Representational change as a way to enha

The experimental research and the theoretical underpinnings gathered under its umbrella convey to the following hypothesis: if the knowledge advancement today is dynamic, interdependent,

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

Introduction 13

Chapter 1: Setting the theme – some prerequisites for representational change 14

1.1 Science development and human development: what do they have in common? 14

1.4 Teaching for representational change: arguments for its sustainability 21 1.4.1 Children’s innate propensities for processing domain-specific information 21

1.4.4 Children’s capacity to shift among representations in problem solving 26

Chapter 2: Theoretical models that support representational change 28

2.1 The Dynamic Infrastructure of Mind as lever for representational change 28

2.2.1 How does the human mind organize itself? The building of structures 33

3.2 Extending the sphere: Representational change within a constructivist approach 48 3.3 Multi-representational teaching as a strategic approach at university level 52

Chapter 4 Problem posing as part of representational change 55

4.2 Developing a conceptual framework for studying problem solving and posing 56 4.3 A new framework for studying creativity through problem posing 58

Chapter 5 Representational change, expertise, and metacognition 63

ANNEX: Resources for representational change – some examples 69

References 97

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 do 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

1 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 do 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

2 Theoretical models that support representational change

Drawing on recent research in cognitive psychology and neuroscience, Chapter 2 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 2

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 general processing mechanisms versus domain-specific processors), innateness (inborn characteristics versus environmentally driven acquisitions), and modularity (the mind consists

(cross-domain-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

1 Singer, F M (2012) Boosting the Young Learners' Creativity: Representational Change as a Tool to Promote Individual Talents (Plenary lecture) In The 7 th 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

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, 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

3 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

multi-representational training methodology is used to demonstrate how multi-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

4 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 posing 4 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

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, 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

4 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

5 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 “over-learning” 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

5 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

6 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.1751-228X.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

1 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.

2 Theoretische Modelle, die Repräsentationsänderung unterstützen

Auf neuere Forschungen in der kognitiven Psychologie und Neurowissenschaften gestützt, präsentiert Kapitel 2 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 2 gibt eine spezifische Antwort auf diese

allgemeine Frage Diese Antwort bezieht sich auf den Theorierahmen Dynamische

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

7

Singer, F M (2012) Boosting the Young Learners' Creativity: Representational Change as a Tool to Promote Individual Talents (Plenary lecture) In the 7 th 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

die gegebene Aufgabe geeignet sind Für einige Fälle werden Beschreibungen dieser Darstellungen, als kognitive Strukturen organisiert, dargestellt Ihre Organisation hilft die Vorteile und Risiken verschiedener Lernwege zu verstehen8

3 Rahmen zur Impementierung des Repräsentationsänderung-Ansatzes

Wie frühere Experimente gezeigt haben, können Lehrkräfte relativ einfach einige Aufgaben benutzen, die mehrere Repräsentationen ermöglichen Die Frage ist, inwieweit der RC-Rahmen in großem Umfang verwendet werden kann bzw wie verallgemeinerbar dieser ist Kurz gesagt, es geht um zwei Dimensionen: Die Entwicklung dynamisch-konzeptioneller Strukturen im Rahmen des Lehrplans und die Organisation der Unterrichtspraxis in einer Weise, die bei den Lernenden dynamischen Denkstrukturen erzeugt In der Praxis bedeutet eine Lehre, die auf der Repräsentationsänderung basiert, mindestens zwei Dinge: Ausrichtung des didaktischen Ansatzes auf Repräsentation als ein mächtiges Werkzeug zum Lernen, und das frühe Einführen abstrakter Begriffe in der Schule (in informeller Weise), um die Abstraktionskapazität der Kinder während ihrer Entwicklung zu stimulieren Das dritte Kapitel der Habilitationsschrift ist der Antwort auf die folge Frage gewidmet: „Wie kann man auf der Basis des Ansatzes der Repräsentationsänderung lehren?" Eine Antwort wird auf drei Ebenen entwickelt: Erstens werden einige strategische Ansätze für eine effektive Lehre, unter

dem Namen Dynamisches strukturelles Lernen dargestellt, welches eine Methodik anbietet,

um die Repräsentationskapazitäten eines jeden Lernenden zu entwickeln Ferner wird ein Rahmen für die Organisation der Interaktionen im Klassenzimmer innerhalb eines

konstruktivistischen Ansatzes vorgestellt In diesem Rahmen wird die Dynamik der

Wechselwirkungen in der kognitiven Entwicklung der Lernenden in einem Modell der

Wissenskonstruktion abgebildet Drittens wird eine Multirepräsentations-Trainingsmethodik

genutzt, um zu zeigen, wie in der Lehrerausbildung Repräsentationsänderungsstrategien domänenspezifisches Lernen von der Expertenlehrkraft über die Novizenlehrkraft effektiv verdoppeln können Momentaufnahmen aus einigen experimentellen Programmen, die im Rahmen der vorliegenden Arbeit entwickelt worden sind, und die sich auf RC für die ersten Schulklassen, für die Sekundarstufe und für die Universitätsebene konzentrieren, und, darüber hinaus, die Philosophie dieser Programme, sollen die Stärke der Anwendung von RC

in der Unterrichtspraxis unter realen Bedingungen zeigen9

4 Problemformulierung als Teil der Repräsentationsänderung

Da die Leistungsstärke der Repräsentationsänderung mit dem Transfer und der Kreativität zu tun hat, ist das Formulieren und Entwickeln mathematischer Problemlöseaufgaben eine Methode, die Kompetenz zur Repräsentationsänderung zu trainieren Dieses Kapitel analysiert die wichtigsten Ergebnisse einiger Studien, die von Singer und ihrem Team

9 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

entwickelt wurden, in denen Lernende verschiedenen Alters über ihre Ansätze bei der Entwicklung von Problemaufgaben befragt worden sind Auch ermöglicht dies Einsichten auf hochbegabte Lernende, die hohe mathematische Leistungen erzielen, indem es die Art ihrer Handhabung von Repräsentationen analysiert.

Dabei wird in diesem Kapitel eine wichtige Frage aufgeworfen: Welche Art von Kreativität sollte / kann in der Schule durch Mathematikunterricht entwickelt werden und was könnte für alle Lernenden verallgemeinert werden? Die Diskussion zeigt die Spezifität mathematischer Kreativität, deckt einige der Grenzen kreativer Ansätze für Lernende auf und stellt Fragen über die Angemessenheit einiger Strategien, um diese Grenzen in verschiedenen sozialen Kontexte zu überwinden10

5 Repräsentationsänderung, Expertise und Metakognition

Kinder werden besser in der Lage sein, ihre eigenen Fähigkeiten zu analysieren, wenn sie das Strukturieren und Umstrukturieren innerhalb des Ansatzes der Repräsentationsänderung lernen Das in Experimenten beschriebene Phänomen des sogenannten „Hyperlernens" oder

"Überlernens" scheint die Hypothese, dass das domänenspezifische Können, mit entsprechender Ausbildung altersübergreifend verbessert werden kann, zu bestätigen Die Diskussion über das Verhältnis zwischen RC und Expertise fokussiert einige Merkmale, die den Unterschied zwischen Experten und Novizen in einer bestimmten Domäne bedeutsam machen, wie z.B.: Adaptive Denkschemata, Komplexität der Repräsentation der zu lösenden Aufgaben, zielorientiertes Prozesswissen, Automatisierung, die die Konzentration der Aufmerksamkeit reduziert, und, vor allem, metakognitive Kapazitäten der Selbstregulierung

RC ist dafür bestimmt, das Lernen durch die Verstärkung des expertenartigen Lernens zu optimieren und explizit die Metakognition zu entwickeln

Das letzte Kapitel unterstreicht diese Verbindungen und beschreibt zusammenfassend, wie die Mathematik als Werkzeug für das rationale Denken beim Vorbereiten flüssiger Denkprozesse, die in einer dynamischen Welt benötigt werden, eine wichtige Rolle spielen kann Das eröffnet die Diskussion innerhalb der zeitgenössischen Debatten um Lehren und Lernen im digitalen Zeitalter, und hebt hervor, dass Entscheidungen anhand interdisziplinärer Forschung getroffen werden sollten Genauer gesagt, analysiert das Kapitel die Situation der Lernenden von heute (und morgen), die mit verschiedenen Informations- und Kommunikationsmittel konfrontiert sind, in einem Ausmaß, das keine der früheren Generationen erlebt hat Im Hinblick auf all diese Aspekte werden mögliche Folgen der systematischen Anwendung des RC - Ansatzes in der Lehre diskutiert Das Kapitel befasst sich auch mit einigen Grenzen der empirischen Forschung zu diesem Thema und öffnet den Weg für neue Hypothesen, die in Verbindung mit dem Repräsentationsänderung - Ansatz in der Lehre und beim Lernen bestätigt werden könnten11

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 DOI: 10.1007/s10649-013-9478-2

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

11 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.

Introduction

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 do children build new knowledge, and develop mathematical understanding? What meaning does the relationship between concrete and abstract have today, when digital media are more and more accessible and pervasive? And finally: How to optimize learning under the pressure of almost exponential increase of information in a changing world?

