An insight into student understanding of functions in a graphing calculator environment

The teaching of functions plays a key role in the secondary mathematics curriculum. Electronic technologies afford deepening student understanding of functions by linking multiple representations. Findings from my previous research which investigated the understanding of functions by students in their final two years of secondary school whilst immersed in a graphing calculator learning environment will be presented and discussed. In the RITEMATHS project I am extending this research to consider a range of technologies and how teachers and students can realise the range of affordances offered by these technologies to support the teaching and learning of functions in a multiple representation environment.

[1]  L. Stenhouse Case Study and Case Records: towards a contemporary history of education , 1978 .

[2]  Nira Granott,et al.  Patterns of interaction in the co-construction of knowledge: Separate minds, joint effort, and weird creatures. , 1993 .

[3]  Jo Boaler,et al.  Experiencing School Mathematics: Teaching Styles, Sex, and Setting , 1997 .

[4]  E. Paul Goldenberg Mathematical, Technical, and Pedagogical Challenges in the Graphical Representation of Functions. [Final] Technical Report 88-4. , 1988 .

[5]  Claudia Marie Giamati The effect of graphing calculator use on students' understanding of variations on a family of equations and the transformations of their graphs. , 1990 .

[6]  Ed Dubinsky,et al.  The Concept of Function: Aspects of Epistemology and Pedagogy [MAA Notes, Volume 25] , 1992 .

[7]  S. Merriam Qualitative research in practice : examples for discussion and analysis , 2002 .

[8]  M. Goos,et al.  Do it this way! Metacognitive strategies in collaborative mathematical problem solving , 1996 .

[9]  Barry Kissane,et al.  The importance of being accessible: The graphics calculator in mathematics education , 1995 .

[10]  Andrea Scarantino,et al.  Affordances Explained , 2003, Philosophy of Science.

[11]  Alice F. Artz,et al.  Development of a Cognitive-Metacognitive Framework for Protocol Analysis of Mathematical Problem Solving in Small Groups , 1992 .

[12]  D. Silverman Interpreting Qualitative Data: Methods for Analysing Talk, Text and Interaction , 1994 .

[13]  Jill Brown Teacher approaches to graphing a difficult cubic function , 2004 .

[14]  Helen M. Doerr,et al.  The Teacher, the Task and the Tool: The Emergence of Classroom Norms , 1999 .

[15]  Paul Cobb,et al.  A method for conducting longitudinal analyses of classroom videorecordings and transcripts , 1996 .

[16]  Ruhama Even,et al.  Subject matter knowledge for teaching and the case of functions , 1990 .

[17]  Peter A. Bibby,et al.  Information technology and multiple representations: new opportunities – new problems , 1997 .

[18]  Kenneth Ruthven The influence of graphic calculator use on translation from graphic to symbolic forms , 1990 .

[19]  M. K. Heid,et al.  The Technological Revolution and the Reform of School Mathematics , 1997, American Journal of Education.

[20]  E. Knuth Student Understanding of the Cartesian Connection: An Exploratory Study , 2000 .

[21]  Susan A. Yoon Thinking Aloud with Thinkback: A Teacher Practitioner’s Guide for Enhancing General Metacognitive Problem Solving Skills , 2002 .

[22]  Harry F. Wolcott Writing Up Qualitative Research... Better , 2002, Qualitative health research.

[23]  A. Su,et al.  The National Council of Teachers of Mathematics , 1932, The Mathematical Gazette.

[24]  Orit Zaslavsky,et al.  Conceptual obstacles in the learning of quadratic functions , 1997 .

[25]  Geoffrey E. Mills,et al.  Educational Research: Competencies for Analysis and Application , 1995 .

[26]  M. Goos,et al.  Reshaping teacher and student roles in technology-enriched classrooms , 2000 .

[28]  Uri Leron,et al.  Being sloppy about slope: The effect of changing the scale , 2002 .

[29]  Anselm L. Strauss,et al.  Qualitative Analysis For Social Scientists , 1987 .

[30]  Rudolf vom Hofe Investigations into Students' Learning of Applications in Computer-Based Learning Environments , 2001 .

[31]  J. Gerring A case study , 2011, Technology and Society.

[32]  The effective use of computers and graphing calculators in college algebra , 1993 .

[33]  C. Hirsch Curriculum and Evaluation Standards for School Mathematics , 1988 .

[34]  M. Kathleen Heid,et al.  Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool. , 1988 .

[35]  Kathleen Hart,et al.  Children's Understanding of Mathematics , 1989 .

[36]  Zvia Markovits,et al.  Functions Today and Yesterday. , 1986 .

[37]  Ann M. Farrell Roles and Behaviors in Technology-Integrated Precalculus Classrooms , 1996 .

[38]  Sasha A. Barab,et al.  Guest Editors' Introduction: Rethinking Methodology in the Learning Sciences , 2001 .

[39]  Kristiina Kumpulainen Children's Talk During Collaborative Writing at the Computer , 1994 .

[40]  J. Maxwell Understanding and Validity in Qualitative Research , 1992 .

[41]  Sue Willis,et al.  A national statement on mathematics for Australian schools , 1991 .

[42]  Bill Gillham,et al.  Case Study Research Methods , 2000 .

[43]  Jerry K. Stonewater Strategies for problem solving , 1980 .

[44]  Jill Patricia Brown Defining moments in determining a complete graph in a graphing calculator teaching and learning environment , 2003 .

[45]  David Tall,et al.  Functions and Calculus , 1996 .

[46]  Cynthia Piez,et al.  Multiple Representations: Using Different Perspectives To Form a Clearer Picture. , 1997 .

[47]  Ian Alexander,et al.  An introduction to qualitative research , 2000, Eur. J. Inf. Syst..

