philosophical theoretical ground mathematics teaching

PHILOSOPHICAL THEORETICAL GROUND MATHEMATICS TEACHING

SEAMEO REGIONAL CENTER FOR QITEP IN MATHEMATICS 2010 i

TABLE OF CONTENTS

CHAPTER 1:

1.1. The Philosophy of Mathematics Education .............................................................................. 1

1.2. The Ideology of Mathematics Education. .................................................................................. 2

1.3. The Foundation of Mathematics Education............................................................................. 3

1.4. The Nature of Mathematics and School Mathematics .......................................................... 4

1.5. The Value of Mathematics ................................................................................................................ 6

1.6. The Nature of Students ..................................................................................................................... 6

1.7. The Aim of Mathematics Education ............................................................................................. 7

CHAPTER 2:

2.1. The Nature of Teaching and Learning of Mathematics ........................................................ 8

2.2. The Nature of Teaching and Learning Resources .................................................................. 9

2.3. The Nature of Assessment ............................................................................................................. 10

2.4. The Relationship between Mathematics Education and Society ................................... 10

2.5. Philosophical Ground of Human Resources Development: Its Implication to

Educational Change ......................................................................................................................... 11

CHAPTER 3:

3.1. Students’ Mathematical Learning ............................................................................................... 14

3.2. Teaching by Fostering Learning Strategies ............................................................................ 15

3.3. Teaching by Problem-Solving Strategies ................................................................................. 16

3.4. Methods, Approaches, and Models of Teaching and Learning ....................................... 25

3.5. Teaching by Problem-Solving Strategies ................................................................................. 27

3.6. Methods, Approaches, and Models of Teaching and Learning ....................................... 28

CHAPTER 4:

4.1. Current Trends of Mathematics Teaching Practice ............................................................. 35

4.2. Teaching Mathematics by Using Information Technology

(IT-Based Teaching) .......................................................................................................................... 43

4.3. Promoting Teachers’ Professional Development ................................................................. 46

References ......................................................................................................................................... 52


SEAMEO REGIONAL CENTER FOR QITEP IN MATHEMATICS 2010 ii


SEAMEO REGIONAL CENTER FOR QITEP IN MATHEMATICS 2010 1

Chapter 1

The Philosophy of Mathematics Education

1.1. Philosophy of Mathematics Education

Philosophy of mathematics education covers a review of some central problems

of mathematics education: its ideology, its foundation and its aim. It also serves a more

insight into the nature of its aspects: the nature of mathematics, the value of

mathematics, the nature of student, the nature of learning, the nature of teaching of

mathematics, the nature of teaching learning resources, the nature of assessment, the

nature of school mathematics, the nature of students’ learn mathematics. In order to

have a clear picture of the role of the study of philosophy of mathematics and its

relationship to workshop activities, it may be discussed about the nature of human

resources development and the nature of lesson study in mathematics education.

According to Paul Ernest (1994), the study of philosophy of mathematics

education implies to the practice of mathematics teaching through the issues reflected

on the following questions:

“What theories and epistemologies underlie the teaching of mathematics? What

assumptions, possibly implicit, do mathematics teaching approaches rest on? Are these

assumptions valid? What means are adopted to achieve the aims of mathematics

education? Are the ends and the means consistent? What methods, resources and

Figure 1.1. Applications of philosophy to mathematics education

(Paul Ernest, 1994)

Aplication

1. PHILOSOPHY OVERALL

2. Philosophy of

Mathematics

3. Philosophy of

Educations

3. Mathematics

Education



techniques are, have been, and might be, used in the teaching of mathematics? What

theories underpin the use of different information and communication technologies in

teaching mathematics? What sets of values do these technologies bring with them, both

intended and unintended? What is it to know mathematics in satisfaction of the aims of

teaching mathematics? How can the teaching and learning of mathematics be evaluated

and assessed? What is the role of the teacher? What range of roles is possible in the

intermediary relation of the teacher between mathematics and the learner? What are the

ethical, social and epistemological boundaries to the actions of the teacher? What

mathematical knowledge does the teacher need? What impact do the teacher’s beliefs,

attitudes and personal philosophies of mathematics have on practice? How should

mathematics teachers be educated? What is the difference between educating, training

and developing mathematics teachers?”

In a more general perspective, it can be said that the philosophy of mathematics

education has aims to clarify and answer the questions about the status and the

foundation of mathematics education objects and methods, that is, ontologically clarify

the nature of each component of mathematics education, and epistemologically clarify

whether all meaningful statements of mathematics education have been objective and

determine the truth. Perceiving that the laws of nature, the laws of mathematics, the laws

of education have a similar status, the very real world of the form of the objects of

mathematics education forms the foundation of mathematics education.

1.2. The Ideology of Mathematics Education

Ideologies of mathematics education cover the belief systems to which the way

mathematics education is implemented. They cover radical, conservative, liberal, and

democracy. The differences of the ideology of mathematics education may lead the

differences on how to develop and manage knowledge, teaching, learning, and schooling.

In most learning situation we are concerned with activity taking place over periods of

time comprising personal reflection making sense of engagement in this activity; a

government representative might understand mathematics in term of how it might be

partitioned for the purpose of testing (Brown, T, 1994).

Comparison among countries certainly reveals both the similarities and the

differences in the policy process. The ideologies described by Cochran-Smith and Fries

(2001) in Furlong (2002), as underpinning the reform process, are indeed very similar.

Yet at the same time, a study of how those ideologies have been appropriated, by whom,

and how they have been advanced reveals important differences. He further claimed

that what that demonstrates, is the complexity of the process of globalization. Furlong



quoted Eatherstone (1993), “One paradoxical consequence of the process of

Cglobalization, the awareness of the finitude and boundedness of the plane of humanity,

is not to produce homogeneity, but to familiarize us with greater diversity, the extensive

range of local cultures”.

Ernest, P (2007) explored some of the ways in which the globalization and the

global knowledge impacts on mathematics education. He has identified four components

of the ideological effect to mathematics education. First, there is the reconceptualization

of knowledge and the impact of the ethos of managerialism in the commodification and

fetishization of knowledge. Second, there is the ideology of progressivism with its

fetishization of the idea of progress. Third, there is the further component of

individualism which in addition to promoting the cult of the individual at the expense of

the community, also helps to sustain the ideology of consumerism. Fourth is the myth of

the universal standards in mathematics education research, which can delegitimate

research strategies that for ground ethics or community action more than is considered

‘seemly’ in traditional research terms.

