Abstract
Teachers notice through the lens of their professional knowledge and views. This study hence focuses not solely on teachers’ noticing, but also on their knowledge and views, which allows insight into how noticing is informed and shaped by professional knowledge. As a discipline-specific perspective for noticing we chose dealing with multiple representations, since they play a double role for learning mathematics: On the one hand they are essential for mathematical understanding, but on the other hand they can also be an obstruction for learning. This comparative study takes into account pre-service as well as in-service teachers in order to explore the role of teaching experience for such professional knowledge, views and noticing. The participants answered a questionnaire addressing different components of specific knowledge and views. For eliciting the teachers’ theme-specific noticing, vignette-based questions were used. The data analysis was done mainly by quantitative methods, but was complemented by a qualitative in-depth analysis focusing on how the teachers’ theme-specific noticing was informed by different components of their professional knowledge. The results suggest that pre-service as well as in-service teachers do not fully understand the key role of multiple representations for learning mathematics in the sense of their discipline-specific significance. The participating in-service teachers distinguished themselves however from the pre-service teachers especially regarding their theme-specific noticing. Moreover, the evidence indicates that teachers’ noticing of critical instances of dealing with multiple representations draws on situated as well as on global knowledge and views.
Similar content being viewed by others
References
Acevedo Nistal, A., van Dooren, W., Clareboot, G., Elen, J., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM the International Journal on Mathematics Education, 41(5), 627–636.
Ainley, J., & Luntley, M. (2007). The role of attention in expert classroom practice. Journal of Mathematics Teacher Education, 10(1), 3–22.
Ainsworth, S. (2006). A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16, 183–198.
Ainsworth, S., Bibby, P., & Wood, D. (1998). Analysing the costs and benefits of multi-representational learning environments. In M. W. Someren, P. Reimann, H. P. A. Boshuizen, & T. de Jong (Eds.), Learning with multiple representations (pp. 120–134). Amsterdam: Pergamon.
Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25–62.
Ball, D. L. (1993). 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, NJ: Erlbaum.
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.
Berliner, D. C. (1994). Expertise: The wonder of exemplary performances. In J. M. Mangier & C. C. Block (Eds.), Creating powerful thinking in teachers and students: Diverse perspectives (pp. 161–186). Fort Worth, TX: Holt, Rinehart, & Winston.
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, 15(6), 1–23.
Brenner, M., Herman, S., Ho, H., & Zimmer, J. (1999). Cross-national comparison of representational competence. Journal for Research in Mathematics Education, 30(5), 541–557.
Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316.
Cobb, P. (2002). Modeling, symbolzing, and tool use in statistical data analysis. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 171–196). Dordrecht: Kluwer Academic Publishers.
Depaepe, F., Verschaffel, L., & Kelchtermans, G. (2013). Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research. Teaching and Teacher Education, 34, 12–25.
Dreher, A. (2012). Den Wechsel von Darstellungsformen fördern und fordern oder vermeiden? Über ein Dilemma im Mathematikunterricht (Fostering or avoiding changes between different forms of representations? A dilemma in the mathematics classroom). 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., Winkel, K., & Kuntze, S. (2012). Encouraging learning with multiple representations in the mathematics classroom. In T. Y. Tso (Ed.), Proceedings of the 36th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 231). Taipei, Taiwan: PME.
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. Hillsdale, NJ: Lawrence Erlbaum Associates.
Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521–544.
Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17, 105–121.
Gerster, H., & Schulz, R. (2000). Schwierigkeiten beim Erwerb mathematischer Konzepte im Anfangsunterricht: Bericht zum Forschungsprojekt, Rechenschwäche – Erkennen, Beheben, Vorbeugen. Retrieved November 23, 2013, from http://nbn-resolving.de/urn:nbn:de:bsz:frei129-opus-161
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, MA: NCTM.
Graham, A. T., Pfannkuch, M., & Thomas, M. (2009). Versatile thinking and the learning of statistical concepts. ZDM The International Journal on Mathematics Education, 41(5), 681–695.
Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2002). Symbolizing, modeling and tool use in mathematics education. Dordrecht: Kluwer Academic Publishers.
Heid, M. K., Blume, G., Zbiek, R. M., & Edwards, B. (1999). Factors that influence teachers learning to do interviews to understand students’ mathematical understandings. Educational Studies in Mathematics, 37, 223–249.
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.
Janvier, C. (1987). Translation processes in mathematics education. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 27–32). Hillsdale, NJ: Erlbaum.
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, VA: 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 (Primary students' strategies in changing representations: an investigation using the example of multiplication in school year two). Springer Spektrum: Schuljahr. Wiesbaden.
