Teachers Facing the Dilemma of Multiple Representations Being Aid and Obstacle for Learning: Evaluations of Tasks and Theme-Specific Noticing

Anika Dreher1, Sebastian Kuntze1
1Ludwigsburg University of Education, Ludwigsburg, Germany

Tóm tắt

Từ khóa


Tài liệu tham khảo

Acevedo Nistal, A., van Dooren, W., Clareboot, G., Elen, J., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathe-matical problem solving and learning: a critical review. ZDM The International Journal on Mathematics Education, 41(5), 627–636.

Ainsworth, S. (2006). A conceptual framework for considering learning with multiple rep-resentations. Learning and Instruction, 16, 183–198.

Ball, D. L. (1993a). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–196). Hillsdale: Erlbaum.

Ball, D. L. (1993b). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal, 93(4), 373–397.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

Baumert, J., & Kunter, M. (2006). Stichwort professionelle Kompetenz von Lehrkräften. Zeitschrift für Erziehungswissenschaften, 9(4), 469–520.

Baumert, J., Kunter, M., Blum, W., Voss, T., Jordan, A., Klusmann, U., Krauss, S., Neubrand, M., & Tsai, Y. (2010). Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133–180.

Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations. Computers in Human Behavior, 22, 27–42.

Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. (2011). Translations among mathematical representations: Teacher beliefs and practices. International Journal of Mathematics Teaching and Learning (June 15th 2011). http://www.cimt.plymouth.ac.uk/journal/bosse4.pdf . Accessed 6 Oct. 2014.

Burns, R., & Burns, R. (2008). Business research methods and statistics using SPSS. London: SAGE Publications Ltd.

Cox, R. (1999). Representation construction, externalized cognition and individual differences. Learning and Instruction, 9, 343–363.

diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10, 105–225.

Dreher, A. (2012). Den Wechsel von Darstellungsformen fördern und fordern oder vermeiden? Über ein Dilemma im Mathematikunterricht. In J. Sprenger, A. Wagner, & M. Zimmermann (Eds.), Mathematik lernen, darstellen, deuten, verstehen—Didaktische Sichtweisen vom Kindergarten bis zur Hochschule (pp. 215–225). Wiesbaden: Springer Spektrum.

Dreher, A., & Kuntze, S. (accepted). Teachers’ professional knowledge and noticing—The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

Elia, I., Panaoura, A., Eracleous, A., & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5(3), 533–556.

English, L. D., & Halford, G. S. (1995). Mathematics education—models and processes. Mahwah: Erlbaum.

van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers’ interpretations of classroom interactions. Journal of Technology and Teacher Education, 10(4), 571–596.

Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17, 105–121.

Font, V., Godino, J. D., Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1), 97–124.

Goldin, G., & Shteingold, N. (2001). Systems of representation and the development of mathematical concepts. In A. A. Cuoco, & F. R. Curcio (Eds.), The role of representation in school mathematics (pp. 1–23). Boston: NCTM.

Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2002). Symbolizing, modeling and tool use in mathematics education. Dordrecht: Kluwer Acad. Publ.

Hasemann, K. (1981). On difficulties with fractions. Educational Studies in Mathematics, 12(1), 71–87.

Helsper, W. (1996). Antinomien des Lehrerhandelns in modernisierten pädagogischen Kulturen. Paradoxe Verwendungsweisen von Autonomie und Selbstverantwortlichkeit. In A. Combe (Ed.), Pädagogische Professionalität (pp. 521–569). Frankfurt a. M.: Suhrkamp.

Helsper, W. (2007). Eine Antwort auf Jürgen Baumerts und Mareike Kunters Kritik am strukturtheoretischen Professionsansatz. Zeitschrift für Erziehungswissenschaft, 10(4), 567–579.

Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., Chui, A. M. Y., Wearne, D., Smith M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., & Stigler, J. (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 Video Study. Washington, DC: U.S. Department of Education, NCES.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.

Jöreskog, K. G. (1971). Statistical analysis of sets of congeneric tests. Psychometrica, 36(2), 109–133.

Kali, Y., Goodyear, P., & Markauskaite, L. (2011). Researching design practices and design cognition: Contexts, experiences and pedagogical knowledge-in-pieces. Learning, Media and Technology, 36, 129–149.

Kalyuga, S., Chandler, P., & Sweller, J. (1999). Managing split-attention and redundancy in multimedia instruction. Applied Cognitive Psychology, 13, 351–371.

Kaput, J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner, & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 167–194). Reston: National Council of Teachers of Mathematics.

Kline, R. B. (2005). Principles and practice of structural equation modeling, 2nd ed. New York: Guilford.

Kuhnke, K. (2013). Vorgehensweisen von Grundschulkindern beim Darstellungswechsel: eine Untersuchung am Beispiel der Multiplikation im 2. Schuljahr. Wiesbaden: Springer Spektrum.

Kultusministerkonferenz (KMK). (2003). Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss. http://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2003/2003_12_04-Bildungsstandards-Mathe-Mittleren-SA.pdf . Accessed 2 April 2014.

Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (2011). Pro-fessionelle Kompetenz von Lehrkräften. Ergebnisse des Forschungsprogramms COACTIV. Münster: Waxmann.

