"Mathematics teachers at all levels are expected ever more frequently to integrate use of emergent technologies (e.g., calculators, software, online tools, device applets) in mathematics teaching and learning. Too often technology is used for technology's sake, rather than in intentional ways to support mathematical reasoning, sense-making, and understanding. Teacher decision-making is necessarily complex as they consider the complex interactions of mathematics, technology, and learning, especially attending to affordances and limitations of technologies in relation to mathematical goals, rigor, representation, and interaction. It is critical to consider ways in which teachers can be supported in developing decision-making in locating, designing, adapting, and integrating technologies for effective mathematics teaching and learning. In this study, I explore teachers' conceptions of the nature of mathematics, and mathematics teaching and learning, teachers' noticing of characteristics of online tools and resources, and potential relationships between their conceptions and noticings. I also describe implications for mathematics education and directions for future research. This study is a qualitative analysis of cases, with six focus teachers: Breanna, Melinda, Jana, Nicole, Heidi, and Alison. Breanna, Melinda, and Jana were grouped in a lower elementary grades team in the course, Nicole and Heidi in an upper elementary team, and Melinda in a middle grades team. Teachers varied in length of teaching experience: Breanna (2 years), Melinda (3), Jana (16), Nicole (5), Heidi (6), and Alison (4). Nicole, Heidi, and Alison had a mathematics credential, while Breanna, Melinda, and Jana had not. Regarding the three conceptions of the nature of mathematics described by Thompson (1992) and Ernest (1989b), findings of the study included the necessity of expanding meanings of aspects of conceptions and focusing expanded aspects to include school mathematics. Meanings of aspects of mathematical conceptions should reflect teachers' meanings of those aspects as well as reflecting researchers' meanings. All teachers showed some evidence of an Instrumentalist conception, but Breanna and Melinda most strongly showed alignment with an Instrumentalist conception. Jana, Nicole, Alison, and Heidi showed close alignment with a Platonist conception. Alison additionally indicated alignment with a Problem-Driven view. Teachers showed rich understanding of aspects of the nature of mathematics that are excluded from Ernest's framework. In considering teachers' noticing of mathematical aspects of online tools, patterns across levels of sophistication and alignment with conceptions emerged. Noticing patterns emerged that supported past research on levels of sophistication in noticing structures of mathematical tasks and aspects of mathematics curricula. For example, teachers with fewer years of experience attended to general goals more easily than specific goals as described by M. G. Sherin and Drake (2009). Teacher with fewer years of experience also attended to non-mathematical aspects (e.g., visuals, reasons for engagement) of online tools more frequently than mathematical aspects (B. Sherin & Star, 2011; M. G. Sherin & van Es, 2009). All teachers expressed expectations for more online tools to provide feedback on correctness than to provide opportunities for interaction with mathematical consequences. Relationships between teachers' conceptions of the nature of mathematics and their noticing of aspects of online tools emerged through teachers' focus on three aspects of online tools: learning goals, strategies for learning, and knowledge structures. For each, language used by teachers was considered in tandem with descriptions of relationships between Instrumentalist and Platonist conceptions of the nature of mathematics and approaches to teaching (Thompson, 1992; Ernest, 1989b). Breanna's noticing showed close alignment with an Instrumentalist conceptions, while Nicole's and Alison's showed close alignment with a Platonist conception. Jana, Melinda, and Heidi showed some variation in their alignments and showed some balance between alignment with Instrumentalist and with Platonist conceptions."--Pages ii-iii.
Read
