In Chapter One, I described how APOS Theory would be the guiding framework for this study. In this chapter, I will show how the APOS Theory framework will guide the methodology for my research. Specifically, I will focus on the context (3.1), the procedure (3.2), and the preliminary genetic decomposition (3.3).
3.1 The Context
Students taking an elementary geometry course during Spring 2013 were recruited to participate in this study. The course is a required course in the sequence of mathematics courses for the Early Childhood Education program. The scope of the course covers topics from Geometry that are normally taught in the upper elementary and middle school levels. The instructor provided content and pedagogical tools that the students can implement in their own classrooms when they are in the teaching profession.
There were two sections of the class that were invited to participate. The study aimed to ask all students from the course to participate yet a subgroup of twenty-six students out of sixty volunteered based on scheduling and availability. The participating students have had widely varying academic backgrounds, knowledge, and skills of mathematical literacy, and ability to express their mathematical and content pedagogical knowledge. All these components are important and may impact their future teaching of elementary mathematics, so they were good candidates for inclusion in the study.
After permission from the IRB, another researcher explained the project to the students and asked for voluntary participation in the project. Course instructors were not involved in the recruitment process. The benefit of the project to the individual student was presented. It was made
clear that participation would not affect any person's grade directly however it may affect their grade only in a positive way since some learning may occur during the interview which then might affect their class performance. An alternative assignment was available for non-participants that required similar time, effort, and grade value to the course. The content and purpose of the study was explained including the consent forms and the rights of students to stop participating at any time during the problem-solving session or the interview (See Appendix A). Students also received an email notification regarding the consent to use their written work in the class for the study.
3.2 Procedure
Eight interview sessions were conducted with one to four participants attending each session. Participants worked individually on a written problem-solving session for 30 minutes. This session was followed by an audio-taped and video recorded hour-long interview with questions related to the problems solved by the students. Students each took turns who would share first on a problem. The interviewer encouraged students to elaborate on their answers to help understand students’ conceptual understanding on the problem. A common protocol was used (see Appendix B) to maintain consistency between the interview sessions. Since I was the instructor of the course, I did not attend any of the sessions to avoid influencing students’ responses.
Additional data consisted of students' written work completed as part of the coursework, specifically problems from two quizzes and the final examination. In addition, course instruction was audio-recorded over the topic of special quadrilaterals to affirm that proper definitions were used and explained during class-time.
3.3 Genetic Decomposition for the Concept of Special Quadrilaterals
As noted in Chapter One, one of the first steps in APOS Theory research is to create an initial genetic decomposition. This genetic decomposition is to show how learners may proceed through the development of understanding the concept. After data has been collected and analyzed a revised genetic decomposition may be created based on the results of this study, if necessary.
The following step is the initial genetic decomposition for the comprehension of the hierarchical properties of special quadrilaterals:
1) The schema of Mathematical Definition is interiorized to conceptualize hierarchical definitions. At this step, students understand that mathematical definitions can be used to differentiate geometric shapes into their appropriate quadrilateral classifications.
2) Hierarchical definitions involve conditional statements. The schema of Logical Reasoning in encapsulated in treating the process of conditional reasoning for if-then statements and applies that to inclusive definitions. At this step students can conceptually reason that quadrilaterals with properties may belong to a higher classification of properties (e.g. if a quadrilateral is a rhombus then it is also a kite).
3) Visualization must be generalized from a prototypical concept image of special quadrilaterals to a more inclusive context. At this step, students use their concept image of a quadrilateral to visualize the properties that overlap with other quadrilaterals (e.g. a student may visualize a square and focus on the properties of four congruent sides to consider that a square has the same properties as a rhombus.)
4) Properties of special quadrilaterals emerge as the inclusive definitions are de-encapsulated back to the characteristics of the more general quadrilaterals. At this step, students take the generalized visualization from Step 3 and the Hierarchical Definitions of Inclusion from Steps 1 and 2 to identify the properties of the quadrilateral they are investigating.
5) The application of the properties of special quadrilaterals to solve problems comes from a de- encapsulation of the general properties of these quadrilaterals. Students can now apply the discovered properties within the context of problem solving.
This preliminary genetic decomposition of the concept of special quadrilaterals above will be investigated through the data collected and analyzed in this study. The following chapter breaks down each of the data that pertains to the main research questions (Section 1.2).