# Planning and Generating Sequences of Exercises for the Assessment and Development of Mathematical Knowledge and Proficiency

## Teaching Problem-solving Skills

Teaching Problem-solving Skills by Alan H. Schoenfeld, Problem Solving: A Handbook for Teachers by Stephen Krulik and Jesse A. Rudnick, Problem Solving by Miriam Bassok and Laura R. Novick, Learning to Solve Problems: An Instructional Design Guide by David H. Jonassen, Toward a Design Theory of Problem Solving by David H. Jonassen, Designing Knowledge Scaffolds to Support Mathematical Problem Solving by Bethany Rittle-Johnson and Kenneth R. Koedinger, Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics by Alan H. Schoenfeld, Thinking, Problem Solving, Cognition by Richard E. Mayer, Cognitive Processes in Well‐defined and Ill‐defined Problem Solving by Gregory Schraw, Michael E. Dunkle and Lisa D. Bendixen, Conceptual Structures in Mathematical Problem Solving by Victor Cifarelli, Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process by Bethany Rittle-Johnson, Robert S. Siegler and Martha W. Alibali, On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin by Anna Sfard and Problem Solving and Cognitive Skill Acquisition by Kurt VanLehn.

## Mathematical Examples

Learning Mathematics from Examples and by Doing by Xinming Zhu and Herbert A. Simon, Learning from Examples: Instructional Principles from the Worked Examples Research by Robert K. Atkinson, Sharon J. Derry, Alexander Renkl and Donald Wortham, Sequencing Examples and Nonexamples to Facilitate Concept Attainment by Osmond S. Petty and Lars C. Jansson, Worked Examples and Concept Example Usage in Understanding Mathematical Concepts and Proofs by Keith Weber, Mary Porter and David Housman, Teaching by Examples: Implications for the Process of Category Acquisition by Judith Avrahami, Yaakov Kareev, Yonatan Bogot, Ruth Caspi, Salomka Dunaevsky and Sharon Lerner, The Subgoal Learning Model: Creating Better Examples so That Students Can Solve Novel Problems by Richard Catrambone, Increasing Mathematics Confidence by Using Worked Examples by William M. Carroll, Facilitating Learning Events through Example Generation by Randall P. Dahlberg and David L. Housman, Structuring Effective Worked Examples by Mark Ward and John Sweller and Generating Tailored Worked-out Problem Solutions to Help Students Learn from Examples by Giuseppe Carenini and Cristina Conati.

## Transitioning from Examples to Exercises

Structuring the Transition from Example Study to Problem Solving in Cognitive Skill Acquisition: A Cognitive Load Perspective by Alexander Renkl and Robert K. Atkinson, Towards Adaptive Generation of Faded Examples by Erica Melis and Giorgi Goguadze, Transitioning from Studying Examples to Solving Problems: Effects of Self-explanation Prompts and Fading Worked-out Steps by Robert K. Atkinson, Alexander Renkl and Mary Margaret Merrill, From Example Study to Problem Solving: Smooth Transitions Help Learning by Alexander Renkl, Robert K. Atkinson, Uwe H. Maier and Richard Staley, Generalizing Solution Procedures Learned from Examples by Richard Catrambone, Generalizing from the Use of Earlier Examples in Problem Solving by Brian H. Ross and Patrick T. Kennedy and Improving Examples to Improve Transfer to Novel Problems by Richard Catrambone.

