From the 1990ties on, large-scale international video studies have significantly advanced the discussion about quality factors of learning support in the mathematics classroom (Hiebert et al., 2003; Stigler et al., 1999). In particular the results of TIMSS 1999 Video (Hiebert et al., 2003) have suggested that very different factors can contribute to the learners’ success. There is – in an international perspective – not a simple observation pattern which allows distinguishing ‘the’ successful mathematics classroom from less successful teaching. However, by these findings, the initial hope of these video studies, namely that culture-overarching quality characteristics can be identified, has not been fulfilled satisfactorily, which calls for deepening research especially in the field of mathematics education.
An example of a study responding to this need of deepening studies is an approach with emphasis on structural clarity (Drollinger-Vetter, 2011) in a bi-national video analysis of whole-class instruction related to the Pythagorean Theorem, which contained among other a focus on the use of representations of mathematical objects in the classroom.
The way how representations are dealt with is indeed a key quality aspect of mathematics instruction (e.g. Duval, 2006; Ainsworth, 2006; Kuntze, 2013): As mathematical objects can only be accessed through representations and being able to handle them as well as to change between different representations is a core aspect of students’ mathematical competency (Ainsworth, 2006; Lesh, Post & Behr, 1987; Duval, 2006; Dreher & Kuntze, 2015a), students should be encouraged and supported to flexibly use multiple representations. However, conversions between representation registers are potentially complex for students (Ainsworth, Dreher & Kuntze 2015b), so that teachers have to help students to connect representations and to translate between them. This can be expected to be the case in particular in so-called learning support situations, in which the teacher has the possibility to react on students’ individual questions or to interact with them individually (Krammer, 2009; Schnebel, 2013). By learning support situations, we understand situations in which there is not a whole-class dialogue, but interactions between teachers and students during seat-work phases of the students, who are working on tasks on their own, in pairs or in small groups. Such learning support situations can be initiated by students, e.g. when they ask a question, or alternatively also by the teacher, e.g. when the teacher asks the student(s) or gives them a hint or feedback related to their working process. It is obvious that such interaction situations are a key opportunity for helping the students with conversions between representation registers (as they are often required by tasks) or to encourage the students to change between representations (in order to support their ability to change flexibly between representations of mathematical objects).
However so far – and despite the international relevance of these quality aspects – there is hardly any empirical evidence beyond case studies about whether and how mathematics teachers encourage their students to use multiple representations and provide them with help in learning support situations.
Consequently, the study reported here focuses on this research need. With this purpose, the study concentrates on the following research questions:
(a) What role does the use of multiple representations play in the teachers’ interaction with students in learning support situations?
(b) Do the secondary school mathematics teachers actively promote the use of multiple representations in learning support situations?
(c) In particular, do they support the change between representation registers of their students by help focused on connecting the different representation registers?
Acknowledgements We acknowledge the contribution of Melanie Mayr and Johannes Massini to the coding process of this study. References Ainsworth, S. E. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198. Batzel, A., Bohl, T., Kleinknecht, M., Leuders, T., Ehret, C., Haug, R. & Holzäpfel, L. (2013). Kognitive Aktivierung im Unterricht mit leistungsschwächeren Schülerinnen und Schülern. Theoretische Grundlagen, methodisches Vorgehen und erste Ergebnisse. In U. Riegel & K. Macha (Hrsg.), Videobasierte Kompetenzforschung in den Fachdidaktiken (S. 97-113). Münster: Waxmann. Dreher, A. & Kuntze, S. (2015a). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114. Dreher, A. & Kuntze, S. (2015b). Teachers Facing the Dilemma of Multiple Representations Being Aid and Obstacle for Learning: Evaluations of Tasks and Theme-Specific Noticing. Journal für Mathematik-Didaktik, 36(1), 23-44. Drollinger-Vetter, B. (2011). Verstehenselemente und strukturelle Klarheit. Fachdidaktische Qualität der Anleitung von mathematischen Verstehensprozessen im Unterricht. Münster: Waxmann. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. Hiebert, J. et al. (2003). Teaching Mathematics in Seven Countries. Results from the TIMSS 1999 Video Study. NCES 2003-013. Washington DC: U.S. Department of Education, National Center for Education Statistics. Kuntze, S. (2013). Vielfältige Darstellungen nutzen im Mathematikunterricht. [Using multiple representations in the mathematics classroom]. In Wagner, A. et al. (Eds.). In J. Sprenger, A. Wagner, M. Zimmermann (Hrsg.). Mathematik lernen, darstellen, deuten, verstehen (S. 17-34). Wiesbaden: Springer. Krammer, K. (2009). Individuelle Lernunterstützung in Schülerarbeitsphasen. Eine videobasierte Analyse des Unterstützungsverhaltens von Lehrpersonen im Mathematikunterricht. [Individual learning support in seatwork phases. A video-based analysis of the support provided by teachers in the mathematics classroom]. Münster: Waxmann. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of math. (pp. 33-40). Hillsdale: Erlbaum. Schnebel, S. (2013). Lernberatung, Lernbegleitung, Lerncoaching – neue Handlungsformen in der Allgemeinen Didaktik? Jahrbuch für Allgemeine Didaktik, 3, 278-296. Stigler, J.W., Gonzales, P., Kawanaka, T., Knoll, S., & Serrano, A. (1999). The TIMSS Videotape Classroom Study: Methods and Findings from an Exploratory Research Project in Eigth-Grade Mathematics Instruction in Germany, Japan, and the United States. NCES 1999-074. Washington DC: U.S. Department of Education, National Center for Education Statistics.
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