Use of ICT for acquiring, practicing and assessing algebraic expertise

For several years the skill level of students leaving secondary education in the Netherlands has been discussed. Lecturers in higher education –for example– complain about their freshmen’s apparent lack of algebraic skills. Another development in recent years is the advent of the use of technology in mathematics education. Combining algebraic expertise and ICT use, the aim of this study is to design an online environment for learning supported by formative assessment of both procedural skills and conceptual understanding in algebra, to investigate the effects of the environment, and to identify decisive factors that influence the outcome. The central research question, therefore, is: in what way can the use of ICT support acquiring, practicing and assessing algebraic expertise? This general question leads to several sub-questions, each related to an appropriate cycle in the study. The theoretical framework is based on the three key perspectives n ICT tool use, algebraic expertise, assessment and feedback. As we aim to design an intervention in several iterations, the research method is based on the principles of design research. Research takes place in one preparatory cycle and three subsequent cycles. The preliminary cycle concerns the design of criteria for an evaluation instrument for digital algebra tools. The instrument was used to choose an appropriate algebra tool for the remainder of the study, and design prototypical digital activities. In the first research cycle one-to-one think-aloud sessions were conducted with five 12th grade students. The results were used to examine the interplay between ICT and the acquisition of algebra, and determine what feedback could be added to the intervention. Based on the initial characteristics, the digital activities and feedback, the intervention was redesigned in an iterative fashion. The revised intervention was field tested in a second cycle for two classes (12th grade, wiskunde B, N=31), after which we made the final improvements based on three design principles all focusing on feedback. The final intervention was field tested in the third and last cycle in nine different schools (N=324). The use of the intervention for an average of five hours has a large effect on improving algebraic expertise, as post-test score is significantly higher than the pretest score. Furthermore, previous knowledge, time spent in self-test and summative test mode, and general attitude towards mathematics are the largest predictors for a high posttest outcome. The fact that these variables have nothing to do with ICT may indicate that indeed conventional pen-and-paper techniques and ICT techniques are reconciled. In line with this, the variables overall quality of the school (operationalized by trend exam grades), total practice time and the home work – school work ratio did not significantly predict the outcome. Discussion points for the study concern the interplay between acquiring skills and understanding, the extrapolation of the findings for a small sub-domain of algebraic knowledge to algebra as a whole and the methodology of the study.

[1]  Celia Hoyles,et al.  Learning mathematics and logo , 1992 .

[2]  Y. Chevallard L'analyse des pratiques enseignantes en théorie anthropologique du didactique , 1999 .

[3]  Paul Drijvers,et al.  Secondary Algebra Education , 2010 .

[4]  Andrew Gelman,et al.  Data Analysis Using Regression and Multilevel/Hierarchical Models , 2006 .

[5]  Slava Kalyuga,et al.  The Expertise Reversal Effect , 2003 .

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

[7]  David Kirshner,et al.  Visual Salience of Algebraic Transformations , 2004 .

[8]  Manu Kapur Productive Failure , 2006, ICLS.

[9]  Jeremy Kilpatrick,et al.  Adding It Up: Helping Children Learn Mathematics , 2013 .

[10]  Michael Beeson,et al.  Logic and Computation in MATHPERT: An Expert System for Learning Mathematics , 1989, Computers and Mathematics.

[11]  Yanghee Kim,et al.  The impact of learner attributes and learner choice in an agent-based environment , 2011, Comput. Educ..

[12]  Andy Field,et al.  Discovering statistics using SPSS, 2nd ed. , 2005 .

[13]  André Heck,et al.  Using SCORM to monitor student performance: experiences from secondary school practice , 2006 .

[14]  Joan Bliss,et al.  Tools for Exploratory Learning , 1989 .

[15]  M.H.A.M. van den Heuvel-Panhuizen,et al.  Assessment and realistic mathematics education , 1996 .

[16]  Reginald S. Lee,et al.  Multilevel Modeling: A Review of Methodological Issues and Applications , 2009 .

[17]  Luc Trouche,et al.  The didactical challenge of symbolic calculators : turning a computational device into a mathematical instrument , 2005 .

[18]  G. Gibbs,et al.  Conditions Under Which Assessment Supports Students’ Learning , 2005 .

[19]  R. Butler ENHANCING AND UNDERMINING INTRINSIC MOTIVATION: THE EFFECTS OF TASK‐INVOLVING AND EGO‐INVOLVING EVALUATION ON INTEREST AND PERFORMANCE , 1988 .

