Framework for Examining the Transformation of Mathematics and Mathematics Learning in the Transition from School to University

An Analysis of German Textbooks from Upper Secondary School and the First Semester
  • Maike Vollstedt
  • Aiso Heinze
  • Kristin Gojdka
  • Stefanie Rach
Chapter
First Online:

Abstract

Throughout the last decade, increasing attention has been given to the discontinuity phenomena of university students in mathematics during their transition from school to university. We hypothesize that two transformations in this transition period have played an important role: the transformation of the character of mathematics and the transformation of the learning strategies necessary at school and at university. Following this hypothesis, we will present a study analyzing and comparing German textbooks at upper secondary level and university level, respectively. We assume that both transformations can be understood more deeply when we examine the way textbooks are designed. Hence, a categorical system has been developed which focuses on the criteria such as “development of concepts”, “deduction of theorems”, “proof” and “tasks” as well as “motivation”, and “structure and visual representation”. This article presents the developed framework and discusses results from two feasibility studies conducted with different widely used German textbooks at both school and university levels.

Keywords

Text book comparison Character of mathematics Learning strategies Secondary school University Double discontinuity High dropout rate Theory-based Development of framework Feasibility studies Development of concepts Deduction of theorems Proof Tasks Motivation Structure and visual representation Reliability Interrater agreement 
This is a preview of subscription content, log in to check access.

References

  1. Alsina, C. (2001). Why the professor must be a stimulating teacher. In D. Holton (Ed.), The teaching and learning of mathematics at university level. An ICMI study (pp. 3–12). Dordrecht: Kluwer.Google Scholar
  2. Ausubel, D. P. (1960). The use of advance organizers in the learning and retention of meaningful verbal material. Journal of Educational Psychology, 51(5), 267–272.CrossRefGoogle Scholar
  3. Beutelspacher, A. (2010). Lineare Algebra: Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen. 7th edition. Wiesbaden: Vieweg+Teubner.Google Scholar
  4. Biermann, H. R., & Jahnke, H. N. (2013). How 18th century mathematics was transformed into 19th century school curricula. (this volume).Google Scholar
  5. Boero, P. (1999). Argumentation of mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof. Retrieved June 3, 2011 from http://www-didactique.imag.fr/preuve/Newsletter/990708Theme/990708ThemeUK.html
  6. Brändström, A. (2005). Differentiated tasks in mathematics textbooks: An analysis of the levels of difficulty. Licentiate Thesis: Vol. 18. Lulea: Lulea University of Technology, Department of Mathematics.Google Scholar
  7. Brandt, D., & Reinelt, G. (2009). Lambacher Schweizer Gesamtband Oberstufe mit CAS Ausgabe B. Stuttgart: Klett Verlag.Google Scholar
  8. Chi, M. T. H., Bassok, M., Lewis, M., Reimann, P., & Glasser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.CrossRefGoogle Scholar
  9. Daniels, Z. (2008). Entwicklung schulischer Interessen im Jugendalter. Münster: Waxmann.Google Scholar
  10. Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York: Plenum.CrossRefGoogle Scholar
  11. Deiser, O., & Reiss, K. (2013). Knowledge transformation between secondary school and university mathematics. (this volume).Google Scholar
  12. Dörfler, W., & McLone, R. (1986). Mathematics as a school subject. In B. Christiansen, A. G. Howson, & M. Otte (eds.), Perspectives on mathematics education (pp. 49–97). Reidel: Dordrecht.CrossRefGoogle Scholar
  13. Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Mathematics education library: Vol. 11. Advanced mathematical thinking (pp. 25–41). Dordrecht: Kluwer.Google Scholar
  14. Drüke-Noe, C., Herd, E., König, A., Stanzel, M., & Stühler, A. (2008). Lambacher Schweizer 9 Ausgabe A. Stuttgart: Klett Verlag.Google Scholar
  15. Engelbrecht, J. (2010). Adding structure to the transition process to advanced mathematical activity. International Journal of Mathematical Education in Science and Technology, 41(2), 143–154.CrossRefGoogle Scholar
  16. Forster, O. (2008). Analysis 1. Differential- und Integralrechnung in einer Veränderlichen. Wiesbaden: Vieweg.Google Scholar
  17. De Guzman, M., Hodgson, B. R., Robert, A., & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. Documenta Mathematica, Extra Volume ICME 1998 (III), 747–762.Google Scholar
  18. Griesel, H., & Postel, H. (Eds.). (2001). Elemente der Mathematik 11 Druck A. Schülerband, 5th edition. Hannover: Schroedel.Google Scholar
  19. Griesel, H., Postel, H., & Suhr, F. (Eds.) (2007). Elemente der Mathematik Leistungskurs Analysis Druck A. 6th edition. Hannover: Schroedel.Google Scholar
  20. Griesel, H., Postel, H., & Suhr, F. (Eds) (2008). Elemente der Mathematik 8 Druck A. Schülerband. Hannover: Schroedel.Google Scholar
  21. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. J. Bishop et al. (Hrsg.), International handbook of mathematics education. Dodrecht: Kluwer.Google Scholar
  22. Heinze, A., & Reiss, K. (2007). Reasoning and proof in the mathematics classroom. Analysis, 27(2–3), 333–357.Google Scholar
  23. Heublein, U., Hutzsch, C., Schreiber, J., Sommer, D., & Besuch, G. (2009). Ursachen des Studienabbruchs in Bachelor- und in herkömmlichen Studiengängen: Ergebnisse einer bundesweiten Befragung von Exmatrikulierten des Studienjahres 2007/08. Hannover: Hochschul-Informations-System GmbH.Google Scholar
  24. Heymann, H. W. (2003). Why teach mathematics? A focus on general education. Dordrecht: Kluwer.CrossRefGoogle Scholar
  25. vom Hofe, R. (1995). Grundvorstellungen mathematischer Inhalte. Heidelberg: Spektrum.Google Scholar
  26. Howson, A. G. (1995). Mathematics textbooks: A comparative study of grade 8 texts. TIMSS monograph: Vol. 3. Vancouver: Pacific Educational Press.Google Scholar
  27. Hoyles, C., Newman, K., & Noss, R. (2001). Changing patterns of transition from school to university mathematics. International Journal of Mathematical Education in Science and Technology, 32(6), 829–845.CrossRefGoogle Scholar
  28. Kaiser, G. (1999). Comparative studies on teaching mathematics in England and Germany. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 140–150). London: Falmer Press.Google Scholar
  29. Kaiser, G., & Buchholtz, N. (2013). Overcoming the gap between university and school mathematics: The impact of an innovative programme in mathematics teacher education at the Justus-Liebig-University in Giessen. (this volume).Google Scholar
  30. Kajander, A., & Lovric, M. (2009). Mathematics textbooks and their potential role in supporting misconceptions. International Journal of Mathematical Education in Science and Technology, 40(2), 173–181.CrossRefGoogle Scholar
  31. Kawanaka, T., Stigler, J. W., & Hiebert, J. (1999). Studying mathematics classrooms in Germany, Japan and the United States: Lessons from the TIMSS videotape study. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 140–150). London: Falmer Press.Google Scholar
  32. Kettler, M. (1998). Der Symbolschock: Ein zentrales Lernproblem im mathematisch-wissenschaftlichen Unterricht. Frankfurt am Main: Lang.Google Scholar
  33. Klauer, K., & Leutner, K. (2007). Lehren und Lernen: Einführung in die Instruktionspsychologie. Weinheim: Beltz.Google Scholar
  34. KMK. (2004). Beschlüsse der Kultusministerkonferenz: Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss. Beschluss vom 4.12.2003. München: Wolters Kluwer.Google Scholar
  35. Königsberger, K. (2004). Analysis 1. 6th edition. Heidelberg: Springer.CrossRefGoogle Scholar
  36. Kunter, M. (2005). Multiple Ziele im Mathematikunterricht. Münster: Waxmann.Google Scholar
  37. Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33, 159–174.CrossRefGoogle Scholar
  38. Langer, I., Schulz von Thun, F., & Tausch, R. (1974). Verständlichkeit in Schule, Verwaltung, Politik und Wissenschaft. München: Reinhardt.Google Scholar
  39. Langer, I., Schulz von Thun, F., & Tausch, R. (2006). Sich verständlich ausdrücken. München: Reinhardt.Google Scholar
  40. Langer, I., Schulz von Thun, F., Meffert, J., & Tausch, R. (1973). Merkmale der Verständlichkeit schriftlicher Informations- und Lehrtexte. Zeitschrift für experimentelle und angewandte Psychologie, 20(2), 269–286.Google Scholar
  41. Mayer, R. E., & Gallini, J. K. (1990). When is an illustration worth ten thousand words? Journal of Educational Psychology, 82(4), 715–726.CrossRefGoogle Scholar
  42. Mayer, R. E., & Johnson, C. I. (2008). Revising the redundancy principle in multimedia learning. Journal of Educational Psychology, 100(2), 380–386.CrossRefGoogle Scholar
  43. Mayer, R. E., & Moreno, R. (1998). A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory. Journal of Educational Psychology, 90(2), 312–320.CrossRefGoogle Scholar
  44. OECD. (2010). PISA 2012 Mathematics Framework. Draft version. Paris: OECD.Google Scholar
  45. Pepin, B. (2013). Student transition to university mathematics education: Transformation of people, tools and practices. (this volume).Google Scholar
  46. Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: A way to understand teaching and learning cultures. Zentralblatt für Didaktik der Mathematik, 33(5), 158–175.CrossRefGoogle Scholar
  47. Rach, S., & Heinze, A. (2011). Studying Mathematics at the University: The influence of learning strategies. In Ubunz, B. (Ed.). Proceedings of the 35rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 9–16). Ankara, Turkey: PME.Google Scholar
  48. Rakoczy, K. (2008). Motivationsunterstützung im Mathematikunterricht: Unterricht aus der Perspektive von Lernenden und Beobachtern. Münster: Waxmann.Google Scholar
  49. Reiss, K., Heinze, A., Kuntze, S., Kessler, S., Rudolph-Albert, F., & Renkl, A. (2006). Mathematiklernen mit heuristischen Lösungsbeispielen. In M. Prenzel & L. Allolio-Näcke (Hrsg.), Untersuchungen zur Bildungsqualität von Schule (pp. 194–208). Münster: Waxmann.Google Scholar
  50. Rezat, S. (2006). The structure of German mathematics textbooks. Zentralblatt für Didaktik der Mathematik, 38(6), 482–487.CrossRefGoogle Scholar
  51. Rezat, S. (2009). Das Mathematikbuch als Instrument des Schülers: Eine Studie zur Schulbuchnutzung in den Sekundarstufen. Wiesbaden: Vieweg+Teubner.CrossRefGoogle Scholar
  52. Ryan, R. M., & Deci, E. L. (2002). Overview of self-determination theory: An organismic dialectical perspective. In E. L. Deci & R. M. Ryan (Eds.), Handbook of self-determination research (pp. 3–33). Rochester: University of Rochester Press.Google Scholar
  53. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H. C., Wiley, D. E., Cogan, L. S., & Wolfe, R. G. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco: Jossey-Bass.Google Scholar
  54. Sweller, J. (2005). Implications of cognitive load theory for multimedia learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 19–30). New York: Cambridge University Press.CrossRefGoogle Scholar
  55. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  56. Valverde, G. A. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer.CrossRefGoogle Scholar
  57. Vollrath, H.-J. (1984). Methodik des Begriffslehrens. Stuttgart: Klett.Google Scholar
  58. Vollstedt, M. (2011). Sinnkonstruktion und Mathematiklernen in Deutschland und Hongkong: Eine rekonstruktiv-empirische Studie. Wiesbaden: Vieweg+Teubner.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  • Maike Vollstedt
    • 1
  • Aiso Heinze
    • 1
    • 2
  • Kristin Gojdka
    • 1
  • Stefanie Rach
    • 1
  1. 1.Institute for Mathematics, Mathematics EducationFree University BerlinBerlinGermany
  2. 2.Department of Mathematics EducationIPN—Leibniz Institute for Science and Mathematics EducationKielGermany

Personalised recommendations