Comparison Processes and Their Benefits for Learning Frequently Confused Concepts in Algebra

........................................................................................................................................ iii Zusammenfassung ......................................................................................................................... iv Acknowledgment ......................................................................................................................... vi

[1]  John R. Anderson.  Retrieval of propositional information from long-term memory , 1974 .

[2]  D. Gentner.  Structure‐Mapping: A Theoretical Framework for Analogy* , 1983 .

[3]  Alexander Jordan,et al.  Der Mathematikunterricht der PISA-Schülerinnen und -Schüler , 2005 .

[4]  J. Sweller,et al.  Structuring Effective Worked Examples , 1990 .

[5]  K. Holyoak,et al.  Mathematical problem solving by analogy. , 1991, Journal of experimental psychology. Learning, memory, and cognition.

[6]  James Hiebert.  Missing Links@@@Conceptual and Procedural Knowledge: The Case of Mathematics , 1987 .

[7]  Robert S. Siegler,et al.  A featural analysis of preschoolers' counting knowledge. , 1984 .

[8]  P. Chandler,et al.  Cognitive Load Theory and the Format of Instruction , 1991 .

[9]  B. Ross.  This is like that: The use of earlier problems and the separation of similarity effects. , 1987 .

[10]  D. Gentner,et al.  Analogical Learning in Negotiation Teams : Comparing Cases Promotes Learning and Transfer , 2003 .

[11]  Ann L. Brown,et al.  Training in Self-Explanation and Self-Regulation Strategies: Investigating the Effects of Knowledge Acquisition Activities on Problem Solving , 1995 .

[12]  Marianna Bogomolny,et al.  The role of example-generation tasks in students' understanding of linear algebra , 2006 .

[13]  B. Rittle-Johnson,et al.  Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. , 2007 .

[14]  James Hiebert,et al.  Instruction, Understanding, and Skill in Multidigit Addition and Subtraction , 1996 .

[15]  J. Flavell.  Development of children’s knowledge about the mental world , 2000 .

[16]  Dedre Gentner,et al.  Comparison Facilitates Children's Learning of Names for Parts , 2007 .

[17]  Michael Schneider,et al.  The developmental relations between conceptual and procedural knowledge: a multimethod approach. , 2010, Developmental psychology.

[18]  P. Moyer.  Are We Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics , 2001 .

[19]  M. Chi.  Constructing Self-Explanations and Scaffolded Explanations in Tutoring , 1996 .

[20]  K. Holyoak,et al.  Overcoming contextual limitations on problem-solving transfer. , 1989 .

[21]  K. Holyoak,et al.  Surface and structural similarity in analogical transfer , 1987, Memory & cognition.

[22]  D. Gentner,et al.  Where Hypotheses Come From: Learning New Relations by Structural Alignment , 2010 .

[23]  W. R. Garner.  The Processing of Information and Structure , 1974 .

[24]  J. Gibson,et al.  Perceptual learning; differentiation or enrichment? , 1955, Psychological review.

[25]  Dietrich Albert,et al.  Comprehension of Preference Graphs , 2002 .

[26]  B. Rittle-Johnson,et al.  Conceptual and procedural knowledge of mathematics: Does one lead to the other? , 1999 .

[27]  Lauretta M. Reeves,et al.  The Role of Content and Abstract Information in Analogical Transfer , 1994 .

[28]  John Sweller,et al.  Cognitive Load During Problem Solving: Effects on Learning , 1988, Cogn. Sci..

[29]  A. Renkl.  The worked-out-example principle in multimedia learning , 2005 .

[30]  J. Piaget,et al.  The Growth of Logical Thinking , 1959 .

[31]  R. Catrambone.  Improving examples to improve transfer to novel problems , 1994, Memory & cognition.

[32]  B. Rittle-Johnson,et al.  Learning from explaining: does it matter if mom is listening? , 2008, Journal of experimental child psychology.

[33]  S. Derry,et al.  Learning from Examples: Instructional Principles from the Worked Examples Research , 2000 .