[20]  Angeliki Kolovou,et al.  Mathematical problem solving in primary school , 2011 .

[21]  P. M. Hiele Structure and Insight: A Theory of Mathematics Education , 1985 .

[22]  J. Voogt,et al.  International handbook of information technology in primary and secondary education , 2008 .

[23]  Paul Drijvers,et al.  The Co-Emergence of Machine Techniques, Paper-and-Pencil Techniques, and Theoretical Reflection: A Study of Cas use in Secondary School Algebra , 2006, Int. J. Comput. Math. Learn..

[24]  David H. Kirshner The Visual Syntax of Algebra. , 1989 .

[25]  M. Elshout-Mohr Feedback in Self-Instruction , 1994 .

[26]  Randolph M. Jones,et al.  Cascade Explains and Informs the Utility of Fading Examples to Problems , 2001 .

[27]  M. Goos,et al.  Teachers and Teaching: Theoretical Perspectives and Issues Concerning Classroom Implementation , 2009 .

[28]  Anthony S. Bryk,et al.  Hierarchical Linear Models: Applications and Data Analysis Methods , 1992 .

[29]  Paul Drijvers,et al.  Algebra Education: Exploring Topics and Themes , 2011 .

[30]  L. Corno,et al.  A Factorial Experiment in Teachers' Written Feedback on Student Homework: Changing Teacher Behavior a Little Rather Than a Lot. , 1985 .

[31]  Kurt VanLehn,et al.  A model of the self-explanation effect. , 1992 .

[32]  Jean-François Nicaud,et al.  Mixing Microworld and Cas Features in Building Computer Systems that Help Students Learn Algebra , 2004, Int. J. Comput. Math. Learn..

[33]  Tommy Dreyfus,et al.  STRUCTURE SENSE IN HIGH SCHOOL ALGEBRA: THE EFFECT OF BRACKETS , 2005 .

[34]  David Klein,et al.  A quarter century of US ‘math wars’ and political partisanship , 2007 .

[35]  Ruth Butler,et al.  Effects of no feedback, task-related comments, and grades on intrinsic motivation and performance , 1986 .

[36]  Paul Drijvers,et al.  Effects of feedback conditions for an online algebra tool , 2011 .

[37]  Larry Ambrose,et al.  The power of feedback. , 2002, Healthcare executive.

[38]  Luc Trouche,et al.  Managing the Complexity of Human/Machine Interactions in Computerized Learning Environments: Guiding Students’ Command Process through Instrumental Orchestrations , 2004, Int. J. Comput. Math. Learn..

[39]  Clare Lee,et al.  Assessment for Learning- putting it into practice , 2003 .

[40]  Chris Sangwin,et al.  Implementing Computer Algebra Enabled Questions for the Assessment and Learning of Mathematics. , 2008 .

[41]  Paul Drijvers,et al.  Digital Tools for Algebra Education: Criteria and Evaluation , 2010, Int. J. Comput. Math. Learn..

[42]  Jarmila Novotná,et al.  How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra , 2008 .

[43]  Joop J. Hox,et al.  Applied Multilevel Analysis. , 1995 .

[44]  D. Nicol,et al.  Formative assessment and self‐regulated learning: a model and seven principles of good feedback practice , 2006 .

[45]  Kaye Stacey,et al.  Developing algebraic insight , 2007 .

[46]  Michael Beeson Design principles of Mathpert: software to support education in algebra and calculus , 1998, Computer-Human Interaction in Symbolic Computation.

[47]  Paul Drijvers,et al.  Effects of a digital intervention on the development of algebraic expertise , 2012, Comput. Educ..

[48]  A. Ellington A Meta-Analysis of the Effects of Calculators on Students' Achievement and Attitude Levels in Precollege Mathematics Classes , 2003 .

[49]  Claude Dee Moran,et al.  Keeping the faith? , 1990 .

[50]  Seymour Papert,et al.  Mindstorms: Children, Computers, and Powerful Ideas , 1981 .

[51]  Qing Li,et al.  A Meta-analysis of the Effects of Computer Technology on School Students’ Mathematics Learning , 2010 .

[52]  Paul Drijvers,et al.  Symbol Sense Behavior in Digital Activities , 2010 .

[53]  Luc Trouche,et al.  The Complex Process of Converting Tools into Mathematical Instruments: The Case of Calculators , 1998, Int. J. Comput. Math. Learn..

[54]  Klaus Krippendorff,et al.  Answering the Call for a Standard Reliability Measure for Coding Data , 2007 .

