Nitsa Movshovitz-Hadar

Preservice Education of Math Teachers Using Paradoxes.

This is a report on a naturalistic study of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers mathematical concepts; (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained in parts of the study through written responses and class-videotapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions and constructive learning.

The effect of using Transparent Pseudo-Proofs in Linear Algebra

In an empirical study, the impact of exposing Linear Algebra students to a ‘Transparent Pseudo-Proof’ (TPP) of theorems was compared to that of exposing a comparable group of students to the formal proof of the same theorems. Following an analysis of the quantitative data collected in 36 sessions, this paper suggests a distinction between the benefit from employing TPPs to the teaching and learning of algorithmic proofs and that of employing TPPs to non-algorithmic proofs. Based upon analyses of the qualitative data gathered employing several tools, this paper discusses the particular advantages of employing TPPs to non-algorithmic proofs.

How to Present It? On the Rhetoric of an Outstanding Lecturer.

This paper analyses a lecture by an excellent teaching award winner professor of mathematics, given to high school mathematics teachers. The analysis is based upon two sources: (i) the lecture plan, as expressed in a series of 29 transparencies, prepared by the lecturer in advance; (ii) the actual implementation of the lecture, as transcribed from its video-taped record. Based on this analysis six principles for planning and giving a good lecture were developed. The paper provides the readers with full details of the content as well as non-verbal communication gestures exemplifying the employment of the six principles.

Nurturing a Community of Practice through a Collaborative Design of Lesson Plans on a Wiki System

Eleven graduate students, experienced mathematics teachers, participated in a semester long activity in which they collaboratively designed and developed lesson plans on a Media Wiki system. The described study examined the processes involved in this collaborative effort and its contribution to the development of mathematics teachers community of practice. Data were collected through tracking each Wiki page written by the participants, their reflective journals, a questionnaire, semi-structured interviews, and researchers notes of the whole class discussions. Evidently, the use of the Wiki system triggered a process of change, and the participants quickly evolved into a small and very Wiki-active community of practice. The results indicate that this teachers community of practice was concerned with two major issues: social ones and technical ones. The social issues related to the teachers consideration as to how to provide and receive feedback, as well as uncertainty about the possibility of losing the ownership over their creative work. The technical issues had to do with various difficulties participating teachers faced while writing in Wiki syntax.

More about mathematical paradoxes in preservice teacher education

Abstract This is a third report on a naturalistic study in Israel of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers mathematical concepts; and (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained through written responses and class-video-tapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions, and constructive learning. Generalizations are discussed.

Infinitely Many Different Quartic Polynomial Curves

Nitsa Movshovitz-Hadar received her B.Sc. at the Hebrew University in Jerusalem, her M.Sc. at Technion-lsrael Institute of Technology, and her Ph.D. at the University of California at Berkeley in 1975, where she was a Fullbright-Hays grantee and an AAUW fellow. Since 1975 she has been on the academic staff at Technion. She is also the mathematics consultant for Israel Educational Television. Her main interest is in the study of cognitive processes and obstacles involved in the learning of mathematics as they are related to curriculum development and to teacher-preparation.

Bridging Between Mathematics and Education Courses: Strategy Games as Generators of Problem Solving and Proving Tasks

Mathematics courses—pure or applied—and education courses, with pedagogical, didactic, and psychological contents are two groups of courses of which programs for training prospective high-school mathematics teachers commonly consist. They often are given in separate departments, with very little, if any, coordination among the instructors. Four independent problem solving centered bridging courses are described. They differ in the context which gives rise to the problems: (1) Mathematics problems that arise in the context of (strategy) games; (2) Mathematics problems that raise cognitive conflicts (paradoxes); (3) Mathematics problems that had a significant impact on the development of mathematics throughout its history; (4) Mathematics problems related to applications of mathematics and mathematical modeling. Ambiguity, contradictions, surprise, and paradoxes are the common thread of all the activities in these courses. This chapter focuses on the details of the first course, illustrating it by two sample tasks: (1) “Who gets first to 100?”, and (2) “Checker Board Jumps”. Emerging problem-solving activities are described and analyzed. The ultimate goal of the four-course series is to provide for a rich context in which prospective teachers can grasp the wide-scope nature of mathematics as a problem-posing/conjecturing and problem-solving/proving discipline, as well as the culture, beauty, and intellectual fulfillment of mathematics, so that they develop an enthusiastic attitude towards communicating these values to high-school students.