## Mathematical Exercises

What Is a Mathematical Question? by Christopher J. Sangwin, Mathematical Question Spaces by Christopher J. Sangwin, The Exercise as Mathematical Object: Dimensions of Possible Variation in Practice by Anne Watson and John Mason, What Do Students Learn While Solving Mathematics Problems? by Elizabeth Owen and John Sweller, Students' Mathematical Reasoning in University Textbook Exercises by Johan Lithner, Designing Problems to Promote Higher-order Thinking by Renee E. Weiss, Writing Problems for Deeper Understanding by Barbara J. Duch, Inquiring Systems and Problem Structure: Implications for Cognitive Development by Phillip K. Wood, Automatic Creation of Exercises in Adaptive Hypermedia Learning Systems by Stephan Fischer and Ralf Steinmetz, Automatically Generating Algebra Problems by Rohit Singh, Sumit Gulwani and Sriram K. Rajamani, Automatic Exercise Generation in Euclidean Geometry by Andreas Papasalouros, Generating Mathematical Word Problems by Sandra Williams, Content Effects in Problem Categorization and Problem Solving by Stephen B. Blessing and Brian H. Ross, Problem Content Affects the Categorization and Solutions of Problems by Stephen B. Blessing and Brian H. Ross, Effects of Semantic Cues on Mathematical Modeling: Evidence from Word-problem Solving and Equation Construction Tasks by Shirley A. Martin and Miriam Bassok, Influence of Rewording Verbal Problems on Children's Problem Representations and Solutions by Erik De Corte, Lieven Verschaffel and Luc De Win, Using Students’ Representations Constructed during Problem Solving to Infer Conceptual Understanding by Daniel Domin and George Bodner, This Is like That: The Use of Earlier Problems and the Separation of Similarity Effects by Brian H. Ross, Towards a Formalization of the Automatic Generation of Exercises by Sergio Gutiérrez, Francisco J. Losa and Carlos Delgado Kloos, Using AI Planning to Enhance E-learning Processes by Antonio Garrido, Lluvia Morales and Ivan Serina and Modeling E-learning Activities in Automated Planning by Antonio Garrido, Eva Onaindia, Lluvia Morales, Luis Castillo, Susana Fernández and Daniel Borrajo.

## Puzzles and Games

Entertaining Mathematical Puzzles by Martin Gardner, Puzzle-based Learning by Zbigniew Michalewicz, Puzzle-based Learning: The First Experiences by Nickolas Falkner, Raja Sooriamurthi and Zbigniew Michalewicz, Puzzle-based Learning for Engineering and Computer Science by Nickolas Falkner, Raja Sooriamurthi and Zbigniew Michalewicz, Towards the Support of Scaffolding in Customizable Puzzle-based Learning Games by Javier Melero, Davinia Hernández-Leo and Josep Blat, Geometric Puzzle Design by Stewart Coffin, The Challenge of Designing Scientific Discovery Games by Seth Cooper, Adrien Treuille, Janos Barbero, Andrew Leaver-Fay, Kathleen Tuite, Firas Khatib and Alex and Learning Mathematics and Critical Thinking via Computer Games Design by Andrey Koptelov and Sylvia Taube.

## Generalization

Schemas in Problem Solving by Sandra P. Marshall, Comparison of Types of Generalizations and Problem-solving Schemas Used to Solve a Mathematical Problem by Tatjana H. Čadež and Vida M. Kolar, The Effects of Information Order and Learning Mode on Schema Abstraction by Renee Elio and John R. Anderson, The Effects of Category Generalizations and Instance Similarity on Schema Abstraction by Renee Elio and John R. Anderson, Principle Explanation and Strategic Schema Abstraction in Problem Solving by Allan B. Bernardo and Does Training Schematic Representation Improve Ability to Solve Math Word Problems? by Jeffery Washington.