1.3. Foundation of Mathematics Education

The foundation of mathematics searches the status and the basis of mathematics

education. Paul Ernest (1994) delivered various questions related to the foundation of

mathematics as follows:

“What is the basis of mathematics education as a field of knowledge? Is mathematics

education a discipline, a field of enquiry, an interdisciplinary area, a domain of extra-

disciplinary applications, or what? What is its relationship with other disciplines such as

philosophy, sociology, psychology, linguistics, etc.? How do we come to know in

mathematics education? What is the basis for knowledge claims in research in

mathematics education? What research methods and methodologies are employed and

what is their philosophical basis and status? How does the mathematics education

research community judge knowledge claims? What standards are applied? What is the

role and function of the researcher in mathematics education? What is the status of

theories in mathematics education? Do we appropriate theories and concepts from other

disciplines or ‘grow our own’? How have modern developments in philosophy (post-

structuralism, post-modernism, Hermeneutics, semiotics, etc.) impacted on mathematics

education? What is the impact of research in mathematics education on other disciplines?

Can the philosophy of mathematics education have any impact on the practices of teaching

and learning of mathematics, on research in mathematics education, or on other

disciplines?”



It may emerge the notions that the foundation of mathematics education serves

justification of getting the status and the basis for mathematics education in the case of

its ontology, epistemology and axiology. Hence we will have the study of ontological

foundation of mathematics education, epistemological foundation of mathematics

education and axiological foundation of mathematics education; or the combination

between the two or among the three.

1.4. The Nature of Mathematics and School Mathematics

Mathematics ideas comprise thinking framed by markers in both time and space.

However, any two individuals construct time and space differently, which present

difficulties for people sharing how they see things. Further, mathematical thinking is

continuous and evolutionary, whereas conventional mathematics ideas are often treated

as though they have certain static qualities. The task for both teacher and students is to

weave these together. We are again face with the problem of oscillating between seeing

mathematics extra-discursively and seeing it as a product of human activity (Brown, T,

1994).

Paul Ernest (1994) provokes the nature of mathematics through the following

questions:

“What is mathematics, and how can its unique characteristics be accommodated in a

philosophy? Can mathematics be accounted for both as a body of knowledge and a social

domain of enquiry? Does this lead to tensions? What philosophies of mathematics have

been developed? What features of mathematics do they pick out as significant? What is

their impact on the teaching and learning of mathematics? What is the rationale for

picking out certain elements of mathematics for schooling? How can (and should)

mathematics be conceptualized and transformed for educational purposes? What values

and goals are involved? Is mathematics value-laden or value-free? How do mathematicians

work and create new mathematical knowledge? What are the methods, aesthetics and

values of mathematicians? How does history of mathematics relate to the philosophy of

mathematics? Is mathematics changing as new methods and information and

communication technologies emerge?”

In order to promote innovation in mathematics education, the teachers need to

change their paradigm of what kinds of mathematics to be taught at school. Ebbutt, S.

and Straker, A. (1995) proposed the school mathematics to be defined and its

implications to teaching as the following:



a. Mathematics is a search for patterns and relationship

As a search for pattern and relationship, mathematics can be perceived as a network of

interrelated ideas. Mathematics activities help the students to form the connections in

this network. It implies that the teacher can help students learn mathematics by giving

them opportunities to discover and investigate patterns, and to describe and record the

relationships they find; encouraging exploration and experiment by trying things out

in as many different ways as possible; urging the students to look for consistencies or

inconsistencies, similarities or differences, for ways of ordering or arranging, for ways

of combining or separating; helping the students to generalize from their discoveries;

and helping them to understand and see connections between mathematics ideas. (ibid,

p.8)

b. Mathematics is a creative activity, involving imagination, intuition and

discovery

Creativity in mathematics lies in producing a geometric design, in making up computer

programs, in pursuing investigations, in considering infinity, and in many other

activities. The variety and individuality of children mathematical activity needs to be

catered for in the classroom. The teacher may help the students by fostering initiative,

originality and divergent thinking; stimulating curiosity, encouraging questions,

conjecture and predictions; valuing and allowing time for trial-and-adjustment

approaches; viewing unexpected results as a source for further inquiry; rather than as

mistakes; encouraging the students to create mathematical structure and designs; and

helping children to examine others’ results (ibid. p. 8-9)

c. Mathematics is a way of solving problems

Mathematics can provide an important set of tools for problems- in the main, on paper

and in real situations. Students of all ages can develop the skills and processes of

problem solving and can initiate their own mathematical problems. Hence, the teacher

may help the students learn mathematics by: providing an interesting and stimulating

environment in which mathematical problems are likely to occur; suggesting problems

themselves and helping students discover and invent their own; helping students to

identify what information they need to solve a problem and how to obtain it;

encouraging the students to reason logically, to be consistent, to works systematically

and to develop recording system; making sure that the students develop and can use

mathematical skills and knowledge necessary for solving problems; helping them to

know how and when to use different mathematical tools (ibid. p.9)



d. Mathematics is a means of communicating information or ideas

Language and graphical communication are important aspects of mathematics

learning. By talking, recording, and drawing graphs and diagrams, children can come

to see that mathematics can be used to communicate ideas and information and can

gain confidence in using it in this way. Hence, the teacher may help the students learn

mathematics by: creating opportunities for describing properties; allocating time for

both informal conversation and more formal discussion about mathematical ideas;

encouraging students to read and write about mathematics; and valuing and

supporting the diverse cultural and the linguistic backgrounds of all students (ibid.

p.10)

1.5. The Value of Mathematics

In the contemporary times, the mathematical backbone of its value has been

extensively investigated and proven over the last ten years. According to Dr. Robert S.

Hartman’s, value is a phenomena or concept, and the value of anything is determined by

the extent to which it meets the intent of its meaning. Hartman (1945) indicated that the

value of mathematics has four dimensions: the value of its meaning, the value of its

uniqueness, the value of its purpose, and the value of its function. Further, he suggested

that these four “Dimensions of Value” are always referred to as the following concepts:

intrinsic value, extrinsic value, and systemic value. The bare intrinsic and inherent

essence of mathematical object is a greater, developed intensity of immediacy.