Logic in Wonderland: Alice's Adventures in Wonderland as the Context of a Course in Logic for Future Elementary Teachers

The teaching experiment described in this chapter assumes at the outset that children’s literature can be a useful context for teaching elementary ideas of logic while bridging the gap between the abstractness of formal logic and its expression in a real world context. Alice’s Adventures in Wonderland, by Lewis Carroll (a unique combination of a logician and a story teller) was chosen for this purpose, based upon a careful examination of its potential. Inspired by The Annotated Alice (Carroll 2000), over 75 additional annotations to Carroll’s book were developed, having in mind their employment in an introductory course in logic for prospective elementary school teachers specializing in mathematics. These annotations are in four categories: Logic, Mathematics, General education, and Science. Sample annotations are included. This chapter describes the tasks and activities developed for the course. Data collection instruments were interwoven in the teaching materials development. A sample is included as well. Several results are reported and discussed.

Studying Student Teachers’ Voices and Their Beliefs and Attitudes

Learning to teach mathematics is a complex undertaking, and in the last twenty years there has been a great deal of research looking at aspects of the process. There are many ways one might structure an analysis of research on learning to teach mathematics. It is clear, though, from all the research on learners in all kinds of situations that what student teachers bring to their teacher education courses in terms of prior knowledge, experience, attitudes, beliefs, goals, fears, hopes, and expectations has to be a key factor in preparing for and teaching those courses and hence for research. This particular focus for research in our field is not new; my own doctoral studies, completed in 1986, looks at connections between student teachers’ beliefs about the nature of mathematics and their perceptions of teaching mathematics (Lerman, 1990).1 It remains, however, of great importance, and there are new insights drawing on a range of theoretical frameworks emerging in the field. Central to research in the study of student teachers’ attitudes and beliefs and any changes in those beliefs during pre-service teacher education courses are issues of methodology. Access to student teachers’ beliefs and experiences is inevitably through their voices, expressed in interviews, conversations, and their writing, but the interaction of beliefs and practice is a necessary consideration too. First, regarding students’ voices, what must concern us is how to read across a number of stories in order to be able to say something about how these voices are produced (Arnot & Reay, 2004). Without that focus we produce a spiral of more and more detailed stories with no possibility of making sense of the data (see Brown & McNamara, 2005, for an excellent example of the struggle for an appropriate theoretical framework for analysing student teachers’ voices). Second, there is always a gap between what people say about what they do and what they actually do in their practice (Lerman, 2002). Research needs, therefore, to be aware of that gap

Science in the service of society: the Israel National Museum of Science

Science as a cornerstone of nation building is a guiding principle of the Israel National Museum of Science, which caters to a culturally diverse public of all ages. Nitsa Movshovitz‐Hadar is director of the museum and professor at the Technion–Israel Institute of Technology and former head of its Department of Education in Technology and Science. Since 1986, she has been academic director of the Israel National Pedagogical Center for Mathematics. For more than ten years, she was mathematics consultant to Israel Educational Television, which produced ‘DraMath’, a series of sixteen videotaped dramatic programmes in mathematics that won the 1985 Japan Prize International Contest of Educational Video Programs. Drora Kass, a psychologist by training, heads a consulting firm that assists institutions to enunciate goals, conceptualize programmes, devise strategies and raise funds. For more than thirty years she has been active in the promotion of peace between Israel and its neighbours and has won numerous awards on behalf of this work. Her previous positions include: director of Public affirs and Resource Development Division, the Technion; special consultant to the Israeli Minister of Education and Culture; and director of the US Office of the International Center for Peace in the Middle East.