Kuntze, S., & Dreher, A. (2014). PCK and the awareness of affective aspects reflected in teachers’ views about learning opportunities—a conflict? In B. Pepin & B. Rösken-Winter (Eds.), From beliefs and affect to dynamic systems: (exploring) a mosaic of relationships and interactions. NY: Springer. Advances in Mathematics Education series.

Kuntze, S., Lerman, S., Murphy, B., Kurz-Milcke, E., Siller, H.-S., & Winbourne, P. (2011). Development of pre-service teachers’ knowledge related to big ideas in mathematics. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 105–112). Ankara: PME.

Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review, 55, 178–194.

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 mathematics (pp. 33–40). Hillsdale: Lawrence Erlbaum.

Malle, G. (2004). Grundvorstellungen zu Bruchzahlen. mathematik lehren, 123, 4–8.

Meira, L. (1998). Making sense of instructional devices: The emergence of transparency in mathematical activity. Journal for Research in Mathematics Education, 29(2), 121–142.

Moser Opitz, E. (2009). Erwerb grundlegender Konzepte der Grundschulmathematik als Voraussetzung für das Mathematiklernen in der Sekundarstufe I. In A. Fritz & S. Schmidt (Eds.), Fördernder Mathematikunterricht in der Sek. I (pp. 29–46). Weinheim: Beltz Verlag.

NCTM—National Council of Teachers in Mathematics. (2000). Principles and Standards for School mathematics. Reston: NCTM.

Rau, M. A., Aleven, V., & Rummel, N. (2009). Intelligent tutoring systems with multiple representations and self-explanation prompts support learning of fractions. In V. Dimi-trova, R. Mizoguchi, & B. du Boulay (Eds.), Proceedings of the 14th International Con-ference on Artificial Intelligence in Education (pp. 441–448). Amsterdam: IOS Press.

Renkl, A., Berthold, K., Große, C. S., & Schwonke, R. (2013). Making better use of multiple representations: How fostering metacognition can help. In R. Azevedo (Ed.), Springer International Handbooks of Education: Vol. 28. International Handbook of Metacognition and Learning Technologies (pp. 397–408). New York: Springer.

Schipper, W. (2005). Rechenstörungen als schulische Herausforderung. Beschreibung des Moduls 4 für das Projekt Sinus-Transfer Grundschule. http://www.uni-bielefeld.de/idm/serv/sinus-modul4.pdf . Accessed 7 Feb. 2014.

Sfard, A. (2000). Symbolizing mathematical reality into being: How mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating: perspectives on mathematical discourse, tools, and instructional design (pp. 37–98). Mahwah: Erlbaum.

Sherin, M. G. (2007). The development of teachers’ professional vision in video clubs. In R. Goldman, R. Pea, B. Barron, & S. Derry (Eds.), Video research in the learning sciences (pp. 383–395). Hillsdale: Erlbaum.

Sherin, M. G., Jacobs, V. R., & Philipp, R. A. (2010). Situating the study of teacher noticing. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 3–13). New York: Routledge.

Steinbring, H. (2000). Mathematische Bedeutung als eine soziale Konstruktion—Grundzüge der epistemologisch orientierten mathematischen Interaktionsforschung. Journal für Mathematik-Didaktik, 21(1), 28–49.

Stern, E. (2002). Wie abstrakt lernt das Grundschulkind. In H. Petillon (Ed.), Individuelles Lernen in der Grundschule—Kinderperspektive und pädagogische Konzepte (pp. 27–42). Opladen: Leske + Budrich.

Syring, M., Bohl, T., Kleinknecht, M., Kuntze, S., & Rehm, M. (in press). Kompetenzen, Antinomien, Fallarbeit. Vorschlag eines integrativen Professionalisierungsansatzes und praktische Implikationen für die erste Phase der Lehrerbildung. In C. Kraler & M. Schratz (Eds.), Interdisziplinäre und integrative Lehrerbildung. Münster: Waxmann.

Tall, D. (1988). Concept image and concept definition. In J. de Lange & M. Doorman (Eds.), Senior Secondary Mathematics Education (pp. 37–41). Utrecht: OW & OC.

Wagner, A., & Wörn, C. (2013). Veranschaulichungs- und Erklärmodelle zum Rechnen mit negative Zahlen—Ein Plädoyer für eine Reduzierung der Vielfalt an Repräsentationen im Unterricht. In J. Sprenger, A. Wagner, & M. Zimmermann (Eds.), Mathematik lernen, darstellen, deuten, verstehen—Didaktische Sichtweisen vom Kindergarten bis zur Hochschule (pp. 191–203). Wiesbaden: Springer Spektrum.

Weideneder, S., & Ufer, S. (2013). Which kinds of tasks do mathematics teachers select for instruction, and why? In A. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 385–392). Kiel: PME.

Zbiek, R. M., Heid, K., & Blume, G. W. (2007). Research on technology in mathematics education. In F. K. Lester (Eds.), Second handbook of research on mathematics teaching and learning—a project of the national council of teachers of mathematics (pp. 1169–1207). Charlotte: Information Age Publ.