It has been argued that teachers do not always teach in the ways their teacher education programs promoted. One cause of this problem has to do with teachers' conceptions about mathematics and ability being incompatible with the visions of mathematics that teacher educators promote. For example, teacher educators may emphasize the need for conceptual understanding but a teacher who equates understanding with being correct and fast will enact this goal differently from someone who expects students to be able to explain and communicate their understanding. The way a teacher understands what it means to do mathematics and be good at mathematics will influence their future teaching. Furthermore, the messages teachers send to their students about who they are as mathematical learners are full of messages about what it means to do and be good at mathematics. Additionally, these messages can have a long-term impact on how those students view themselves and the decisions they make for their future. This study uses qualitative methods to better understand pre-service teacher (PST) talk, how their talk relates, and how their talk changes over time. I describe PST talk as it relates to mathematics (M), ability (A), and themselves as mathematical learners (P) (the three components of the MAP framework). This study took place with the PSTs in two sections of a mathematical content course for elementary school teachers taught by the researcher. First, using grounded theory, I developed codes to understand how these PSTs talked in regards to the three components and applied these codes to written reflections at the beginning and end of our course. After analyzing the data I selected and interviewed 14 PSTs one year after our course. Again, I applied the same codes to their talk in the interview to see how their talk continued to change. Select interview PSTs were then chosen to represent common and uncommon examples of PST talk. Findings from this study show that talk across the MAP framework was related and that this talk became more standards-aligned by the end of the content course. However, the findings also provide a much more nuanced insight into different relationships and changes in talk. One finding shows that when variations in PST talk existed between framework components it was most commonly due to PSTs talking about mathematics in a more traditional way than when they talked about ability or themselves as mathematical learners. Another finding shows that during the interviews (one year after our course) PSTs continued to talk about themselves and abilities in mostly standards-aligned ways but reverted towards more traditional talk when discussing mathematics and how someone demonstrates their mathematical understanding. These findings have important implications for future research and for teacher educators. First, the relationships between the components of the MAP framework suggest that addressing PSTs conceptions of mathematics and their conceptions of ability may affect how they talk about individuals as mathematical learners. Second, the findings show which aspects in the MAP framework PSTs more readily talk about in standards-aligned ways. This provides insights into which areas teacher educators may want to emphasize more in trying to promote changes in PST talk. Lastly, these findings also show which aspects of PST talk maintain over a longer time frame and which aspects need a greater sustained emphasis. All of this is necessary as we support PSTs to think and talk about mathematics and mathematical abilities in standards-aligned ways that are truly supportive of all students.

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.