[48]  Kenneth Ruthven,et al.  Calculators in the Mathematics Curriculum: the Scope of Personal Computational Technology , 1996 .

[49]  Melvin Wilson,et al.  The impact of graphics calculators on students' understanding of function , 1994 .

[50]  John D. Smith An Intuitive Approach to Calculus , 1994 .

[51]  Jean-Baptiste Lagrange,et al.  Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus , 1999, Int. J. Comput. Math. Learn..

[52]  Ruhama Even,et al.  Factors involved in linking representations of functions , 1998 .

[53]  Dave Pratt,et al.  Mediation of Mathematical Meaning Through the Graphic Calculator , 1997 .

[54]  K. Norwood,et al.  The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students' Understanding of Function. , 1999 .

[55]  Henk van der Kooij Functional Algebra with the Use of the Graphing Calculator , 2004 .

[56]  M. Ellett,et al.  Introduction to qualitative research. , 2002, Gastroenterology nursing : the official journal of the Society of Gastroenterology Nurses and Associates.

[57]  Kaye Stacey,et al.  Teaching with CAS in a Time of Transition , 2002 .

[58]  N. Movshovitz-Hadar A Constructive Transition From Linear To Quadratic Functions , 1993 .

[59]  J. M. Allen Mathematical Association of Victoria , 1949, The Mathematical Gazette.

[60]  R. Teese Undemocratic Schooling: Equity and Quality in Mass Secondary Education in Australia , 1999 .

[61]  David Steele The Wesley College technology enriched graphing project , 1994 .

[62]  E. Paul Goldenberg The difference between graphing software and educational graphing software , 1991 .

[63]  C. G. Williams Looking over Their Shoulder: Some Difficulties Students Have with Graphing Calculators. , 1993 .

[64]  Alan R. Osborne,et al.  Learning How to See: Students Graphing Difficulties. , 1991 .

[65]  M. Miles,et al.  The Qualitative Researcher's Companion , 2002 .

[66]  B. Macdonald The Portrayal of Persons as Evaluation Data , 1976 .

[67]  Gwendolyn M. Lloyd,et al.  Supporting Innovation: The Impact of a Teacher's Conceptions of Functions on His Implementation of a Reform Curriculum. , 1998 .

[68]  J. Morse Qualitative data analysis (2nd ed): Mathew B. Miles and A. Michael Huberman. Thousand Oaks, CA: Sage Publications, 1994. Price: $65.00 hardback, $32.00 paperback. 238 pp , 1996 .

[69]  R. Pea Practices of distributed intelligence and designs for education , 1993 .

[70]  J. Hiebert,et al.  Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis , 1986 .

[71]  C. Hirsch,et al.  Student preferences for representations of functions , 1992 .

[72]  J. Myers,et al.  Development of a professional school counselor identity: A grounded theory. , 1999 .

[73]  P. Galbraith,et al.  Applying mathematics with real world connections: metacognitive characteristics of secondary students , 1998 .

[74]  Rose Mary Zbiek Influences on Mathematics Teachers' Transitional Journeys in Teaching with CAS. , 2002 .

[75]  K. Eisenhardt Building theories from case study research , 1989, STUDI ORGANIZZATIVI.

[76]  W. Firestone Alternative Arguments for Generalizing From Data as Applied to Qualitative Research , 1993 .

[77]  T. Wood What Does it Mean to Teach Mathematics Differently , 2002 .

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

[79]  Elliot W. Eisner,et al.  The New Frontier in Qualitative Research Methodology , 1997 .

[80]  Harry F. Wolcott,et al.  Writing Up Qualitative Research , 1990 .

[81]  Barbara Hayes-Roth,et al.  A Cognitive Model of Planning , 1979, Cogn. Sci..

[82]  Graphical Representations of Speed: Obstacles Preservice K-8 Teachers Experience. , 2000 .

[83]  Egbert G. Harskamp,et al.  The graphics calculator and students’ solution strategies , 2000 .

[84]  Alan H. Schoenfeld,et al.  On Paradigms and Methods: What Do You Do When the Ones You Know Don't Do What You Want Them To? issues in the Analysis of Data in the Form of Videotapes , 1992 .

[85]  Michael Cavanagh,et al.  Students’ difficulties in operating a graphics calculator , 2000 .

[86]  A. Schoenfeld Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint) , 2009 .

[87]  Marina Penglase,et al.  The graphics calculator in mathematics education: A critical review of recent research , 1996 .

[88]  Ruhama Even Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept. , 1993 .

[89]  Tine Wedege To know or not to know – mathematics, that is a question of context , 1999 .

[90]  Tommy Dreyfus,et al.  Intuitive Functional Concepts: A Baseline Study on Intuitions. , 1982 .

[91]  Michal Yerushalmy,et al.  Effects of Computerized Feedback on Performing and Debugging Algebraic Transformations , 1991 .

[92]  Merrilyn Goos Technology as a tool for transforming mathematical tasks , 1998 .

[93]  Gaea Leinhardt,et al.  Functions, Graphs, and Graphing: Tasks, Learning, and Teaching , 1990 .

[94]  Sasha A. Barab,et al.  Constructing Networks of Action-Relevant Episodes: An In Situ Research Methodology , 2001 .

[95]  Robert F. Testa,et al.  Educational Research: Competencies for Analysis and Application , 1979 .

[96]  Franklin Demana,et al.  Calculators in Mathematics Teaching and Learning: Past, Present, and Future. Part 2: Technology and the Mathematics Classroom. , 2000 .

[97]  H. Doerr,et al.  You have printed the following article : Creating Meaning for and with the Graphing Calculator , 2008 .

[98]  Tommy Dreyfus,et al.  QuadFun—a case study of pupil computer interaction , 1991 .

[99]  R. Stake The art of case study research , 1995 .