The present habilitation thesis tries to synthesize answers to the above questions based on the research I conducted, situated at the crossing point of cognitive psychology, neuroscience, mathematics, and mathematics education The experimental research and theoretical underpinnings addressed by this work 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

More specifically, a representational-change approach in mathematics teaching will:

1 Offer deeper connections for making mathematics meaningful to learners, by drawing on children’s innate propensities, their endowment for recursive processes, their spontaneous ability for cognitive bridging, and their natural capacity to shift among representations in problem solving;

2 Stress the continuity between old and new knowledge in students, through both valuing students’ adaptive representations and providing students with a variety of new representations that engender a dynamic view of mathematical knowledge;

3 Accommodate children with abstractions through gradual processes of understanding based on successive cycles of integrating complexity into subsequent levels of abstracting, which is typical for mathematics as a domain of human knowledge

These claims are sustained along the paper at conceptual and empirical levels with evidence from theoretical models and practical applications developed during almost twenty years of research

Chapter 1: Setting the theme – some prerequisites for

representational change

This chapter tries to shed new light on convergences and divergences that 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 Some limits of its applications into mathematics teaching are unraveled, which allow hypothesizing that an extension of the conceptual change theory

towards a representational change (RC) approach may compensate these limitations Then, a

closer look to these aspects brings new studies into the scene, showing how they can be undertaken from the representational-change perspective The question to be answered next is

if a RC approach is feasible for learners of various ages, in other words: if students have the needed 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 discuss: 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 solving12

1.1 Science development and human development: what do they have in common?

Throughout human history, scientific theories crossed deep changes, which fundamentally restored even the basics of science domains What mechanisms are causing such shifts in the evolution of a science? Is there an inexorable progression towards the truth? Is there a "true theory" to describe the "reality" in which we live? How does scientific progress happen? By the middle of the 20th century, these questions have been consistently addressed by the philosopher Thomas Kuhn

Paradigm as a key to explain theory change in science

The deep changes in the representations of excellence and in practices that secure scientific knowledge at a certain moment of time and in a certain research area were called by Kuhn (Kuhn, 1962/1996) scientific revolutions Kuhn claims that the basis of consensus in a science that has reached a maturity stage is not a scientific theory, but something more complex, which is the paradigm – a theoretical, instrumental and methodological entity that offers models for problem solving to a scientific community The knowledge contained in a paradigm is rather tacit, developed and agreed by traditions and habits that are practiced within that scientific community, and it determines the general way of thinking and solving problems

For Kuhn, a change of paradigm (such as the one from the geocentric to the heliocentric universe) is a transition between frameworks that are incommensurable in important ways Incommensurability between two theories essentially means that the concepts and procedures

of a theory cannot be expressed by means of concepts and procedures of the other theory, that

is they cannot be translated into the terms of the other theory For this reason, the paradigm change cannot be done step by step and constrained by a neutral experience — it needs a

“revolution”, a radical swift Moreover, every scientific community is resistant against a new paradigm – the history of science shows that it takes decades, or even centuries, until a new paradigm pervades a scientific community (For example, the passage from the conception that the Earth is in the center of the universe to accepting that the Earth just moves around the Sun needed almost two millennia)

Radical shifts in children’s cognitive development

From the history of science, we move to a different domain: the cognitive development of children from birth to adulthood Systematic observations of children’s behavior have shown that all normal children go through several qualitatively different levels of thought during the period from infancy to adolescence These shifts (such as switching from the incapacity of grasping the idea that the number of objects does not change when the distance between objects changes, to the spontaneous use of conservation of number) happen across any cultural-educational and social-economic contexts (e.g Case, 1992; Commons, Trudeau,

Stein, Richards, & Krause, 1998; Feldman, 1980/1994; Dawson-Tunik, Commons, Wilson, &

Fischer, 2005) However, children do not need to be explicitly taught these properties; they are naturally achieved over time

Many observations of this type led cognitive psychologists to the idea that children cross developmental stages Piaget remarked, in some of his last papers, as well as the “new Piagetians”, that each developmental stage is characterized by a recurrent advancement of the child’s cognitive system, in which the first part is an active construction phase that has the culminating point in what was called ‘‘taking of consciousness’’ of the system as a whole, and the second is an active extension and elaboration phase (Piaget, 1954; 1972; Bringuier, 1980; Feldman, 2004) These stages are incommensurable one against the other (for example,

a 3-4 year old will not be able to accept the idea of conservation of number, while 1-2 years later he/she will find this property natural and obvious)

It seems that history of science and cognitive development of children from birth to adulthood are both characterized by incommensurable leaps engendering changes that take place (keeping the proportions) slowly and with great difficulty “The process of paradigm change is closely tied to the nature of perceptual (conceptual) change in an individual Novelty emerges only with difficulty, manifested by resistance, against a background provided by expectation” (Kuhn, 1996, p 64)

1.2 The Conceptual Change framework

The idea of conceptual change entered education as an analogy drawn from the history and philosophy of science that proved helpful in understanding the difficulties people experience

in changing conceptions from one explanatory framework to another Most of the studies on conceptual change emerged from analyzing domain-specific conceptual changes by means of structural analogies to Kuhnian paradigmatic change of a scientific community (e.g Carey,

combined with the Kuhnian ideas of theory change in the history of science In education, the conceptual-change framework crossed a variety of approaches Starting from analogies with the paradigmatic change of a scientific community, the classical conceptual change view emphasizes an epistemological position in the teaching practice, with a definite focus on the

students’ knowledge In showing that some of the child's concepts are incommensurable with

the ones of adults, Carey (1985, 1999) argued for strong knowledge restructuring during childhood, and Vosniadou (1994) called similar changes radical restructuring and explained that revisions to central "framework theories" involve both ontological and epistemological changes

Strike and Posner (1992) revised the epistemological approach emphasizing that a wider range of factors (such as motives, goals, social sources of them) and their interactions needs

to be taken into account to describe a learner’s conceptual ecology, where developmental and interactionist views are coexisting On short, they recommended a more dynamic view of students’ conceptions Other studies use specific ontological terms to explain changes to the way students conceptualize science entities (Chi, Slotta, & De Leeuw, 1994; Thagard, 2000; Vosniadou, 1994)

Briefly posed, the conceptual-change approach in education supposes that the teacher makes students’ alternative frameworks explicit prior to designing a teaching approach consisting of conceptions that do not fit the students’ existing ideas and thereby promoting dissatisfaction

in order to provoke a cognitive conflict A new framework is then introduced based on formal results obtained in that domain of knowledge, which will explain the anomaly (Duit & Treagust, 2003)

Obviously, development resolves some of the incommensurabilities between previous and new knowledge in individuals, but other incommensurabilities need to be solved by school instruction The research on domain-specific knowledge has frequently focused on the issue

of misconceptions, especially asking what characteristics would make developmental transitions to be more (or less) susceptible to engender misconceptions Summarizing years

of misconceptions research in Newtonian mechanics, Clement (2008) reported that many preconceptions are deep seated and resistant to change Duit (2009) has compiled a bibliography with over 8,000 references focused on student conceptions and sense-making in science learning It seems clear from the large amount of research that student learning in science must take into account students’ pre-existing ideas Since students have existing conceptions, these ideas must be engaged and modified, they cannot be ignored as is traditionally done (Brown, 2014) In order to deal with the flaws of a unitary misconceptions

perspective, a number of theorists started to discuss students’ conceptions as systems of

conceptual elements rather than simple, uncorrelated bits of information (Gopnik and Schulz, 2004; Gopnik and Wellman, 1994; Vosniadou, 2013; Vosniadou et al., 2008)

In general terms, the conceptual-change approach evolved from a radical constructivist view,

to then merged with social constructivist and social cultural orientations that more recently resulted in recommendations to employ multi-perspective epistemological frameworks in order to adequately address the complex process of learning (Duit & Treagust, 2003)

Changing the paradigm implies changing habits of mind of well-trained researchers, habits of mind that are difficult to influence, as they are not accessible to introspection and are automatized at the level of experts practicing a domain Margolis (1993) explained the resistance of a scientific community to changing the paradigm through the habits of mind that govern scientific belief, considering that the habits of mind, like those of behavior, are neural programming that engender mental barriers

Habits of mind are, therefore, the deep source of the attachment of practicians from a scientific community to a research tradition and a barrier in understanding a different research tradition (Singer, 2007a) What about children’s habits of mind? Although a focus

on a cognitive conflict could challenge learning in various situations, there are many instances

when such focus could undermine children’s natural propensities that favor effective learning

Moreover, gaps and spurts, anticipations of the new knowledge and simultaneities have been recorded in cognitive development by many researchers, new Piagetians included (e.g.: Fischer & Rose, 2001; Granott & Parziale, 2002; Perner, Rendl, and Garnham, 2007) Therefore, a new question emerges: What about if, instead of focusing the learning process

on discontinuities and incomensurabilities, we focus it on continuities, connections and a feed-forward approach?

In a paper focused on reconceiving misconceptions, Smith, diSessa, & Roschelle (1993) summarize the classical approach in the conceptual-change theory under the following statements: Students have misconceptions (that can be characterized in terms of preconceptions, alternative conceptions, naive beliefs, alternative frameworks, naive theories, etc.) which originate in prior learning and interfere with learning because of their strength and flawed content Misconceptions should be identified through research, confronted (to help students to see the disparity between their misconceptions and expert concepts), and finally replaced through a shift to expert concepts Analyzing the limitations of this view, they propose a change of perspective by looking at students’ prior knowledge in terms of complex systems of knowledge Thus, they emphasize the following statements: casting misconceptions as mistakes is too narrow a view of their role in learning; misconceptions are faulty extensions of productive prior knowledge; misconceptions are not always resistant to change; strength is a property of knowledge systems; replacing misconceptions is neither plausible nor always desirable; instruction that confronts misconceptions is misguided and unlikely to succeed; it is time to move beyond the identification of misconceptions (Smith, diSessa, & Roschelle, 1993) With an écart of more than 20 years, Brown (2014) makes also

a plea for the use of complex systems in analyzing conceptual changes In a trial to conceive the variety of theories inside the conceptual-change paradigm in education, Brown (2014) proposes as unifying tool the complex system perspective This perspective brings in the first place the representations students have regarding their own knowledge

1.3 The Representational Change Framework13

Conceptual change as a theory of explaining children’s conceptual shifts in cognitive development benefits from a strong analogy with the history of science and this makes it very popular among researchers As the recent studies above presented suggest, a larger perspective should allow overpassing the limitations induced by the classical view of conceptual change

In mathematics teaching, this view based on deepening cognitive conflicts in order to create levers for students’ understanding of new knowledge overemphasizes the discontinuity between old and new knowledge in students, making the acquisition of mathematical

13

The concept of teaching for representational change (Singer, 2007a) has been developed in a paper published in Mind,

Brain and Education (a journal born at Harvard University which was ranked in 2007 as the Best New Journal in the Social

Sciences & Humanities, and which rapidly evolved in a leading journal for the integrated domain of cognition, neuroscience and education) This topic has been also developed in a plenary lecture given at the International Mathematical Creativity and Giftedness Conference in South Korea in 2012 (Singer, 2012a)

expertise difficult to conceptualize The present paper tries to identify and emphasize the benefices of stressing the continuity between learners’ old and new knowledge through both valuing students’ adaptive representations and providing students with a variety of new representations that engender a dynamic view of mathematical knowledge

