Exploring the Relationship Between Orpda and Teachers' Conceptual Understanding of Place Value

The purpose of this case study was to understand whether or not the use of an invented number system, called Orpda, helped teachers develop a deeper understanding of place value in hopes that this will translate into their own teaching of place value concepts. Thirteen teachers enrolled in a graduate mathematics education course served as the participants for this study. Data were collected from teachers’ reflections on various activities related to Orpda, pre- and post-Orpda concept maps teachers created, online discussions between the teachers, teacher demographic sheets, and an interview with the instructor of the course.

Analysis of the teachers' reflections revealed that Orpda increased teachers’ attention to three critical components necessary for developing a conceptual understanding of place value, namely unitizing, regrouping, and recognizing the meaning of different place values within a multi-digit number. In addition, Orpda encouraged teachers to reflect on their own teaching of place value. Comparing the structures of the teachers' pre-Orpda and post-Orpda concept maps showed changes in some cases but did not reveal clear patterns. Analysis of the categories teachers included in pre- and post-Orpda maps revealed that teachers were moving from a procedural to a more conceptual view of place value, as did the analysis of squared adjacency matrices created from each teacher's pre- and post-Orpda concept maps.

Four conclusions can be drawn from this study: (a) Orpda increased teachers' attention to the importance of unitizing in place value, (b) Orpda encouraged teachers to reflect deeply on their thinking, (c) Concept maps show promise for revealing and documenting changes in conceptual understanding, and (d) Orpda increased teachers' attention to the importance of patterns in understanding place value. Further research is needed using Orpda with different groups and numbers of teachers, and in different settings, e.g., longer full semesters and teacher professional development meetings. Research exploring the use of follow-up interviews to accompany concept maps and enhance the assessment of conceptual understanding is also recommended. This study indicates two recommendations for practice in teacher education, the importance of a classroom environment that supports reflection, and the careful choosing of activities to provide appropriate challenge.