This document discusses effective teaching of mathematics. It outlines three phases of mathematical inquiry: (1) abstraction and symbolic representation, (2) manipulating mathematical statements, and (3) application. It also discusses the nature and principles of teaching mathematics, including that mathematics relies on both logic and creativity. Effective teaching requires understanding what students know and challenging them, as well as using worthwhile tasks to engage them intellectually. Teachers must have mathematical knowledge and commit to students' understanding.

1. Prepared by: Math majors

2. The heart of education is the
education of the heart.
-EFA Act

3. 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

4. 3. Develop a sense of independent critical
thinking, resourcefulness, and responsibility

5. Nature and Principles of Teaching
and Learning Mathematics

6. 
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.

7. 
For others, including many scientists and
engineers, the chief value of mathematics is how
it applies to their own work.

8. 
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.

9. 
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).

10. 
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.

11. 
The results of theoretical and applied mathematics
often influence each other.

12. 
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).

13. Phase 1: Abstraction and Symbolic Representation

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.

14. Phase 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.

15. Phase 2: Manipulating Mathematical Statements

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.

16. Phase 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.

17. Phase 2: Manipulating Mathematical 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.

18. Phase 2: Manipulating Mathematical 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.

19. Phase 3: Application

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.

20. Phase 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.

21. Phase 3: Application

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.

22. Phase 3: Application

Mathematics is essentially a process of thinking
that involves building and applying
abstract, logically connected networks of ideas.

23. 
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.

24. 
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.

25. 
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.

26. 
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).

27. 
Effective teaching math involves observing
students, listening carefully to their ideas, having
mathematical goals, and using the information to
make instructional decisions.

28. 
Learning the “basics” is important.

Learning with understanding also helps students
become autonomous learners.

29. 
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).