An investigation into students' understandings of linear relationships when using dynamic mathematical software as an exploration tool

Abstract

Numerous studies have provided substantial evidence of the benefits of technology when learning mathematical concepts. Despite this evidence, various factors inhibit successful implementation and navigation of technology. As a result, when students require technology to further develop their understanding of concepts, they may fall short because they lack the consolidated skills enabling them to competently advance. This study investigated students' understandings of Linear Relationships concepts using the dynamic software GeoGebra. The purpose was to identify developmental hurdles and issues that could be addressed by teachers in their teaching sequence. The design involved a tightly focused investigation of the understanding of Linear Relationships concepts through analysing the students' responses during the unit. Empirical evidence is provided to explain the difficulties faced by students when using technology to support their understanding of Linear Relationships. This evidence has theoretical and practical implications for the instruction of the Linear Relationships unit and the incorporation of GeoGebra as a pedagogical tool. The theoretical base for this study was the van Hiele Teaching Phases and the SOLO model. The five Teaching Phases that form part of the van Hiele Theory presented a framework for teaching and learning in which to sequence Linear Relationships activities that facilitate students' cognitive development for the transition between van Hiele levels. While the Teaching Phases have been widely studied, particularly with respect to Geometry, this study extends research by utilising activities for the teaching sequence aligned with the Teaching Phases, and based on the Linear Relationships unit for the Australian Curriculum Stage 5.3 (approximately 14–16 years old) incorporating GeoGebra as a technological tool to support understanding. To provide a deeper insight with which to view students' understandings of Linear Relationships concepts, the SOLO model was used as an analytical tool. Qualitative data was collected during the investigation. This comprised three Google Form Tests: a Pre-test, an End of Topic test, and a Delayed Post-test, to provide the main set of data, along with video and audio footage, photographs and workbook samples. The tests were completed by 26 students from a Year 9 Stage 5.3 class (approximately 14–16 years old), all from one secondary school. A central finding of this study is the identification of the developmental pathway for Linear Relationships concepts. This pathway characterises student growth in understanding and recognised two cycles of responses within the concrete symbolic SOLO mode. Whilst first cycle responses indicated possible levels of support from the ikonic mode, students were mainly operating within the concrete symbolic mode. This study also identified student difficulties associated with dynamic technology, such as GeoGebra, when used to support the understanding of Linear Relationships concepts. The research highlights the difficulties students encounter when attempting to utilise technology to develop conceptual understanding with mathematical topics such as Linear Relationships. Hence, the characterisation of a developmental pathway for conceptual understanding of Linear Relationships provides a valuable tool for teaching. In addition, this study highlights the use of the SOLO model as an interpretive tool for research in Mathematics education and the van Hiele Teaching Phases as a pedagogical tool for developing comprehensive learning experiences.