Research
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.
A Linking Visual Active Representation DHLP for Students 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.