## Analogical Reasoning

Mathematical Problem Solving by Analogy by Laura R. Novick and Keith J. Holyoak, Exploring the Relationship between Similar Solution Strategies and Analogical Reasoning by Peter Liljedah, Role of Analogical Reasoning in the Induction of Problem Categories by Denise D. Cummins, Learning by Analogy: Discriminating between Potential Analogs by Lindsey E. Richland and Ian M. McDonough, Analogical Problem Construction and Transfer in Mathematical Problem Solving by Allan B. Bernardo, Priming, Analogy, and Awareness in Complex Reasoning by Christian D. Schunn and Kevin Dunbar, Conceptual Priming in a Generative Problem-solving Task by Richard L. Marsh, Martin L. Bink and Jason L. Hicks, Analogy as Relational Priming: A Developmental and Computational Perspective on the Origins of a Complex Cognitive Skill by Robert Leech, Denis Mareschal and Richard P. Cooper, Analogical Priming via Semantic Relations by Barbara A. Spellman, Keith J. Holyoak and Robert G. Morrison, Relational Processing in Conceptual Combination and Analogy by Zachary Estes, Lara L. Jones, Robert Leech, Denis Mareschal and Richard P. Cooper, Surface and Structural Similarity in Analogical Transfer by Keith J. Holyoak and Kyunghee Koh, Context-dependent Effects on Analogical Transfer by R. Mason Spencer and Robert W. Weisberg and Analogy in Context by Brian Falkenhainer.

## Comparing and Selecting Strategies

The Development of Problem-solving Strategies by Deanna Kuhn and Erin Phelps, Does Comparing Solution Methods Facilitate Conceptual and Procedural Knowledge? An Experimental Study on Learning to Solve Equations by Bethany Rittle-Johnson and Jon R. Star, The Benefits of Comparing Solution Methods in Solving Equations by Zuya Habila Elisha, Learning through Case Comparisons: A Meta-analytic Review by Louis Alfieri, Timothy J. Nokes-Malach and Christian D. Schunn, The Importance of Prior Knowledge When Comparing Examples: Influences on Conceptual and Procedural Knowledge of Equation Solving by Bethany Rittle-Johnson, Jon R. Star and Kelley Durkin, How Children Change Their Minds: Strategy Change Can Be Gradual or Abrupt by Martha W. Alibali, The Importance of Metacognition for Conceptual Change and Strategy Use in Mathematics by Martha Carr and Metacognition and Mathematics Strategy Use by Martha Carr, Joyce Alexander and Trisha Folds‐Bennett.

## Mathematical Discovery

Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving by George Polya, How Children Discover New Strategies by Robert S. Siegler and Eric Jenkins, The "AHA Moment": Students Insights in Learning Mathematics by Peter Liljedahl, Mathematical Discovery and Concept Formation by Charles S. Chihara, A Theory of the Discovery and Predication of Relational Concepts by Leonidas A. Doumas, John E. Hummel and Catherine M. Sandhofer, On the Spontaneous Discovery of a Mathematical Relation during Problem Solving by James A. Dixon and Ashley S. Bangert, Investigating Insight as Sudden Learning by Ivan K. Ash, Benjamin D. Jee and Jennifer Wiley, A Computational Model of Conscious and Unconscious Strategy Discovery by Robert Siegler and Roberto Araya, Visualizing as a Means of Geometrical Discovery by Marcus Giaquinto and Creativity: Flow and the Psychology of Discovery and Invention by Mihaly Csiksentmihalyi.

## Metacognition

Metacognition and Mathematics Education: An Overview by Annemie Desoete and Brigitte de Craene, Metacognition and Mathematics Education by Wolfgang Schneider and Cordula Artelt, Evaluating and Improving the Mathematics Teaching-learning Process through Metacognition by Annemie Desoete, Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics by Alan H. Schoenfeld, The Importance of Metacognition for Conceptual Change and Strategy Use in Mathematics by Martha Carr and Metacognition and Mathematics Strategy Use by Martha Carr, Joyce Alexander and Trisha Folds‐Bennett.

## Cognitive Load

Some Examples of Cognitive Task Analysis with Instructional Implications by James G. Greeno, Implications of Cognitive Theory for Instruction in Problem Solving by Norman Frederiksen, Evidence for Cognitive Load Theory by John Sweller and Paul Chandler, Cognitive Load during Problem Solving: Effects on Learning by John Sweller and Cognitive Load Theory: Instructional Implications of the Interaction between Information Structures and Cognitive Architecture by Fred Paas, Alexander Renkl and John Sweller.

