PowerPoint Presentation

(Some) PsychologyofLearning & Doing Mathematics

The Open UniversityMaths Dept

University of OxfordDept of Education

Promoting Mathematical Thinking

Oxford N1A 3 2012

#1Think back to previous session: were the relationships we worked on necessary and extant or were they created by us?Seeing & Believeing

#Psychology: study of the psycheCognition, intellectEnaction, behaviourAffect, emotionAttention & willHow these are influenced by the social#Cutural influences on practices, beliefs, dispositionsExpectations (personal, peer, significant others)

3What does it mean to understand in mathematics?To carry out procedures unaidedBy templatingBy re-constructingTo have come to mind appropriate procedures and conceptsTo explain adequately to someoneOwn narrativeTo a NoviceTo a Peer/ColleagueTo an ExpertTo modify to meet new circumstancesTo be familiar with a viable model (structural relationships treated as properties)Understand or Overview or Appreciate?#Recall the coke can balancing act I appealed to continuity to justify the need for the CofG to move from the middle to the lowest point to the middle again without a sudden jump. What was needed to understand or appreciate continuity in that context?4Concept Image all the cognitive structure in the individual's mind that is associated with a given concept (Tall & Vinner 1981)the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures.

Concept Image Concept Definition#It is quite hard to say what you mean by understanding something.

Number; angle; function; continuity; group; ring; 5Concept Images: examplesWhat is A fractionAngle between two linesDifferentiabilityTangent to a curve#ZigZagsImagine the graph of f1(x) = |x 1|

What does it mean to understand or appreciate x > |x|?Imagine the graph of f2(x) = |f1(x) 2|Imagine the graph of f3(x) = |f2(x) 1|#What happens in the long run.What about other sequences?

7Floored

Floor(x) = = greatest integer x

(X 0)

Sketch the function Sketch the function #Floor & Ceilingfloor (x) = = greatest integer x

Ceiling (x) =

= smallest integer x#Onion Model of UnderstandingSusan Pirie & Tom KierenStructuringInventisingFormalisingObservingImagehavingPropertynoticingPrimitiveImagemaking#Templating & Re-ConstructingInstrumental & Relational Understanding (Skemp 1971)Following a Template & Reconstructing to Meet CircumstancesProcedural & ConceptualFormal & Intuitive insight/comprehensionTrained Behaviour & Educated Awareness#Templating & Reconstruction: examplesReverse Number NotationWhat can be said about f if

#Challenging automatised proceduresTurning a doing into an undoing 12DevelopmentManipulating 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)

#Stinulated by Bruner but rephrased to try to capture experience and provide a practical framework for teaching13

Structure of the PsycheImageryAwareness (cognition)WillBody (enaction)Emotions (affect)HabitsPractices#14From the Upanishad; reflected in modern psychologyStructure of a TopicLanguage Patterns& prior SkillsImagery/Sense-of/Awareness; ConnectionsDifferent Contexts in which likely to arise;dispositionsTechniques & IncantationsRoot Questionspredispositions

Standard Confusions & Obstacles

Only Behaviour is TrainableOnly Emotion is HarnessableOnly Awareness is EducableBehaviourEmotionAwareness

#15SoaT is NOT a checklist; it is a source of inspiration when ideas dry up; it is a reminder of dimensions of significance to draw upon.Coming to Know in MathematicsObservingWhat strikes us (disturbance revealing expectation )Probing and elucidating underlying structure(Modelling, Axiomatising, Proving)#Present or Absent?

#Attention (Will) in MathematicsHolding Wholes (gazing)Discerning DetailsRecognising Relationships (in the situation)Perceiving PropertiesReasoning on the basis of agreed properties#Worlds of ExperienceMaterial WorldWorld of SymbolsInner World of imageryenactiveiconicsymbolic#Bruner19SummaryConcept Image & Concept DefinitionTemplating & ReconstructingAwareness (ability to act) Emotion BehaviourOnly awareness is educableOnly behaviour is trainableOnly emotion is harnessableStructure of a TopicOnion model of coming to understandManipulating getting-a-sense-of ArticulatingEnactive Iconic Symbolic modes or worldsAttention:Gazing (holding wholes) Discerning DetailsRecognising Relationships Perceiving PropertiesReasoning on the basis of agreed properties#