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Representational Change as a Way to Enhance Mathematics Learning with Understanding Habilitation Thesis Defended at the University of Hamburg, Faculty of Education, Germany Prof Dr Florence Mihaela Singer UPG University of Ploiesti, Romania 2016 Contents Summary Zusammenfassung Introduction 13 Chapter 1: Setting the theme – some prerequisites for representational change 14 1.1 Science development and human development: what they have in common? Paradigm as a key to explain theory change in science Radical shifts in children’s cognitive development 1.2 The Conceptual Change framework 1.3 The Representational Change Framework 1.4 Teaching for representational change: arguments for its sustainability 1.4.1 Children’s innate propensities for processing domain-specific information 1.4.2 Children’s endowment for recursive processes 1.4.3 Representational strategies and bridging 1.4.4 Children’s capacity to shift among representations in problem solving Chapter 2: Theoretical models that support representational change 2.1 The Dynamic Infrastructure of Mind as lever for representational change 2.2 Mental structures and representational change 2.2.1 How does the human mind organize itself? The building of structures 2.2.2 Types of Spontaneous Structures 2.2.3 Aggregate Structures and Knowledge Construction 28 Chapter 3: Frameworks for implementing 41 14 14 15 15 17 21 21 22 25 26 28 33 33 34 39 the representational-change approach 41 3.1 Dynamic Structural Learning 3.2 Extending the sphere: Representational change within a constructivist approach 3.3 Multi-representational teaching as a strategic approach at university level 42 48 52 Chapter Problem posing as part of representational change 55 4.1 A vision on school problem posing in the knowledge society 4.2 Developing a conceptual framework for studying problem solving and posing 4.3 A new framework for studying creativity through problem posing 4.4 A few findings of the problem-posing studies 4.5 Problem-posing: how teachers interact with? 55 56 58 59 62 Chapter Representational change, expertise, and metacognition 5.1 Operations and dynamics 5.2 Viewing the student as an expert-like learner 5.3 Looking ahead into the research of learning effectiveness 63 ANNEX: Resources for representational change – some examples 69 References 63 65 66 97 Acknowledgements 108 APPENDIX: Full Papers 109 Representational Change as a Way to Enhance Mathematics Learning with Understanding Summary The general aim of the present thesis is to draw on possible strategies to improve learning for new generations of students who are exposed to the actual phenomenon of almost exponential increase of information in terms of both amount and accessibility Can domain-specific expertise be enhanced with appropriate training, across ages? How children build new knowledge, and develop mathematical understanding? What meaning does the relationship between concrete and abstract have in a digital world? These are some of the questions addressed by the present theoretical framework The experimental research and the theoretical underpinnings gathered under its umbrella convey to the following hypothesis: if the knowledge advancement today is dynamic, interdependent, mediated by technology, and mostly unpredictable, and the school learning needs to be better connected to everyday-life contexts, then a possible solution for more effective teaching of mathematics might be training students’ cognitive capabilities through processes that emphasize representational change The theoretical approach of representational change refers to how individuals’ existing knowledge is adapted to allow facing new intellectual challenges, and particularly, to how new knowledge is generated in individuals From a pragmatic view, representational change offers contexts for more effective teaching through processes that stimulate flexibility of mind by operating with adaptive representations The present thesis is structured in five chapters, respectively: Chapter 1: Setting the theme – some prerequisites for representational change; Chapter 2: Theoretical models that support representational change; Chapter 3: Frameworks for implementing representational change; Chapter 4: Problem posing as part of representational change; and Chapter 5: Representational change, expertise, and metacognition In addition, an Annex titled Resources for representational change – some examples offers a few sets of multiple representations and strategies for school contexts at various ages, which have been shown as having potential for enhancing mathematics learning Setting the theme – prerequisites for representational