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.