## Context

Context in Mathematics Curricula by Margaret R. Meyer, Truus Dekker and Nanda Querelle, Context in Problem Solving: A Survey by Patrick Brézillon, The Role of Situational Context in Solving Word Problems by Elisabeth Stern and Anne Lehrndorfer, Effects of Problem Context on Strategy Use within Functional Thinking by Katherine L. McEldoon, Caroline Cochrane-Braswell and Bethany Rittle-Johnson, Making Connections in Math: Activating a Prior Knowledge Analogue Matters for Learning by Pooja G. Sidney and Martha W. Alibali, The Effect of Context on the Structure of Categories by Emilie M. Roth and Edward J. Shoben, Context-independent and Context-dependent Information in Concepts by Lawrence W. Barsalou, Feature Availability in Conceptual Combination by Ken Springer and Gregory L. Murphy, Feature Accessibility in Conceptual Combination: Effects of Context-induced Relevance by Sam Glucksberg and Achary Estes, Constraints on Access in a Problem Solving Context by Greg A. Perfetto, John D. Bransford and Jeffery J. Franks, Memory in Context: Context in Memory by Graham M. Davies and Donald M. Thomson, Feeling of Knowing in Memory and Problem Solving by Janet Metcalfe, The Combined Contributions of the Cue-familiarity and Accessibility Heuristics to Feelings of Knowing by Asher Koriat and Ravit Levy-Sadot, Context and Structure in Conceptual Combination by Douglas L. Medin and Edward J. Shoben and Contextual Influences on the Comprehension of Complex Concepts by Richard J. Gerrig and Gregory L. Murphy.

## Multitasking and Task Switching

Toward a Unified Theory of the Multitasking Continuum: From Concurrent Performance to Task Switching, Interruption, and Resumption by Dario D. Salvucci, Niels A. Taatgen and Jelmer P. Borst, Creativity under Concurrent and Sequential Task Conditions by Deepika Rastogi and Narendra K. Sharma, Task Switching by Stephen Monsell, The Effects of Recent Practice on Task Switching by Nick Yeung and Stephen Monsell, The Strategic Use of Preparation Cues in the Task Switching Paradigm by Mike Wendt, Aquiles Luna-Rodriguez, Renate Reisenauer and Gesine Dreisbach, Intuition, Incubation, and Insight: Implicit Cognition in Problem Solving by Jennifer Dorfman, Victor A. Shames and John F. Kihlstrom, Modeling the Aha! Moment: A Computational Mechanism for Structuring and Incubation in Creative Problem Solving by Zhao Cheng, Laura Ray, Ha T. Nguyen and Jerald D. Kralik, Cues to Solution, Restructuring Patterns, and Reports of Insight in Creative Problem Solving by Patrick J. Cushen and Jennifer Wiley and Investigating the Effect of Mental Set on Insight Problem Solving by Michael Öllinger, Gary Jones and Günther Knoblich.

## The Aesthetics of Mathematical Exercises

What Makes a Good Problem? An Aesthetic Lens by Nathalie Sinclair and Sandra Crespo, The Roles of the Aesthetic in Mathematical Inquiry by Nathalie Sinclair, Aesthetic Influences on Expert Mathematical Problem Solving by Edward A. Silver and Wendy Metzger, Which Problems Do Teachers Consider Beautiful? A Comparative Study by Alexander Karp and Mathematical Beauty and Its Characteristics: A Study of the Students' Points of View by Astrid Brinkmann.