[34]  P. Chandler,et al.  THE SPLIT‐ATTENTION EFFECT AS A FACTOR IN THE DESIGN OF INSTRUCTION , 1992 .

[35]  Randall P. Dahlberg,et al.  Facilitating Learning Events Through Example Generation , 1997 .

[36]  Daniel L. Schwartz,et al.  A time for telling , 1998 .

[37]  Karen C. Fuson,et al.  Conceptual Structures for Multiunit Numbers: Implications for Learning and Teaching Multidigit Addition, Subtraction, and Place Value , 1990 .

[38]  Alexander Renkl,et al.  Computer-supported example-based learning: When instructional explanations reduce self-explanations , 2006, Comput. Educ..

[39]  Kenneth D. Forbus,et al.  MAC/FAC: A Model of Similarity-Based Retrieval , 1995, Cogn. Sci..

[40]  ames Hiebert and Diana Wearne.  Procedures Over Concepts: The Acquisition of Decimal Number Knowledge , 2013 .

[41]  D. Gentner,et al.  Reasoning and learning by analogy. , 1997, The American psychologist.

[42]  B. Rittle-Johnson,et al.  Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process. , 2001 .

[43]  J. Sweller,et al.  Effects of schema acquisition and rule automation on mathematical problem-solving transfer. , 1987 .

[44]  John R. Anderson.  A Spreading Activation Theory of Memory , 1988 .

[45]  M. Chi.  Three Types of Conceptual Change: Belief Revision, Mental Model Transformation, and Categorical Shift , 2009 .

[46]  A. Renkl.  Worked-out examples: instructional explanations support learning by self- explanations , 2002 .

[47]  J. Sweller.  COGNITIVE LOAD THEORY, LEARNING DIFFICULTY, AND INSTRUCTIONAL DESIGN , 1994 .

[48]  Mitchell J. Nathan.  Learning through Self-Explanation of Mathematics Examples: Effects of Cognitive Load. , 1994 .

[49]  K. VanLehn,et al.  Cognitive skill acquisition. , 1996, Annual review of psychology.

[50]  Carolyn Kieran.  The learning and teaching of school algebra. , 1992 .

[51]  B. Rittle-Johnson,et al.  Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving , 2009 .

[52]  Elsbeth Stern,et al.  Spontaneous use of conceptual mathematical knowledge in elementary school children , 1992 .

[53]  D. Gentner,et al.  Avoiding Missed Opportunities in Managerial Life: Analogical Training More Powerful Than Individual Case Training , 2000 .

[54]  P. Kirschner,et al.  Taking the Load Off a Learner's Mind: Instructional Design for Complex Learning , 2003 .

[55]  John Sweller,et al.  Efficiency in Learning: Evidence-Based Guidelines to Manage Cognitive Load , 2005 .

[56]  L. R. Novick.  Analogical transfer, problem similarity, and expertise. , 1988, Journal of experimental psychology. Learning, memory, and cognition.

[57]  K. Holyoak,et al.  Children's development of analogical reasoning: insights from scene analogy problems. , 2006, Journal of experimental child psychology.

[58]  Dedre Gentner,et al.  Bootstrapping the Mind: Analogical Processes and Symbol Systems , 2010, Cogn. Sci..

[59]  J. Tuovinen,et al.  A comparison of cognitive load associated with discovery learning and worked examples , 1999 .

[60]  Ann L. Brown,et al.  How people learn: Brain, mind, experience, and school. , 1999 .

[61]  R. Siegler,et al.  Conscious and unconscious strategy discoveries: a microgenetic analysis. , 1998, Journal of experimental psychology. General.

[62]  D. Gentner,et al.  Structure mapping in analogy and similarity. , 1997 .

[63]  Lucia Mason,et al.  Fostering Understanding by Structural Alignment as a Route to Analogical Learning , 2004 .

[64]  Dedre Gentner,et al.  Learning by Analogical Bootstrapping , 2001 .