In addition, there are at least three reasons for introducing a distinction between the teaching for conceptual change in science and the teaching for conceptual change in mathematics I will discuss these reasons below

First, the procedural difference between the way in which students’ misconceptions are developing in science and the way in which misunderstandings, mistakes, and false principles are built in mathematics learning needs slightly different treatments to achieve expert knowledge in each of these domains Mathematics works with formal representations to a larger extent than science, which is grounded in conceptualizing various experiences and experiments, and in validating theoretical assumptions through new experiments While misconceptions analyzed in science emerge from students’ interaction with the physical and social every-day life contexts, mathematics misconceptions mostly arise from students' prior learning Most students’ mathematical mistakes develop within the learning process, and they might become misconceptions for further learning

In elementary mathematics, such misconceptions usually originate in prior instruction, based

to the fact that students tend to incorrectly generalize learned rules to grapple with new tasks (Nesher, 1987; Resnick et al., 1989) To clarify these aspects, some examples are needed Thus, in their efforts to understand the ordering of decimal fractions with fractional parts (e.g., 3.4 and 3.176), many middle-school students apply prior knowledge of either whole numbers or common fractions (e.g Nesher & Peled, 1986; Resnick et al., 1989) Students who apply whole-number knowledge to compare 3.4 and 3.176 ignore the decimal points and treat each as whole numbers, concluding that 3.176 is greater than 3.4 Similarly, the 1999 TIMSS study records that the question: which of the numbers: A) 0.625; B) 0.25; C) 0.375; D) 0.5; E) 0.125 is the smallest? was correctly solved by only 46% of the 8th graders participating at international level These results allow inferring that an interference with the set of natural numbers prevented students’ correct reasoning An insufficient concern to explaining the operational context within the two sets of numbers generated rigid structures

in students’ thinking, with long-term negative effects

In addition, students have difficulties to understand that numbers in decimal notation like –0.7 and improper fractions like 13/8, along with integers, can all be placed on a number line because all of them are real numbers, representing quantities along a continuum (Richland, Stigler & Holyoak, 2012) The belief that “multiplication makes larger, division makes smaller”, which only happens for natural numbers, affects the correct reasoning of many students; in their analysis of students' notions of multiplication, Fischbein, Deri, Nello, & Marino (1985) noted that the correct concept competes with prior learning, and this might be

a reason for arbitrary answers Moreover, college students frequently make a reversal error in translating multiplicative relationships into equations (e.g., translating "there are four people ordering cheesecake for every five people ordering strudel" into "4C = 5S"), in different conditions in which the initial relations were stated in sentences, pictures, or data tables (Clement, 2008)

Second, the epistemological differences between mathematics and science require different domain-specific processes to effectively acquire domain-specific knowledge While physics and other natural sciences are based on evidence gathered in experiments and observations, theories that move from the particular evidence to the general extrapolation, and the requirement that contradicting evidence falsifies a theory (e.g Lakatos, 1963; Popper, 1962),

mathematics offers to students a corpus of knowledge that explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world; a central line of investigation in theoretical mathematics is identifying a limited set of basic axioms in each field of study from which all other ideas and rules in that field can

be logically deduced; even more, part of the sense of beauty that many people have perceived

in mathematics lies not in finding the greatest complexity but in discovering the greatest simplicity of representation and proof (Horsten, 2015) While classical sciences gather evidence from experiments and observations, and develop theories based on empirical evidence that is extrapolated and then validated through non-contradicting evidence, mathematics explores patterns and relationships among abstractions and develop theories based on deductive systems that should be (or at least tend to be) consistent, coherent, and complete, which is not characteristic of everyday knowledge Even if, especially physics among natural sciences is becoming more and more mathematized, most of the scientific part that is approached in school keeps the features presented above

Another series of reasons for which failure in mathematics may have different causes compared to the ones in science come from the different nature of mathematics and science Thus, Mathematics is a formal system, within which the key relations are not “causal” in any straightforward sense (Richland, Stigler & Holyoak, 2012) Thus, students may assume that there are no real “reasons” why various procedures work For example, in a recent study, Richland, Stigler and Holyoak (2012) observed that, lacking any meaningful model of what multiplication “means” outside of the procedure itself, college students involved in the research lacked a reliable basis for finding relevant relationships between juxtaposed problems such as “if you know that 10×13 =… then 20×13=…” able to lead to a correct answer

Third, the review of conceptual-change literature clearly shows a focus on science learning, whilst examples from mathematics are, in most cases, just peripheral The domain under study in the conceptual-change research is usually science (ranging from studies of learning about the human circulatory system to key physics concepts, such as heat and light); sporadically, some emphasis was placed on mathematics learning (e.g Carey, 2001; Le Corre, Van de Walle, Brannon, & Carey, 2006; Smith, Solomon, & Carey, 2005) However, studies related to mathematics either combine mathematics with science, making imperceptible transfers between domains in building argumentation (e.g Smith, diSessa and

La Roschelle, 1993), or use contexts that involve science concepts For example, analyzing students’ conceptions about division, Smith, Solomon, and Carey actually base their argumentation on students’ conceptions of matter (Smith, Solomon, & Carey, 2005) When the reference is strictly to mathematics, the term that appear more consequent is representational change, not conceptual change (e.g Opfer & DeVries, 2008; Opfer & Siegler, 2007; Siegler, 1996; 2000; Varma & Schwartz, 2011)

The differences between mathematics and science epistemologies require a different approach as more effective solution Diminishing the epistemology in favor of ontology will not solve the problem of a unifying theory of conceptual change valid for effective learning During the school years, children go through transitions between different concepts and even different forms of the same concept This may be seen as a consequence of the need to cover the milestones of history of knowledge domains during the school years As we have seen, the knowledge development process itself has crossed series of incommensurable transitions

We face a dilemma related to school learning: the process of shifting from one concept to

another has necessarily to pass through conceptual changes in children’s minds, or does representational change play a more important role than previously assumed? Do children necessarily learn mathematics by moving through changes in theories analogous to scientific

revolutions, or do they learn better by switching between representations to flexibly develop concepts? The focus of this paper is to go beyond this dilemma and to show how by using adequate representations, domain-specific structural models can be internalized for in-depth learning of mathematics

Within the present paper, representational-change approach focuses on organizing the inputs

in a way that misconceptions induced through instruction are diminished Teaching for representational change facilitates transfer by avoiding rigid connections through exposing students to a variety of structured representations

Most of the discussion above took into account stages of cognitive development from early childhood up to adult age There is also another dimension of cognitive development that takes place in minutes or even seconds while solving an everyday-life problem This has been called micro-development (Fischer & Bidell, 2006; Granott, 2002; Yan & Fischer - ‎2002) The research in the area of changes in micro-development correlates also better with the representational-change framework For this reason, a different type of literature connected to

RC has to be discussed Representational change is also used in relation to insight problems

to explain the blockage a problem solver would face to find a solution (Jones, 2003; Kaplan

& Simon, 1990; Knoblich, Ohlsson, Haider, & Rhenius, 1999; Ohlsson, 1984; Sternberg & Davidson, 1995; Wertheimer, 1959) Here, representational change is seen as an extension of the Gestalt term “Restructuring” According to this view, accomplishing a change in one’s representation (seeing the problem in another way –”outside the box”, or changing the representational goal) is the main source of difficulty in achieving insight For example, an inappropriate problem representation that over-constrains the search space for a possible solution could be to search for a solution within a 2D space when a 3D representation is re-quired, as in the six-matchstick-problem where the goal is to form four equilateral triangles from six matchsticks (Katona, 1940; Scheerer, 1963) The resulting unexpected appearance of

a solution in consciousness, together with the associated “Aha” feeling, has been termed the

“insight experience” and an “insight problem” is one whose solution is likely to be accompanied by this experience (MacGregor & Cunningham, 2009; Smith, 1995).Representational change may appear as the result of unconscious processes that occur when problem solvers get stuck in an impasse To overcome it, a representational change is necessary Insight depends on an underlying representational change, which acts as the door opener that ensures that the appropriate heuristics can be applied to the problem representation (Bowden & Jung-Beeman, 2003; Jones, 2003; Öllinger, Jones, & Knoblich, 2006; Wegner, 2002)

Well trained problem solvers learn how to overcome such impasses Is it possible to bring into school a type of learning that help developing fluid thinking so that decision taking in everyday-life situations to be optimal according to individual potential? Within the RC framework, this is less about changing a concept, but about building the concept by alternating representations in order to help students internalize the meaningfulness of that concept Ultimately, it is about seeing the domain under study in successive frames that finally compose the big picture, or, better – the big movie (because it is about a dynamic, multifaceted construction) of the concept

By its system of dynamic multiple representations, the process of RC places emphasis on mathematics as a meaningful system, governed by an interconnected set of relations Though not “causal” per se, these relations are seen as having relevance to mathematical goals (Bartha, 2010) As Bartha argued, the general notion of functional relevance (of which causal relations are a special case) governs mathematics-based inferences Clearly, simply solving sequences of mathematics problems does no guarantee that the student will end up deeply

understanding the conceptual structure of mathematics (Richland, Stigler & Holyoak, 2012)

RC places emphasis on functional relations that serve to explain why various mathematical inferences are valid, while others are not

Various studies have been shown that if two examples are juxtaposed but processed independently, without relational comparison, learning is severely limited (Gentner, Loewentein, & Thompson, 2003; Loewenstein, Thompson, & Gentner, 2003) Even when comparison is strongly encouraged, some people will fail to focus on the goal-relevant functional relations and subsequently fail on transfer tasks (Gick & Holyoak, 1983) As response, RC introduces source analogs that “ground out” formal mathematical operations in domains that provide a clear semantic interpretation (as, for example, introducing the number line as a basic model for concepts and operations involving continuous quantities)

Moreover, even if a good source for analogy is provided, relational comparisons are affected

by the limited working memory capacity (Halford, 1993; Hummel & Holyoak, 2003; Waltz, Lau, Grewal, & Holyoak, 2000) RC is meant to reduce working-memory demands and helps people focus on goal-relevant relations This will aid learning of effective problem schemas and thereby improve subsequent transfer to new problems RC supports relational representations and comparative thinking to understand the potential role of analogical reasoning