## The Aesthetics of Mathematical Thought, Reasoning and Proof

The Mathematical Experience by Philip J. Davis and Reuben Hersh, On the Aesthetics of Mathematical Thought by Tommy Dreyfus and Theodore Eisenberg, The Phenomenology of Mathematical Beauty by Gian-Carlo Rota, The Phenomenology of Mathematical Proof by Gian-Carlo Rota, Syntax and Meaning as Sensuous, Visual, Historical Forms of Algebraic Thinking by Luis Radford and Luis Puig, Computer Presentations of Structure in Algebra by Patrick W. Thompson and Alba G. Thompson, Can a Computer Proof Be Elegant? by Steve Seiden, Granularity by Jerry R. Hobbs, Granularity-adaptive Proof Presentation by Marvin Schiller and Christoph Benzmüller, Representation of Mathematical Concepts for Inferencing and for Presentation Purposes by Helmut Horacek, Armin Fiedler, Andreas Franke, Markus Moschner, Martin Pollet and Volker Sorge, Mathematics as Story by George Gadanidis and Cornelia Hoogland, The Aesthetic in Mathematics as Story by George Gadanidis and Cornelia Hoogland, Mathematical Beauty and the Evolution of the Standards of Mathematical Proof by James W. McAllister and Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics by Ulianov Montano.

## Motivation and Affect

Cognitive, Metacognitive, and Motivational Aspects of Problem Solving by Richard E. Mayer, Motivation and Ability as Factors in Mathematics Experience and Achievement by Ulrich Schiefele and Mihaly Csikszentmihalyi, The Concept of Flow by Jeanne Nakamura and Mihaly Csikszentmihalyi, Arousal of Flow Experience in a Learning Setting and Its Effects on Exam Performance and Affect by Julia Schüler, The Intricate Dance between Cognition and Emotion during Expert Tutoring by Blair Lehman, Sidney D’Mello and Natalie Person, Emotions during the Learning of Difficult Material by Arthur C. Graesser and Sidney D'Mello, The Effect of Perceived Challenges and Skills on the Quality of Subjective Experience by Giovanni B. Moneta and Mihaly Csikszentmihalyi, Affect and Mathematics Learning by Gilah C. Leder, Positive Affect Facilitates Creative Problem Solving by Alice M. Isen, Kimberly A. Daubman and Gary P. Nowicki, Positive Affect Increases the Breadth of Attentional Selection by Gillian Rowe, Jacob B. Hirsh and Adam K. Anderson and Positive Emotions Broaden the Scope of Attention and Thought-action Repertoires by Barbara L. Fredrickson and Christine Branigan.

## Assessment

Automatic Assessment of Problem-solving Skills in Mathematics by Cliff E. Beevers and Jane S. Paterson, Automatic Assessment in University-level Mathematics by Jarno Ruokokoski, Mathematics Exercise System with Automatic Assessment by Matti Harjula, Automatic Assessment of Mathematics Exercises: Experiences and Future Prospects by Antti Rasila, Matti Harjula and Kai Zenger, Linking On-line Assessment in Mathematics to Cognitive Skills by Jane S. Paterson, Exploring Mathematical Creativity Using Multiple Solution Tasks by Roza Leikin, Multiple Solutions for a Problem: A Tool for Evaluation of Mathematical Thinking in Geometry by Anat Levav-Waynberg and Roza Leikin, Techniques for Plan Recognition by Sandra Carberry, Evaluating E-assessment for Exercises That Require Higher-order Cognitive Skills by Tim A. Majchrzak and Claus A. Usener and An E-assessment System for Mathematical Proofs by Susanne Gruttmann, Herbert Kuchen and Dominik Böhm.

## Modeling Students and Problem Solving

A Review of Student Models Used in Intelligent Tutoring Systems by Philip I. Pavlik, Keith Brawner, Andrew Olney and Antonija Mitrovic, Modelling Student's Problem Solving by Derek H. Sleeman and Michael J. Smith, Cognitive Models of Learning by Chad H. Lane and Student Modeling: Supporting Personalized Instruction, from Problem Solving to Exploratory Open-ended Activities by Cristina Conati and Samad Kardan.