[65]  Dedre Gentner,et al.  Relations, Objects, and the Composition of Analogies , 2006, Cogn. Sci..

[66]  Christof Körner,et al.  Concepts and misconceptions in comprehension of hierarchical graphs , 2005 .

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

[68]  Shaaron Ainsworth,et al.  The effects of self‐explaining when learning with text or diagrams , 2003 .

[69]  D. Gentner,et al.  Comparison and Categorization in the Development of Relational Similarity , 1996 .

[70]  B. Ross.  Distinguishing Types of Superficial Similarities: Different Effects on the Access and Use of Earlier Problems , 1989 .

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

[72]  Rolene Liebenberg,et al.  LAYING THE FOUNDATION FOR ALGEBRA: DEVELOPING AN UNDERSTANDING OF STRUCTURE , 1998 .

[73]  D. Gentner,et al.  Comparison in the Development of Categories , 1999 .

[74]  Dedre Gentner,et al.  Using structural alignment to facilitate learning of spatial concepts in an informal setting , 2009 .

[75]  Carolyn Kieran.  The Core of Algebra: Reflections on its Main Activities , 2004 .

[76]  James D. Slotta,et al.  Helping Students Understand Challenging Topics in Science Through Ontology Training , 2006 .

[77]  Alexander Renkl,et al.  Learning from Worked-Out-Examples: A Study on Individual Differences , 1997, Cogn. Sci..

[78]  J. Star.  Reconceptualizing procedural knowledge. , 2005 .

[79]  S. Krauss,et al.  Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress , 2010 .

[80]  Anne Watson,et al.  Student-Generated Examples in the Learning of Mathematics , 2002 .

[81]  Christian Rogge,et al.  Misconceptions or Missing Conceptions , 2010 .

[82]  D. Gentner,et al.  Respects for similarity , 1993 .

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

[84]  Richard Catrambone,et al.  Reducing Cognitive Load and Fostering Cognitive Skill Acquisition: Benefits of Category-Avoiding Instructional Examples , 2019, Proceedings of EuroCogSci 03.

[85]  Alexander Renkl,et al.  LEARNING FROM WORKED-OUT EXAMPLES: THE TRANSITION FROM INSTRUCTIONAL EXPLANATIONS TO SELF-EXPLANATION PROMPTS , 2004 .

[86]  Dedre Gentner,et al.  Systematicity and Surface Similarity in the Development of Analogy , 1986, Cogn. Sci..

[87]  M. Alibali.  How children change their minds: strategy change can be gradual or abrupt. , 1999, Developmental psychology.

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

[89]  Vladimir M. Sloutsky,et al.  Relevant Concreteness and Its Effects on Learning and Transfer , 2005 .

[90]  E. Stern.  Wie abstrakt lernt das Grundschulkind , 2002 .

[91]  David F. Treagust,et al.  Teaching with analogies: A case study in grade-10 optics , 1993 .

[92]  B. Rittle-Johnson,et al.  It pays to compare: an experimental study on computational estimation. , 2009, Journal of experimental child psychology.

[93]  Michelene T. H. Chi,et al.  Eliciting Self-Explanations Improves Understanding , 1994, Cogn. Sci..

[94]  J. Greeno,et al.  Transfer of situated learning , 1996 .

[95]  Paul J. Feltovich,et al.  Categorization and Representation of Physics Problems by Experts and Novices , 1981, Cogn. Sci..

[96]  D. Gentner,et al.  Analogical encoding facilitates knowledge transfer in negotiation , 1999, Psychonomic bulletin & review.

[97]  B. Rittle-Johnson,et al.  The Importance of Prior Knowledge When Comparing Examples: Influences on Conceptual and Procedural Knowledge of Equation Solving , 2009 .

[98]  P. Hayes,et al.  Analogy , 2000, Encyclopedic Dictionary of Archaeology.

[99]  Elsbeth Stern,et al.  Knowledge restructuring as a powerful mechanism of cognitive development: how to lay an early foundation for conceptual understanding in formal domains , 2005 .

