The analysis reported in this chapter focuses on a second-grade classroom in which the children typically displayed positive emotional acts as they attempted to solve personally challenging mathematics problems. We argue that, within the microculture established in a particular classroom, certain emotional acts but not others are appropriate in situations such as solving challenging mathematical tasks. The emotional acts that are warranted in particular situations can differ significantly from one classroom to another depending on the nature of the social norms that have been established. We illustrate how the negotiation of social norms that contrasted sharply with those established in most US mathematics classroom made possible the children’s generally positive responses to mathematical problem solving. In the final section of the chapter, we discuss the implications for the development of productive classroom learning environments in which debilitating emotions such as frustration while solving mathematical problems are not warranted.

The Handbook of Mathematics Teacher Education, the first of its kind, addresses the learning of mathematics teachers at all levels of schooling to teach mathematics, and the provision of activity and programmes in which this learning can take place. It consists of four volumes. Volume 2, Tools and Processes in Mathematics Teacher Education, focuses on the “how” of mathematics teacher education. In this volume, the authors share with the readers their invaluable experience in employing different tools in mathematics teacher education. This accumulated experience could assist teacher educators, researchers in mathematics education and those involved in policy decisions on teacher education in making decisions about both the tools and the processes to be used for various purposes in mathematics teacher education. There are four sections. The first describes and discusses four successful ways of using cases in mathematics teacher education, including narratives, mathematics case discussions, video-recordings, and lesson studies. The second presents predominant tools that are used in mathematics teacher education, two textual tools (written tasks and examples) and two physical tools (manipulatives and machines). The third section suggests ways in which the accumulated research on common students’ ways of thinking contributes to the development of tools and processes in mathematics teacher education. The last section provides critical response and general perspective, raising questions such as: How can the teaching of mathematics be used as a tool to promote general educational values? What are the dimensions of proficient teaching? The concluding chapter offers a provisional framework consisting of a set of seven dimensions of proficiency for teaching mathematics. Together, the chapters provide various promising tools and processes for facilitating the acquisition of major proficiencies needed for teaching mathematics, and principles that could guide the selection and use of such tools. Bibliographical Information for the complete set: VOLUME 1: Knowledge and Beliefs in Mathematics Teaching and Teaching Development Peter Sullivan, Monash University, Clayton, Australia and Terry Wood, Purdue University, West Lafayette, USA (eds. ) paperback: 978-90-8790-541-5, hardback: 978-90-8790-542-2, ebook: 978-90-8790-543-9 VOLUME 2: Tools and Processes in Mathematics Teacher Education Dina Tirosh, Tel Aviv University, Israel and Terry Wood, Purdue University, West Lafayette, USA (eds. ) paperback: 978-90-8790-544-6, hardback: 978-90-8790-545-3, ebook: 978-90-8790-546-0 VOLUME 3: Participants in Mathematics Teacher Education: Individuals, Teams, Communities and Networks Konrad Krainer, University of Klagenfurt, Austria and Terry Wood, Purdue University, West Lafayette, USA (eds. ) paperback: 978-90-8790-547-7, hardback: 978-90-8790-548-4, ebook: 978-90-8790-549-1 VOLUME 4: The Mathematics Teacher Educator as a Developing Professional Barbara Jaworski, Loughborough University, UK and Terry Wood, Purdue University, West Lafayette, USA (eds. ) paperback: 978-90-8790-550-7, hardback: 978-90-8790-551-4, ebook: 978-90-8790-552-1

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and I conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.

Seven elementary teachers participated in a project designed to help them learn to teach mathematics according to reform recommendations. Teachers were provided opportunities to learn through both private reflection and public inquiry about their teaching and children's learning. The teachers’ instruction, reflection, and beliefs were studied. All of the teachers adopted some reform-based procedures including having children report problem-solving strategies. However, only three of them developed more complex practice in which children were involved in inquiry into one another's strategies. The groups had different beliefs about the autonomy of children to construct mathematics and their own autonomy to make instructional decisions.

