Metacognition in Mathematical
Problem Solving
Workshop P8
AME-SMS Conference 2016
Dr Loh Mei Yoke
Curriculum Planning and Development Division
MOE Singapore
Mathematics Curriculum Framework
Beliefs
Interest
Appreciation
Monitoring of one’s own thinking
Self-regulation of learning
Confidence
Perseverance
Numerical calculation
Algebraic manipulation
Reasoning, communication
and connections
Spatial visualisation
Applications and modelling
Thinking skills and heuristics
Data analysis
Measurement
Use of mathematical
tools
Estimation
Numerical
Algebraic
Geometric
Statistical
2
Probabilistic
Analytical
MOE, 2012, p. 16
21CC
Metacognition
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Definition of Metacognition
‘‘Metacognition’ refers to one’s knowledge
concerning one’s own cognitive processes and
products or anything related to them...
metacognition refers, among other things, to the
active monitoring and consequent regulation and
orchestration of these processes in relation to the
cognitive objects on which they bear, usually in
the service of some concrete goal or objective.’
(Flavell, 1976, p.232)
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Definition of Metacognition
‘Metacognition, or ‘thinking about thinking’,
refers to the awareness of, and the ability to
control one’s thinking processes. In particular the
selection and use of problem-solving strategies. It
includes monitoring of one’s own thinking, and
self-regulation of learning.’
(MOE, 2012, p.19)
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‘To develop metacognitive awareness and
strategies, and know when and how to use the
strategies, students should have opportunities to
solve non-routine and open-ended problems, to
discuss their solutions, to think aloud and reflect
on what they are doing, and to keep track of how
things are going and make changes when
necessary.’
(MOE, 2012, p.19)
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The following activities may be used to develop the
metacognitive awareness of students and to enrich their
metacognitive experience:
 Expose students to general problem solving skills,
thinking skills and heuristics, and how these skills can be
applied to solve problems.
 Encourage students to think aloud the strategies and
methods they use to solve particular problems.
 Provide students with problems that require planning
(before solving) and evaluation (after solving).
 Encourage students to seek alternative ways of solving
the same problem and to check the appropriateness and
reasonableness of the answer.
 Allow students to discuss how to solve a particular
problem and to explain the different methods that they
use for solving the problem.
(MOE, 2006, p. 15)
Definition of Metacognition in mathematical
problem solving (Loh, 2015)
Metacognitive Knowledge refers to an individual's
awareness of his or her own cognitive and affective
resources in relation to the task.
Metacognitive Monitoring refers to periodic
engagement in understanding the task performance
while executing the cognitive actions.
Metacognitive Regulation refers to re-evaluation of
cognitive and metacognitive activities throughout the
problem solving process.
Cognitive Strategies vs
Metacognitive Strategies
A cognitive strategy is ‘to help you achieve the
goal of whatever cognitive enterprise you are
engaged in’;
A metacognitive strategy is ‘to provide you with
information about the enterprise or your
progress in it’.
(Flavell, 1985, p.106)
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A Presentation of Pólya’s model for
Problem Solving
(MOE, 1990)
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‘Trying to find the solution, we may repeatedly
change our point of view, our way of looking at
the problem. We have to shift our position again
and again. Our conception of the problem is
likely to be rather incomplete when we start the
work; our outlook is different when we have
made some progress; it is again different when
we have almost obtained the solution.’
(Pólya, 1957, p.5)
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Metacognitive Strategies by Phases of
Problem Solving (Loh, 2015)
Understanding
Strategic behaviour to assess and understand a problem:
a)Comprehension strategies
b)Analysis of information and conditions
c)Assessment of familiarity of the problem
d)Initial and subsequent representation
e)Assessment of level of difficulty
Planning
Choice of approach, heuristics and plan to solve the problem.
Execution
Strategic behaviour to assess:
a)execution of plan (e.g. computation, procedure)
b)progress
c)trade-off decisions (e.g. speed vs accuracy, degree of elegance)
Reflection
Evaluation of decisions made on
a)Processes involved during understanding, planning and execution phases
b)Product e.g. answers and procedure, reasonableness of answers
Reference
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Flavell, J. H. (1976). Metacognitive Aspects of Problem Solving. In
Resnick, L. (Ed.), The Nature of Intelligence (pp. 231-235). New
Jersey: Lawrence Erlbaum Associates.
Garofalo, J. and Lester, F. K. Jr. (1985). Metacognition, cognitive
monitoring and mathematical performance. Journal for Research
in Mathematics Education, 16(3), 163-176.
Loh, M. Y. (2015). Metacognitve Strategies Secondary One
Students Employed While Solving Mathematics Problems.
Unpublished doctoral thesis, Nanyang Technology University,
Singapore.
Ministry of Education (2012). Primary Mathematics Teaching and
Learning Syllabus. Singapore: Curriculum Planning and
Development Division, Ministry of Education.
Pólya, G. (1945, 1957, 1973). How to Solve it. Princeton:
Princeton University Press.