change Within this chapter, the focus is on explaining how the concept of representational change appeared and developed The explanations try to shed new light on convergences and divergences that first guided the conceptual change approach in cognitive science and later on in education and learning More specific, some facts in the history of science and some relevant changes in human cognitive development are analyzed in parallel, emphasizing aspects that generated the conceptual change approach Revealing some limits of its application into mathematics teaching, an extension of the conceptual change theory towards a representational change (RC) approach that may compensate those limitations is proposed Then, a closer look brings new studies into the scene, showing the usefulness of a representational-change perspective for effective learning The question to be answered next is if a RC approach is feasible for students of various ages, in other words: if children have the necessary capacities to process a curriculum that is focused on representational change The answer to this question is given by reviewing a gamut of studies that discusses: children’s innate propensities; children’s endowment for recursive processes; children’s spontaneous bridging ability; and their intuitive capacity to shift among representations in problem solving1 Theoretical models that support representational change Drawing on recent research in cognitive psychology and neuroscience, Chapter provides brief descriptions of two theoretical models developed by the author, which suggest possible explanations on how the human mind works for acquiring knowledge and how it builds structures To a large extent, learning mathematics suppose practicing However, given the time constraints, what to practice in order to enhance learning? The first section of Chapter gives a particular answer to this general question This answer refers to the Dynamic Infrastructure of Mind seen as an information processing mechanism responsible for the cognitive representational power The clusters of operations that constitute the Dynamic Infrastructure of Mind (DIM) offer modalities for training representational capacities in students, in domain-specific situations The features of this model are discussed in the context of recent debates in cognitive psychology related to universalities of mind (cross-domaingeneral processing mechanisms versus domain-specific processors), innateness (inborn characteristics versus environmentally driven acquisitions), and modularity (the mind consists of pre-defined encapsulated modules versus progressive modularization), but the accent is put on the model functionality for effective learning The dynamic infrastructure of mind acts as a domain-general mechanism that progressively specializes with development through learning within environmental interactions The second section of this chapter, titled Mental structures and representational change, goes further in explaining the mechanisms that allow, enhance, or hinder the representational power of mind by describing various types of cognitive structures identified in mathematics learning This chapter reviews a few studies that explore how students activate and use various structures while solving or posing problems These structures have been revealed by interviewing and observing students of various ages The conclusions obtained allowed inferring that, while solving challenging problems, students navigate among concurrent representations and select the most adaptive one for the task at hand Descriptions of these representations – organized as cognitive structures – are provided in a few cases Their organization helps understanding advantages and risks of different learning pathways2 Singer, F M (2012) Boosting the Young Learners' Creativity: Representational Change as a Tool to Promote Individual Talents (Plenary lecture) In The 7th International Group for Mathematical Creativity and Giftedness (MCG) International Conference Proceedings Busan, South Korea: MCG, p 3-26 Singer, F M (2010) Children’s Cognitive Constructions: From Random Trials to Structures In Jared A Jaworski (Ed.), Advances in Sociology Research, vol 6, 1-35 Singer, F M (2009) The Dynamic Infrastructure of Mind - a Hypothesis and Some of its Applications, New Ideas in Psychology, 27(1), 48–74, DOI:10.1016/j.newideapsych.2008.04.007 Singer, F M., Voica, C (2008) Between perception and intuition: thinking about infinity, The Journal of Mathematical Behavior, 27, pp 188-205 Frameworks for implementing the representational change approach Previous experiments have shown that teachers can relatively easy apply some tasks that allow