In this chapter we address questions of the interrelationship of the processes involved in working together and learning, mainly in the field of mathematics. In the first section we present a process model, which is suitable to analyse episodes in which students collaborate on tasks that aim at level raising. The model focuses on key activities in the level raising process and incorporates social activities that affect the occurrence of these key activities. In the second section we present a multiple analysis of a collaborative learning episode in a broader context. We analysed the episode from three different perspectives: social interaction, division of time and mathematical level raising. Integrating the three perspectives brings into focus the complexity of settings, in which students regulate their social and cognitive activities

In the research conducted, the relationship between teaching complexity and children's mathematical thinking was investigated in 4 'reform' classes and 1 conventional elementary class (7-8 years). Forty lessons were analyzed for the type of teaching and children's mathematical thinking revealed during class discussion. The results indicate increased complexity in teaching and level of children's thinking was highly related to the kinds of interaction that distinguished three class cultures. These findings complete a previously proposed theoretical framework that integrates teaching and learning by detailing acts of teaching in relation to complexity of children's thinking. It is well documented that the concerted effort in the U. S. to change conventional mathematics teaching to forms of pedagogy that coincide with learning for conceptual understanding is more difficult than initially anticipated. One reason may be, in part, due to the fact that this requires the development of far more complex and sophisticated pedagogy than was understood or even known at the onset of the effort (Wood, Nelson & Warfield, 2001). Although research on learning over the past century has influenced our knowledge of learning, similar transformation in our understanding of the teaching practices has yet to occur. Educators, such as Darling-Hammond (1996) believe the challenge for education in this century is the advancement of " knowledge for a different kind of teaching. .. that goes far beyond dispensing information, giving a test, and giving a grade " (p. 7). From recent studies such as Askew et al. (1999) and Franke et al. (1998) we are beginning to understand what characterizes the complexity in new forms of teaching and how this relates to student learning. However, it still remains that " only a few studies exist which empirically examine teaching in these classes with the same detail and attention to theory building as found in the investigations of learning " (Wood, 1998, p. 193). In previous research we have examined class cultures for differences in pedagogy and found that teaching for conceptual understanding does not consist of a singular practice, but rather varies on two dimensions—expectations for class members' participation and the breadth of pupils' thought (Wood & Turner-Vorbeck, 2001). While these two dimensions differentiate the nature of teaching in 'reform-oriented' class cultures, the relationship of teaching to children's mathematical thinking was only theoretically conjectured. Therefore, the purpose of this research report is to present the results of an investigation into the relationship between teaching …

transformation in the ways students learn and teachers teach mathematics. These changes aim for ways of learning and teaching that result in students knowing a "different" kind of school mathematics. As adults with experience in traditional mathematics pedagogy, it is not easy for us to envision how teaching mathematics could be different. Therefore, I offer a brief excerpt to illustrate what is meant by "teaching differently," but first I give an example of traditional pedagogy (see McNeal, 1991) as a point of contrast. In a third grade class, the teacher (following the directions given in the textbook) has made 45 tally marks on the board and has circled four groups of ten.

The widespread acceptance in the view that learning is an active constructive process requires teaching that is fundamentally different from classical pedagogy. It is generally accepted that teaching must consist of highly interactive and discursive situations. However, these differences in teaching are not well understood. In this paper, examples of teaching and the ways these distinctions influence children’s opportunities for learning.

This paper examines the use of the Internet, particularly online discussion forums, as a means of promoting discourse among educators. Internet forums offer key advantages for promoting discourse including: accessibility, flexibility, equitable participation, and storage of exchanges for reflection and response. Quantitative and qualitative methods are used to examine the discourse generated as part of: (1) an online course, and (2) a teacher professional development project. Some results were disappointing. Maintenance of consistent participation was difficult. In the online course, participation peaked early but declined toward the end of the course, and, in the professional development project, overall participation levels fell below expectations. In addition, reflective dialog was difficult to obtain in the professional development project. However, other results were promising. Most participants readily adapted to the technology. The online course yielded extensive, in-depth dialog about course, topics. In the professional development project, common problems successfully stimulated discussions about the classroom, and teacher educators successfully promoted dialog and reflection through focused questions and comments. The positive findings suggest that the Internet has considerable promise as a tool for discourse and community-building among educators.

The manner in which a horizontal addition and subtraction number sentence activity was constituted in one second grade classroom is analyzed for the purpose of discussing and illustrating how mathematical meaning is interactively constituted in the classroom. In particular, the teacher's emphasis on different solutions contributed to students' development of increasingly sophisticated concepts of ten. In turn, students' solutions contributed to the teacher's development of an increasingly sophisticated understanding of the children's mathematical activity and their concepts of ten.