# Orientations toward Mathematical Processes of Prospective Secondary Mathematics Teachers as Related to Work with Tasks

Open Access

- Graduate Program:
- Curriculum and Instruction
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- October 10, 2016
- Committee Members:
- Rose Mary Zbiek, Dissertation Advisor
- Rose Mary Zbiek, Committee Chair
- Edith Frances Arbaugh, Committee Member
- Mary Kathleen Heid, Committee Member
- Andrew Leonard Belmonte, Outside Member
- Gwendolyn Monica Lloyd, Committee Member

- Keywords:
- mathematical tasks
- mathematical processes
- discovery
- prospective teachers
- teacher beliefs

- Abstract:
- Mathematics can be conceptualized in different ways. Policy documents such as the National Council of Teachers of Mathematics (NCTM) (2000) and the Common Core State Standards Initiative (CCSSI) (2010), classify mathematics in terms of mathematical content (e.g., quadratic functions, Pythagorean theorem) and mathematical activity in the form of mathematical processes or mathematical practices (e.g., justifying, representing). A situated cognition perspective (Lave & Wenger, 1991) positing that how one learns mathematics and what one learns about mathematics are dependent on the context in which one engages in mathematical activity (Boaler, 2002; J. S. Brown, Collins, & Duguid, 1989) is used to think about learning mathematics and learning to teach mathematics. Mathematical activity in which learning is situated can be framed using mathematical processes. Four targeted processes of defining, generalizing, justifying, and representing from a processes and actions framework (Heid et al., 2015; Zbiek & Heid, 2012; Zbiek et al., 2014) are used to study opportunities for mathematical activity in classrooms. Students’ experiences with mathematics in the classroom, particularly with the mathematical tasks in which they engage, shape their views about mathematics. Teachers can help students form positive process conceptualizations of mathematics by selecting, modifying, and sequencing mathematical tasks (collectively referred to as task work) that provide opportunities for students to engage in mathematical processes. Yet, not all teachers provide such opportunities. Of particular interest are prospective mathematics teachers (PMTs), who might have personal experiences engaging in mathematical processes yet struggle to include such activity in their classrooms. The construct of orientations toward mathematical processes (OMP) is used to understand the nature of PMTs work with tasks. A mathematics teacher’s orientation toward mathematical processes (OMP) in school mathematics is a set of conceptions about mathematics, its teaching and learning related to mathematical processes, and engagement in mathematical processes that influences how a teacher provides opportunities for students to engage in mathematical processes. A situated view of teacher education (Borko et al., 2000) serves as a lens for investigating PMTs’ OMPs and how they might be related to PMTs’ task work. Specific research questions, briefly stated, are: What are PMTs’ OMPs, and how are PMTs’ OMPs related to how PMTs’ select, modify and sequence mathematical tasks? Five secondary mathematics prospective teachers who had completed at least some of their student teaching experiences participated in the study. Data were collected using three qualitative interviews focused on components of OMP and the teaching activities of selecting, modifying and sequencing mathematical tasks to gather data for each of the participants. The definition of OMP with the processes and actions framework (Heid et al., 2015; Zbiek & Heid, 2012; Zbiek et al., 2014) guided the data analysis, which involved the creation of OMP profiles. An OMP profile is a description of an individual participants’ OMP organized around a theme and paralleling research questions about the nature of OMP and its relationship to task work. An organization of profiles into types based on pedagogical and mathematical differences structures findings to the first research question. The construct of task template as a means of simplifying task work forms the basis for findings to the second research question. The notion of discovery figured prominently in participants’ OMPs and explains observations about PMTs’ OMPs. Insights into PMTs’ OMPs and their task work has potential implications for undergraduate teacher education and secondary mathematics curricula.