[100]  Gerald W. Bracey.  Facing the consequences: Using TIMSS for a closer look at US mathematics and science education , 2002 .

[101]  P. Chandler,et al.  Why Some Material Is Difficult to Learn , 1994 .

[102]  S. Vosniadou.  Capturing and modeling the process of conceptual change. , 1994 .

[103]  Jeffrey Bisanz,et al.  Strategic and nonstrategic processing in the development of mathematical cognition. , 1990 .

[104]  John R. Anderson.  Cognitive Psychology and Its Implications , 1980 .

[105]  Vincent Aleven,et al.  An effective metacognitive strategy: learning by doing and explaining with a computer-based Cognitive Tutor , 2002, Cogn. Sci..

[106]  A. Karmiloff-Smith.  Précis of Beyond modularity: A developmental perspective on cognitive science , 1994, Behavioral and Brain Sciences.

[107]  Michelene T. H. Chi,et al.  Self-Explanations: How Students Study and Use Examples in Learning To Solve Problems. Technical Report No. 9. , 1987 .

[108]  John R. Anderson,et al.  The role of examples and rules in the acquisition of a cognitive skill. , 1997, Journal of experimental psychology. Learning, memory, and cognition.

[109]  Andy P. Field,et al.  Discovering Statistics Using SPSS , 2000 .

[110]  Camilla K. Gilmore,et al.  Patterns of Individual Differences in Conceptual Understanding and Arithmetical Skill: A Meta-Analysis , 2009 .

[111]  Danielle S. McNamara,et al.  Self-Explanation Reading Training: Effects for Low-Knowledge Readers , 2004 .

[112]  Elsbeth Stern,et al.  Children's understanding of successive divisions in different contexts , 1996 .

[113]  L. Oakes,et al.  A Comparison of Infants' Categorization in Paired and Successive Presentation Familiarization Tasks. , 2005, Infancy : the official journal of the International Society on Infant Studies.

[114]  John D. Bransford,et al.  New approaches to instruction: because wisdom can't be told , 1989 .

[115]  Hironari Nozaki,et al.  Effects of self-explanation as a metacognitive strategy for solving mathematical word problems† , 2007 .

[116]  E. Stern,et al.  The Nature of Teachers' Pedagogical Content Beliefs Matters for Students' Achievement Gains: Quasi-Experimental Evidence from Elementary Mathematics. , 2002 .

[117]  D. Gentner,et al.  Learning and Transfer: A General Role for Analogical Encoding , 2003 .

[118]  Juan Carlos Bou-Llusar,et al.  Strategic knowledge transfer and its implications for competitive advantage: an integrative conceptual framework , 2006, J. Knowl. Manag..

[119]  Laura R. Novick,et al.  Component processes in analogical transfer: Mapping, pattern completion, and adaptation. , 1994 .

[120]  Elsbeth Stern,et al.  The development of mathematical competencies: Sources of individual differences and their developmental trajectories. , 2009 .

[121]  Slava Kalyuga,et al.  When problem solving is superior to studying worked examples. , 2001 .

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

[123]  L. Schauble,et al.  Beyond Modularity: A Developmental Perspective on Cognitive Science. , 1994 .

[124]  Michael McCloskey,et al.  Catastrophic Interference in Connectionist Networks: The Sequential Learning Problem , 1989 .

[125]  Michelene T. H. Chi,et al.  The Cambridge Handbook of Multimedia Learning: The Self-Explanation Principle in Multimedia Learning , 2005 .

[126]  Ana Stephens.  What “counts” as algebra in the eyes of preservice elementary teachers? , 2008 .

[127]  R. Mayer.  Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. , 2004, The American psychologist.

[128]  John R. Anderson.  Learning and memory: An integrated approach, 2nd ed. , 2000 .

[129]  D. Carnine,et al.  Instructional Design in Mathematics for Students with Learning Disabilities , 1997, Journal of learning disabilities.