multiple representations The question is to what extent the RC framework can be used on a large scale, how generalizable it is Briefly, it is about two dimensions: developing dynamic conceptual structures within the curriculum, and organizing the teaching practice in a way that generates dynamic structures of thinking in students’ minds In practice, a representational-change-based teaching means at least two things: centering the didactical approach on representation as a powerful tool in learning, and bringing abstract concepts to school very early (in an informal way), in order to stimulate the children’s abstracting capacity during development The third chapter of this synthesis is devoted to answer the question: “How to teach based on representational change?” This answer is three-fold First, some strategic approaches for effective teaching are gathered under the name of Dynamic structural learning, which offers a methodology meant to develop representational capacities of each student Further, a framework for organizing the classroom interactions within a constructivist approach is presented Within this framework, the dynamics of interactions in cognitive development are mirrored in a model of the knowledge construction by the learner Third, a multirepresentational training methodology is used to demonstrate how representational-change strategies can duplicate effectively domain-specific learning from the expert teacher to the novice-prospective teacher Snapshots into some experimental programs developed within the present research, which focus RC in primary grades, in secondary school, and at university level, as well as the philosophy beyond these programs are meant to show the power of applying RC in teaching practice in real settings3 Problem Posing as part of representational change As the capacity of changing representations deals with transfer and creativity, a way to train representational capacities is problem posing4 This chapter brings into attention the main aspects revealed by a couple of studies developed by Singer and her team, in which students of various ages have been questioned about their approaches in devising problems It also allows looking at mathematical high-achieving and gifted students, analyzing how they are dealing with representations An important question addressed in this chapter is: What type of creativity should/can be developed in school through mathematics lessons and what might be generalizable to all students? The discussion reveals the specificity of mathematical creativity, uncovers some of Singer, F M., & Moscovici, H (2008) Teaching and learning cycles in a constructivist approach to instruction Teaching and Teacher Education, Vol 24/6, pp 1613-1634, DOI: 10.1016/j.tate.2007.12.002 Singer, M., Sarivan, L (2009) Curriculum Reframed Multiple Inteligences and New Routes to Teaching and Learning in Romanian Universities In J.Q Chen, S Moran, H Gardner (Eds.), Multiple intelligences around the world Pp 230-244 New York: Jossey-Bass, ISBN: 978-0-7879-9760-1 Singer, F M., Ellerton, N., Cai, J (2013) Problem-Posing Research in Mathematics Education: New Questions and Directions Educational Studies in Mathematics 83(1), 1-7 DOI: 10.1007/s10649-013-9478-2 the limits of students’ creative approaches, and raises questions on the appropriateness of some strategies to overcome these limits in social contexts5 Representational change, expertise, and metacognition By learning to structure and restructure within a representational-change approach, children become more able to analyze their own capabilities The so-called “hyper-learning” or “overlearning” phenomenon recorded within the experiments addressing RC approaches in learning seems to confirm the hypothesis that domain-specific expertise can be enhanced with appropriate training, across ages The discussion concerning the relationship between RC and expertise focuses some traits that make meaningful difference between experts and novices in a specific domain, such as: adaptive thinking schemes, complexity of problem-to-solve representation, goal-oriented procedural knowledge, automation that reduces the concentration of attention, and, above all, metacognitive capacities of self-regulation RC is meant to optimize the learning process through building expert-type cognitive behavior and explicitly developing metacognition The last chapter emphasizes these connections and tries to summarize how mathematics as a tool for rational thinking can play an important role in preparing the fluent thinkers needed in a dynamic world This opens up the discussion within contemporary debates related to teaching and learning in the digital era, highlighting that