Mathematical object is genuinely independent of either consciousness or other things;

something for itself. In and for itself belongs to the absolute alone, mathematical objects

can be perceived as the developed result of its nature and as a system of internal

relations in which it is independent of external relations.

1.6. The Nature of Students

Understanding the nature and characteristics of young adolescent development

can focus effort in meeting the needs of these students. The National Middle School

Association (USA, 1995) identified the nature of students in term of their intellectual,

social, physical, emotional and psychological, and moral. Young adolescent learners are

curious, motivated to achieve when challenged and capable of problem-solving and

complex thinking. There is an intense need to belong and be accepted by their peers

while finding their own place in the world. They are engaged in forming and questioning

their own identities on many levels. The students may be mature at different rates and

experience rapid and irregular growth, with bodily changes causing awkward and



uncoordinated movements. In term of emotional and psychological aspect, they are

vulnerable and self-conscious, and often experience unpredictable mood swings. While

in the case of moral, they are idealistic and want to have an impact on making the world

a better place.

Most of the teachers always pay much attention to the nature of student’s ability.

We also need to have an answer how to facilitate poor and low-ability children in

understanding, learning and schooling. Intellectual is really important to realize mental

ability; while, their work depends on motivation. It seems that motivation is the crucial

factor for the students to perform their ability. In general, some teachers are also aware

that the character of teaching learning process is a strong factor influencing student’s

ability. We need to regard the pupils as central to our concerns if our provision for all the

pupils is to be appropriate and effective; some aspects of teaching for appropriateness for

students might be: matching their state of knowledge, identifying and responding to their

particular difficulties, extending them to develop their potential in mathematics, providing

some continuities of teaching with a demonstrated interest in progress, developing an

awareness of themselves as learners using the teacher as a resource, and providing regular

feedback on progress (Ashley, 1988). Those who teach mathematics must take into account

the great variations which exist between pupils both in their rate of learning and also in

their level of attainment at any given age (Cockroft Report, 1982, para. 801).

1.7. The Aim of Mathematics Education

Philosophically, the aims of mathematics education stretch from the movement

of back to basic of arithmetic teaching, certification, transfer of knowledge, creativity, up

to develop students understanding. Once upon a time, a mathematics teacher delivered

his notion that the objective of his mathematical lesson was to use better mathematical,

more advanced terminology and to grasp a certain concept of mathematics. Other

teacher claimed that the objective of his mathematical lesson was to achieve notions

stated in the syllabi. While others may state that his aim was to get the true knowledge

of mathematics. So the purpose of mathematics education should be enable students to

realize, understand, judge, utilize and sometimes also perform the application of

mathematics in society, in particular to situations which are significant to their private,

social and professional lives (Niss, 1983, in Ernest, 1991). Accordingly, the curriculum

should be based on project to help the pupil's self-development and self-reliance; the life

situation of the learner is the starting point of educational planning; knowledge acquisition

is the part of the projects; and social change is the ultimate aim of the curriculum (Ernest,

1991).



Exercises:

Discuss the following aspects and their implication to mathematics teaching practice.

Which one of them is the most favorable to you and your teaching? Explain.

1. Ideologies of education: Radical, Conservative, Liberal, Humanist, Progressive,

Socialist, Democracy

2. Natures of education: Obligation, Preserving, Exploiting, Transforming, Liberating,

Needs Others, Democracy.

3. Natures of mathematics: Body of Knowledge, Science of truth, Structure of truth,

Process of Thinking, Social Activities.

4. Natures of school mathematics: Search for pattern and relation, Problem Solving,

Investigation, Communication.

5. Moral of mathematics education: Good versus Bad, Pragmatism, Hierarchy,

Paternalistic, Humanity, Justice-Freedom, Others.

6. Values of mathematics education: Intrinsic, Extrinsic, Systemic.

7. Natures of students: Empty Vessel, Character Building, Creativity, Growing like a

seed, Constructing, Others.

8. Natures of students’ ability: Talent Given, Effort, Need, Competency, Culture,

Contextual, Others.

9. Aims of mathematics education: Back to Basic Arithmetic), Certification, Transfer of

Knowledge, Creativity, to develop people comprehensively.



Chapter 2

The Nature of Mathematics Teaching

2.1. The Nature of Teaching Learning of Mathematics Some students learn best when they see what is being taught, while others

process information best auditorily. Many students will prefer movement or touching to

make the learning process complete. The best approach to learning styles is a

multisensory approach. This type of environment allows for children, who are primarily

kinesthetic or motor learners, to be able to learn through touch and movement; it allows

the visual learner to see the concept being taught, and the auditory learner to hear and

verbalize what is being taught. Ideally, the best learning takes place when the different

types of processing abilities can be utilized. Constructivists have focused more on the

individual learner’s understanding of mathematical tasks which they face (von

Glasersfeld, 1995 in Brown, 1997).

Educationists use the terms ‘traditional’ and ‘progressive’ as a shorthand way of

characterizing educational practices. The first is often associated with the terms ‘classical/

whole class’, ‘direct’, ‘transmission’, ‘teacher-centered/subject-centered’, ‘conventional’, or

‘formal’; and the second is sometimes associated with the terms ‘individual’, ‘autonomy’,

‘constructive’, ‘child-centered’, ‘modern’, ‘informal’, and/or ‘active learning’. The lack of any

clear definition of what the terms mean is one of the sources of misleading rhetoric of the

practices. Bennett (1976) found evidence that the loose terms ‘traditional’ and

‘progressive’ are symbolic of deep conflicts about some of the aims of education. The main

sociological point is that the terms ‘progressive’ and ‘traditional’ are emotionally loaded but

lack any consensual meaning among practitioners or researchers (Delamont, 1987). He

found that, in the UK, ever since 1948 there has been a division between those exposing

traditional and progressive ideals, and that feelings about these ideals are bitter and

vehemently held. Then, since 1970, there have been some investigations on how the

teachers’ behaviors attributed by the term of ‘traditional’ or ‘progressive’. The most

persuasive prescriptive theory of teaching was that reflected in the Plowden Report (1967)

which, influenced by the educational ideas of such theorists as Dewey and Froebel, posited

a theory of teaching which distinguished between progressive and traditional teachers.

