The role of representations in mathematical understanding: Virtual and physical manipulatives

The purpose of this qualitative case study was (a) to investigate the emerging mathematical understanding of 10th grade students about plane transformations and (b) to explore the role of manipulatives in this understanding. Data were gathered through a variety of sources. The pre-post-delayed tests, weekly task-based semi-structured interview forms, and participant classroom observation notes of the lessons were used to reach the research purpose. After the six-week long pilot study, the pre-test were administered in the classroom where the research was performed. Participants, four students who were selected purposively from the classroom, were interviewed by using the same questions used in the pre-test. During the next four weeks, lessons about translation, rotation, reflection and dilation transformations were conducted respectively. The lessons were enriched with virtual and physical manipulatives in addition to verbal, graphical and algebraic representations of mathematical concepts. Weekly task-based semi-structured interviews were carried out with participants after each transformation lesson. In order to investigate students' mathematical understanding and the role of manipulatives in understanding during the whole research process, the same set of questions used in the pre-test was readministered as a post-test following the instructional interventions, and as delayed test sixteen weeks after the post test. Data collected through participant observations were used for triangulation. The data were analyzed according to Pirie-Kieren and representation theory that determined the conceptual framework of the study. The components of these theories were followed during reporting the findings. The nature and the growth of students' understanding and the role of manipulatives were examined in the context of different mathematical understanding levels and representational systems. The results indicated that students worked within different levels of mathematical understanding while constructing meaning about transformations. The folding back movements helped them to strengthen their understandings, and the interventions in the context gave direction to their understandings. Students differed in comprehending and applying the multiple representations of mathematical concepts. Netting movements among these representations played a significant role in the students' growth of mathematical understanding. Virtual and physical manipulatives, as complementary learning tools, supported students' understandings. Both types of manipulatives helped students to comprehend and apply the different representations in different mathematical understanding levels. Key Words: Mathematical Understanding, Pirie-Kieren Theory, Representation Theory, Virtual Manipulative, Physical Manipulative, Concrete Material, Plane Transformations, Translation, Rotation, Reflection, Dilation