250 likes | 456 Views
The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Discerning In and Between Theories in Mathematics Education. John Mason Oxford Nov 2010. Outline. ‘Theory’ as ‘perceiving’ (discernment) Notable Theorists in Maths Edn Attention
E N D
The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking Discerning In and Between Theories in Mathematics Education John Mason Oxford Nov 2010
Outline • ‘Theory’ as ‘perceiving’ (discernment) • Notable Theorists in Maths Edn • Attention • Structuralist View of Theories • Appreciating Theories Specific to Mathematics Education
Theory as Perceiving • ‘Theory’ • From Greek: a way of seeing … perceiving … hence ‘perspective’ • We see what we expect‘seeing’ is believing • “I see”: • comprehend, understand, appreciate, make sense of … • “I see what you are saying”
More on Theory as Perceiving • ‘Theoretical perspective’ predisposes us to attend to certain ‘things’ and not others … … with lots of consequences • On multiple levels • Macro, Micro & Meso Theories
Examples Macro Micro • Skemp: Intelligence, Learning & Action • Dubinsky: APOS • Sfard: Commognition • Pirie & Kieren: Growth of Understanding • Tall et al: proceptsConcept image • Bruner: enactive-iconic-symbolic Meso • Van Hiele levels • Vertical & Horizontal Mathematization • RBC + C Grain Size Background Foreground
Notable Grand Theorists Plato (Socrates) 500 BCE Dewey (1859-1952) Jesuits Gattegno (1911-1988) Comenius (1592-1671) Skemp (1919-1995) Froebel (1782-1852) Brousseau (1933) Montessori (1870-1952) Chevallard Freudenthal (1905-1990)
Different Worlds • World of External Facts • Repeatable experiments; sciences, psychology • World of Opinion and Belief • Surveys & Questionnaires; sociology • World of Others’ Experience • Participant Observation; ethnographic • World of Involvement in Action • Change & prediction; action research • World of Personal Experience • Sensitising to notice; phenomeno-logic & -graphic
Features of Worlds • Epistemology • Ontological commitment • Value system: what is researchable • Psychology of researcher & subjects • How environment influences actors
Attention • Holding Wholes (Gazing) • Discerning Details (Hyparchic acts) • Recognising Relationships • Perceiving Properties • Reasoning on the basis of agreed properties
Development • Manipulating familiar confidence inspiring objects (specialising, particularising) • In order to get a sense of underlying structural relationships (modelling, axiomatising, justifying, proving … • Bringing this experience to articulation, which over time, becomes more succinct and useable (manipulable)
Purposes & Uses • Descriptive • Illustrative • Explanatory • Informative • Predictive • Evaluative Grain Size Using theories to make sense of experience … Experiencing the sense made as a result of theories
An Overtly Structural View of Theories • Neo-Pythagorean (qualities of number) • Provides a justification for lists
Six Modes of Interaction with Theory Describing Illustrating; Probing Explaining Informing Predicting Evaluating
Situation Theory Situation Enquirer Theory Enquirer Evaluating Explaining Theory Situation Theory Enquirer Situation Enquirer Predicting Informing Six Modes of Interaction with Theory Enquirer Theory Situation Describing Enquirer Situation Theory Illustrating Probing
Student Content Teacher Content Teacher Student Expressing Examining Content Student Teacher Student Teacher Content Exercising Exploring Teacher Student Content Expounding Teacher Content Student Explaining
Activity Goal Insight Influence Want Tasks Resources Current State Distinctions Relationships Properties Actions Having come to mind Choosing (?) Exploiting Draw Upon Introduce, Try Know Unfamiliar Situation Background Distinguish: -‘placing in scheme’ - gaining insight from Foreground
Essence: theories as webs Format for findings Action View of validation Insight Methods: view of data; analysis Basis ofperception Coherent view of objects Vocabulary Formulate enquiry Desire to know, to understand to act Background Foreground
Possibilities in the Moment Mechanicalapplication Informative insight Conservationof practices InspirationIdeals CreativeResponses RestrictionsBlinkers Background Foreground
Appreciating a Theory Bricolage Immersion Combining Theories: completeness consistency
Structure of TheoriesAssociated Actions • Distinctions • Labelled ––> constructs • Assembled into collections ––> Frame(works) • Supporting recognition of relationships • Articulated and taken-as properties Resulting in: Phenomena (Hyparxis & Ontology) Leading to actions
Follow Up • mcs.open.ac.uk/jhm3 (Presentations) • Sources • The Role of Theory in Mathematics Education and Research (1996) in Bishop et a • Theories of Mathematics Education (2010) Sriraman & English • PME research group on Theory in Maths Education • Systematics (J. G. Bennett 1960s)