anne watson hong kong 2011. grasp formal structure think logically in spatial, numerical and...
DESCRIPTION
‘Higher achievement was associated with: ◦ asking ‘what if..?’ questions ◦ giving explanations ◦ testing conjectures ◦ checking answers for reasonableness ◦ splitting problems into subproblems Not associated with: ◦ explicit teaching of problem-solving strategies ◦ making conjectures ◦ sharing strategies Negatively associated with use of real life contexts for older studentsTRANSCRIPT
Shifts of understanding necessary to learn
mathematics
Anne WatsonHong 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 appreciate clarity and rationality switch from direct to reverse trains of thought memorise mathematical objects
◦ (Krutetski)
What good maths students do
‘Higher achievement was associated with: ◦ asking ‘what if..?’ questions◦ giving explanations◦ testing conjectures◦ checking answers for reasonableness◦ splitting problems into subproblems
Not associated with: ◦ explicit teaching of problem-solving
strategies◦ making conjectures◦ sharing strategies
Negatively associated with use of real life contexts for older students
Working mathematically (Australia)
What activities can/cannot change students’ ways of thinking or objects of attention?
What activities require new ways of thinking?
Why?
35 + 49 – 35
a + b - a
From number to structure From calculation to relation
Shifts
grasp formal structure think logically in spatial, numerical and symbolic
relationships generalise rapidly and broadly curtail mental processes be flexible with mental processes appreciate clarity and rationality switch from direct to reverse trains of thought memorise mathematical objects
What good maths students do
28 and 34280 and 3402.8 and 3.4
.00028 and .000341028 and 1034
38 and 44-38 and -4440 and 46
Find the number mid-way between
From physical to models From symbols to images From models to rules From rules to tools From answering questions to seeking
similarities
Shifts
grasp formal structure think logically in spatial, numerical and symbolic
relationships generalise rapidly and broadly curtail mental processes be flexible with mental processes appreciate clarity and rationality switch from direct to reverse trains of thought memorise mathematical objects
What good maths students do
From visual response to thinking about properties
From ‘it looks like…’ to ‘it must be…’
Shifts
grasp formal structure think logically in spatial, numerical and symbolic
relationships generalise rapidly and broadly curtail mental processes be flexible with mental processes appreciate clarity and rationality switch from direct to reverse trains of thought memorise mathematical objects
What good maths students do
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 naturally
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)
Discuss implications
Integrate and connect
AffirmThis is where shifts can be made, talked
about, embedded
What else do mathematics teachers do? (CMTP)
Vary the variables, adapt procedures, identify relationships, explain and justify, induction and prediction, deduction
Discuss implications
Associate ideas, generalise, abstract, objectify, formalise, define
Integrate and connect
Adapt/ transform ideas, apply to more complex maths and to other contexts, prove, evaluate the process
Affirm
Learning shifts in maths lessons
Remembering something familiar Seeing something new Public orientation towards concept, method and
properties Personal orientation towards concept, method or
properties Analysis, 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..)
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 objects Characteristics & properties of classes of
objects Classification of objects Variables; variation; covariation
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 view Between deduction and induction Between domains of meaning and extreme
values as sources of structural knowledge
Shifts (van Hiele levels of understanding) Visualise, seeing whole things Analyse, describing, same/different Abstraction, distinctions, relationships
between parts Informal deduction, generalising, identifying
properties Rigour, formal deduction, properties as new
objects
generalities - examples making change - thinking about mechanisms making change - undoing change making change - reflecting on the results following rules - using tools different points of view - representations representing - transforming induction - deduction using domains of meaning - using extreme
values
Shifts of focus in mathematics
Shifts (Watson: work in progress) Methods: from proximal, ad hoc, and sensory
and procedural methods of solution to abstract concepts
Reasoning: 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 tasks
Representations:from ideas that can be modelled iconically to those that can only be represented symbolically
Thankyou [email protected]
Watson, A . (2010) Shifts of mathematical thinking in adolescence Research in Mathematics Education 12 (2) Pages 133 – 148