[55]  Johan van Braak,et al.  ICT integration in the classroom: Challenging the potential of a school policy , 2008, Comput. Educ..

[56]  Pierre Vérillon,et al.  Cognition and artifacts: A contribution to the study of though in relation to instrumented activity , 1995 .

[57]  Christian Bokhove,et al.  Use of ICT in formative scenarios for algebraic skills , 2008 .

[58]  Martin Tessmer,et al.  Planning and conducting formative evaluations , 1993 .

[59]  Celia Hoyles,et al.  Looking back and looking forward , 1992 .

[60]  P. Black,et al.  Inside the Black Box: Raising Standards through Classroom Assessment , 2010 .

[61]  Richard K. Staley,et al.  From Example Study to Problem Solving: Smooth Transitions Help Learning , 2002 .

[62]  A. Schoenfeld Cognitive Science and Mathematics Education , 1987 .

[63]  Anno Van Streun Representations in applying functions , 2000 .

[64]  Mykola Pechenizkiy,et al.  Feedback adaptation in web-based learning systems , 2007 .

[65]  Tammy Schellens,et al.  Content analysis schemes to analyze transcripts of online asynchronous discussion groups: A review , 2006, Comput. Educ..

[66]  Margot Berger,et al.  Using CAS to solve a mathematics task: A deconstruction , 2010, Comput. Educ..

[67]  Paul A. Kirschner,et al.  Effects of attitudes and behaviours on learning mathematics with computer tools , 2010, Comput. Educ..

[68]  Chen-Lin C. Kulik,et al.  The Instructional Effect of Feedback in Test-Like Events , 1991 .

[69]  Philip Lambert,et al.  The Journal of Experimental Education , 1970 .

[70]  Chris DiGiano,et al.  IDEA: Identifying design principles in educational applets , 2005 .

[71]  Jacob Cohen,et al.  A power primer. , 1992, Psychological bulletin.

[72]  Kaye Stacey,et al.  Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software , 2010, Int. J. Comput. Math. Learn..

[73]  A. Schoenfeld The Math Wars , 2004 .

[74]  A. Arcavi Symbol sense : Informal sensemaking in formal mathematics Prologue , 2001 .

[75]  T. Crooks The Impact of Classroom Evaluation Practices on Students , 1988 .

[76]  Roy B. Clariana,et al.  A comparison of answer until correct feedback and knowledge of correct response feedback under two conditions of contextualization , 1990 .

[77]  Michèle Artigue,et al.  Learning Mathematics in a CAS Environment: The Genesis of a Reflection about Instrumentation and the Dialectics between Technical and Conceptual Work , 2002, Int. J. Comput. Math. Learn..

[78]  J. Sweller,et al.  The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra , 1985 .

[79]  Frank N. Dempster Synthesis of Research on Reviews and Tests. , 1991 .

[80]  L. Vygotsky Mind in Society: The Development of Higher Psychological Processes: Harvard University Press , 1978 .

[81]  Cornelia S. Große,et al.  How Fading Worked Solution Steps Works – A Cognitive Load Perspective , 2004 .

[82]  Jean-Baptiste Lagrange,et al.  L'intégration d'instruments informatiques dans l'enseignement: Une approche par les techniques , 2000 .

[83]  T. P. Carpenter,et al.  Using Knowledge of Children’s Mathematics Thinking in Classroom Teaching: An Experimental Study , 1989 .

[84]  Christian Bokhove,et al.  Implementing Feedback in a Digital Tool for Symbol Sense , 2010 .

[85]  Roel Bosker,et al.  Multilevel analysis : an introduction to basic and advanced multilevel modeling , 1999 .

[86]  Francine M Cheater,et al.  Expert consensus on the desirable characteristics of review criteria for improvement of health care quality , 2001 .

[87]  Celia Hoyles,et al.  Mathematics education and technology-rethinking the terrain : the 17th ICMI study , 2009 .

[88]  André Heck,et al.  Mathematics on the threshold , 2006 .

[89]  David Tall,et al.  Cognitive Conflict and the Learning of Mathematics , 1977 .

[90]  综合社会科学 The London Mathematical Society , 2012, From Servant to Queen: A Journey through Victorian Mathematics.

[91]  Paul Black,et al.  The Formative Purpose: Assessment Must First Promote Learning , 2004, Teachers College Record: The Voice of Scholarship in Education.

[92]  Gary Natriello The Impact of Evaluation Processes on Students , 1987 .

[93]  Susan McKenney,et al.  Educational design research , 2014 .