1.4 Teaching for representational change: arguments for its sustainability

The classical view of conceptual change supposes developing a curriculum in which concepts are step-by-step integrated and conceptual progress is made by resolving cognitive conflicts between old and new knowledge Teaching for representational change, instead, lacks this linearity and, perhaps it lacks also a certain formal coherence given by an explicit succession

of information (However, this coherence frequently happens to remain only at the curriculum level documents – it is not internalized by the students, or even by the teachers, in some cases.) Therefore, a curriculum based on representational change has to meet certain criteria, and we have to take some precautions before launching the RC teaching hypothesis These precautions are essentially related to the question: Are children endowed with the necessary capacities to process a RC-based curriculum? In other words, is a RC teaching approach sustainable in education? The answers to this question are based on empirical studies and literature review related to the following four dimensions: children’s innate propensities, children’s endowment for recursive processes, children’s spontaneous bridging ability, and children’s capacity to shift among representations in problem solving

1.4.1 Children’s innate propensities for processing domain-specific information

During the last three decades, a large body of research has been devoted to analyzing infants’ cognitive capacities Important findings concern the roots of inductive reasoning and categorical exclusion, even if the mechanisms underlying categorization are not entirely known Many studies bring evidence that the mechanism of categorical learning – essential in building concepts – is active in preverbal infants (e.g Quinn et al., 1993; 1996; 2001; Behl-Chadha, 1996; Baillargeon, Needham, & DeVos, 1992)

A large gamut of literature on early cognition refers, in the last three decades, to the innate propensity for numerical competence This propensity can be described in terms of subitizing

- recognizing numerosity limited up to four items at a glance, without counting (e.g Gallistel

and Gelman, 1992; Mandler and Shebo, 1982; Benoit, Lehalle, & Jouen, 2004), or of

‘bootstrapping’ a generative understanding of number out of the productive syntactic and morphological structures available in the counting system (e.g Bloom, 1994), or of an embodied number sense (e.g Dehaene, 1997; Dehaene, 2007; Dehaene et al., 2003; Spelke, 2003; Dotan & Dehaene, 2013), or of a weaker nativist condition – the use of one-to-one correspondence to measure set identity (e.g Izard, Streri, & Spelke, 2014)

A large category of experimental data brings evidence for the number sense in infants Thus, Wynn (1990, 1992), and Starkey (1992) showed that 5-month-old infants are able to compare two sets of up to three objects and to react when the result of putting together or taking away one object is falsified These experiments were followed by many replications and extensions This research area of neuroscience is important from an educational perspective because it shows that, far from being ”tabula rasa” at birth, children have predispositions that allow them later to construct domain specific knowledge, in particular mathematical knowledge

1.4.2 Children’s endowment for recursive processes

Although children usually begin verbal counting in their second or third year of life, they arrive to understand the meaning of the counting routine at least two or even three years later (Wynn, 1992, 1995) Deepening the meaning of counting (which is based on the underlying connections between numbers and various sets of objects) and understanding numerical operations are processes that take long time and are frequently accompanied by failure In spite of this, however, there is converging evidence that after learning to count, children are able to develop sequences of natural numbers indefinitely long, without special learning/training Even more, there is converging evidence from different clinical interviews that many grade 1 and grade 2 students can articulate the principle that there is no biggest integer (e.g Hartnett & Gelman, 1998; Sansavini, Bertoncini, & Giovanelli, 1997)

This result was confirmed by a series of surveys with primary students arbitrary selected from Romanian schools Thus, six-seven-year-old children in school context have been found able and eager to continue indefinitely a sequence of increasing numbers, even if they never formally learned such numbers Singer and Voica (2003; 2008b) assumed that this is because

of their core dynamic knowledge, which includes the use of concepts and rules recursively What was striking in these experiments was the fact that, when asked to say a bigger number than a given one (for example 23 billions), many 1st graders did not actually use the Peano’s axiom (adding one to get the next bigger natural number), rather they have used a construction that seems to be of a linguistic nature (“a billion of billions” or “quadrillions of quadrillions”)

Another interesting example that may stress on the connection between language and mathematics emerged from students in grades 5 and 6 Interviews with these students captured unexpected representations For example, the drawing in Figure 1 resulted from a response of a 6th grader to the question related to how many numbers might be in the interval [2; 3]

Fig 1 The structure identified by Alice (grade 6) to argue that the rational numbers between 2 and 3 are infinitely many Reproduced from (Singer, 2010).

Her argumentation, which is not based on something already taught (similar to other interviewed students of 12 years old) is interesting because it naturally connects to another tree-type graph used in linguistics An example is provided in Figure 2, in which the syntax

of the sentence “The young child creates new knowledge.” is shown This type of representation was used by Noam Chomsky to show the grammar structure behind a sentence For Chomsky, the study of language concentrates on the investigation of the structures of syntax, an investigation which is almost logical-mathematical The task to uncover the set of rules or principles that could account for all of the grammatical sentences

of a language involves the identification of underlying syntactical processes and the search for counter-examples – in an effort to delineate the nature of the formal system underlying all the languages spoken by human beings By designing tree-graphs for sentences, Chomsky (1980) changed the perception of language: while communicating, we are not producing linear strings of sounds/words interrupted by small gaps of silence; we are actually building

“tree structures” that bear syntactic and semantic complexities What is special in connection

to this is the fact that an untrained 12th year-old mind emphasized such a tree-structure associated with the infinity of numbers

Figure 2 A tree representation for the syntax of the sentence “The young child creates new knowledge.”

The above remarked similarity between the two schemata, similarity that was not commented

in the literature up to now as for my knowledge, is relevant from the present thesis because it shows that a common cognitive mechanism may govern producing sentences and generating numbers This might be an argument for the universal structures of mind, but it may be as well an argument for a general-processing mechanism I tried to model, and which will be presented in Chapter 2

The similarities remarked between a sequence of rational numbers and a sentence structure may suggest that the underlying cognitive mechanisms might be similar to some extent for meaningfully generating numbers or for meaningfully organizing words into sentences The

connection point seems to be recursion The role of recursion is essential for the way we

combine letters/utterances into written/spoken words and words into sentences in a language, taking a finite set of elements (words, sentences) and yielding an array of discrete expressions (Hauser, Chomsky, and Fitch, 2002), which could be considered potentially infinite Children’s capacity to utter bigger and bigger numbers deals with this combinatorial endowment, analyzed mostly for language Yet, it seems that the recursion property acts as well for the “mathematical” mind Children are able to generate, from a set of a few digits, infinitely many natural numbers through a recursive procedure given, essentially, by the Peano’s axioms (Singer, 2010) They are also able to identify and use consistently recurrence rules for numerical and geometrical patterns (Singer & Voica, 2003; 2007) without any special training

Similar conclusions have been obtained from a survey of a representative statistical sample for the Romanian school population made with the purpose to observe structures that spontaneously emerge in children In this idea, a test with items that asked the continuation of some sequences of numbers and sequences of geometric figures was presented to the sample consisted of 3,837 students in 4th grade (10-11 years old) Patterns are not used in the current practice of Romanian teachers; therefore such a test provides more significant data about children’s natural potentialities Moreover, because the sample is representative, the use of quantitative statistics is relevant for making assumptions Most of students tried to find a completion for the given patterns (less than 3% did not answer) A vast majority of children (75%) gave meaningful continuations for the sequences Among these, about one third of students even found two-dimensional regularities and applied them consequently for developing patterns The results obtained during this research are confirmed by similar outcomes obtained for relatively similar items and other samples (for example the samples used in (Singer & Voica, 2003; 2008a)

From an epistemological view, recursive processes are pervasive in mathematics, from elementary arithmetic (such as: the building of base ten counting system that uses units, tens, hundreds, etc.; the principle of developing other numerical bases; the division algorithm for rational numbers) to mathematical analysis The similarities above discussed refer to the human mind capacity to build that mathematics

It seems, however, that the connections are even deeper The embodied metaphors theory in linguistics (Lakoff, 1987; Lakoff and Johnson, 1999) extends the properties of syntax to human conceptual systems (that led to developing the domains of knowledge or the disciplines as we know them today) and thus helps illuminate children’s behaviors in relation

to numbers For Lakoff and his colleagues, language is embodied, which means that its structure reflects our bodily experience, which in turn creates concepts that are then abstracted into syntactic (and semantic) categories For example, to express the conceptual

metaphor "time flies" we actually understand time in terms of space, and this allows us a

more concrete and accessible representation about time They conclude that grammar is shared (to some degree) by all humans for the simple reason that we all share roughly the same bodily experience Moreover, the core of our conceptual systems – including mathematics – is directly grounded in perception, body movement, and experience, which integrate both physical and social contexts (Lakoff & Nunez, 2000) Concluding, children are endowed with a capacity for recursion, which seems to be pervasive for at least a few knowledge areas, and even more, this seems to be a feature of the way our mind internalizes knowledge

1.4.3 Representational strategies and bridging

To better explain the ideas, a short story of a second grader will show how a change in representation can trigger the use of knowledge never formally studied and deepen the understanding of new knowledge (Singer, 2007a) The interviewer asked Andrei (2nd grade, 7½ years old) to color one half and then one tenth, and then even one hundredth of a rectangle drawn on a grid paper This question was asked in spite of the fact that the curriculum for second grade does not require division, or fractions, not even multiplication While the child was able to answer the first question, for the next, he got stuck Then, a change in representation interfered by changing the paper sheet with which the student was working from a grid to a blank sheet (on which an identical rectangle was drawn) This determined a radical change of perspective activating a different thinking in the child: the second grader successfully identified one-tenth, and even one hundredth of the rectangle Therefore, while the grid encouraged a focus on numeric elements (counting of the number of squares in the grid), the blank rectangle allowed a focus on spatial relationships, and this shift enabled the child to accomplish the task far beyond the curriculum requirements (Singer, 2007a)

Relevant here is the fact that this type of scaffolding, by moving from a discrete numerical approach (counting the squares) to a topological approach (assessing globally the figure and its limits) led the child to use different layers of understanding It seems that he found a

“marker shell” The metaphor of marker shells was used to indicate targets for development and learning in bridging as a process that helps to generate new skills on the basis of the existing ones The marker shells serve as place-holders that people use to direct their own learning and development toward achieving new, more powerful mental structures (Granott, 1994; Granott, Fischer & Parziale, 2002)

