Günter Törner

Beliefs A hidden variable in Mathematics Education

Tables and Figures. Acknowledgements. Contributors. 1. Setting the Scene G.C. Leder, et al. Part 1: Beliefs: Conceptualization and Measurement. 2. Framing Students Mathematics-Related Beliefs. A Quest for Conceptual Clarity and a Comprehensive Categorization P. Opt Eynde, et al. 3. Rethinking Characterizations of Beliefs F. Furinghetti, E. Pehkonen. 4. Affect, Meta-Affect, and Mathematical Belief Structures G.A. Goldin. 5. Mathematical Beliefs A Search for Common Ground: Some Theoretical Considerations on Structuring Beliefs, Some Research Questions, and Some Phenomenological Observations G. Toerner. 6. Measuring Mathematical Beliefs and Their Impact on the Learning of Mathematics: A new Approach G.H. Leder, H.J. Forgasz. 7. Synthesis Beliefs and Mathematics Education: Implications for Learning, Teaching, and Research B.B. McLeod, S.H. McLeod. Part 2: Teachers Beliefs. 8. Mathematics Teacher Change and Development. The Role of Beliefs M. Wilson, T.J. Cooney. 9. Mathematics Teachers Beliefs and Experiences with Innovative Curriculum Materials. The Role of Curriculum in Teacher Development G.M. Lloyd. 10. A Four Year Follow-up Study of TeachersBeliefs after Participating in a Teacher Enhancement Project L.C. Hart. 11.Belief Structure and Inservice High School Mathematics Teacher Growth O. Chapman. 12. Participation and Reification in Learning to Teach: The Role of Knowledge and Beliefs S. Llinares. 13. A Study of the Mathematics Teaching Efficacy Beliefs of Primary Teachers G. Philippou, C. Christou. 14. A Study ofthe Mathematics Teaching Efficacy Beliefs and on Change S. Lerman. Part 3: StudentsBeliefs. 15. Beliefs about Mathematics and Mathematics Learning in the Secondary School: Measurement and Implications for Motivation P. Kloosterman. 16. The Answer is Really 4.5: Beliefs about Word Problems B. greer, et al. 17. Beliefs about the Nature of Mathematics in the Bridging of Everyday and School Mathematical Practices N. Presmeg. 18. Beliefs and Norms in the Mathematics Classroom E. Yackel, C. Rasmussen. 19. Intuitive Beliefs, Formal Definitions and Undefined Operations: Cases of Division by Zero P. Tsamir, D. Tirosh. 20. Implications of Research on Students Beliefs for Classroom Practice F.K. Lester Jr. Index

Mathematical Beliefs — A Search for a Common Ground: Some Theoretical Considerations on Structuring Beliefs, Some Research Questions, and Some Phenomenological Observations

A range of research and theory from different sources is reviewed in this chapter, in an attempt to understand better the construct of mathematical beliefs. Definitions of mathematical beliefs in the literature are not consistent and thus working out the core elements of a definition is one aspect of the chapter. Specifically, a four-component definition of beliefs is presented. The model focuses on belief object, range and content of mental associations, activation level or strength of each association, and some associated evaluation maps. This framework is not empirically derived but is based on common characteristics of the literature on didactics, particularly mathematics didactics. This effort towards achieving a precise definition can provide new understandings of fundamental issues in research on mathematical beliefs and give rise to new research questions. In particular, it allows description of the term “belief systems” allowing clustering of individual beliefs into a system across each of the four components. Furthermore, it makes sense to distinguish between global beliefs, domain-specific beliefs and subject-matter beliefs. The question immediately arises as to what interdependencies exist between the individual beliefs. Some observations from a survey of mathematical beliefs of students studying calculus are also included.

Understanding a Teacher’s Actions in the Classroom by Applying Schoenfeld’s Theory Teaching-In-Context: Reflecting on Goals and Beliefs

The theory teaching-in-context, first introduced by Schoenfeld in 1998, has the objective of making the actions of a teacher in mathematics lessons rationally understandable. According to this theory, it suffices to locate the behavior as a function of the following three parameters: available teacher knowledge, goals, and beliefs. In the following, we discuss a particular videoed classroom lesson with a remarkable turning point on the background of this approach. A teacher, who recently attained an in-service training course about the use of open tasks, tried to adapt the imparted issues to the topic of linear functions. While the lesson did not develop as desired, she shifted back to her hitherto established traditional teaching repertoire. Schoenfeld’s hypothesis implies that such spontaneous alterations in the teaching trajectories can be explained through shifts in the interplay of knowledge, goals and beliefs.