Specifically, Ernest (1994) elaborated issues of mathematics education as

follows:

a. Mathematical pedagogy – problem solving and investigational approaches to

mathematics versus traditional, routine or expository approaches? Such

oppositions go back, at least, to the controversies surrounding discovery methods

in the 1960s.



b. Technology in mathematics teaching – should electronic calculators be permitted or

do they interfere with the learning of number and the rules of computation? Should

computers be used as electronic skills tutors or as the basis of open learning? Can

computers replace teachers, as Seymour Papert has suggested?

c. Mathematics and symbolization – should mathematics be taught as a formal

symbolic system or should emphasis be put on oral, mental and intuitive

mathematics including child methods?

d. Mathematics and culture – should traditional mathematics with its formal tasks and

problems be the basis of the curriculum, or should it be presented in realistic,

authentic, or ethnomathematical contexts?”

2.2. The Nature of Teaching Learning Resources

John Munchak (2004) indicated that in order to provide lessons that are both

engaging and challenging to each individual, it is necessary to know the students as

people. Each individual will come to my class with their own set of abilities, motivations,

attitudes, goals, and cultural background. Further he stated the following:

“Getting to know these various facets of my students will allow me to excel as a teacher

because I can tap into their talents, resources, and knowledge to make the classroom more

interesting, dynamic, and personal. Establishing a familiar bond and some trusts between

me and the students, as well as among the students themselves, contributes to a safe and

caring learning environment. Learning their interests, the activities they enjoy, their

academic strengths and weaknesses, their future plans and motivations informs how I will

teach each individual. This personal information is important in order to differentiate

learning in a classroom with students of various levels of motivation, career goals, and

academic abilities. Caring for my students means I will "honor their humanity, hold them in

high esteem, expect high performance from them, and use strategies to fulfill their

expectations"

Related to the resources of teaching, Ernest (1991) suggested that due to the

learning should be active, varied, socially engaged and self-regulating, the theory of

resources has three main components:

(1) the provision of a wide variety of practical resources to facilitate the varied and active

teaching approaches;

(2) the provision of authentic material, such as newspaper, official statistics, and so on for

socially relevant and socially engaged study and investigation; and

(3) the facilitation of student self-regulated control and access to learning resources.



2.3. The Nature of Assessment

Traditionally, assessments have been concerned with assessing the student’s

production of “correct” mathematical statements as evidence of a broader mathematical

understanding. An alternative of these places emphasizes on the “story” told about the

event of a mathematical activity (Mason, 1989, in Brown, 1997). In assessing

mathematics we seem at first to be caught between on the hand, working with a style of

mathematics where we assert a field of mathematics as if devoid of human and, on the

other hand, speaking of mathematics as a depth interpretation of a certain style of

human activity; this is not a satisfactory dichotomy, if only because we never have a

choice of one over-arching symbolic framework (ibid.).

2.4. The relationship between mathematics education and society

Nobody argues that mathematics education is closely related to society.

However, we may learn the extent of its relationship. Paul Ernest (1994) delivered some

questions:

"Are the aims of mathematics education valid and valued for their society? What are the

aims? To whom that mathematics is taught? Who will participate in the practice of

mathematics teaching? Who supports, who takes the benefits, who decides, who

dominates, who gains and who loses? To what extent the social, cultural and historical

contexts relate to mathematics education? Further he exposed about: what values

underpin different sets of aims? How does mathematics contribute to the overall goals

of society and education? What is the role of the teaching and learning of mathematics in

promoting or hindering social justice conceived in terms of gender, race, class, ability

and critical citizenship? Is anti-racist mathematics education possible and what does it

mean? He also highlighted on how is mathematics viewed and perceived in society?

What impact does this have on education? What is the relationship between

mathematics and society? What functions does it perform? Which of these functions are

intended and visible? Which functions are unintended or invisible? To what extent do

mathematical metaphors permeate social thinking? What is their philosophical

significance? To whom is mathematics accountable?



2.5. Philosophical Ground of Human Resources Development: Its implication to Educational Change

a. Human Resources Development

According to Swanson, R.A. and Holton III, E.F. (2009) the philosophical

ground of human resources development covers (1) a shift to the human

resources school of thought, (2) the growth of laboratory training, (3) the use of

survey research and feedback, (4) an increased use of action research

(problem-solving) techniques, (5) an acknowledgment of socio-technical

systems and quality of work life, and (6) a new emphasis on strategic change.

Further, they suggested that developed mostly in response to serious concerns

about the viability of traditional and bureaucratic organizations, the human

relations model attempted to move away from these classical assumptions and

focused more heavily on individual identities, their needs, and how to facilitate

stronger interpersonal communication and relationships.

Accordingly, much of the concepts of human resources development are

currently focusing on the increase of the effectiveness of strategic change. The use of

open-systems planning was one of the first applications of strategic change methods. An

educational institutions’ demand and response systems could be described and

analyzed, the gaps reduced, and the performance improved. In the case of education, this

work represents a shift in teachers’ professional development, away from a sole focus

on the individual and the supporting assumption, that it is completely mediated through

individuals, to a more holistic and open systems view of educational institution. This

shift continues to this day and is evidenced in key revelations stemming from strategic

change work including the importance of educational leadership support, multilevel

involvement, and the criticality of alignment between organizational strategy, structure,

culture, and systems (ibid)

b. Managing Educational Change

Fullan (1982, 1991) proposed that there are four broad phases in the

educational change process: initiation, implementation, continuation, and outcome.

1) Initiation

The factors that affecting the initiation phases include: existence and quality of

innovations, access to innovations, advocacy from central administration, teacher

advocacy, and external change agents

2) Implementation

Three areas of the major factors affecting implementation: characteristics of



change, local characteristics and external factors (government and other

agencies). They identified different stakeholders in local, and federal and

governmental levels. They also identified characterizations of change to each

stakeholder and the issues that each stakeholder should consider before

committing a change effort or rejecting it.