Kultusministerkonferenz (KMK). (2003). Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss (Education Standards in mathematics for an intermediate school-leaving certificate). Retrieved November 23, 2013, fromhttp://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2003/2003_12_04-BildBildungsstand-Mathe-Mittleren-SA.pdf
Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (2011). Professionelle Kompetenz von Lehrkräften. Ergebnisse des Forschungsprogramms COACTIV (Professional competence of teachers. Results from the COACTIV Project). Münster: Waxmann.
Kuntze, S. (2012). Pedagogical content beliefs: Global, content domain-related and situation-specific components. Educational Studies in Mathematics, 79(2), 273–292.
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. Advances in Mathematics Education series. NY: Springer.
Leinhardt, G., & Greeno, J. (1986). The cognitive skill of teaching. Journal of Educational Psychology, 78(2), 75–95.
Lerman, S. (2001). A review of research perspectives on mathematics teacher education. In F.-L. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 33–52). Dordrecht: Kluwer.
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, NJ: Lawrence Erlbaum.
Malle, G. (2004). Grundvorstellungen zu Bruchzahlen (Basic ideas “Grundvorstellungen” of rational numbers). Mathematik lehren, 123, 4–8.
Mason, J. (1987). Representing representing: Notes following the conference. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 207–214). Hillsdale, NJ: Erlbaum.
Mason, J. (2002). Researching your own practice: The discipline of noticing. New York: Routledge.
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.
Miller, K., & Zhou, X. (2007). Learning from classroom video: What makes it compelling and what makes it hard. In R. Goldman, R. Pea, B. Barron, & S. J. Derry (Eds.), Video research in the learning sciences (pp. 321–334). Mahwah, NJ: Erlbaum.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Pajares, F. M. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.
Pepin, B. (1999). Epistemologies, beliefs and conceptions of mathematics teaching and learning: The theory, and what is manifested in mathematics teachers’ practices in England, France and Germany. In B. Hudson, F. Buchberger, P. Kansanen, & H. Seel (Eds.), Didaktik/Fachdidaktik as science (s) of the teaching profession (pp. 127–46). Umeå: TNTEE.
Prediger, S. (2011). Why Johnny can’t apply multiplication? Revisiting the choice of operations with fractions. International Electronic Journal of Mathematics Education, 6(2), 65–88.
Rau, M. A., Aleven, V., Rummel, N., & Rohrbach, S. (2012). Sense making alone doesn't do it: Fluency matters too! ITS support for robust learning with multiple representations. In S. Cerri, W. Clancey, G. Papadourakis, & K. Panourgia (Eds.), Intelligent Tutoring Systems, 7315 (pp. 174–184). Berlin: Heidelberg: Springer.
Rau, M. A., Aleven, V., & Rummel, N. (2009). Intelligent tutoring systems with multiple representations and self-explanation prompts support learning of fractions. In V. Dimitrova, R. Mizoguchi, & B. du Boulay (Eds.), Proceedings of the 14th international conference 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.
Schifter, D. (2010). Examining the behavior of operations: Noticing early algebraic ideas. In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 204–220). New York: Routledge.
Schoenfeld, A. H. (2010). Noticing matters. A lot. Now what? In M. G. Sherin, V. R. Jacobs, & R. A. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 223–238). New York: Routledge.
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, NJ: 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.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
Siegler, R. S. (2010). Developing effective fractions instruction: A practice guide. Washington: National Center for Education Evaluation and Regional Assistance, IES, U.S. Department of Education.
Stern, E. (2002). Wie abstrakt lernt das Grundschulkind? Neuere Ergebnisse der entwicklungspsychologischen Forschung (How abstract does the primary student learn? Recent findings in developmental psychological research). In H. Petillon (Ed.), Individuelles Lernen in der Grundschule – Kinderperspektive und pädagogische Konzepte (pp. 27–42). Opladen: Leske + Budrich.
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.
Törner, G. (2002). Mathematical beliefs – A search for a common ground. In G. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 73–94). Dordrecht: Kluwer.
van Es, E. A. (2010). A framework for learning to notice student thinking. In M. G. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 134–151). Routledge: New York.
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.
Zbiek, R. M., Heid, K., & Blume, G. W. (2007). Research on technology in mathematics education. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: (pp. 1169–1207). Charlotte, NC: Information Age Publishing.
Acknowledgments
The data gathering phase of this study has been supported in the framework of the project ABCmaths which was funded with support from the European Commission (503215-LLP-1-2009-1-DE-COMENIUS-CMP). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.
Moreover, this study is closely connected to the work in the project La viDa-M which is funded by a research grant from Ludwigsburg University of Education.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dreher, A., Kuntze, S. Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educ Stud Math 88, 89–114 (2015). https://doi.org/10.1007/s10649-014-9577-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-014-9577-8