Other children's patterns for solving similar tasks also showed that a small change in the representational strategy leads to a higher level of performing a task The mechanism of

cognitive bridging, in which a person uses a conceptual shell to guide his or her own activity

in order to solve a problem or learn something new, can be identified in many other situations For example, the bridging mechanism could explain the children’s dramatic progress in the learning of a second language trough simple social interaction in informal communities Other experimental studies show that young children can develop early algebraic thinking from the first years of schooling (e.g Ainley, 1999; Carraher, Schliemann, Brizuela, 2000) if provided with adequate support that offer bridging to the unknown

From another perspective, a series of studies on cognition uncovers many potentialities that children possess, beyond the stages of cognitive development Recent findings confirm what educators and parents frequently observed: children come spontaneously to invent new knowledge, anticipating further development with a décalage in advance of one or even two-three years Thus, for example, three-or-four-year olds begin to conserve simple number and continuous quantity transformations well before the stage of concrete operations emerges at six or seven years old (e.g Carey & Spelke, 1994; Feldman, 2004) These actions take several forms with a similar underlying mechanism, all of which being characterized by setting tentative targets for an unknown skill to be constructed at a developmental level higher than the level of the person's current activity (Granott & Parziale, 2002; Granott, Fischer, & Parziale, 2002) The phenomenon of bridging is visible in many occasions, even when children get very weak inputs Actually, bridging is a transition mechanism that people use spontaneously at a wide range of ages

1.4.4 Children’s capacity to shift among representations in problem solving

In the last decades, many cognitive-research studies revealed that children spontaneously use various strategies to solve problems in domains like arithmetic (e.g De Corte, Verschaffel, Lowyck, Dhert, Vandeput, 2002), and causal and scientific reasoning (e.g Schauble, 1996; Kuhn, 1996) The adaptiveness of these strategies increases as children gain experience, though it is obvious even in early years The overlapping-waves theory (Siegler, 1996) depicts children as generally knowing and using multiple, co-existing representations These representations compete with one another for use, and the adaptive choice is made depending

on the problem situation (Siegler, 2000; Siegler et al, 2011) In relation to this, the purpose beyond RC is to answer the question: If this propensity already exists in young children, could it be mobilized for further learning?

Questioning what underlies mathematics achievement, Sasanguie, Göbel, Moll, Smets, and Reynvoet (2013) propose to contrast approximate number sense, symbolic number processing, or number-space mappings performance of students of ages 6 to 8 Results showed that performance on the mathematics achievement tests was best predicted by how well children compared 1-digit numbers In addition, an association between performance on the symbolic number line estimation task and mathematics achievement scores for the general curriculum-based mathematics test measuring a broader spectrum of skills was found Together, these results emphasize the importance of learning experiences with symbols for later mathematical abilities Yet, such approach contradicts the conceptual-change classical vision, when, in order to stress the contradiction between the old and the new knowledge, the difference between numerical computing and symbolic computing should be made acute

In a study on children’s and adults’ understanding of magnitudes, Opfer and DeVries (2008) gave children and adults the task of estimating the value of salaries given in fractional notation The representational-change hypothesis generated the surprising (and accurate) prediction that when estimating the magnitude of salaries given in fractional notation, young children would outperform adults (Opfer & DeVries, 2008), meaning that based on their own wits, children are able to build complex intuitive representations

Somehow from an opposite view, Varma and Schwartz (2011) observed an concrete shift in the understanding of mathematical concepts in the case of the mental representation of integers Their experiments confirm the hypothesis that mastering a new

abstract-to-symbol system restructures the existing magnitude representation to encode its unique

properties Children, who have yet to restructure their magnitude representation to include negative magnitudes, use intuitive rules to compare negative numbers Taken together, these experiments suggest an abstract-to-concrete shift: symbolic manipulation can transform an existing magnitude representation (of natural numbers) so that it incorporates additional perceptual-motor structure, in this case symmetry about a boundary (for negative integers) (Varma & Schwartz, 2011) This again can be interpreted as a capacity to operate over representations without the need of a paradigmatic change Moreover, a study of Opfer and Bulloch (2007) indicates that young children possess the cognitive control to choose the similarities that best predict accurate generalizations that drive young children’s induction, naming, and categorization (Opfer & Bulloch, 2007) The young children’s representational capacity may be also revealed by their analogical problem solving (e.g Chen & Siegler, 2013) These results illuminate the development of early representation and processes involved in analogical problem solving, showing that a goal-oriented context facilitates representational change We interpret these results as shown that when teaching takes into account children’s representational potential in a specific domain and uses it as purpose-

oriented support for learning, the avoidance of a cognitive conflict is more profitable for learning

As previously shown, children are active constructors of their own knowledge in a variety of situations, even at very early ages (Singer, 2010), and they often generate useful new

representational strategies in solving encountered problems without mastering conceptual

understanding of a domain

Concluding this chapter, representational change is part of natural cognitive processes recently identified by cognitive and neuroscience research: children employ various ad-hoc layers of understanding and bridging techniques, combined with the intuitive use of multiple representations, to increase problem solving abilities Representational change is a good candidate to stimulate these capabilities by adequate educational tools and practices because

it allows an operational focus that moves beyond tasks, domains, and cross-domain connections to focus on dynamic knowledge structures (e.g Case, Okamoto, et al., 1996; Singer, 2001; 2007a) There are premises for implementing a RC-based teaching model because it builds on existing capacities and facilitates transfer by avoiding rigid connections (Singer, 2002) Moreover, the implications of this view refer to a teaching process that meets the cognitive needs of children (Singer, 2010; 2012a) Based on the above premises, representational change might be a good solution to optimize the learning process (less effort and more effectiveness) under the constraints of limited time and informational overload

Chapter 2: Theoretical models that support representational

change

Drawing on recent research in cognitive psychology and neuroscience, Chapter 2 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 improve and enhance learning? The first section of Chapter 2 gives a particular answer to this general question This answer refers to the

Dynamic Infrastructure of Mind14 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 contexts The features of this model are discussed

in the context of recent debates in cognitive psychology related to universalities of mind (cross-domain-general 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 and allows learning through 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 pathways

2.1 The Dynamic Infrastructure of Mind as lever for representational change

As argued in the previous chapter, representational change is supported by the natural propensities of mind Because we are talking about representations and strategies to generate

various ad-hoc representations, a question still remains: How could a mechanism that allows

the human mind build knowledge look like? Put in this way, the question invites to a definite

answer As result of a long process of conceptualizations, the answer to this question

appeared to be the Dynamic Infrastructure of Mind (DIM) This model assumes that the mind

14 The foundations of this concept have been presented in an article published in New Ideas in Psychology

(Singer, 2009), and are developed in a paper published in Advances in Sociology Research (Singer, 2010)

However, aspects related to the dynamics of mind are among my old research interests, being discussed in the

Ph.D thesis (1996), as well as in other articles, subsequently published (e.g Singer, 2001; 2003)

is generically endowed with categories of basic mental operations responsible for a general information-processing mechanism that specializes with development A basic mental

operation is seen as the simplest mind activity that can be identified by differentiating from

others, and which expresses forms of actions that are directly learnable when a specified content is associated

The DIM consists in categories of mental operations foundational for learning that contain inborn components called inner operations, which are self-developing in the interaction mind-

environment DIM is a complex mechanism underlying the computational properties of the

cognitive architecture of mind through its minimal set of operational clusters

The human mind is able to process a multitude of operations; part of trying to optimize this process is to determine which operations are foundational, that is, which operations can be the basis for the others The identified operational clusters as foundational have been each

denominated based on their major component, as follows: Associating, Relating,

Proto-quantitative (arithmetical) operations, Logical operations, Topological operations, Iterating,

and Generating Within the assumed model, these are the elements that constitute a general

information-processing mechanism responsible for initiating and sustaining cognitive development

A brief excursion into recent research on mind, brain and education will help clarifying the definitions and properties associated with this mechanism To answer some fundamental questions related to how the mind works, a huge amount of empirical data was collected through methods ranging from very innovative techniques used to depict infants’ cognitive behavior facing various stimuli, to non-invasive complex technology such as functional Magnetic Resonance Imaging (fMRI) There is a large debate in cognitive science and neuroscience literature concerning modularity (e.g Fodor, 1983; 2000), weak modularity (e.g Pinker, 1997), progressive modularity (e.g Karmiloff-Smith, 1992; 1994), specialization (e.g Barrett and Kurzban, 2006), on the one hand and universals of mind, innateness, and domain specificity on the other hand

One of the oldest reasons for disagreement among theorists is the “tabula rasa” hypothesis

versus the innate-mental-capacities models While the first perspective seems to be tacitly

accepted by many teachers – in the sense that they do not pay attention to students’ prior learning (e.g Gardner, 1999), the second is sustained by new research studies Chomsky is the one who re-launched the old debate between Locke and Hume, by assuming that children are born with knowledge of the fundamental principles of grammar According to the ‘Core

Cognition’ or ‘Core Knowledge’ hypothesis (Carey, 2004; Carey & Spelke, 1994; Spelke,

2003), evolution has equipped our species (and other species as well) with an innate

repertoire of conceptual representations Numerous experiments with many replications and extensions suggest this hypothesis, by showing that infants possess cognitive mechanisms at birth that allow them to learn very early in life, eventually in some “privileged” domains

among which: perception of physical objects, number, and theory of mind (e.g Bransford, Brown, and Cocking, 2000) From the perspective of the present thesis, these experiments

reveal a dynamics that might be modeled through innate adaptive mechanisms rather than

through domain-specific innate knowledge The knowledge domains are too recent (if we refer to human species evolution) to shape the mind, while arguments for an innate dynamics come from a larger spectrum of cognitive research

