Implementing the Dynamic Geometry Approach in Classrooms

A research project funded by NSF examines the efficacy of an approach to high school geometry that utilizes Dynamic Geometry (DG) software and supporting instructional materials to supplement ordinary instructional practices. It compares effects of that intervention (the DG approach) with standard instruction that does not make use of computer investigation/drawing tools. This presentation reports a study conducted during the second year of the project.

An integrative framework (Olive and Makar, 2009) drawing from Constructivism, Instrumentation Theory and Semiotic Mediation was used to guide the study. Central to Instrumentation Theory is the process of Instrumental Genesis – How a tool changes from an artifact to an instrument in the hands of a user, and how both the tool and user are transformed in the process (Olive, 2011). The notion of semiotic mediation was introduced by Vygotsky (1978). According to this notion, cognitive functioning is closely linked to the use of signs and tools, and affected by it. Olive and Makar (2009) focus on the mathematical knowledge and practices that may result from access to digital technologies. They put forward a new tetrahedral model that integrates aspects of instrumentation theory and the notion of semiotic mediation. This new model illustrates how interactions among the didactical variables: student, teacher, task and technology (that form the vertices of the tetrahedron) create a space within which new mathematical knowledge and practices may emerge.

The research questions of the study include: 1) How do students taught in a DG oriented instructional environment perform in comparison with students in the control condition? 2) How does the DG intervention contribute to narrowing the achievement gap between students receiving free or reduced price lunch and other students? 3) How is students' learning related to the fidelity and intensity with which the teachers implement the DG approach in their classrooms? and 4) What characterizes the learning communities in the experimental and control classes?

As the first efficacy study on the DG approach at a moderately large scale in the nation, its answers to these research questions will significantly contribute to the knowledge base of how the dynamic geometry approach can really enhance our students' geometry learning and how we can effectively help our geometry teachers to develop and improve their technological pedagogical content knowledge.

The population from which the participants of this study were sampled are the 10^th grade geometry teachers and their students. Based on a power analysis, 76 geometry teachers were selected from those who applied to the project with support from their principals.

The research study follows a mixed methods, multi-site randomized cluster design. The 76 teachers selected were randomly assigned to the Experimental Group and the Control Group. For each of them, one geometry class that he/she taught participated in the study.

Student learning was assessed by a geometry test, a conjecturing-proving test, and a measure of student beliefs about the nature of geometry. Teachers in both treatment and control groups received relevant professional development. To determine how to capture the critical features of the DG approach, we designed measures of fidelity of implementation - a DG implementation questionnaire and a classroom observation instrument. To probe more deeply into the teachers' and students' thinking processes, and to gather evidence about the range and variability of participants' development of the most important abilities that the DG approach fosters, this study used in-depth interviews of selected students and teachers.

For all project-developed measures, the Cronbach's Alpha statistical values are within the acceptable ranges for reliability. Other psychometric properties were examined for some of the instruments and provide evidence supporting the validity of each.

The project team has completed its year 2 data collection. Two-level hierarchical linear modeling and constant comparison (Glaser & Strauss, 1967) methods are employed to analyze the quantitative and qualitative data respectively. Some initial data analysis has been conducted, but the more thorough and complete analysis of the collected data is still on going and will be continued during project year 3.

Guided by the theoretical framework described above, in the data collection and analysis processes, we pay close attention to the interactions among teachers (including the researchers), tasks (the conjecturing/proving tasks presented to the participating teachers or students), technology (GSP tools) and students. These interactions brought forth new mathematical thinking of the participating teachers and students through the processes of instrumental genesis and semiotic mediation.

Below is a summary of the main findings from the initial data analysis:

In the geometry post-test, the students in the Regular Geometry classes in the DG group significantly outperformed their counterparts in the control group even after adjusting for pre-test scores.

The teachers' fidelity and intensity of implementing the DG approach was positively related to student performance.

The participating teachers, students, and school districts have benefited from the various components of the project.  Teachers have been increasing their level of conjecturing about geometric relationships, proving those conjectures, and rethinking the way they present geometry concepts to their students.  Students have been able to develop and retain understanding of geometric relationships through their explorations and also apply what they have learned in new contexts.  Conjecturing and proving are re-emerging as areas of focus in the geometry curriculum of participating school districts.

The intended lessons produced by the DG group teachers were significantly better designed than those of the control group.  In particular, lesson plans for the DG group teachers had, in general, appropriate objectives and were designed to move students from initial conjecture, to investigation, to more thoughtful conjecture, to verification and proof.

(Due to limited space here, more findings will be presented at the session.)

The first part of the session will be a 15-minute overview of the study, and the second part will be engaging two 15-minute roundtable rotations. The speaker will lead the participants to concentrate discussion on issues such as research ideas/evidence regarding teachers' and/or students' conjecturing and proving facilitated by the use of DG tools and research ideas/evidence about providing teachers with effective technology-centered professional development.

References

Olive, J. (2011). Fractions on a dynamic number line. In the 35th Conference of the International Group for the Psychology of Mathematics Education (PME-35).

Olive, J. & Makar, K. (2009). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles & J-B. Lagrange (Eds.), Mathematics Education and Technology: Rethinking the Terrain, 133-178. The Netherlands: Springer.

Glaser, B. & Strauss, A. (1967). The discovery of grounded theorem: Strategies for qualitative research. New York: Aldine De Gruyter.

Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.