[130]  D. Gentner,et al.  Splitting the Differences: A Structural Alignment View of Similarity , 1993 .

[131]  B. A. van Amerom,et al.  Reinvention of early algebra : developmental research on the transition from arithmetic to algebra = Heruitvinden van aanvankelijke algebra : ontwikkelingsonderzoek roud de overgang van rekenen naar algebra , 2002 .

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

[133]  James P. Byrnes,et al.  Role of Conceptual Knowledge in Mathematical Procedural Learning , 1991 .

[134]  Constance Kamii,et al.  To teach or not to teach algorithms , 1997 .

[135]  F. Paas,et al.  Cognitive Architecture and Instructional Design , 1998 .

[136]  K. Holyoak,et al.  The analogical mind. , 1997, The American psychologist.

[137]  John Sweller,et al.  Cognitive technology: Some procedures for facilitating learning and problem solving in mathematics and science. , 1989 .

[138]  D. Gentner.  The development of relational category knowledge , 2005 .

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

[140]  Miriam Bassok,et al.  OBJECT-BASED REPRESENTATIONS : TRANSFER BETWEEN CASES OF CONTINUOUS AND DISCRETE MODELS OF CHANGE , 1995 .

[141]  B. Rittle-Johnson,et al.  Promoting transfer: effects of self-explanation and direct instruction. , 2006, Child development.

[142]  Michael Schneider,et al.  Mental number line, number line estimation, and mathematical achievement : Their interrelations in grades 5 and 6 , 2009 .

[143]  T. S. Hilbert,et al.  Learning to teach with worked-out examples: a computer-based learning environment for teachers , 2008, J. Comput. Assist. Learn..

[144]  Early training: who, what, when, why, and how? , 1997 .

[145]  Kelly S. Mix,et al.  Preschoolers' use of surface similarity in object comparisons: taking context into account. , 2006, Journal of experimental child psychology.

[146]  D. Gentner,et al.  Making a silk purse out of two sow's ears: young children's use of comparison in category learning. , 2002, Journal of experimental psychology. General.

[147]  Jerry M. Mendel,et al.  Generating fuzzy rules by learning from examples , 1992, IEEE Trans. Syst. Man Cybern..

[148]  Kenneth D. Forbus,et al.  The Roles of Similarity in Transfer: Separating Retrievability From Inferential Soundness , 1993, Cognitive Psychology.

[149]  K. Holyoak,et al.  Schema induction and analogical transfer , 1983, Cognitive Psychology.

[150]  H. Braathe.  COMMUNICATIVE POSITIONINGS AS IDENTIFICATIONS IN MATHEMATICS TEACHER EDUCATION , 2010 .

[151]  A. Renkl.  Learning mathematics from worked-out examples: Analyzing and fostering self-explanations , 1999 .

[152]  Dedre Gentner,et al.  Spatial Mapping in Preschoolers: Close Comparisons Facilitate Far Mappings , 2001 .

[153]  R. Atkinson,et al.  Transitioning From Studying Examples to Solving Problems: Effects of Self-Explanation Prompts and Fading Worked-Out Steps. , 2003 .

[154]  D. McNamara.  SERT: Self-Explanation Reading Training , 2004 .

[155]  Alexander Renkl,et al.  From Studying Examples to Solving Problem: Fading Worked-Out Solution Steps Helps Learning , 2000 .

[156]  Katharina Scheiter,et al.  Goal Configurations and Processing Strategies as Moderators Between Instructional Design and Cognitive Load: Evidence From Hypertext-Based Instruction , 2003 .

[157]  Alexander Renkl,et al.  Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations , 2009 .

[158]  Carolyn Kieran.  Algebraic Thinking in the Early Grades: What Is It? , 2004 .

[159]  Robert S. Siegler,et al.  The relation between conceptual and procedural knowledge in learning mathematics: A review , 2021, The Development of Mathematical Skills.

[160]  Elsbeth Stern,et al.  Abstract reasoning with mathematical constructs , 1997 .