HOW STUDENTS LEARN AND
UNDERSTANDING OF MATHEMATICAL
STRANDS
ITLM TOPIC 4
OBJECTIVES
Be aware and describe the principles of how
students learn and acquire mathematical
knowledge as a basis for designing an
instruction.
Understand and describe the mathematical
strands in learning mathematics.
HOW PEOPLE LEARN
Principle 1: Engaging Prior Understandings
Principle 2: The Essential Role of Factual
Knowledge and Conceptual Frameworks in
Understanding
Principle 3: The Importance of Self-
Monitoring
PRINCIPLE 1
A fundamental insight about learning is that
new understandings are constructed on a
foundation of existing understandings and
experiences
The understandings children carry with them
into the classroom, even before the start of
formal schooling, will shape significantly how
they make sense of what they are taught
Learning with understanding affects our
ability to apply what is learned
This concept of learning with understanding
has two parts:
factual knowledge must be placed in a
conceptual framework to be well understood;
and
concepts are given meaning by multiple
representations that are rich in factual detail.
PRINCIPLE 2
PRINCIPLE 3
“You are the owners and operators of your
own brain, but it came without an instruction
book. We need to learn how we learn.”
A “metacognitive” or self-monitoring
approach can help students develop the
ability to take control of their own
learning, consciously define learning
goals, and monitor their progress in
achieving them.
MATHEMATICAL PROFICIENCY
MATHEMATICAL PROFICIENCY
Conceptual understanding — comprehension
of mathematical concepts, operations, and
relations
Procedural fluency — skill in carrying out
procedures
flexibly, accurately, efficiently, and
appropriately
Strategic competence — ability to
formulate, represent, and solve mathematical
problems
MATHEMATICAL PROFICIENCY
Adaptive reasoning—capacity for logical
thought, reflection, explanation, and
justification
Productive disposition—habitual inclination to
see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence
and one’s own efficacy
PRECONCEPTIONS ABOUT MATHEMATICS
Preconception 1: Mathematics is about
learning to compute
Preconception 2: Mathematics is about
“following rules” to guarantee correct
answers
Preconception 3: Some people have the
ability to “do math” and some don’t
INSTRUCTIONAL CHALLENGES
First, how can we teach mathematics so
students come to appreciate that it is not
about computation and following rules, but
about solving important and relevant
quantitative problems?
Second, how can we link formal mathematics
training with students’ informal knowledge
and problem-solving capacities?
INSTRUCTIONAL CHALLENGES
There is surely no single best instructional
approach,
But it is possible to identify certain features of
instruction that support the above goals:
Allowing Multiple Strategies
Encouraging Math Talk
Designing Bridging Instructional Activities
ASSIGNMENT
Read the articles in Constructivism, Situated Learning, and Other Learning Theories
Work in pair, and choose one learning theory for each pair.
Write summary of the learning theory you have chosen.
Choose one mathematics topic and apply the learning theory for teaching the topic chosen.
Present the summary in front of the class.
LEARNING ENVIRONMENT
LEARNING ENVIRONMENT
The learner-centered lens encourages
attention to preconceptions, and begins
instruction with what students think and
know.
The knowledge-centered lens focuses on
what is to be taught, why it is taught, and
what mastery looks like.
LEARNING ENVIRONMENT
The assessment-centered lens emphasizes
the need to provide frequent opportunities to
make students’ thinking and learning visible
as a guide for both the teacher and the
student in learning and instruction
The community-centered lens encourages a
culture of questioning, respect, and risk
taking.
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