# Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom

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## 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.

## Keywords

Teacher professional knowledge Teacher noticing Multiple representations In-service teachers Pre-service teachers## Notes

### 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.

## 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.Google Scholar - Ainley, J., & Luntley, M. (2007). The role of attention in expert classroom practice.
*Journal of Mathematics Teacher Education, 10*(1), 3–22.CrossRefGoogle Scholar - Ainsworth, S. (2006). A conceptual framework for considering learning with multiple representations.
*Learning and Instruction, 16*, 183–198.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations.
*Computers in Human Behavior, 22*, 27–42.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics, 61*, 103–131.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - English, L. D., & Halford, G. S. (1995).
*Mathematics education: Models and processes*. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Even, R. (1990). Subject matter knowledge for teaching and the case of functions.
*Educational Studies in Mathematics, 21*(6), 521–544.CrossRefGoogle Scholar - Even, R. (1998). Factors involved in linking representations of functions.
*Journal of Mathematical Behavior, 17*, 105–121.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2002).
*Symbolizing, modeling and tool use in mathematics education*. Dordrecht: Kluwer Academic Publishers.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Kline, R. B. (2005).
*Principles and practice of structural equation modeling*(2nd ed.). New York: Guilford.Google Scholar - 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.Google Scholar - 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.Google Scholar - Kuntze, S. (2012). Pedagogical content beliefs: Global, content domain-related and situation-specific components.
*Educational Studies in Mathematics, 79*(2), 273–292.CrossRefGoogle Scholar - 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.Google Scholar - Leinhardt, G., & Greeno, J. (1986). The cognitive skill of teaching.
*Journal of Educational Psychology, 78*(2), 75–95.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - Malle, G. (2004). Grundvorstellungen zu Bruchzahlen (Basic ideas “Grundvorstellungen” of rational numbers).
*Mathematik lehren, 123*, 4–8.Google Scholar - 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.Google Scholar - Mason, J. (2002).
*Researching your own practice: The discipline of noticing*. New York: Routledge.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: NCTM.Google Scholar - Pajares, F. M. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct.
*Review of Educational Research, 62*(3), 307–332.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Shulman, L. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher, 15*(2), 4–14.CrossRefGoogle Scholar - 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.Google Scholar - 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.CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar