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

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.

[1]  J. R. Landis,et al.  The measurement of observer agreement for categorical data. , 1977, Biometrics.

[2]  Tommy Dreyfus,et al.  Advanced Mathematical Thinking Processes , 2002 .

[3]  Gabriele Kaiser,et al.  International Comparisons in Mathematics Education , 1999 .

[4]  E. Deci,et al.  Overview of self-determination theory: An organismic-dialectical perspective. , 2002 .

[5]  Miroslav Lovric,et al.  Mathematics textbooks and their potential role in supporting misconceptions , 2009, The Best Writing on Mathematics 2010.

[6]  R. Mayer,et al.  A Split-Attention Effect in Multimedia Learning: Evidence for Dual Processing Systems in Working Memory , 1998 .

[7]  D. Ausubel.  The use of advance organizers in the learning and retention of meaningful verbal material. , 1960 .

[8]  Friedrich Sauvigny.  Differential- und Integralrechnung in einer Veränderlichen , 2014 .

[9]  Hans Niels Jahnke,et al.  How Eighteenth-Century Mathematics Was Transformed into Nineteenth-Century School Curricula , 2014 .

[10]  Michael Otte,et al.  Perspectives on Mathematics Education , 1986 .

[11]  Hans Werner Heymann.  Why Teach Mathematics?: A Focus on General Education , 2003 .

[12]  J. Sweller.  Implications of Cognitive Load Theory for Multimedia Learning , 2005, The Cambridge Handbook of Multimedia Learning.

[13]  Mareike Kunter,et al.  Multiple Ziele im Mathematikunterricht , 2005 .

[14]  Rudolf vom Hofe,et al.  Grundvorstellungen mathematischer Inhalte , 1995 .

[15]  E. Deci,et al.  Handbook of Self-Determination Research , 2002 .

[16]  Joan K. Gallini,et al.  When Is an Illustration Worth Ten Thousand Words , 1990 .

[17]  Cheryl I. Johnson,et al.  Revising the Redundancy Principle in Multimedia Learning. , 2008 .

[18]  Celia Hoyles,et al.  Changing patterns of transition from school to university mathematics , 2001 .

[19]  Edward L. Deci,et al.  Intrinsic Motivation and Self-Determination in Human Behavior , 1975, Perspectives in Social Psychology.

[20]  Matthew W. Lewis,et al.  Self-Explonations: How Students Study and Use Examples in Learning to Solve Problems , 1989, Cogn. Sci..

[21]  Otto Forster,et al.  Analysis 1 : Differential- und Integralrechnung einer Veränderlichen , 2008 .

[22]  Detlev Leutner,et al.  Lehren und Lernen : Einfuehrung in die Instruktionspsychologie , 2007 .

[23]  Claudi Alsina,et al.  Why the Professor Must be a Stimulating Teacher , 2001 .

[24]  W. Dörfler,et al.  Mathematics as a School Subject , 1986 .

[25]  Martina Kettler.  Der Symbolschock: Ein zentrales Lernproblem im mathematisch-naturwissenschaftlichen Unterricht , 2001 .

[26]  David Tall,et al.  Concept image and concept definition in mathematics with particular reference to limits and continuity , 1981 .

[27]  Hans Werner Heymann,et al.  Why Teach Mathematics , 2003 .

[28]  Mogens Niss.  Discussion Document for a forthcoming ICMI Study On the Teaching and Learning of Mathematics at University Level , 1998 .

[29]  Aiso Heinze,et al.  Reasoning and proof in the mathematics classroom , 2007 .

[30]  Sebastian Rezat.  Das Mathematikschulbuch als Instrument des Schülers — Eine empirische Studie zur Schulbuchnutzung in den Sekundarstufen , 2009 .

[31]  William H. Schmidt,et al.  According to the Book , 2002 .

[32]  David Tall,et al.  Advanced Mathematical Thinking , 1994 .

[33]  A. G. Howson,et al.  Mathematics textbooks : a comparative study of grade 8 texts , 1995 .

[34]  Anna Brändström,et al.  Differentiated tasks in mathematics textbooks : an analysis of the levels of difficulty , 2005 .

[35]  Gilbert A. Valverde.  According to the Book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks , 2002 .

[36]  Birgit Pepin,et al.  Mathematics textbooks and their use in English, French and German classrooms: a way to understand teaching and learning cultures , 2001 .

[37]  Reinhard Tausch,et al.  Sich verständlich ausdrücken , 2002 .

[38]  Maike Vollstedt.  Sinnkonstruktion und Mathematiklernen in Deutschland und Hongkong , 2011 .

[39]  Johann Engelbrecht,et al.  Adding structure to the transition process to advanced mathematical activity , 2010 .

[40]  Richard Mayer,et al.  Multimedia Learning , 2001, Visible Learning Guide to Student Achievement.

[41]  Jeremy Kilpatrick,et al.  International handbook of mathematics education , 1997 .

[42]  Sebastian Rezat,et al.  Das Mathematikbuch als Instrument des Schülers , 2009 .

[43]  Gabriele Kaiser,et al.  Overcoming the Gap Between University and School Mathematics , 2014 .

[44]  William H. Schmidt,et al.  Why Schools Matter: A Cross-National Comparison of Curriculum and Learning , 2001 .

[45]  G. Hanna,et al.  Proof and Proving , 1996 .

[46]  Beu Birgit Pepin,et al.  Student transition into university mathematics education: transformation of people, tools and practices , 2014 .

[47]  Sebastian Rezat,et al.  The Structures of German Mathematics Textbooks , 2006 .

[48]  Zoe Daniels,et al.  Entwicklung schulischer Interessen im Jugendalter , 2008 .

[49]  Kristina Reiss,et al.  Knowledge Transformation Between Secondary School and University Mathematics , 2014 .