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Making Implicit Algebraic Thinking Explicit: Exploiting National Characteristics of German Approaches

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Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

German mathematics teaching-units in primary school lack explicit algebra learning environments. Then again, many national characteristics of teachers’ attitudes and beliefs, everyday school life in mathematics classes, and deep-seated approaches that expect children to communicate and argue about mathematical findings, provide favorable prerequisites for algebra. Moreover, the contents taught have the potential to address algebraic thinking if approached from a new perspective. Yet, teachers and children are mostly unaware of the algebraic potential of certain tasks. This chapter includes three studies with a special explicit focus on possible key ideas, children’s abilities, and challenges offered by tasks. These evaluated ideas illustrate in interweaving perspectives feasible approaches that enable teachers to integrate algebraic thinking into their classroom culture. Moreover, the implicitly given opportunities revealed by the special focus of each study are hoped to lead to a sensible acceptance of algebraic thinking in primary math classes and its curriculum.

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References

  • Affolter, W., Baerli, G., Hurschler, H., Jaggi, B., Jundt, W. Krummacher, R., …, Wieland, G. (2003). Mathbu.ch 7 [mathematics book 7]. Zug, Bern: Klett & Balmer.

    Google Scholar 

  • Akinwunmi, K. (2012). Zur Entwicklung von Variablenkonzepten beim Verallgemeinern mathematischer Muster [Development of variable concepts by generalization of patterns]. Wiesbaden: Vieweg + Teubner.

    Google Scholar 

  • Banerjee, R., & Subramaniam, K. (2012). Evolution of a teaching approach for beginning algebra. Educational Studies in Mathematics, 80(3), 351–367.

    Google Scholar 

  • Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 165–184). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Bauersfeld, H. (1983). Subjektive Erfahrungsbereiche als Grundlage einer Interaktionstheorie des Mathematiklernens und –lehrens [Subjective fields of experience as foundation of an interaction theory of teaching and learning mathematics]. In H. Bauersfeld et al. (Eds.), Lernen und Lehren von Mathematik (pp. 1–56). Köln: Aulis.

    Google Scholar 

  • Brownell, J., Chen, J.-Q., & Ginet, L. (2014). Big ideas of early mathematics. Boston: Pearson.

    Google Scholar 

  • Carpenter, T.P., Levi, L., Franke, M.L., & Koehler Zeringue, J. (2005). Algebra in elementary school: Developing relational thinking. ZDM, 37(1), 53–59.

    Google Scholar 

  • Chick, H., & Harris, K. (2007). Grade 5/6 teachers’ perceptions of algebra in the primary school curriculum. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 121–128). Seoul: PME.

    Google Scholar 

  • Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalise: Models, representations and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187–214). Berlin: Springer.

    Google Scholar 

  • Devlin, K. (1997). Mathematics: The science of patterns – The search for order in life, mind, and the universe (2nd Edition). New York: Scientific American Library.

    Google Scholar 

  • Drijvers, P., Goddijn, A., & Kindt, M. (2011). Algebra education: Exploring topics and themes. In P. Drijvers (Ed.), Secondary algebra education: Revisiting topics and themes and exploring the unknown (pp. 5–26). Rotterdam: Sense Publishers.

    Google Scholar 

  • Franke, M., & Wynands, A. (1991). Zum Verständnis von Variablen – Testergebnisse in 9. Klassen Deutschlands [Understanding variables: Test results of grade 9 students in Germany]. Mathematik in der Schule, 29(10), 674–691.

    Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

    Google Scholar 

  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.

    Google Scholar 

  • Fujii, T., & Stephens, M. (2001). Fostering an understanding of algebraic generalization through numerical expressions: The role of quasi-variables. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12 th ICMI Study Conference: The future of the teaching and learning of algebra (pp. 258–364). Melbourne: University of Melbourne.

    Google Scholar 

  • Gallin, P. (2012). Die Praxis des Dialogischen Mathematikunterichts in der Grundschule [Practice of dialogical mathematics education in primary school]. Retrieved from http://www.sinus-an-grundschulen.de/fileadmin/uploads/Material_aus_SGS/Handreichung_Gallin_final.pdf.

  • Gerhard, S. (2013). How arithmetic education influences the learning of symbolic algebra. In B. Ubuz et al. (Eds.), CERME8: Proceedings of the 8 th Congress of the European Society of Research in Mathematics Education (pp. 430–439). Ankara: CERME.

    Google Scholar 

  • Kaput, J.J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaput, J.J., Blanton, M.L., & Moreno, L. (2008a). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19–55). New York: Routledge.