A second issue is related to searching for universal structures of mind Since the early 1970’s,

logic, language, and mathematics seemed successively being able to express the universal structure(s) of mind To be more specific, Piaget (1968/1970) developed many of his ideas around the Universal Logic of Mind and – connectedly – around the synchrony in

development; Chomsky (1980) triggered an international chain reaction in stating that Universal Grammar is a kind of linguistic genotype; Case (1992; 1996) proposed a central conceptual structure for numbers, space and event Further research brought nuances in these conclusions, emphasizing aspects “beyond universals” (e.g Feldman, 1980/1994), a view that

is more consistent with the present approach

A third issue revolves around general versus modular distinction Modularity represents a

critical feature of the cognitive architecture of mind Chomsky (1980) and Fodor (1983) stated that the mind is composed of an array of specialized subsystems (modules) with limited flows of inter-communication Modular systems have some typical properties: they are domain specific in the sense that they operate on, and have a computational architecture that is unique to certain stimuli (for example, different mechanisms processes visual or acoustic inputs), they are cognitively impenetrable, fast, self-contained, informationally encapsulated, and have shallow outputs (Fodor, 1983; 2000) However, some cognitive mechanisms might have access to large amounts of information in the mind but only process information that meets certain input criteria (For example, we recognize familiar faces from a large mass of human beings; this is happening because we normally possess a module specialized for face recognition) While the classical modularity theory claims that modules are encapsulated, innate, predetermined and consequently impossible to be changed, within the view of weak modularity (e.g Pinker, 1997), the specialization of the cognitive system is

in some way heuristic (Barrett & Kurzban, 2006) Barrett and Kurzban (2006) argued that many systems in the mind, including central ones, might have wide input access but narrow processing criteria Thus, only information of certain types or formats is processable by a specialized input system For example, systems specialized for speech perception will process only transduced representations of sound waves; or, more concretely, although eyes and ears are exposed to both light and sound, eyes process only light, and ears process only sound The DIM model navigates among these theories and takes some stances which are sustained based on a careful review of a large body of specific literature The model offers clear descriptions of components and examples of possible application A pragmatic assumption on which the model is based is that the implementation of a DIM-based-learning model in the teaching practice in school, especially in primary grades can be taken as a validation of the DIM concept The main ideas behind DIM are concentrated in the following features, which makes it distinctive from other cognitive models in the literature:

a) DIM has operational specificity, not core-knowledge specificity

b) DIM is a domain-general mechanism, not domain-specific

c) DIM is endowed with progressive modularization and functional specialization; it

is not modular per se

Brief explanations on these aspects follow

a) As mentioned above, within the DIM paradigm, the initial cognitive systems are not domain specific (e.g Fodor, 1983; Chomsky, 1980), or core-knowledge specific (e.g Spelke,

2003), they are operation specific Its operational capacity induces predispositions to further developments that become domain specific while experiencing the environment This change

of perspective is relevant for an entire field of research in cognitive science because it restructures the concept of domain specificity

A necessary condition for considering an inner operation is to identify it in at least one experiment at very young ages Consequently, the definition of the basic operational categories was made with the help of various examples from studies on infants and toddlers The inner operations allow infants to build classes of objects based on similarity, and to

develop the extensions of these classes to more abstract categories The model tries to capture the invariance in the variability of human cognition The DIM’s components concur to generate behaviors that respond adequately to environmental stimuli In the process of adapting the response to environmental stimuli, DIM procedures specialize for the problem domain by recruiting specific mechanisms (operation plus domain/sub-domain) to solve

specific problems In addition, the DIM creates the dynamic context that predisposes the

human mind to develop through language Language augments the existing computational abilities by externalizing and recombining the information used by pre-linguistic computations in several ways (Clark, 1998)

b) Premises for DIM as a domain-general system come from a few sources identified in the

literature First, there exist common brain mechanisms for some distinct processes (Petitto,

1993; Petitto, Zatorre, Gauna, Nikelski, Dostie, & Evans, 2000) Second, cognitive models are implemented in distributed brain areas (e.g Bruer, 1997; 1999; Butterworth, 1999; Dehaene, 2007; Dehaene, Piazza, Pinel, & Cohen, 2003) Third, still, children develop in stages that are universal, although there are asynchronies among individuals’ ages and across domains

c) Briefly, the DIM hypothesis states that a general inborn mechanism composed of a few operational clusters processes data and organizes domain specific inputs via progressive modularization Chiappe and MacDonald’s (2005) claim that domain-general mechanisms are central to human cognition in that they allow for the solution of non-recurrent problems (problems that were never solved before) Barrett and Kurzban (2006) orient the discussion of domain-general abilities to the question of how a domain should be construed As they have suggested, “domains should be construed in terms of the formal properties of information that render it processable by some computational procedures” (Barrett and Kurzban, 2006, p 634)

The categories of operations identified as foundational sustain the architecture of cognitive development In order to fulfill its function, the DIM model satisfies a condition of optimality: the minimum necessary elements that predict the potential of development The other operations the mind processes result from the basic categories through relating and combining them The operational clusters that participate in composing the dynamic infrastructure of mind are listed in Table 1

Table 1: The main components of the Dynamic Infrastructure of Mind Reproduced from (Singer, 2009)

A detailed description of these operational categories can be found in (Singer, 2009; 2010) I

will only detail here the category of logical operations because this operational category

allows stressing on the connection between language and reasoning by deriving types of logical inferences used spontaneously (more or less correctly) in daily life

Logical operations refer to the capacity to use basic connectors: conjunction, disjunction,

negation, quantifiers as main composites for combining actions, or propositions Logical operations play an important role in language development, supporting the ‘‘scaffolding’’ (Vygotsky, 1934/1986) function of language This leads to building meta-systems of thought

in which the logical operations play the role of connection-agents The category of Logical

operations extends the use of basic inner operators (conjunction, disjunction, negation) to the

capacity of formulating logical inferences As already stressed before, the language (language

developed in interactive social contexts) has a decisive role in this capacity For example,

different patterns are involved in inductive, analogical, abductive and deductive reasoning

Inductive reasoning manifests when a conclusion is formulated about the entire class based

on examination of a number of cases from a class of objects (e.g “My friend is usually late

He will be late today, too.”) Analogical reasoning applies when a conclusion about an

individual case is formulated as a consequence of observing one or more individual cases (e.g “Just as airplanes could not imitate birds’ flight, computers are a mere fake of the human

mind.”) In this type of reasoning, the iterating capacity also interferes Abductive inference

(Fodor, 1983, 2000) can be characterized as “inference to the best explanation” For example, scientists are performing abductive inference when they decide what is the best possible explanation to account for a set of observed phenomena Provoking abductive type inferences

in science classes might be a modality to diminish misconceptions Abductive inference is said to be global, in the sense that any piece of information could, in principle, bear on the

inference Here, topological category has also a contribution Deductive reasoning may involve different patterns in: conditional, consecutive, causal, modal, normative, or

procedural reasoning The daily reasoning and argumentation actually mix together many of

these logical categories In addition, we have to take into account that many of our decisions are based on judgments that involve logic, but that are not necessarily expressed in specific words Yet, this description becomes necessary when analyzing the mechanisms that underlie understanding, in order to develop appropriate training

The interplay of the DIM operations functions at a micro-level (for solving specific problems), as well as at a macro-level of development In this last case, its expression is a whole preliminary labor (of which the child had no consciousness) of preparing a stage of development (Piaget, 1954; 1976) The interplay of the DIM operations within the stages of cognitive development is of a cyclical nature in two aspects On the one hand, a phase is

initiated by the generating and iterating categories and is ended also by these two operations,

because they allow bootstrapping into a new phase; on the other hand, the process replicates

on higher levels of complexity and abstraction (Singer, 2007b) Within the DIM, each phase

in the process of growth creates conditions that specify and constrain the next phase Cyclically, along development, the cognitive system arrives at correlating the dynamic mechanisms activated by the basic operations of its infrastructure These are the moments of

“taking of consciousness” As Piaget remarked, “the transformation is slow What is sudden

is the final comprehension when the structure is completed Yes, and of course it presupposes

a whole preliminary labor, underneath, of which the child had no consciousness; but taking consciousness [prise de conscience] is sudden Suddenly he sees things in the external world

in a whole new way That’s what’s sudden—not the construction, but the taking of consciousness.” (Bringuier, 1980, p 45)

2.2 Mental structures15 and representational change

A theoretical construct in my Ph.D thesis analyses types of cognitive structures The identification of structures is an old research interest, which led to various explorations at both theoretical and empirical levels In the last decade, within an empirical research program, samples of students of various ages have been tested and interviewed on a topic that

is marginal to the school curriculum although it is fundamental in mathematics This is the

concept of infinity This concept has a paradoxical situation: the perception-intuition of

infinity is natural, yet it is a challenging concept when it comes to learn it By exposing students to challenging tasks related to the concept of infinity, spontaneous mental structures have been revealed, further allowing to experimenters to analyze and classify them Later on, more consistence for these structures has been found (and in particular, a validation of these findings) in their pervasiveness in mathematics thinking and beyond I will go further in explaining the mechanisms that allow, enhance, or hinder the representational power of mind

by describing various types of mental structures identified in mathematics learning

2.2.1 How does the human mind organize itself? The building of structures

The description starts with a brief presentation of theoretical approaches related to defining

cognitive structures Piaget (1952) defined a schema as 'a cohesive, repeatable action

sequence possessing component actions that are tightly interconnected and governed by a core meaning' For Piaget, a schema is the basic element of intelligent behavior that ensure

the knowledge organization Indeed, it is useful to think of schemas as “units” of knowledge each relating to one aspect of the world, including objects, actions and theoretical concepts When Piaget was talking about the development of a person's mental processes, he was referring to increases in the number and complexity of the schemata that a person had learned Wadsworth (2004) suggests that schemata (the plural of schema) is an 'index cards' filed in the brain, each one telling an individual how to react to incoming stimuli or information An individual’s reaction is based on the mental representations of that individual As Goldin (2002) remarked, children’s representations evolve to the construction

of autonomous representational systems However, there have been few studies with children, which describe general characteristics of structural development and how structures are integrated into concept development Brown (2014) tried to unify different apparently

irreconcilable visions on conceptual change under the view of dynamically emergent

structures A certain level of understanding allows learners to think generatively within that

content area, enabling them to select appropriate procedures for each step when solving new problems, make predictions about the structure of solutions, and construct new understandings and problem-solving strategies (Huang & Bargh, 2009) The term “conceptual understanding” has been given many meanings, which in turn has contributed to difficulty in changing teacher practices (e.g Skemp, 1976)

