nature and principles of teaching and learning math

Module I Nature and Principles of Teaching and Learning in the Subject Areas

Prepared by: Math majors

The heart of education is the

education of the heart.

-EFA Act

At the end of this module, you are expected to:

1. Discuss the elements that constitute the concept

of effective teaching of Mathematics, Natural

Science, Social Science, and the Language Arts

2. Explain the concepts of mathematical inquiry and

scientific inquiry in problem solving, and the

concepts of communicative competence in

language arts learning

Enabling Objectives

3. Develop a sense of independent critical

thinking, resourcefulness, and responsibility

Enabling Objectives

Nature and Principles of Teaching and Learning

Mathematics

Chapter 1

Mathematics relies on both logic and creativity.

It is studied both for a variety of practical

purposes and for its intrinsic interest.

For some people, and not only professional

mathematicians, the essence of mathematics

lies in its beauty and its intellectual challenge.

The Nature of Mathematics

For others, including many scientists and

engineers, the chief value of mathematics is

how it applies to their own work.

Nature of Mathematics

Mathematics is the science of patterns and

relationships (Mahaniski, 2003).

As a theoretical discipline, mathematics

explores the possible relationships among

abstract numerical formulas without concern

for whether or not those abstractions have

applicative representations in the real world.

Patterns and Relationships in Mathematics

Previously unrelated parts of mathematics are

found to be derivable from one another, or from

some more general theory.

The sense of beauty of math lies not in finding

the greatest elaborateness or complexity but on

the contrary, in finding the greatest economy and

simplicity of representation and proof (Miller &

Alexander, 1996).


Mathematics is an applied science (Simon, 1995).

Many mathematicians focus their attention on

problem solving that originate in the world of

experience.

In contrast to theoretical mathematicians, applied

mathematicians might study the interval pattern of

prime numbers to develop a new system for coding

numerical information, rather than as an abstract

problem.


The results of theoretical and applied

mathematics often influence each other.


Using mathematical inquiry to express ideas and

solve problems involves at least three phases:

(1) Representing some aspects of things abstractly

(2) Manipulating the abstractions by rules of logic to

find new relationships between them

(3) Seeing whether the new relationships say

something useful about the original things

(Leitzil, 1991).

Mathematical Inquiry

Mathematical thinking often begins with the process

of abstraction---that is, noticing a similarity between

two or more objects or events.

Aspects that they have in common, whether

concrete or hypothetical, can be represented by

symbols such as numbers, letters, other marks,

diagrams, geometrical constructions, or even words.

Mathematical InquiryPhase 1: Abstraction and Symbolic

Representation

Such abstraction enables mathematicians to

concentrate on some features of things and

relieves them of the need to keep other

features continually in mind.

Mathematical InquiryPhase 1: Abstraction and Symbolic

Representation

Simon (1995) explains that after

abstractions have been made and symbolic

representations of them have been

selected, those symbols can be combined

and recombined in various ways according

to precisely defined rules.

Mathematical InquiryPhase 2: Manipulating Mathematical

Statements

Sometimes that is done with a fixed goal in mind; at

other times it is done in the context of experiment.

Sometimes an appropriate manipulation can be

identified easily from the intuitive meaning of the

constituent words and symbols; at other times a

useful series of manipulations has to be worked out

by trial and error.


Statements

Typically, strings of symbols are combined

into statements that express ideas or

propositions.

Example: the symbol A for the area of any

square mat be used with the symbol s for

the length of the square’s side to form the

proposition A=s^2.


Statements

In a sense, then, the manipulations of

abstractions is much like a game: Start with

some basic rules, then make any moves

that fit those rules---which includes

inventing additional rules and finding new

connections between old rules.


Statements

Mathematical processes can lead to a kind of model

of a thing, from which insights can be gained about

the thing itself (Cole, Coffey, & Goldman, 1994).

Any mathematical relationships arrived at

manipulating abstract statements may or may not

convey something truthful about the thing being

molded.

Mathematical InquiryPhase 3: Application

For example, if 2 cups of water are added to

3 cups of water and the abstract

mathematical operation 2+3=5 is used to

calculate the total, the correct answer is 5

cups of water.


However, if 2 cups of sugar are added to 3

cups of hot tea and the same operation is

used, 5 is an incorrect answer, for such an

addition actually results in only slightly

more than 4 cups of very sweet tea.


Mathematics is essentially a process of

thinking that involves building and applying

abstract, logically connected networks of

ideas.


Students learn mathematics through the

experiences that teachers provide.

Teachers must understand deeply the

mathematics they are teaching and be

committed to their students as learners and

as human beings.

There is no one “right way” to teach

mathematics.

The Principles of Teaching Mathematics

The teacher is responsible for creating an

intellectual environment in the classroom where

serious engagement in mathematical thinking is

the norm.

Teachers need to increase their knowledge about

math and pedagogy, learn from their students,

and colleagues, and engage in professional

development and self-reflection.

The Principles of Teaching Mathematics

Effective math teaching requires understanding what

students know and need to learn and then challenging

and supporting them to learn it well (Davidson, 1990).

Teaching math well is a complex endeavor, and there

are no easy recipes for helping all students learn or for

helping all teachers become effective.

Effective teaching requires reflection and continual

efforts.

Effective Mathematics Teaching

Teachers need several different kinds of mathematical

knowledge.

Effective math teaching requires a serious

commitment to the development of students’

understanding of math.

In effective teaching, worthwhile mathematical tasks

are used to introduce important mathematical ideas

and to engage and challenge students intellectually

(Cole, Coffey, & Goldman, 1994).


Effective teaching math involves observing

students, listening carefully to their ideas,

having mathematical goals, and using the

information to make instructional decisions.


Learning the “basics” is important.

Learning with understanding also helps

students become autonomous learners.

The Principles of Learning Mathematics

When challenged with appropriately chosen

tasks, students can become confident in

their ability to tackle difficult problems,

eager to figure things out in their own,

flexible in exploring mathematical ideas,

and willing to persevere when tasks are

challenging (Clarke & Wilson, 1994).

The Principles of Learning Mathematics

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