3) Characteristics of Change Local Factors External Factors

Fullan (1999) characterized educational changes and their factors at different

levels as follows:

Characteristics of Change Local Factors External Factors

1. Need of change

2. Clarity about goals and needs

3. Complexity: the extent of

change required to those

responsible for

implementation

4. Quality and practicality of the

program

1. The school district

2. Board of community

3. Principal

4. Teacher

1. Government and

other agencies

4) Continuation/ Sustainability

According to Fullan (1999), continuation is a decision about institutionalization

of an innovation based on the reaction to the change, which may be negative or

positive. Continuation depends on whether or not:

a) The change gets embedded/built into the structure (through policy/budget/

timetable)

b) The change has generated a critical mass of administrators or teachers who

are skilled and committed to

c) The change has established procedures for continuing assistance

Fullan (1999) pointed out the importance of the recognition that the educational

change process is complex. To deal with such complexity is not to control the

change, but to guide it. He provides eight new lessons about guiding educational

change:

a) Moral purpose is complex and problematic

b) Theories of education and theories of change need each other

c) Conflict and diversity are our friends

d) Understanding the meaning of operating on the edge of chaos

e) Emotional intelligence is anxiety provoking and anxiety containing



f) Collaborative cultures are anxiety provoking and anxiety containing

g) Attack incoherence connectedness and knowledge creation are critical

h) There is no single solution. Craft your own theories and actions by being a

critical consumer.

Exercises:

Discuss the following aspects of mathematics teaching practices and their implication. Which

one of them is the most favorable for you and your teaching? Explain.

1. Theories of learning: (a) Work Hard, Exercises, Drill. Memorize, (b) Thinking and Practice,

(c) Understanding and Application, (d) Exploration, (e) Discussion, Autonomy,

2. Theories of teaching: (a) Transfer of knowledge, (b) External Motivation, (c) Internal

Motivation, (d) Construction, (e) Discussion, (f) Investigation, (g) Development, (h)

Facilitating, (i) Expository.

3. Theories of teaching mathematics: (a) Expository, (b) Problem Solving, (c) Memorize, (d)

Drill, (e) Discussion, (f) Practical Work, (g) Development, (h) Facilitating.

4. Natures of teaching learning resources: (a) White board, Chalk, Anti Calculator, (b)

Teaching Aids, (c) Visual Teaching Aid for Motivation, (d) Various Resources/Environment,

(e) Social Environment.

5. Natures of assessment: (a) External Test, (b) Portfolio, (c) Social, (d) Contextual.

6. Natures of society: (a) Diversity, (b) Monoculture, (c) Decentralization, (d) Competency, (e)

Multiple Solution, (f) Heterogenomous, (g) Social Capital, (h) Local Culture.

7. Natures of curriculum: (a) Instrument Curriculum, (b) Subject-based Curriculum, (c)

Integrated Curriculum, (d) Knowledge-based Curriculum, (e) Competence-based

Curriculum, (f) Individual Curriculum, (g) Interactive Curriculum, (h) ICT-based

Curriculum.

8. Natures of students’ learn mathematics: (a) Individual, (b) Competition, (c) Motivation, (d)

Readiness, (e) Scaffolding, (f) Collaborative, (g) Constructing, (h) Contextual, (i)

Enculturing.

9. Natures of mathematical thinking: (a) Subjective, (b) Objective, (c) Producing, (d)

Reflecting, (e) Criticizing, (f) Constructing, (g) Social Activity, (h) Attitude, (i) Content, (j)

Method, (k) Conjecture, (l) Embodiment.



Chapter 3

A sound knowledge of mathematics is a necessity for mathematics teachers in their teaching

activities. However, the understanding of mathematics content does not guarantee that the

teaching will be excellent. An excellent mathematics teacher is a teacher who comprehend all

concepts of school mathematics and a variety of teaching and learning strategies and

determines which strategy should be implemented in his/her classrooms based on students’

characteristics, learning situations/environment, and the topics which will be presented.

3.1. Students’ Mathematical Learning

Learning can be defined as a relatively permanent change in someone’s knowledge

based on the person’s experience (Mayer, 2002). Learning is a long-term process. So, a change

that disappears after a few hours can not be considered as learning. Learning is also a change.

There is no learning when there is no change. Learning is an experience-based process, and it

depends on the experience of the learner.

In learning we recognise previous learning and new learning. When there is an effect

from previous learning on new learning, then it means a transfer is resulted. Here transfer

occurs successfully when someone uses knowledge in previous learning to help learning

something new.

In learning there can be also a problem-solving transfer, which occurs when someone

uses knowledge from previous learning to come to a new idea, recommendation, or a solution to

a new problem. So, a problem-solving transfer is the ability to use someone has learned when

he/she is faced with a new problem.

In meaningful learning, the first step is selecting relevant material from what has been

presented. Teaching for meaningful learning consists of guiding the learner’s attention towards

relevant aspect of the instructional activity. The second step is organizing, which means

guiding the students to organise their teaching material in working memory into coherent

mental structures. The third step is integrating knowledge in the working memory with existing

knowledge in the long-term memory

3.2. Understanding Students’ Individual Differences

Students as human being are very complex and almost unpredictable, in terms of their

appearance, ability, and personality. This means that thinking about students’ learning is also

thinking about individual differences, which also means that in teaching, teachers will face with

many problems of treating students who have individual differences.



Students are different in some aspects. Teachers should be aware that students have

different mental ability, reasoning, and problem solving. One of these traits is different from one

to another, as depicted by the so-called Intellectual Aptitude Test (IQ).

Students also have different mathematical abilities, including reasoning, understanding,

communication, and representation. Some of them may lack of ability in knowledge of

mathematical concepts, structures, and processes. It is important that teachers check their

students in these three aspects, as the previous educational experience of students will largely

determines the readiness of the students in learning a new lesson.

Other than students’ ability in cognition, students’ motivations, interests, attitudes, and

appreciations are essential factors that teachers should put into account. Further, students’

physical, emotional, and social maturity should also put into the teachers’ consideration, as all

these things contribute significantly to students’ achievements.

Some other factors that can contribute to the students’ performance and achievements

are talents, creativities, retention span, learning habits, self-discipline, attentions, reading skills,

and writing skills.

Since students are unequal in the above ability and skills, teachers should treat them in

different way, not only in mathematics lessons, assignments, instructions, allocation of time, but

also in achievement requirements.

To understand individual differences means that teachers should have broad

information on their students’ characteristics, which can be collected through a number of tests

and non test techniques (quantitative or qualitative techniques).

3.3. Understanding Learning Theories in Teaching Mathematics

(a) Piagetian Theory of Intellectual Development

Learning theories and its applications are of importance for teachers to understand, as

these theories can support successful mathematics teaching. The various approaches in the

study of intellectual development and the nature of learning in different ways have resulted in a

number of learning theories.

