Paper 2: Design Considerations for Visually-Aided Discussion Prompts: Emphasizing Mathematical Reasoning in Teacher Education

The availability and familiarity of online discussion tools create new instructional options that teacher educators can use to foster prospective teachers’ understanding of mathematics. In particular, online discussion blogs provide an avenue through which teacher educators can press prospective teachers to explore mathematical concepts and share their mathematical reasoning with peers. Furthermore, by incorporating visual stimulations as a design component of these discussion blogs, prospective teachers can make sense of and respond to others’ ideas about mathematical concepts with greater clarity. This paper shares preliminary findings of a research study that examined the extent to which the design of a series of visually-aided online discussion prompts facilitated prospective elementary teachers’ (PSTs) use of mathematical reasoning in a geometry and measurement course. Results suggest that (a) the wording of discussion prompts influences the nature of mathematical justifications that PSTs focus on in their responses and (b) social norms for communicating in online forums may influence the ways in which PSTs interact with peers in an online discussion blog about their mathematical reasoning. GAMTE Proceedings 2014 16 Design Considerations for Visually-Aided Discussion Prompts: Emphasizing Mathematical Reasoning in Teacher Education The National Council of Teachers of Mathematics ([NCTM], 2000; 2014) identifies technology as an essential resource for teaching and learning mathematics. Similarly, the Conference Board of the Mathematical Sciences (2012) promotes technology as a strategic tool that should be used in professional development and teacher preparation programs. Both organizations view technology as a tool for attaining particular mathematical goals or expanding ways to engage in mathematics learning. Among the many educational technologies available, online discussion blogs continue to be an accessible option for many post-secondary faculty through online Learning Management Systems platforms (e.g., Blackboard, D2L, and Moodle), which are exhibiting an annualized market penetration rate of 25 percent (Marketsandmarkets.com, 2013). As a mathematics professor, Offenholley (2006) used online “discussion[s] to: (a) encourage higher-order thinking, (b) monitor students’ progress, and (c) encourage peer collaboration” (p. 8)—all skills that are important elements of teacher education programs. This paper shares preliminary findings of a research study that examined the extent to which the design of a series of visually-aided online discussion prompts facilitated prospective elementary teachers’ (PSTs) use of mathematical reasoning in a geometry and measurement course. Theoretical Perspective The constructivist approaches to formative assessment espoused by both Popham (2011) and Wiliam (2011) informed our views on designing online discussion blogs as an evidencebased exploration of mathematical concepts. Through the lens of this theoretical perspective, we examined digital discussions as a venue for sharing mathematical evidence that was “elicited, GAMTE Proceedings 2014 17 interpreted, and used by both [instructors] and learners” (p. 43). This perspective aligns with this study’s purpose to examine the ways that an online discussion prompt – intended to provoke reflection and critique as it pertains to classifying quadrilaterals based on particular characteristics – fostered PSTs’ usage of mathematical evidence to support their thinking in response to the prompt. Context and Methods The research team investigated 11 PSTs’ experiences using an online discussion blog (DB) in a geometry and measurement course, the third in a series of required mathematics content courses for this population. The eight-week summer course met twice a week for two hours and thirty minutes and was partitioned into two content sections. The first section was devoted to geometry with an overarching mathematical goal for PSTs to learn characteristics of various shapes and then classify and group those shapes based on particular shared characteristics. The focus of the second section was on geometric measurement with an emphasis on understanding and distinguishing between measurement characteristics of perimeter and area of various shapes. The researchers designed a DB-Prompt (the second of four DBPrompts) instructing PSTs to provide mathematical evidence to support their reasoning about classifying quadrilaterals based on possessing the characteristics of (1) one pair of parallel sides and/or (2) one line of symmetry. Leading up to the second DB assignment, PSTs worked on several tasks (see Appendix) related to identifying different attributes of quadrilaterals and common characteristics across quadrilaterals. Initially, during an in-class activity, PSTs cut out paper shapes for different quadrilaterals and explored their geometric properties using right-angle testers, paper-folding, and tracing paper. PSTs noticed that certain quadrilaterals shared some common characteristics, GAMTE Proceedings 2014 18 while others possessed characteristics that were unique. This difference led to a discussion of finding a way to show relationships between trapezoids 1 , parallelograms, and rhombuses. Unexpectedly, all three groups of students chose to use the same incorrect Venn diagram with three overlapping circles to show relationships between these quadrilaterals for an exit question. The PSTs’ exit question responses indicated a need for the instructor to discuss what different areas of a Venn diagram represent and how to organize the shapes into those circles in meaningful ways. Thus, in the subsequent class meeting, PSTs created the quadrilaterals using AngLegs (ETA hand2mind) and worked in small groups to determine how to arrange the shapes within physical circles to represent their Venn diagram placement. The second DB-Prompt followed these discussions and asked PSTs to consider four different Venn diagrams and decide which one would be the most appropriate for describing which of eight quadrilaterals had the characteristics of a pair of parallel sides or a line of symmetry (see Appendix). Data from this study took on one of two forms: (1) PSTs’ postings in response to the discussion blog prompt (DB-Entries) and (2) PSTs’ responses to two of their peers’ postings (DB-Responses). The pedagogical decision to include DB-Responses served as a feedback mechanism, a core element of formative assessment protocol (Heritage, 2010). Data analysis involved a two-cycle grounded-theory technique that began with open-coding the data for emergent themes and segued into a second-cycle of pattern coding that further parsed the data into one of two categories, strong or weak, reflecting PSTs’ usage of mathematical evidence (Saldana, 2009). Both researchers separately analyzed the data using this dichotomy and identified most entries with the same code, that being weak – lacking mathematical reasoning or supporting evidence. Any differences were reconciled through consensus (Harry, Sturgis, and 1 Note that an inclusive definition of trapezoids was used for this exploration. Thus, a trapezoid was considered to be a quadrilateral with at least one pair of parallel sides. GAMTE Proceedings 2014 19