As a complementary approach to the above, I use the term mental structure or shortly,

structure, for both conceptual and cognitive structures: a conceptual structure (as product of

human knowledge) needs to be perceived by a human mind in order to be identified as such, while a cognitive structure (an individual cognitive sub-mechanism) has to act on a specific concept/content in order to manifest its existence In a broad sense, a structure supposes relations among its components that assure a behavior that is invariant across situations This

15 Articles that approach the topic of cognitive structures have been published in ZDM (Singer, 2001), The Journal of Mathematical Behavior (Singer, & Voica, 2008a), and Mind, Brain and Education (Singer & Voica,

2010) This topic was also discussed in a few papers that have been presented in International conferences (Singer, 2002; 2003)

invariance allows the cognitive system to recognize a structure and to match it with the appropriate context A structure can have as referential: a concept/ a procedure or an interaction concept-procedure, a set of concepts/ procedures bound together, an aggregate of substructures, a series of interdependent aggregates

For reasons of clarity, I operate with two theoretical categories of structures: spontaneous structures, which are ad-hoc mental representations activated in specific contexts, and aggregate structures, which are elaborated within (formal) learning activities around concepts

or procedures that pertain to a domain or are cross-domain Spontaneous structures reveal innate endowment, while aggregate structures are results of learning acquisitions

The practice of structuring starts in the first years of child’s life, while she/he explores the environment within categorical learning, and extends along cognitive development through the organization of spontaneous and aggregate structures This stands for connecting the organization of structures to the DIM model The environment confronts the innate dynamic infrastructure of mind and its categories of operations are self-developing within this confrontation In this process, the proto-operations diversify and specialize while structuring mental representations Conversely, DIM’s operations capture environmental stimuli through inputs that specialize over time This explains the acquisition of cultural tools and tacit knowledge in individuals even if this acquisition is not the result of an explicit teaching-learning process

2.2.2 Types of Spontaneous Structures

To analyze children’s cognitive constructions when they are exposed to challenging situations, a series of studies have been developed (e.g Singer, 2007a, 2012b; Singer & Voica, 2003, 2007, 2009) Among these, in a study focused on students’ perceptions on infinity that involved 209 students from grades 1 to 12 (i.e 6-7 to 18-19 years old), Singer and Voica (2008a) found that, when arguing if a set has a finite or infinite number of elements, and when comparing the cardinals of two infinite sets, students inevitably arrive at emphasizing structures (structure) of those/ that set For this study, data have been collected through: successive questionnaires, interviews, a focus-group discussion, and classroom observations The experimenters carefully recorded not only students’ answers, but especially their arguments and ways of reasoning They concluded that students naturally search for structures in challenging learning contexts and apply solving procedures according to the structure they found most preeminent within a task

In solving problems involving the comparison of infinite sets, children have detached, from those sets, more abstract representations (i.e structures) involving relations inside the sets The idea to look at a set via structures was not at all suggested by the interviewers, it appeared ad-hoc in children’s trials They were able to operate with these abstractions in specific ways, while “leaving” the sets in the background This shift brings new evidence in favor of the existence of separated cognitive sub-mechanisms for processing objects (sets) and relations among objects (structures) Moreover, the identification of some structures in mathematical problems (i.e relations; dynamics) seems to be responsible for progressing in solving – a result that is similar to Anderson et al.’s (2008) conclusion that spatial imagery is more related to success in mathematical problem solving than object imagery

Four types of structures (which have been called a-structures, g-structures, d-structures and

f-structures) have been identified while exploring children’s understanding of infinite sets (Singer, 2010; Singer & Voica, 2010) In describing these structures we found that they have distinct features depending on the nature of the context and on the kinematics of the

processes involved A brief description of each type of identified structures follows, stressing

on how it relates to mathematics and how (if) it is generalizable to other contexts, even beyond mathematics

Algebraic structures An a-structure (algebraic structure) appears as a means of interrelating

the components of a system according to its algebraic properties Characteristic for an

a-structure is the transfer of operations between the sets through one-to-one correspondences

An a-structure operates with the decomposition of the elements of a set in their constituent parts This decomposition uncovers functional transfer between two sets Thus, a-structures

are activated in any process that supposes input-output

In general, each input-output mechanism capable of formal properties might have its origin in this capacity of the human mind of developing arithmetic-type structures It is probable that this endowment made possible the design and development of “in-out” computing machines However, although based on a simple mechanism of in-out type, the concept of function is highly difficult for students (e.g Breidenbach, Dubinsky, Hawks, & Nichols, 1992) The difficulties in understanding the concept of function might be overcome by stressing on the

mental existence of a-structures The teaching of the concept of function needs a long-term

projection, starting from early ages At an elementary level, a function transforms the elements of an input-set into the elements of an output-set by applying a unique procedure to all the elements of the first set Figure 3 presents a suggestive representation of an additive function (adding 3) defined on the set of natural numbers 4, 6, 8, 10 Children in the first

grades can successfully play with such “machineries” based on their capacity to activate

a-structures They can find the output set when the input set and the “operation” are given; they can find the input when the output and the procedure that “moves” the elements of the first set into the second are known, etc

Fig 3 A notational example for the functioning of an additive a-structure Reproduced from Singer &

Voica (2010)

Geometric (geonic) structures When asked to compare infinite sets, some students intuitively

appeal to the number line, even if the task does not mention it at all The representation of numerical sets on the number line and the identification of their geometric properties that are

relevant for a given task grants a g-structure ("geometric structure") to the sets of numbers A

g-structure was defined as a means to interrelate the components of a system which highlight

its visual, graphical, iconic properties Once a g-structure is identified, students are able to

use geometrical transformations and congruence as a way to show the cardinal equivalency of some infinite sets Congruence might intuitively appear as superposition through a slide or

through a rotation (that leads to symmetry) G-structures suppose a transfer between algebra and geometry, during which the initial configuration is mentally modified G-structures also

suppose a holistic vision of the set, which is transferred through representation For these

reasons, a g-structure endows the system (or its subsystems) with an organization that is

global and dynamic

Consequently, for example, because of g-structures, a natural way of representing the graphs

of functions of the type: x « x 2

+1, or x « (x+3)2

(which is an algebraic task), could be as

follows: to start from the “generic” function x « x 2

and to apply a translation to its graph (meaning a geometric approach) In this way, the algebraic properties of a numerical function can be studied on another (simpler) numerical function via a geometrical transformation, a

procedures that is only occasionally used in school Because g-structures are dynamic and

global, they allow the human mind to make transfers among representations that have an iconic component However, students have difficulties in making this transfer (e.g Zazkis, Liljedahl, & Gadowsky, 2003); it is expected that, when systematically making students aware of these structures through adequate practices, the quality of learning can be enhanced due to the support offered by this natural endowment

Fractal structures When, in order to describe the elements of a set, a procedure is

sequentially repeated at different scales, the set was granted an f-structure ("fractal-type structure") A f-structure is seen as a general means to organize a system which highlights the sequential generating of its subcomponents by repetition at different scales Because an f- structure works with different scales, it has a local character An f-structure is in the same time self-generated in a sequential mode, through the application of a rule This is why an f-

structure endows the system (or its subsystems) with an organization that is local and dynamic

F-structures are represented via configurations that self-generate through successive

homotheties based on scale change An important example refers to the transformation between measure systems These structures seem to be also specific for the way in which we understand the numerical systems: the numeric magnitude orders of base ten (units, tens, hundreds, etc.), as well as of other bases, are defined through grouping groups (ten units make a ten; ten tens make a hundred, etc.), which supposes changing the scale at each grouping of different magnitudes

F-structures are also typical for division When dividing two integers, for example, 37 and 8,

we recursively use the grouping and the division of a unit in ten units of the next lower order

of magnitude Thus, when divide 37 by 8 one gets 4 and a remainder 5 In order to continue the division, we transform the 5 remained units into 50 tenths The 50 tenths divided by 8 gives 6 and remainder 2 The 2 remained tenths should be transformed into 20 hundredths The 20 hundredths divided by 8 gives 2 and remainder 4 The 4 remained hundredths should

be transformed into 40 thousandths One finally gets 37:8 = 4,625 doing three changes of scale along the process The difficulty the students encounter when they learn the division algorithm might be given by the passage among different numerical scales while internalizing

an f-structure

Density structures When comparing infinite sets, students frequently referred in interviews

and questionnaires to the density of sets, seen as a degree of piling up the elements of the set Thus, for example, when asked which numbers are more: integers or fractions, some students notice that the natural numbers are “rarer”, while the set of rational numbers is more

“crowded” The idea of piling up, crowdedness, the step of succession, or density seen as an

intuitive measure of the set – how ”crowded” the set is – grants a d-structure ("density-based

structure") to a set of numbers

A d-structure was defined as a means to interrelate the components of a system that

emphasizes its topological properties (concerning vicinity, approximation, border) In dealing

with a d-structure, the child evokes density/ jam/ accumulation of the elements In general, the endowment of a set with a d-structure favors extrapolations from local to global Thus,

topological structures in a wider sense are emphasized, i.e structures that suppose the

invariance at changing the shape Within d-structures, the topological perception overlaps

recursiveness characteristic for discrete sets of numbers In this case, the child focuses on the local description of intervals of the sets This description does not suppose any change of

configurations For these reasons, a d-structure endows the system (or its subsystems) with an

organization that is local and static

The existence of d-structures can be valued in understanding motion in physics The fact that

this motion develops in time, which is perceived as continuous, allows us to trace the trajectory described by the moving ball through successively unifying the ball images that appear in a stroboscopic photo This extrapolation can be valued in explaining the procedure usually used for drawing the graph of a function (i.e representing a finite number of points in

a Cartesian system and join them through a continuous curve) To avoid the formal use of this procedure, a good introduction for drawing the graph of continuous functions is to consider firstly functions that describe a time-dependent movement In this way, because time is perceived as being continuous, unifying the discrete points of the graph becomes natural

Summarizing, as referred to the context they act on, d-structures and f-structures are of local nature, while a-structures and g-structures are global On the other hand, from a kinematic view, d-structures and a-structures are static, while f-structures and g-structures are dynamic

Table 2 summarizes the characteristics of structures based on these criteria

Table 2: Criteria to classify structures

Modalities of processing the context static dynamic

Reproduced from (Singer, 2010)