In most of the cases in the past, mathematics teachers and teacher educators did not

neglect the application of theories about the nature of learning. They merely focused their

teaching strategies on the concepts of the subjects. However, nowadays, teachers and teacher

educators can determine and choose teaching strategies which are based on the information



about the nature of learning and make the use of recent findings in learning theory, mental

development, and new applications of the theory of learning to teaching activities.

In his theory of Intellectual Development, Jean Piaget states that human intellectual

development progresses chronologically in four sequential stages: (i) Sensory-Motor Stage,

which starts form birth until about two years of age. In this stage, learning process consists of

developing and organizing physical and mental activities into well-defined sequences of actions

called schemas. During this stage, learning process includes coordinating sense and movements,

understanding that an object removed from sight does not means lost, learning about attaching

symbols to an object, recognizing the sound of objects, and learning how to walk and talk.

In the second stage, called preoperational stage, extends from about age two until age

seven, children assimilate most experiences in the world into schemas developed from their

immediate environment and view everything in relation to themselves. They think that what they

think and experience also happen to someone else. They find difficulty to see the distinction

between one, two, and many in terms of consequence. Children in this stage cannot think two

aspects of an object at the same time, and cannot conduct an inductive and deductive thinking,

they cannot separate the real world from the world of their imagination. When they come to the

end of this stage, they have an ability to explain the reasons about what they do when they are

asked to express them.

Children in this stage have more focuses to static aspects, rather than transformation of

objects from one state to another. For example, when they are given 2 balls which are made of

candle in the equal quantity, the first ball is converted into a sauce shape and the other one

remains unchanged. When they are asked “Are they now the same?”, they respond “No, the ball

in sauce shape is bigger”. In this case the children keep the idea of candle shape in their mind

and ignoring transformation which convert the ball into sauce shape.

In the third stage (concrete operational stage), from age seven to age twelve or later,

children start to play with other children, and moves from isolated or individualized play. They

also can classify objects which have several characteristics, and put it in a set or subsets and

expressing their characteristics. Although in this stage they start to have abilities in

understanding jokes, they, however, still have difficulties in understanding proverbs or some

hidden meanings of expressions. Different from their ability in the second stage, where they

cannot reverse thought, operations, or procedures, in the third stage they can do so. In the third

stage, they also can deal with complex relationships between two set of objects, understand and

visualize intermediate states of transformation. They also can understand other children’s ideas

and viewpoints. They start to understand how to do inductive and deductive reasoning,

although the inference is still about specific to specific things. At the end of this stage children



can understand verbal abstractions, perform complex operations, with difficulties in carrying

out some symbolic operations and drawing a conclusion using logical reasoning.

The fourth stage is Formal Operational Stage, which is reached when someone come to

adult state. In this stage they do not need to use concrete object representations for

representing or illustrating mental abstractions. They can consider many viewpoints

simultaneously to judge about their own ideas and actions objectively, and to do reflection on

their thought processes. They also can understand definitions and rules, formulate theory,

construct hypotheses and test them. Further, they can use implications in their inferences,

particularly when they come to an inductive or deductive reasoning. In some mathematics

concepts, people who are in the fourth stage can do more complex concepts in counting,

including enumerations, permutations, and combinations. They also can understand proportion,

correlation, and probability.

In Piagetian theory, intellectual development which covers process of assimilation and

accommodation can be categorized into mental structures. Assimilation can be considered as a

process through which new information and experiences are incorporated into mental

structure, while accommodation can be expressed as the resulting restructuring of the mind as a

consequence of new information and experiences. New information is restructured in mind, and

together with old information it is accommodated in the mind. Learning process adds new

information to the structure of odd information, and each piece of new information implies

The process of assimilation and accommodation are related to the process of equilibrium.

The process of equilibrium can be considered as the process where one’s mental structure loses

its stability as a consequence of new experiences and returns to equilibrium through the

processes of assimilation and accommodation. Equilibration process results in developed and

mature mental structures. Piaget believes that maturation, physical experience, logico-

mathematical experience, social transmission, and equilibration affect one’s intellectual

development.

Some seventh graders are twelve or thirteen years of age. However, some of them are

still in the concrete operational stage. The others have just entered the stage of formal

operations or still in transition between these two stages of intellectual development. This

means that many seventh grader students’ intellectual development has not yet progressed to

the point where they have the mental structure necessary for constructing formal mathematical

proofs. Some of these students do not yet see the difference between a single instance of a

general principle and a proof of that principle.

A teacher should expect certain complex abilities, skills, and behaviours from a student

who is in the formal stage and should be concerned if formal operational mental processes are



not exhibited. At every secondary school level there are students who have not completely

entered the formal operational stage, and teachers should be aware of their behaviours than can

be expected from these students.

Every mathematics teacher, especially those who teach in grades six through nine,

should expect many students to be in the concrete operational stage, should be understanding

of students’ mental inabilities in this stage, should provide learning strategies appropriate for

concrete operations, and should plan activities to help students’ progress to the stage of formal

operations.

(b) J.P. Guilford Structure of Intellect Model

Guilford has developed a three-dimensional model consisting of 120 distinct types of

intellectual abilities, which can be specified and measured. Guilford has defined and structured

general intelligence into a variety of very specific mental aptitudes. In his theory, intelligent

students may have difficulty in carrying out certain mental tasks, whereas other students who

have attained low scores on general intelligence tests may do surprisingly well at some types of

mental activities. Individual students may possess a variety of specific mental strengths and

weaknesses.

When a student is unable to attain a minimal level of mastery of certain skills, it is

possible to determine which intellectual abilities which are poorly developed in that students,

and what kind of activities to improve those abilities. Teacher may recognize certain

inadequately developed mental skill faced by a student and can assist them in developing these

skills, when they have severe intellectual or emotional handicaps. Teachers should recognize

and encourage the formation of self image students possess, and encourage students who have

those unique talents.

Mathematics teachers should realize and appreciate the value, the beauty, and the

wonder of mathematics, and they should encourage their students to understand mathematics

in an enjoyable situation.

In The Structure of Intellect Model, Guilford identifies and classifies various mental

abilities. This model comprises three variables: operations (the set of mental processed in

learning), contents (the nature of the material being learned), and products (the manner in

which information is organized in the mind).

In Operations of Mind, five types of mental activities are identified: memory, cognition,

evaluation, convergent production, and divergent production. The following are the elaboration

of them:



(1) Cognition: the ability to recognize various forms of information and to understand

information.

