Ruhama Even

Factors involved in linking representations of functions

Abstract This article focuses on the intertwining between the flexibility in moving from one representation to another, and other aspects of knowledge and understanding. During the first phase of data collection, 152 college mathematics students who were also prospective secondary mathematics teachers completed an open-ended questionnaire. In the second phase, an additional ten subjects were interviewed after responding to the questionnaire. The results illustrate how knowledge about different representations is not independent, but is interconnected with knowledge about different approaches to functions, knowledge about the context of the presentation and knowledge of underlying notions.

The Development of Teacher Leaders and Inservice Teacher Educators

This article discusses the development of teacher leaders and inservice teacher educators whose role it is to promote teacher learning about mathematics teaching in the process of changing school mathematics. The Manor Program for the development of teacher leaders and teacher educators is used as a vehicle for addressing this issue. The article focuses on aspects of curriculum design, discusses the theoretical rationale for the learning opportunities provided by the program, and considers several problematic aspects encountered.

The decimal point situation: a close look at the use of mathematics-classroom-situations in teacher education

Abstract Mathematics classroom situations (MCS) are real or hypothetical classroom situations involving mathematics, in which the teacher has to respond to a students hypothesis, question or idea. This study investigates the use of one such situation, the decimal point situation, in an in-service course for 20 elementary teachers. Data sources include: written questionnaire, individual interview, whole class discussion and student interview. The article examines the potential of MCS to evoke analyses and reflections about teaching and learning mathematics.

Using Assessment to Inform Instructional Decisions: How Hard Can It Be?.

In this article, two problems associated with the expectation that teachers use contemporary assessment techniques are examined. The first problem relates to teachers’ sense-making of assessment data. Illustrative cases revealed that teachers’ processes of interpretation of students’ understanding, knowledge and learning of mathematics draws on a rich knowledge base of understandings, beliefs, and attitudes. Consequently, the process of sense-making of students’ mathematical understandings involves ambiguity and difficulty. The second problem relates to ways of helping teachers adopt contemporary assessment approaches. A professional development activity served as the example examined. Three aspects of what the course instructor promoted with respect to contemporary assessment were analysed: (1) the assessment methods and tools advocated in the course, (2) the degree to which the integration of assessment with instruction was promoted, and (3) the purposes for assessment highlighted in the course. It appeared that attention was paid to the use of contemporary assessment tools, but this was associated with traditional assessment purposes. Learning to use the new assessment tool did, however, influence instruction and fostered greater integration of assessment and instruction than before-a characteristic of contemporary assessment. The article concludes with a discussion of the current expectation that teachers use assessment data to improve instruction.

Between student observation and student assessment: A critical reflection

This paper examines the problematic associated with the current expectation that teachers assess their students’ mathematics learning and understanding by listening to their talk and observing their actions. The paper suggests that the problems are of two different kinds. One kind involves difficulties that could be overcome; the other kind is related to the intrinsic nature of understanding what someone else is saying or doing and therefore cannot be overcome. Several cases that highlight the two different kinds of problems associated with teachers’ making sense of students’ talk and actions are analysed. Implications for teacher education are identified.

The Work of Providers of Professional Development for Teachers of Mathematics: Two Case Studies of Experienced Practitioners

This paper is an initial investigation into the practice of providers of professional development for teachers of mathematics. The study examines the work of two providers of professional development for teachers of mathematics. Both provided professional development while working with teachers on implementing a new mathematics curriculum for seventh grade. Although their work conditions were quite different from each other the study reveals that there were similar characteristics in their practices. The most salient ones were acting out lessons, analyzing principles of the new curriculum, encouraging the teachers to explicate their concerns, and asking teachers to solve concrete practical problems related to the reservations they have about specific components of the new curriculum. The role of a program that prepared the two participants to be providers of professional development for teachers is also discussed.

Strengthening Practice in and Research on the Professional Education and Development of Teachers of Mathematics: Next Steps

This Study brought together 147 scholars and practitioners from 35 countries to discuss the professional formation of teachers of mathematics. Their individual contributions have been assembled to create a volume filled with descriptions of programs and projects, as well as concepts and data. Readers interested in the development of teachers of mathematics will find in this book many ideas and people relevant to their own work, much as was discovered by those who participated in the Study. The conference was lively and intense as ideas and people from around the world interacted around this fundamental problem: Mathematics education works virtually nowhere as well as it needs to if we are to prepare students for life in world where practical, intellectual, and critical quantitative competence will matter more than ever before, and for more people. Although our systems, resources, and results differ, no country is satisfied with the quality or the reach of contemporary mathematics education. This Study was founded on the premise that teachers are central to the improvement of mathematics education, a premise that should be obvious, but too often is not. Since what students learn is a function of the opportunities they have and how those opportunities are managed, improvements that aim solely on curriculum or standards are unlikely to make the sorts of impact on students’ learning that some assume. Despite the importance of teachers, however, the approaches to their education often do not help them develop the skills and insights needed for practice. This Study began an international conversation about what we do and the questions we have. Equally important will be the next steps that follow. We suggest below three main problems that could profit from stronger and more systematic international connections focused on improving the education and professional development of teachers. First is the need to focus teachers’ education on practice—-and the problem of doing it effectively. This was a focus of the Study—-to learn what is done around the world, and what the challenges are. On one hand, it should be obvious that teaching is a practice and that, therefore, teachers’ education must provide sustained, systemic opportunities for teachers to learn and develop their effectiveness with

To Define Or Not To Define: The Case Of (-8)1/3

This paper discusses two possible approaches to (-8)1/3. The first is that (-8)1/3 = 3√(-8) = -2. The second is that (-8)1/3 is undefined. The pros and cons of each of these approaches are considered and some implications to teacher education are specified.

Preparing Mathematics Education Researchers for Disciplined Inquiry: Learning from, in, and for Practice

In this chapter we consider what it takes to learn to conduct research in mathematics education. We argue that learning any complex practice requires opportunities to unpack its components in order to see what underlies competence performance. Many of the components of successful research remain implicit and are left to new researchers to glean from finished products. In this chapter we consider: What is it that accomplished scholars do as they conduct research? What are the practices in which they engage? What is it that they have to mobilize, consider, try, and carry out? We unpack the work of research, considering such domains as reading, writing, moving from particular observations to general insights, and communicating through multiple media. We propose that focusing on these practices of research offers a promising avenue for the preparation of beginning scholars.

Connectedness in Teaching Equivalent Algebraic Expressions: Novice Versus Expert Teachers

This study examines differences in connectedness in instruction between two novice teachers and an expert teacher. Three types of data related to lessons on equivalent algebraic expressions were collected: lesson plans, lesson observations, and post-lesson interviews. Although connectedness is an important characteristic of mathematics teaching and learning, only the expert teacher used both lesson and content connections to guide her teaching. Differences in the teachers’ views and uses of connectedness in instruction are discussed and illustrated.