    Google Scholar 

  • Kaput, J.J., Carraher, D.W., & Blanton, M.L. (2008b). A skeptic’s guide to algebra in the early grades. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. xvii–xxi). New York: Routledge.

    Google Scholar 

  • Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.

    Google Scholar 

  • Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 11–49). Rotterdam: Sense Publishers.

    Google Scholar 

  • Kieran, C., Pang, J., Schifter, D., & Ng, S.F. (2016). Early algebra: Research into its nature, its learning, its teaching. New York: Springer.

    Google Scholar 

  • KMK [Kultusministerkonferenz] (2004). Bildungsstandards im Fach Mathematik für den Primarbereich [Standards in primary school mathematics]. Retrieved from http://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/2004/2004_10_15-Bildungsstandards-Mathe-Primar.pdf.

  • Krauthausen, G., & Scherer, P. (2007). Einführung in die Mathematikdidaktik [Introduction to mathematics education]. Heidelberg: Springer.

    Google Scholar 

  • Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: John Murray.

    Google Scholar 

  • Kühnel, J. (1916/1966). Neubau des Rechenunterrichts [Building a new mathematics education] (Original 1916, 11th Edition). Bad Heilbrunn: Klinkhardt.

    Google Scholar 

  • Lenz, D. (2016). Relational thinking and operating on unknown quantities. In T. Fritzlar et al. (Eds.), Problem solving in mathematics education. Proceedings of the 2015 joint conference of ProMath and the GDM working group on problem solving (pp. 173–181). Münster: WTM.

    Google Scholar 

  • Malle, G. (1993). Didaktische Probleme der elementaren Algebra [Didactical problems of elementary algebra]. Braunschweig: Vieweg.

    Google Scholar 

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289.

    Google Scholar 

  • Mason, J., Graham, A., Pimm, D., & Gowar, N. (1985). Routes to/Roots of Algebra. Milton Keynes: The Open University Press.

    Google Scholar 

  • Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in algebra. London: Sage.

    Google Scholar 

  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structures for all. Mathematics Education Research Journal, 21(2), 10–32.

    Google Scholar 

  • Müller, G., & Wittmann, E. (1984). Der Mathematikunterricht in der Primarstufe. [Mathematics education in primary school]. Braunschweig, Wiesbaden: Vieweg.

    Google Scholar 

  • Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.

    Google Scholar 

  • NCTM - National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics: Algebra. Reston, VA: The Council. Retrieved from http://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Algebra/.

  • Nührenbörger, M., & Schwarzkopf, R. (2016). Processes of mathematical reasoning of equations in primary mathematics lessons. In N. Vondrová (Ed.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (CERME 9) (pp. 316–323). Prague: CERME.

    Google Scholar 

  • Orton, A. (1999). Pattern in the teaching and learning of mathematics. London: Cassell.

    Google Scholar 

  • Radatz, H., Schipper, W., Ebeling, A., & Dröge, R. (1996). Handbuch für den Mathematikunterricht [Handbook for mathematics education]. Hannover: Schroedel.

    Google Scholar 

  • Radford, L. (2003). Gestures, speech, and the sprouting of signs. A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.

    Google Scholar 

  • Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai & E. Knuth (Eds.), Early algebraization. A global dialogue from multiple perspectives (pp. 303–322). Berlin: Springer.

    Google Scholar 

  • Rivera, F. (2013). Teaching and learning patterns in school mathematics: Psychological and pedagogical considerations. New York: Springer.

    Google Scholar 

  • Sawyer, W. W. (1964). Vision in elementary mathematics. Harmondsworth: Penguin Books.

    Google Scholar 

  • Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practice. Mahwah, N.J.: Lawrence Erlbaum Associates.

    Google Scholar 

  • Schütte, S. (2008). Qualität im Mathematikunterricht der Grundschule [Quality of mathematics lessons in primary school]. München: Oldenbourg.

    Google Scholar 

  • Selter, C. (1998). Building on children´s mathematics: A teaching experiment in grade 3. Educational Studies in Mathematics, 36(1), 1–27.

    Google Scholar 

  • Selter, C., & Spiegel, H. (1997). Wie Kinder rechnen [How children calculate]. Leipzig: Klett.

    Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.

    Google Scholar 

  • Specht, B.J. (2009). Variablenverständnis und Variablen verstehen [Understanding variables]. Hildesheim: Franzbecker.

    Google Scholar 

  • Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective. New York: Springer.

    Google Scholar 

  • Steinweg, A.S. (2001). Children’s understanding of number patterns. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 203–206). Utrecht: PME.