(2) Evaluation: the ability to process information in order to make judgements, draw

conclusions, and arrive at decisions.

(3) Convergent production: the ability to take a specified set of information and draw a

universally accepted conclusion or response based upon the given information.

(4) Divergent production: the creative ability to view given information in a new way so that

unique and unexpected conclusions are the consequence.

In Content of Learning, Guilford identifies 4 types of content involved in learning: figural

content (shapes and forms such as triangles, cuboids, hyperbola, etc.), symbolic content

(symbols or codes representing concrete objects or abstract concepts), semantic contents

(words and ideas which evoke a mental image when they are presented as stimuli, behavioral

contents (the manifestation of stimuli and responses in people, the way people behave as a

consequence of their own desires and the actions of other people).

In Products of Learning, Guilford mentioned 6 products of learning, the way the

information is identified and organized in the mind: units (a single symbol, figure, word, object,

or idea), classes(set of units and one mental ability is that of classifying units), relations

(connections among units and classes), systems (a composition of units, classes, and

relationships into a larger and more meaningful structure), transformations (the process of

modifying, reinterpreting, and restructuring existing information into new information; the

transformation ability is usually thought to be a characteristic of creative people), and

implications (a prediction or a conjecture about the consequences of interactions among units,

classes, relations, systems, and transformations).

The following table gives an example:

MATHEMATICAL CONCEPTS PRODUCT OF LEARNING

Real number Unit

The entire set of real numbers Class

Equality and inequality Relations

The set of real numbers with the operations of addition,

subtraction, multiplication, and division and algebraic

properties of these operations

System



Function defined on the real numbers system Transformations

Theorem about functions on the real numbers Implication

(c) Robert Gagne’s Theory of Learning

The phase of learning sequence and the type of learning is particularly relevant for

teaching mathematics.

In his theory, the objects of mathematics learning are those direct and indirect things

which we want students to learn in mathematics. The direct objects of mathematics learning are

facts (arbitrary conventions in mathematics, such as symbols of mathematics), skills (operations

and procedures which students and mathematicians are expected to carry out with speed and

accuracy), concepts (an abstract idea which enables people to classify objects or events and to

specify whether the objects and events are examples or non examples of the abstract idea), and

principles (sequence of concepts together with relationships among these concepts). The

indirect objects are transfer of learning, inquiry ability, problem-solving ability, self-discipline,

and appreciation for the structure of mathematics.

Learning sequence stated by Gagne consists of 8 sets of conditions that distinguish 8

learning types:

(a) Signal learning (involuntary and emotional learning resulting from either a single

instance or a number of repetitions of a stimulus which will evoke an emotional

response in an individual; ),

(b) stimulus-response learning (voluntary and physical learning to respond to a signal),

(c) chaining, (the sequential connection of two or more precisely learned non verbal

stimulus-response actions)

(d) verbal association (chaining of verbal stimuli; that is the sequential connection of two or

more precisely learned verbal stimulus-response actions).

(e) discrimination learning (learning to differentiate among chains; tat is to recognize

various physical and conceptual objects)

(f) concept learning (learning to recognize common properties of concrete objects or

events and responding to these objects or events as a class)

(g) rule learning (the ability to respond to an entire set of situations (stimuli) with a whole

of actions (responses), and

(h) problem solving (selecting and chaining sets of rules in a manner unique to the learner

which results in the establishment of a higher order set of rules which was previously

unknown to the learner).



Each of these 8 learning types occurs in four sequential phases: the apprehending phase

(the learner’s awareness of a stimulus or a set of stimuli which are present in the learning

situation), the acquisition phase (attaining or possessing the fact, skill, concept, or principle

which is to be learned), the storage phase (retaining or remembering fact, concept or

principle), and the retrieval phase (the ability to call out the information that has been

acquired and stored in memory).

(d) Dienes on Learning Mathematics

Dienes considered mathematics as the study of structures, the classification of structures,

sorting out relationships within structures, and categorizing relationships among structures. He

believes that each mathematical concept (or principle) can be properly understood only if it is

first presented to students through a variety of concrete, physical representations).

According to Dienes, there are 3 types of mathematics concepts: pure mathematical

concepts (deal with classifications of numbers and relationships among numbers, and are

completely independent of the way in which the numbers are represented), notational concepts

(properties of numbers which are a direct consequence of the manner in which numbers are

represented), and applied concepts (the applications of pure and notational mathematical

concepts to problem solving in mathematics and related fields).

Ways to learn mathematics (according to Dienes):

(1) Analyze mathematical structures and their logical relationships.

(2) Abstract a common property from a number of different structures or events and

classify the structures or events as belonging together.

(3) Generalize previously learned classes of mathematical structures by enlarging them to

broader classes which have properties similar to those found in the more narrowly

defined classes, and

(4) Use previously learned abstractions to construct more complex, higher order

abstractions.

According to Dienes, there are 6 stages in learning mathematical concepts: (1) Free play, (2)

Games, (3) Searching for Communalities (searching for common properties in the

representations), (4) Representation (diagrammatic representation or verbal representation, or

inclusive example), (5) Symbolization (to describe the representation of the concepts), and (7)

formalization (order the properties of the concept and consider the consequences).



(e) Ausubel’s Theory of Meaningful Verbal Learning

Ausubel’s theory of meaningful verbal learning contain a rationale for expository

teaching and shows how lecture-type of lessons can be organized to teach the structure of a

discipline to make learning more meaningful to students. In Reception Learning the

principal content of what is to be learned is presented to the learner in more ore or less final

form.

Ausubel explained the distinction between rote learning and meaningful learning as

follows: The meaningful learning refers primarily to a distinctive kind of learning process, and

only secondarily to a meaningful learning outcome –attainment –that necessarily reflects the

completion of such a process. Meaningful learning as a process presupposes, in turn, both the

learner employs a meaningful learning set and that the material he learn is potentially

meaningful to him.

Ausubel said that there are 2 preconditions for meaningful reception learning: (1)

meaningful reception learning can only occur in a student who has a meaningful learning set;

(2) the learning task be potentially meaningful through its relation to the learner’s existing

cognitive structure.

(f) Jerome Bruner on Learning and Instruction

Bruner stated that a theory of instruction should be prescriptive and normative;

prescriptive if it contains principles for the most effective procedures for teaching and learning

facts, skills, concepts, and principles, and normative if it contains general criteria of learning and

states the conditions for meeting the criteria.