Teachers' professional development : what are the key change factors for mathematics teachers?

SUMMARY Our purpose is to answer the question: what have been the key factors causing a discontinuity in teachers’ professional development? We gathered data from experienced German teachers (N = 13) during the spring of 1994, with a brief questionnaire and interview. During the interviews, a total of 49 statements about change were mentioned. We could compress these statements into 15 change factors. New findings, that is, change factors which were not earlier reported in the literature, were as follows: experiences and observations with teachers’ own children, teachers’ experiences with other forms of schooling, and changes in society.

Problemlösen als integraler Bestandteil des Mathematikunterrichts — Einblicke und Konsequenzen

As an intensive explicit training in obtaining competence in problem-solving usually does not take place in math teaching in FRG, the authors focuses the performance of non-instructed problem-solver (students). Firstly, the paper offers an subjective description of various attitudes emphasizing problem-solving; subjective positions of the authors are fitted in. Hereby the aspect of a teaching regarding problem-solving is an integral part is characterized. These issues are then justified by the evaluations of 88 solutions to the so-called ‘lish-problem’ (grade 5 / grade 6). A final survey of the solutions will point out the difficulties of an integrative standardization of the students’ attempts, especially if such an analysis is based on a global approach via mathematical patterns.

Underqualified Math Teachers or Out-of-Field-Teaching in Mathematics - A Neglectable Field of Action?

This attempt at stock-taking deals with out-of-field teaching in mathematics - examined both from our German perspective and from an international perspective. It is obvious that there is much more out-of-field teaching in math in schools than generally assumed. Therefore, the community of mathematicians as well as the mathematics lecturers at university have to notice a field of action which requires our permanent and full attention: On the one hand there is no way to avoid enhanced training courses for certain groups and types of school, on the other hand you have to set a suitable course via regulations for the education of teachers so that the teaching staff who are subsequently responsible for the math classes in schools has obtained a basic education in the first phase of their qualification. Since all in all an increasing shortage of qualified math teachers is anticipated, measures to enhance the attraction of this specialist teaching profession - as far as possible in combination with another affine subject - are essential.

Mentale Repräsentationen von Irrationalzahlen — eine Analyse von Schülerinnenaufsätzen

ZusammenfassungAusgangspunkt sind 55 Aufsätze, in denen Schülerinnen aus der Jahrgangsstufe 9 eines Gymnasiums das Thema,Irrationalzahlen’ aus der subjektiven Sicht des erlebten Unterrichts reflektieren. Diese Aufsätze geben Einblick in grundlegende Schwierigkeiten dieses Unterrichtsinhalts. Sie legen individuelle Auffassungen („mentale Repräsentationen”) über Zahlbegriffe frei, mit besonderer Betonung von irrationalen Zahlen. Diese Zahlbegriffsauffassungen werden analysiert hinsichtlich der Kriterien Wissensstruktur, Wissenserzeugung, Ontologie, Relevanz und Emotionalität. In der Mehrzahl der Stellungnahmen wird ein eher,mysteriöses’ Bild dieser Zahlen gezeichnet, wobei die begrifflichen Defizite nicht ausschließlich in der Thematik als solche wurzeln, sondern vielfach schon vorgängig in der Behandlung der Dezimalbrüche angelegt erscheinen. Die traditionelle Verknüpfung dieses Themas mit geometrisch repräsentierten Irrationalzahlen scheint diese Schwierigkeiten an manchen Stellen eher zu verstärken.

Looking Back and Ahead – Some Very Subjective Remarks on Research in Mathematics Education

Let me first express my great thanks for the undertaking of this book, which humbled me. I am aware of the enormous amount of work accomplished by Yeping Li and Judit Moschkovich. I very much appreciate their work and efforts. In addition, I am excited that my scientific work is part of a book in honor of Alan Schoenfeld. Our birthdays are so close that he could not be my brother: however, I regard him not only as my personal friend, but also as my teacher and I see myself as his auditor since I have learned so much from him.