decisions should be taken based on interdisciplinary research More specific, the chapter analyses the situation of the today (and tomorrow) students that are exposed to various information and communication tools as has never happened with previous generations Having this in view, possible consequences of systematically using a RC approach in teaching are discussed The chapter also addresses some limitations of empirical research on this topic and opens the way to new hypotheses that might be validated in relation to the representational-change approach in teaching and learning6 Singer, F.M & Voica, C (2013) A problem-solving conceptual framework and its implications in designing problem-posing tasks Educational Studies in Mathematics DOI: 10.1007/s10649-012-9422-x, 83(1), 9-26 Singer, F.M & Voica, C (2015) Is Problem Posing a Tool for Identifying and Developing Mathematical Creativity? In F.M Singer, N.F Ellerton & J Cai (Eds.) Mathematical Problem Posing: From Research to Effective Practice, NY: Springer, 141-174 Singer, F.M (2007) Beyond Conceptual Change: Using Representations to Integrate Domain-Specific Structural Models in Learning Mathematics Mind, Brain, and Education, 1(2), pp 84-97, DOI: 10.1111/j.1751228X.2007.00009.x Repräsentationsänderung als Möglichkeit das Mathematiklernen mit Verstehen zu verbessern Zusammenfassung Die vorliegende Habilitationsarbeit verfolgt als Hauptziel die Beschreibung und Analyse möglicher Strategien, um die Lernprozesse einer neuen Schüler- und Studentengenerationen, die dem aktuellen Phänomen der fast exponentiellen Informationszunahme, sowohl quantitativ als auch bzgl der Zugänglichkeit ausgesetzt sind, zu verbessern In diesem Zusammenhang stellen sich folgende Fragen: Kann diese domänenspezifische Kompetenz für Jugendliche jedes Alters durch angemessenes Training, verbessert werden? Wie bauen Kinder neues Wissen auf und wie entwickeln sie mathematisches Verständnis? Welche Bedeutung hat die Beziehung zwischen dem Konkreten und dem Abstrakten in einer digitalen Welt? Diese sind einige der Fragen, die im vorliegenden Forschungsrahmen gestellt und untersucht werden Die experimentelle Forschung und die verschiedenen theoretischen Rahmungen, die sie untermauern, führen zu folgender Hypothese: Da heute die Erweiterung des Wissens dynamisch, verflochten, durch Technologie vermittelt, und vor allem unvorhersehbar ist, und da das schulische Lernen besser mit dem Alltagskontext verbunden werden muss, könnte eine mögliche Lösung für einen effektiveren Mathematikunterricht das Training der kognitiven Fähigkeiten der Lernenden durch Prozesse, die Repräsentationsänderungen (representational change) betonen, sein Der theoretische Ansatz der Repräsentationsänderung bezieht sich auf die Art, wie das vorhandene Wissen durch Individuen bei der Bewältigung neuer geistiger Herausforderungen adaptiert wird, insbesondere bzgl der Erzeugung von neuem Wissen durch die Individuen Aus pragmatischer Sicht bietet die Repräsentationsänderung Kontexte für eine effektivere Lehre durch Prozesse, die Flexibilität des Geistes stimulieren, indem sie mit sich anpassenden Repräsentationen arbeiten Die vorliegende Arbeit ist in fünf Kapitel eingeteilt, und zwar wie folgt: Kapitel 1: Vorstellung des Themas - Voraussetzungen für Repräsentationsänderung; Kapitel 2: Theoretische Modelle, die Repräsentationsänderung unterstützen; Kapitel 3: Rahmen für die Umsetzung von Repräsentationsänderung; Kapitel 4: Problemformulierung als Teil der Repräsentationsänderung; Kapitel 5: Repräsentationsänderung, Expertise und Metakognition Darüber hinaus bietet ein Anhang mit dem Titel Ressourcen für Repräsentationsänderung - einige Beispiele einige Zusammenstellungen von mannigfaltigen Repräsentationen und Strategien für schulische Kontexte in verschiedenen Altersstufen, die das Potential für eine Verbesserung des Mathematiklernens haben Vorstellung des Themas - Voraussetzungen für Repräsentationsänderung Das Hauptanliegen dieses Kapitels ist zu darzulegen, wie das Konzept der Repräsentationsänderung (representational change) entstanden ist und entwickelt wurde Die Erklärungen versuchen, neues Licht auf Konvergenzen und Divergenzen zuerst im Ansatz des Konzeptes des begrifflichen Wandels (conceptual change approach) in den Kognitionswissenschaften, und später in den Bereichen Bildung und Lernen, zu werfen Genauer gesagt, werden einige Fakten in der Geschichte der Wissenschaft und einige relevante Veränderungen in der menschlichen kognitiven Entwicklung analysiert, die den Ansatz des begrifflichen Wandels hervorbrachten Einige Einschränkungen der Anwendung dieses Ansatzes im Mathematikunterricht werden dargestellt, und es wird vorgetragen, dass