This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R³ and to their flexibility in the use of different representations.

The general background of this study is an interest in how digital tools contribute to structuring learning activities. The specific interest is to explore how such tools co-determine students’ reasoning when solving word problems in mathematics, and what kind of learning that follows. Theoretically the research takes its point of departure in a sociocultural perspective on the role of cultural tools in thinking, and in a complementary interest in the role of the communicative framing of cognitive activities. Data have been collected through video documentation of classroom activities in secondary schools where multimedia tools are integrated into mathematics teaching. The focus of the analysis is on cases where the students encounter some kind of difficulty. The results show how the tool to a significant degree co-determines the meaning making practices of students. Thus, it is not a passive element in the situation; rather it invites certain types of activities, for instance iterative computations that do not necessarily rely on an analysis of the problems to be solved. For long periods of time the students’ activities are framed within the context of the tool, and they do not engage in discussing mathematics at all when solving the problems. It is argued that both from a practical and theoretical point of view it is important to scrutinize what competences students develop when using tools of this kind.

• Prospective teachers’ reasoning and response to a student’s non-traditional strategy when dividing fractions, by Ji-Won Son and Sandra Crespo. Abstract: Recognizing meaning in students’ mathematical ideas is challenging, especially when such ideas are different from standard mathematics. This study examined, through a teaching-scenario task, the reasoning and responses of prospective elementary and secondary teachers to a student’s non-traditional strategy for dividing fractions. Six categories of reasoning were constructed, making a distinction between deep and surface layers. The connections between the participants’ reasoning, their teaching response, and their beliefs about mathematics teaching were investigated. We found that there were not only differences but also similarities between the prospective elementary and secondary teachers’ reasoning and responses. We also found that those who unpacked the mathematical underpinning of the student’s non-traditional strategy tended to use what we call “teacher-focused” responses, whereas those doing less analysis work tended to construct “student-focused” responses. These results and their implications are discussed in relation to the influential factors the participants themselves identified to explain their approach to the given teaching-scenario task. 
• Working with mathematics teachers and immigrant students: an empowerment perspective, by Núria Planas and Marta Civil. Abstract: This article centers on a professional development project with a group of high school mathematics teachers in Barcelona. The eight participating teachers taught in low-income schools with a high percentage of immigrant students. Our model of professional development is based on the involvement of the teachers as co-researchers of their local contexts and practices. In this approach, our concept of social justice is tied to the notion of empowerment, both for teachers and for their immigrant students. Our analysis of data from twelve sessions with the teachers shows the development of a shared awareness of their local situation that leads to their questioning of their practices followed by a reconstruction of those. Teachers worked together to move from talking to action. Our analysis of data from the implementation of one lesson in a classroom shows that action, and illustrates signs of empowerment in the teacher and the students, such as students’ challenging of aspects of the task and taking on a more participatory role and the teacher’s reflection on the overall experience. 
• Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge, by Jon D. Davis. Abstract: This study examines the influence of reading and planning from two differently organized mathematics textbooks on prospective high school mathematics teachers’ pedagogical content knowledge and content knowledge of exponential functions. The teachers completed a pretest and two posttests. On the pretest, the teachers possessed an incomplete understanding of content and pedagogical content knowledge related to exponential functions. The teachers’ understanding of how to translate from table to closed-form and recursive equations grew as a result of their use of the Mathematics: Modeling Our World textbook, while the Discovering Algebra textbook appeared to be more beneficial in terms of pedagogical content knowledge. Teachers read from the student lessons in both textbooks, but read differently from the sections of both textbooks intended for the teacher. They focused more on the purpose of the Mathematics: Modeling Our World lesson and more on the places where students might experience difficulties in the margins of the Discovering Algebra lesson. The teachers’ learning was influenced by their own personal characteristics (e.g., previous textbook experiences) as well as textbook qualities (e.g., organization).

In most teacher-preparation programs, the preservice students are faced with some sort of field experiences (in Norway we call this “practice”, or “praksis” in Norwegian). A focus on the quality of field experiences received a lot of attention when U.S. teacher education was reformed in the 1980s, and there were several recommendations indicating that preservice teachers should have more “authentic experiences to prepare them to handle the complexity and challenges of the school context” (p. 124). Several teacher education programs, including ours at the University of Stavanger (Norway) have strong emphasis on field experiences. According to Santagata et al., two assumptions are underlying:

1. “exposure to examples of teaching creates learning opportunities for prospective teachers”
2. “through field experiences preservice teachers meld theory into practice” (ibid.).

Both these assumptions are questioned in the introductory part of this article. A problem is that preservice teachers often focus on irrelevant features when observing practice, and their classroom observations might thereby be of little use. Another issue is that teaching practices appear to be stable, and “It is thus not likely that preservice teachers will encounter reform-minded practices in the lessons they observe. On the contrary, field experiences may expose student teachers to a limited repertoire of strategies and to a narrow and unrepresentative sample of students” (ibid.).

The authors of this article propose the use of videos of classroom instruction as an alternative approach. Videos can be studied over and over, and this allows for a deeper and more reflected analysis than during ordinary “live observations”. The authors argue: “Teaching is a cultural activity, and cultural routines are more easily unveiled when the teaching process is slowed down and critically analyzed” (p. 125).

The use of video is not only “a means to expose preservice teachers to specific behaviors to be imitated” but it is also (or is becoming) “a tool for the development of teachers’ professional judgment” (p. 126). In the article they report from two studies in Italy, where videos from the TIMSS 1999 Video Study (report) have been used in preservice teacher education. I will not go into the results from these studies here, but I recommend reading the entire article for further information!

Reference:
Santagata, R., Zannoni, C., and Stigler, J. (2007). The role of lesson analysis in pre-service teacher education: an empirical investigation of teacher learning from a virtual video-based field experience. Journal of Mathematics Teacher Education, 10(2):123-140.