STUDENTS’ COMPETENCIES IN WORKING WITH FUNCTIONS IN SECONDARY MATHEMATICS EDUCATION—EMPIRICAL EXAMINATION OF A COMPETENCE STRUCTURE MODEL

In the subject matter of functional relationships, a student’s ability to translate from one form of representation to another is seen as a central competence. In the course of the HEUREKO project (heuristic work with representations of functional relationships and the diagnosis of mathematical competencies of students), a theoretical competence structure model comprising five competence dimensions was developed. These are based on four types of representation (graph, numerical table, algebraic equation and situational description) and correspond to the skill to translate from one type of representation to another. The following study was aimed to examine the facets of the model. The 5-dimensional model was empirically tested with a sample of N = 645 students of grades 9 and 10. This was accomplished by comparing competing item response models with regard to model fit using information criteria measures. In comparison with other possible model structures, our postulated 5-dimensional model showed the best model fit, suggesting that all translations are relevant for competence assessment and development. Furthermore, in order to allow for the identification of structural components of cognitive actions, the employed tasks used in the empirical testing were divided into categories with regard to the different demands of cognitive action. Our results suggest that cognitive actions may have a specific dimensional structure. These findings can contribute to a better diagnostic approach concerning specific strengths and weaknesses and can therefore foster students’ competencies.

[1]  Patrick W Thompson,et al.  Images of Rate and Operational Understanding of the Fundamental Theorem of Calculus , 1994 .

[2]  M. R. Novick,et al.  Statistical Theories of Mental Test Scores. , 1971 .

[3]  K. A. Ericsson,et al.  Verbal reports as data. , 1980 .

[4]  Baruch Schwarz,et al.  Measuring Integration of Information in Multirepresentational Software , 1993, Interact. Learn. Environ..

[5]  Shelton Ford,et al.  The Effect of Graphing Calculators and A Three-Core Representation Curriculum on College Students' Learning of Exponential and Logarithmic Functions , 2008 .

[6]  Eckhard Klieme,et al.  The concept of competence in educational contexts. , 2008 .

[7]  A. Gagatsis,et al.  The Effects of Different Modes of Representation on Mathematical Problem Solving. , 2004 .

[8]  Edward A. Silver,et al.  Teaching and Learning Mathematical Problem Solving : Multiple Research Perspectives , 1985 .

[9]  R. Glaser.  Advances in Instructional Psychology , 1978 .

[10]  H. Moosbrugger,et al.  Testtheorie und Fragebogenkonstruktion , 2008 .

[11]  Shaaron Ainsworth,et al.  Examining the Effects of Different Multiple Representational Systems in Learning Primary Mathematics , 2002 .

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

[13]  A. Raftery.  Bayesian Model Selection in Social Research , 1995 .

[14]  Shaaron Ainsworth,et al.  The functions of multiple representations , 1999, Comput. Educ..

[15]  Rita Borromeo Ferri,et al.  Theoretical and empirical differentiations of phases in the modelling process , 2006 .

[16]  Caroline Yoon,et al.  Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function , 2010 .

[17]  Kenneth A. Bollen,et al.  Structural Equations with Latent Variables , 1989 .

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

[19]  Kwaku Adu-gyamfi,et al.  Connections among Representations: The Nature of Students' Coordinations on a Linear Function task , 2007 .

[20]  G. Goldin.  Representational systems, learning, and problem solving in mathematics , 1998 .

[21]  Mark Wilson.  Cognitive diagnosis using item response models , 2008 .

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

[23]  Martha W. Alibali,et al.  Constructing Graphical Representations: Middle Schoolers' Intuitions and Developing Knowledge About Slope and Y-intercept , 2012 .

[24]  K. Hart,et al.  Children's understanding of mathematics: 11-16 , 1981 .

[25]  Michael J. Bossé,et al.  Lost in Translation: Examining Translation Errors Associated With Mathematical Representations , 2012 .

[26]  S. Gass,et al.  Stimulated Recall Methodology in Second Language Research , 2000 .

[27]  Mark Wilson,et al.  Validating a Learning Progression in Mathematical Functions for College Readiness , 2011 .

[28]  Michael J. Bossé,et al.  Assessing the Difficulty of Mathematical Translations: Synthesizing the Literature and Novel Findings , 2011, International Electronic Journal of Mathematics Education.

[29]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[30]  J. Confrey,et al.  Splitting, covariation, and their role in the development of exponential functions , 1995 .

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

[32]  Tina Seufert.  Supporting Coherence Formation in Learning from Multiple Representations , 2003 .

[33]  M. Bannert,et al.  Construction and interference in learning from multiple representation , 2003 .

[34]  Eckhard Klieme,et al.  Assessment of Competencies in Educational Contexts , 2008 .

[35]  J. Biggs,et al.  Teaching For Quality Learning At University , 1999 .

[36]  Melvin R. Novick,et al.  Some latent train models and their use in inferring an examinee's ability , 1966 .