eine Erweiterung der Theorie des begrifflichen Wandels zum Ansatz der Repräsentationsänderung - representational change (RC) approach - diese Einschränkungen kompensieren kann Eine detailliertere Analyse macht die Relevanz der Perspektive der Repräsentationsänderung deutlich Weiter ist die Frage zu beantworten, ob ein RC-Ansatz für Lernende verschiedener Altersstufen möglich ist, mit anderen Worten, ob Kinder über die notwendigen Fähigkeiten zur Arbeit entlang eines Lehrplans verfügen, der auf Repräsentationsänderung aufgebaut ist Die Antwort auf diese Frage wird durch die Überprüfung eine ganzer Reihe von Studien gegeben, die folgende Themen behandeln: angeborene Neigungen der Kinder; die Begabung der Kinder für rekursive Prozesse, ihre spontane Überbrückungsfähigkeit und ihre intuitive Fähigkeit, bei der Problemlösung zwischen verschiedenen Darstellungen zu wechseln7 Theoretische Modelle, die Repräsentationsänderung unterstützen Auf neuere Forschungen in der kognitiven Psychologie und Neurowissenschaften gestützt, präsentiert Kapitel kurze Beschreibungen zweier von der Verfasserin entwickelter theoretischer Modelle, die mögliche Erklärungen bieten, wie der menschliche Geist beim Erwerb von Wissen funktioniert und wie er Strukturen aufbaut Mathematiklernen bedeutet zu einem großen Teil Übung Da zeitliche Einschränkungen jedoch gegeben sind, stellt sich die Frage, was geübt werden soll, um das Lernen zu verbessern Der erste Abschnitt von Kapitel gibt eine spezifische Antwort auf diese allgemeine Frage Diese Antwort bezieht sich auf den Theorierahmen Dynamische Infrastruktur des Geistes (Dynamic Infrastructure of Mind) als Informationsverarbeitungsmechanismus, das als verantwortlich für die kognitive Repräsentationsleistung gesehen wird Die Cluster von Operationen, die die Dynamic Infrastructure of Mind (DIM) darstellen, bieten Modalitäten für das Trainieren der Repräsentationskapazitäten von Lernenden, in domänenspezifischen Kontexten Die Merkmale dieses Modells werden in Zusammenhang mit neueren Diskussionen zur kognitiven Psychologie diskutiert, u.a zu Universalien des Geistes (domänenübergreifende Verarbeitungsmechanismen vs domänenspezifische Prozessoren), zu Angeborenheit (angeborene Eigenschaften im Vergleich zum umweltbedingten Erwerb) und Modularität (der Geist besteht aus vordefinierten eingekapselten Modulen vs progressive Modularisierung), allerdings liegt das Schwergewicht auf der Funktionalität des Modells für die Effektivität des Lernens Die dynamische Infrastruktur des Geistes wirkt als domänenübergreifender Mechanismus, der sich schrittweise während der Entwicklung spezialisiert, durch das Lernen unter verschiedenen Umweltwechselwirkungen Der zweite Teil dieses Kapitels, mit dem Titel Mentale Strukturen und Repräsentationsänderung, vertieft die Erklärung der Mechanismen, die die Repräsentationsleistung des Geistes ermöglichen, verbessern oder verhindern, indem verschiedene Arten von kognitiven Strukturen, die das Mathematiklernen kennzeichnen, beschrieben werden In diesem Kapitel werden einige Studien, die untersuchen, wie die Lernenden verschiedene mentalen Strukturen aktivieren und verwenden, während sie Aufgaben lösen oder stellen, vorgestellt Diese Strukturen wurden durch Befragung und Beobachtung von Lernenden verschiedenen Alters rekonstruiert Die Ergebnisse weisen darauf hin, dass Lernende, während sie anspruchsvolle Aufgaben lösen, sich gleichzeitig zwischen konkurrierenden Darstellungen bewegen und diejenigen wählen, die am besten für Singer, F M (2012) Boosting the Young Learners' Creativity: Representational Change as a Tool to Promote Individual Talents (Plenary lecture) In the 7th MCG Conference Busan, South Korea: MCG, p 3-26 Singer, F.M (2010) Children’s Cognitive Constructions: From Random Trials to Structures In Jared A Jaworski (Ed.), Advances in Sociology Research, vol 6, 1-35 10 Example 15: Using representations to understand irrational numbers  The image below shows how to use the compass to draw a segment of length √2  The drawings below help consolidating the understanding of the magnitude of different irrational numbers such as √2 or √8 They also help connecting rational and irrational numbers, as well as algebra and geometry Students are invited to represents many other irrational numbers 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Boston, MA: Pearson Education Waltz, J A., Lau, A., Grewal, S K., & Holyoak, K J (2000) The role of working memory in analogical mapping Memory & Cognition, 28, 1205–1212 Wegner, D M (2002) The illusion of conscious will Cambridge, MA: MIT press Wertheimer, M (1959) Productive thinking New York: Harper & Row Wynn, K (1990) Children’s understanding of