    Google Scholar 

  • Steinweg, A. S. (2003). ‘…the partner of 4 is plus 10 of this partner’ - Young children make sense of tasks on functional relations. In M. A. Mariotti (Ed.), CERME3: Proceedings of 3 rd Conference of the European Society for Research in Mathematics Education. Bellaria, Italy: CERME. Retrieved from http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG6/TG6_steinweg_cerme3.pdf.

  • Steinweg, A. S. (2006). Mathematikunterricht einmal ‚ohne‘ Rechnen [Mathematics lessons without calculating]. Die Grundschulzeitschrift, 20(191), 22–27.

    Google Scholar 

  • Steinweg, A. S. (2013). Algebra in der Grundschule [Algebra in primary school]. Heidelberg: Springer-Spektrum.

    Google Scholar 

  • Steinweg, A. S. (2014a). Mathematikdidaktische Forschung im Grundschulbereich [Research in primary school mathematics education]. Zeitschrift für Grundschulforschung, 7(1), 7–19.

    Google Scholar 

  • Steinweg, A. S. (2014b). Muster und Strukturen zwischen überall und nirgends [Pattern and structures every- and nowhere]. In A. S. Steinweg (Ed.), Mathematikdidaktik Grundschule (Vol. 4, pp. 51–66). Bamberg: University of Bamberg Press.

    Google Scholar 

  • Steinweg, A. S. (2017). Key ideas as guiding principles to support algebraic thinking in German primary schools. In T. Dooley & G. Gueudet (Eds.), CERME10: Proceedings of the 10 th Congress of the European Society of Research in Mathematics Education, Dublin, Ireland: CERME.

    Google Scholar 

  • Stephens, M., & Wang, X. (2008). Investigating some junctures in relational thinking: A study of year 6 and year 7 students from Australia and China. Journal of Mathematics Education, 1(1), 28–39.

    Google Scholar 

  • Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., …, Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics & Technology Education, 1(1), 81–104.

    Google Scholar 

  • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 8–19). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Walther, G., Neubrand, J., & Selter, Ch. (2008). Die Bildungsstandards Mathematik [The national standards in mathematics]. In G. Walther, M. van den Heuvel-Panhuizen, D. Granzer, & O. Köller (Eds.), Bildungsstandards für die Grundschule. Mathematik konkret (pp. 16–41). Berlin: Cornelsen Scriptor.

    Google Scholar 

  • Winter, H. (1991). Entdeckendes Lernen im Mathematikunterricht – Einblicke in die Ideengeschichte und ihre Bedeutung für die Pädagogik. 2. Aufl. [Learning by discovery in mathematics lessons]. Braunschweig, Wiesbaden: Vieweg.

    Google Scholar 

  • Wittmann, E. Ch. (1985). Objekte-Operationen-Wirkungen [Objects-operations-effects]. Mathematik lehren, (11), 7–11.

    Google Scholar 

  • Wittmann, E. Ch. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics, 29(4), 355–374.

    Google Scholar 

  • Wittmann, E. Ch. (1998). Design und Erforschung von Lernumgebungen als Kern der Mathematikdidaktik [Design of and research on learning environments as a core of mathematics education]. Beiträge zur Lehrerbildung, 16(3), 329–342.

    Google Scholar 

  • Wittmann, E. Ch., & Müller, G. N. (2007). Muster und Strukturen als fachliches Grundkonzept [Patterns and structures as fundamental subject-concept]. In G. Walther, M. van den Heuvel-Panhuizen, D. Granzer, & O. Köller (Eds.), Bildungsstandards für die Grundschule: Mathematik konkret (pp. 42–65). Berlin: Cornelsen.

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Bamberg, Mathematics Education, Markusplatz 3, 96047, Bamberg, Germany

    Anna S. Steinweg

  2. TU Dortmund, IEEM, 44221, Dortmund, Germany

    Kathrin Akinwunmi

  3. Faculty of Philosophy III, Martin Luther University Halle-Wittenberg, 06099, Halle, Germany

    Denise Lenz

Authors
  1. Anna S. Steinweg
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  2. Kathrin Akinwunmi
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  3. Denise Lenz
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Correspondence to Anna S. Steinweg .

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  1. Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada

    Carolyn Kieran

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Steinweg, A.S., Akinwunmi, K., Lenz, D. (2018). Making Implicit Algebraic Thinking Explicit: Exploiting National Characteristics of German Approaches. In: Kieran, C. (eds) Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68351-5_12

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  • DOI: https://doi.org/10.1007/978-3-319-68351-5_12

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