We wondered why these criteria that have appeared during the process of identifying structures would be relevant for the structures description and classification The answer might be that the human mind is determined by the continuum space-time The local-global features refer to spatial perception, while static-dynamic criterion is of temporal nature The space-time system of reference might be interpreted as evidence that the structures with which children endow the infinite sets are actually propensities of the human mind We thus retrieve the view of embodied cognition Among others, Lakoff and Núñez (2000) argue that formal mathematical ideas emerge from concrete sensory-motor experiences For example, a fundamental tenet of embodied cognition is that the understandings of basic ideas such as space, motion, and force, as well as more abstract ideas grounded in these intuitive ideas, arise because of sensorimotor, bodily interactions with one’s surroundings In other words, bodily interactions with one’s environment are intimately intertwined with fundamental intuitive understandings Gallese and Lakoff (2005) provide evidence that similar neural pathways are employed in actual bodily interactions as in more abstract conceptualizations of space, motion, and force

The above assumptions are strengthened by our findings Although we identified the mentioned structures as propensities of students confronting with challenging tasks, we further identified these types of structures in larger contexts, in mathematics and beyond A few examples are given below; others can be found in (Singer & Voica, 2010; Singer, 2010)

above-An a-structure is responsible with the identification and separation of constants and variables;

it also deals with the variation of quantities within an input-output process In a broad sense, such a structure allows understanding/representing simple mechanisms that keep performing the same operation while varying the inputs For example, understanding that if 1 kilo of apples costs 2 euro, then 3 kilos of the same quality will cost 6 euro, 4 kilos will cost 8 euro,

etc Moreover, a-structures help to overcome the constraints of subitizing through a

composition-decomposition process and in-out mechanisms that allow additive and multiplicative thinking

At least two reasons concur to activate a-structures within DIM First, the DIM mechanisms

specialize while processing various inputs, therefore the system should be able to detect out procedures and to develop the knowledge of using them; second, DIM is meant to act economically, therefore to automatize processes, and the in-out procedure of function type is the simplest mechanism of automatizing

in-G-structures refer to the recourse to an iconic element that can be described and understood

through its conversion in a representational system G-structures allow the use of notational

schemes for clarifying and simplifying descriptions of situations, contexts, actions (for example the use of maps, or graphic organizers in order to facilitate information processing) Karmiloff-Smith’s metaphor (1992) of the child as notator uncovers the development of this type of structure It also seems to have relevant components in the prehistory of humans, as it contributes to space orientation, which was very important for survival reasons

An interesting case in which a fractal structure is activated is the one of currency change

Some articles in cognitive psychology examine how numerical intuition for prices develops after a major change in currency Reporting a study carried out in France, Portugal, and Ireland before the switch to the euro, Dehaene & Marques (2002) concluded that estimation was less precise in euros, and also took longer to produce, compared to the national familiar currency In a second study, involving University students in Portugal and Austria, after these countries switched to the euro, Marques & Dehaene (2004) concluded that price estimates become progressively more accurate by a process that is related to buying frequency, which favors a relearning hypothesis An alternative rescaling hypothesis was also getting some empirical support The authors concluded that, as in other domains of thought (such as reading, for example), a dual route of processing might be responsible for the different approaches I would like to stress that, if it is either about relearning or rescaling, both reflect the adaptation to a change of scale and the difficulties of this adaptation Fractal structures are difficult to process and their internalization within different types of content needs systematic practice Making aware the learner about the invariants in this process may improve the quality of learning

The phenomena of convergence, limit, or, more generally, the recursive processes in which

the scale is preserved suppose the activation of a d-structure Thus, the fact that students

show a local topological perception (expressed through understanding the density properties) can be used (and it is actually) in the design of the functional graphs at the ‘endpoints’ of the

definition intervals, before the in-depth study of calculus D-structures are also typical for

statistical thinking and statistical methods: we extrapolate conclusions having a local character (obtained on a sample) to an entire population

The studies above discussed brought evidence that students of various ages naturally operate with various structures when they are confronted with challenging topics Up to now, the four types of spontaneous structures identified in empirical studies focused on how children deal with infinite sets seem to be independent Some other basic structures might be revealed in other empirical studies What can be inferred at least, in consensus with the models presented

in this paper, is that the dynamic infrastructure of mind specializes procedures that generate spontaneous cognitive structures The development of these structures is constrained by the individual neural endowment and is driven by environment

2.2.3 Aggregate Structures and Knowledge Construction

The DIM mechanisms capture internal constraints and thus develop embodied knowledge They capture external constraints as well and thus develop adaptive knowledge Part of the embodied knowledge is the recorded propensity of infants and toddlers for the gradually-specializing domains of knowledge Internal constraints and external stimuli coexist and action in conjunction on the cognitive system and they create an interference between embodied knowledge and adaptive knowledge Therefore, adaptive-knowledge learning results from interactions

According to Piaget, horizontal décalage begins with an assumption of general synchrony across domains and explains asynchrony in terms of objects’ resistance to people’s activities

A series of studies on cognition uncovers various asynchronies within the stages of cognitive development which are deeper than the ones emphasized by Piaget Thus, for example, three-or-four-year olds begin to conserve simple number and continuous quantity transformations well before the stage of concrete operations emerges at six or seven (e.g Feldman, 2004) The “new Piagetians” have been shown that the ladder metaphor for cognitive development is

no longer valid: children do not develop in stages that evolve linearly across domains and individuals, so as climbing the stairs A more complex metaphor that better model cognitive

development is the web of skills (e.g Fischer, 2008) Although skills develop according to a standard sequence of levels, children vary substantially in the developmental pathways they

take, and each individual child varies greatly in skill levels across domains

The Dynamic Infrastructure of Mind could account for the fractal model of the developing mind recognoscible in the web of skills proposed by Fisher and his team at Harvard University (e.g Fischer & Rose, 1994; Fischer & Rose, 2001; Fischer, Yan, & Stuart, 2003) Within the web of skills that dynamically act in the human development, acquisitions are based on periodical scale changes; because the DIM’s operations have similar functioning mode, whatever the level of abstraction involved, they can be seen as the dynamic elements that pattern the repeatability in variability (Commons, Trudeau, Stein, Richards, & Krause, 1998) of the phases of the developmental stages (Singer, 2009; 2010)

The DIM model raises some new explanations about asynchronies in development and the apparition of new knowledge out of the existing one Within the mind’s trials to organize the internal world, the operations search for content, in an attempt to “make sense” of the external world As we are endowed with mental operations that are searching for a content to become functional and from this interaction the operations evolve, the increase of the operatorial combinatorics of mind has a decisive contribution to various mind changes, including the ones of a developmental type (for example, understanding conservation of matter) On the other hand, because the operations are embodied in various contents, they create rigid entities, which are difficult to change This could be an explanation within the DIM model for the resistance of early intuitive theories of mind

The efficiency of brain functioning makes a person to tend to develop new structures on “the routes” already used The degree of stability depends upon the training – ad-hoc training environmentally driven, or educational formal training To change various robust intuitive theories of mind, some of which has proved as being wrong, an external intervention is needed

to undermine the intricacies operations-content This is efficient – that is new codes and procedures are internalized – only when a secondary intuition becomes active According to Fischbein (1987), intuitive knowledge, or primary intuition is a self-explanatory cognition; it is

a type of immediate, coercive, self-evident cognition, which leads to generalizations going

beyond the known data, while secondary intuitions are “those that are acquired, not through

natural experience, but through some educational interventions”, when formal knowledge

becomes immediate, obvious, and accompanied by confidence (Fischbein, 1987, p 202), in other words, they become beliefs, self-explanatory conceptions The construction of a secondary intuition in mathematics learning is a challenging application of the DIM model However, aggregate structures are not just a matter of students’ intuitions They are intrinsic

to the children’s cognitive development as they are essential for the human knowledge progress The construction of aggregate structures is the “clock” that engages developmental transformations School learning could highlight the natural propensity of human mind for processing structures, making conceptual learning more efficient Similarly, inappropriate school learning can neglect this propensity, generating ineffective learning acquisitions

A simple system for classifying aggregate structures refers to their mobility Thus, the following types of aggregate structures might be differentiated as distinct theoretical entities: rigid structures, flexible structures, and dynamic structures (Singer, 2001, 2004)

Rigid aggregate structures A rigid aggregate structure is characterized by: (a) oversized,

very stable nuclei, (b) a poorly developed network, sometimes totally lacking, and (c) associations that function in the area of the recognition of a standard situation and its reproduction As an example, this phenomenon emerges frequently in learning classical geometry: a student recognizes the isosceles or right-angled triangle only if the given triangle

is in a certain position; any other position is perceived as a new learning element that requires

a new nucleus in the structure Such rigid mental configurations often become fixations On the positive side, a rigid structure is needed for the practice of a skill or for the learning of algorithms Beyond its positive role in assuring the stability of the acquired knowledge, a rigid structure is usually responsible for the emergence of typical errors Such a structure develops because of two kinds of errors in teaching One error occurs when isolated information is taught without highlighting its connection with previously learned information

or when insufficient time is given to the internalization processes needed to create a conceptual network A second error occurs when excessive focus is placed on the already-taught information This, too, hinders the development of a network Some of the “drill and practice” procedures that developed out of a behaviorist approach can be responsible for this result A mental structure has a regenerative tendency to organize itself, tendency that can be blocked only by the second above-mentioned constraint In fact, that regenerative tendency explains the progress in learning even with the most inappropriate teaching

Flexible aggregate structures A flexible structure is characterized by: (a) stable nuclei, (b) a

developed network, and by (c) associations based on recognizing invariant elements in

various environments A flexible structure allows problem solving through analogy and inductive or deductive inferences when the context is relatively familiar Such a structure might enter into relations with other structures, ensuring a coherence of the reaction A flexible structure is activated when solving problems that are based on “short distance” transfer, such as applying algorithms, identifying particular cases, and, possibly, using analogical reasoning

Dynamic aggregate structures A dynamic structure implies: (a) flexible nuclei that are or

could become structures in their turn; (b) complex networks with ramifications and hierarchies; and (c) dynamic associations that facilitate quick mobilization of the structure

through the discovery of critical paths These associations stimulate the self-development of the structure, highlight underlying relations among different structures, and give rise to links between various structures within the cognitive system A dynamic structure can also behave

as either flexible or rigid, depending on the task to be solved While the flexible structures are mostly adaptive, the dynamic ones are mostly creative