learn mathematics Anne Watson Hong Kong 2011 grasp formal structure think logically in spatial numerical and symbolic relationships generalise rapidly and broadly curtail mental processes be flexible with mental processes ID: 480660
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Slide1
Shifts of understanding necessary to learn mathematics
Anne Watson
Hong Kong 2011Slide2
grasp formal structurethink logically in spatial, numerical and symbolic relationshipsgeneralise rapidly and broadlycurtail mental processes
be flexible with mental processes
appreciate clarity and rationalityswitch from direct to reverse trains of thoughtmemorise mathematical objects(Krutetski)
What good maths students doSlide3
‘Higher achievement was associated with: asking ‘what if..?’ questions
giving explanations
testing conjectureschecking answers for reasonablenesssplitting problems into subproblems
Not associated with:
explicit teaching of problem-solving strategiesmaking conjecturessharing strategiesNegatively associated with use of real life contexts for older students
Working mathematically (
Australia)Slide4
What activities can/cannot change students’ ways of thinking or objects of attention?What activities require new ways of thinking?
Why?Slide5
35 + 49 – 35 a + b - aSlide6
From number to structureFrom calculation to relation
ShiftsSlide7
grasp formal structurethink logically in spatial,
numerical
and symbolic relationshipsgeneralise rapidly and broadlycurtail mental processesbe flexible with mental processesappreciate clarity and rationality
switch from direct to reverse trains of thought
memorise mathematical objectsWhat good maths students doSlide8
28 and 34280 and 3402.8 and 3.4
.00028 and .00034
1028 and 103438 and 44-38 and -44
40 and 46
Find the number mid-way betweenSlide9
From physical to modelsFrom symbols to imagesFrom models to rulesFrom rules to tools
From answering questions to seeking similarities
ShiftsSlide10
grasp formal structurethink logically in spatial, numerical and symbolic relationshipsgeneralise rapidly and broadly
curtail mental processes
be flexible with mental processesappreciate clarity and rationalityswitch from direct to reverse trains of thoughtmemorise mathematical objects
What good maths students doSlide11Slide12Slide13
From visual response to thinking about propertiesFrom ‘it looks like…’ to ‘it must be…’
ShiftsSlide14
grasp formal structurethink logically in spatial, numerical and symbolic relationshipsgeneralise rapidly and broadly
curtail mental processes
be flexible with mental processesappreciate clarity and rationalityswitch from direct to reverse trains of thoughtmemorise mathematical objects
What good maths students doSlide15
Describe
Draw on prior experience and repertoire
Informal induction
Visualise
Seek pattern
Compare, classify
Explore variation
Informal deduction
Create objects with one or more features
Exemplify
Express in ‘own words’
What nearly all learners can do naturallySlide16
Make or elicit statements
Ask learners to do things
Direct attention and suggest ways of seeing
Ask for learners to respond
What teachers
do (CMTP)Slide17
Discuss implications
Integrate and connect
Affirm
This is where shifts can be made, talked about, embedded
What else do mathematics teachers do
? (CMTP)Slide18
Vary the variables, adapt procedures, identify relationships, explain and justify, induction and prediction, deduction
Discuss implicationsSlide19
Associate ideas, generalise, abstract, objectify, formalise, define
Integrate and connectSlide20
Adapt/ transform ideas, apply to more complex maths and to other contexts, prove, evaluate the process
AffirmSlide21
Learning
shifts in
maths lessons
Remembering something familiar
Seeing something newPublic orientation towards concept, method and propertiesPersonal orientation towards concept, method or propertiesAnalysis, focus on outcomes and relationships, generalising
Indicate synthesis, connection, and associated language
Rigorous restatement (note reflection takes place over time, not in one lesson, several experiences over time)
Being familiar with a new object
Becoming fluent with procedures and repertoire (meanings, examples, objects..)Slide22
Lesson analysis: the basics are the focus of attention
Repertoire: terms; facts; definitions; techniques; procedures
Representations and how they relate
Examples to illustrate one or many features
Collections of examples Comparison of objectsCharacteristics & properties of classes of objectsClassification of objects
Variables; variation; covariationSlide23
Shifts (mentioned by Cuoco et al. but not explicitly – my analysis)
Between generalities and examples
From looking at change to looking at change mechanisms (functions)Between various points of viewBetween deduction and inductionBetween domains of meaning and extreme values as sources of structural knowledgeSlide24
Shifts (van Hiele levels of understanding)
Visualise, seeing whole things
Analyse, describing, same/differentAbstraction, distinctions, relationships between partsInformal deduction, generalising, identifying properties
Rigour, formal deduction, properties as new objects
Slide25
generalities - examples
making
change - thinking about mechanismsmaking change - undoing change
making change - reflecting on the results
following rules - using toolsdifferent points of view - representationsrepresenting - transforminginduction -
deduction
using domains of meaning
-
using
extreme values
Shifts of focus in
mathematicsSlide26
Shifts (Watson: work in progress)
Methods:
from proximal, ad hoc, and sensory and procedural methods of solution to abstract conceptsReasoning: from inductive learning of structure to understanding and reasoning about abstract relations
Focus of responses:
to focusing on properties instead of visible characteristics - verbal and kinaesthetic socialised responses to sensory stimuli are often inadequate for abstract tasksRepresentations:from ideas that can be modelled iconically to those that can only be represented symbolicallySlide27
Thankyou
anne.watson@education.ox.ac.uk
Watson, A . (2010) Shifts of mathematical thinking in adolescence
Research in Mathematics Education
12 (2) Pages 133 – 148