in sequential actions: the implementation of the activities, effective teaching and research into students, exploring what they know, which provides the teacher with the feedback they need to adapt the activities

Summary

Introduction

For example: Figures 2 and 3 illustrate two different instrumental paths of constructing the original /initial right triangles, and include transformations which will subsequently be used to construct the custom tools. Students of the experimental team had only worked on the Sketchpad v4 using the pre-constructed custom www.iiste.org tools I had created

Results

Conclusion