counting Cognition, 36, 155-193 Wynn, K (1992) Children’s acquisition of the number words and the counting system Cognitive Psychology, 24, 220–251 Wynn, K (1995) Origins of numerical knowledge Mathematical Cognition 3, 35-60 Yan, Z & Fischer, K.W (2002) Always Under Construction - Dynamic Variations in Adult Cognitive Microdevelopment Human Development; 45, 141-160 Yuan, X., & Sriraman, B (2011) An exploratory study of relationships between students’ creativity and mathematical problem posing abilities – Comparing Chinese and U.S students In B Sriraman, K Lee (Eds.), The Elements of Creativity and Giftedness in Mathematics (pp.5-28) Rotterdam, Netherlands: Sense Publishers Zazkis, R., Liljedahl, P., & Gadowsky, K (2003) Conceptions of function translation: Obstacles, intuitions, and rerouting The Journal of Mathematical Behavior, 22, 435–448 107 Acknowledgements I would like to express my gratitude to my colleagues at the University of Hamburg I would like to thank prof dr Eva Arnold, Dean of the Faculty of Education, for inviting me as a visiting professor at the University of Hamburg This was for me an extremely rich and rewarding learning experience I am grateful to prof dr Gabriele Kaiser, for accepting to accompany me in the complex process of developing the habilitation thesis and procedures I am truly thankful for her right-to-the-point guidance in the conditions of an extremely overloaded program as organizer of the 13th International Congress on Mathematical Education – Hamburg 2016, where more than 3000 researchers confirmed participation I very much appreciated the opportunity to be involved in the vivid exchange of ideas with professors and doctoral students in dr Kaiser’s group I have been honored to be, for a while, part of the team, in a hardworking, highly professional and, in the same time, very friendly community of practice I also wish to thank all the colleagues from the Mathematics Education Department, and especially to prof dr Marianne Nolte for the stimulating discussions we had with various occasions I am grateful to my colleagues from UPG University of Ploiesti for their confidence in my efforts, for their ability to create a healthy and fruitful cooperation in the development of our educational programs, and for the hope that we will succeed to build a competitive doctoral school that can learn a lot from our counterparts from University of Hamburg Finally, I would like to thank my family, who gave me the energy and time to go through the habilitation process in a very dense period of many other duties, my husband Boris for his patience and love, and especially my grandchildren for being a source of motivation and support 108 APPENDIX: Full Papers Full Paper 1: Singer, F.M (2007) Beyond Conceptual Change: Using Representations to Integrate Domain-Specific Structural Models in Learning Mathematics Mind, Brain, and Education, 1(2), pp 84-97 110 Full Paper 2: Singer, F M (2009) The Dynamic Infrastructure of Mind - a Hypothesis and Some of its Applications, New Ideas in Psychology, 27(1), 48–74 124 Full Paper 3: Singer, F M., & Moscovici, H (2008) Teaching and learning cycles in a constructivist approach to instruction Teaching and Teacher Education, Vol 24/6, pp 1613-1634 151 Full Paper 4: Singer, M., Sarivan, L (2009) Curriculum Reframed MI and New Routes to Teaching and Learning in Romanian Universities In J.Q Chen, S Moran, H Gardner (Eds.), Multiple intelligences around the world Pp 230-244 New York: Jossey-Bass 173 Full Paper 5: Singer, F.M (2010) Children’s Cognitive Constructions: From Random Trials to Structures In Jared A Jaworski (Ed.), Advances in Sociology Research, vol 6, 1-35 188 Full Paper 6: Singer, F M (2012) Boosting the Young Learners' Creativity: Representational Change as a Tool to Promote Individual Talents (Plenary lecture) 7th International Mathematical Creativity and Giftedness Conference Proceedings Busan, South Korea: MCG, p.3-26 222 Full Paper 7: Singer, F.M., Voica, C (2008) Between perception and intuition: thinking about infinity, The Journal of Mathematical Behavior, 27, pp 188-205 246 Full Paper 8: Singer, F.M & Voica, C (2013) A problem-solving conceptual framework and its implications in designing problem-posing tasks Educational Studies in Mathematics, 83(1), 9-26 264 Full Paper 9: Singer, F.M & Voica, C (2015) Is Problem Posing a Tool for Identifying and Developing Mathematical Creativity? Mathematical Problem Posing: From Research to Effective Practice, NY: Springer, 141-174 282 Full Paper 10: Singer, F M., Ellerton, N., Cai, J (2013) Problem-Posing Research in Mathematics Education: New Questions and Directions Educational Studies in Mathematics 83(1), 1-7 316 109