Therefore, a theory of instruction should consist of prescribed process and methods for

attaining the learning objectives of instruction, and it contains general learning objectives or

goals and description of how the objectives can be accomplished.

According to Bruner, there are 4 major features that should be covered by any theory of

instruction:

a. Specify the experiences which predispose or motivate various types of students to learn.

b. Specify the manner in which general knowledge and particular discipline must be

organized and structured so that they can be most readily learned by different types of

students.

c. Specify the most effective ways of sequencing material and presenting it to students in

order to facilitate learning.

d. Specify the nature, selection, and sequencing of appropriate rewards and punishments

in teaching and learning a discipline.



In learning mathematics, Bruner formulated four general theorems:

a. Construction Theorem (the best way for a student to begin to learn a mathematical

concept, principle, or rule is by constructing a representation of it).

b. The notation theorem (early constructions or representations can be made cognitively

simpler and can be better understood by students if they contain notation which is

appropriate for students’ level of mental development.

c. The Contrast and Variation Theorem (the procedure of going from concrete

representations of concepts to more abstract representations involves the operations of

contrast and variation).

d. Connectivity Theorem (each concept, principle, and skill in mathematics is connected to

other concepts, principles, or skills).

(g) B.F. Skinner on Teaching and Learning

Related to the nature of human being, there are 2 generally models of human actions:

the behaviourist model (regard people as being somewhat passive organism who are primarily

controlled by stimuli from their environments) and the phenomenological models (people’s

behaviour can be controlled by properly controlling their environments, and that scientific

methods are appropriate for the study of human behaviour.

According to Skinner, there are 2 categories of human behaviour: respondent behaviour

(involuntary or reflex behaviours and result from special environmental stimuli) and operant

behaviour (which are neither automatic, predictable, nor related in any known manner to easily

identifiable stimuli).

Based on the first idea, in order for a respondent behaviour to occur, it is first necessary

that a stimulus be applied to the organism. Based on the operant behaviour, certain behaviours

merely happen, and even if they are caused by specific (but hard to identify) stimuli, this stimuli

are inconsequential to the study of behaviour.

Skinner stated that classical respondent conditioning for respondent learning results

when a new stimulus is presented simultaneously with an older stimulus which elicits the

expected response. Operant conditioning for operant learning is controlled by following

abehaviour with a stimulus. This stimulus, which is presented after the response, is usually

called reinforcement. In reinforcement, reinforcers are stimuli that follow a response and tend to

increase the probability of that response. This reinforcer can facilitate learning and changes in

behaviour. Reinforcers can be positive (if it tends to increase the probability that particular

behaviour will be repeated), and negative (if their removal tends to strengthen behaviours).



If a learned behaviour is not used for along period of time it will be forgotten and will

have to be learned. In forgetting, the effects of operant conditioning are simply lost with the

passage of time. Skinner defines extinction as the process through which conditioned responses

become less and less frequent when reinforcements are no longer forthcoming.

A negative reinforcer is a stimulus whose withdrawal results in the strengthening of

response. This type of reinforcer is called aversive stimulus. We can escape form aversive

stimulus by removing the stimulus after we contact with it. Or by leaving the environment

where the aversive stimulus exists. We can also avoid aversive stimulus by anticipating its

occurrence and staying away from it.

Skinner also discussed the effects and by-products of punishment, and regards

punishment as the deliberate presentation of a negative reinforcer or the deliberate removal of

positive reinforcer. Skinner has identified 3 effects of punishment upon he person being

punished: (a) punishment suppresses behaviour; (2) punishment is to evoke incompatible

behaviour resulting in anxiety and accompanying physiological changes such as increased heart

rate, higher blood pressure, and muscle tension; (3) punishment is to condition the punished

person to do something other than the act for which he or she is being punished,

(h) Thorndike’s Theory on Learning

According to Thorndike, learning to make an appropriate response involves

strengthening the relevant association and weakening the other association. This principle is

called the law of effect, because the learning depends on the effect of each response.

Basically this law is an idea that if a behaviour is followed by a pleasing state of affairs, it

is more likely to occur again in the future under the same circumstances, and if a behaviour is

followed by displeasing state of affairs, it is less likely to be given again in the future (Mayer,

2002).

Thorndike’s theory was based on his experiment on a cat which was put inside a cage.

This cage can be opened from inside if the cat touch the string inside the cage. This experiment

shows that the cat did hard work in the first trial, but just a little effort in the following efforts.

One of the main instructional applications of Thorndike’s work has been on the

emphasis on drill and practice with feedback. In his illustration, a student was asked to spell a

certain word. Feedback which states good climate will be given if the student gives correct

spells, and, the teacher gives the response “no”, if the student gives wrong answers. Thorndike’s

work suggests that practice and response learning will be improved if there is a feedback. So,

feedback is viewed as a way of reinforcing correct responses.



Research also indicates that feedback may serve information rather than reinforcement.

If a feedback is intended as a reinforcer, the students should be informed whether his work is

correct or no. If a feedback is just a kind of information for students, then the feedback should

be made even more obvious and effective than simple right or wrong feedback.

Behaviourist theory sees that learning process occurs when feedback is given, whereas

the cognitive theory will see learning as “all-or-none”. Behaviourist theory sees that learner is

passive, but cognitive theory sees that learning is actively forming hypotheses.

3.4. Teaching by Fostering Learning Strategies

The success of meaningful learning can be achieved by improving the way of presenting

material to students and enhancing students’ learning process. Both efforts are related to

students’ learning strategies. In this part, a number of learning strategies will be discussed,

including mnemonic strategies (for memorizing purposes) , structure strategies (for assisting

students in organizing material), and generative strategies (assisting students in integrating

their new concepts and their existing knowledge).

Learning strategies can be viewed as students’ knowledge processing which are carried

out to enhance their ability in learning process. The objective of learning strategies is to assist

students in order that they become cognitively active learners by doing some efforts intended to

understand the material. This is why learning strategies are so important for students’ success

in their learning process. However, only a little attention has been emphasized into teaching

students how to learn. Most of teachers expect their students to remember a huge number of

facts without teaching them how to memorize.

Applying learning strategies means recalling specific facts, organizing the material into a

coherent structure, or integrating new material with the existing knowledge. In a learning

strategy called as “mnemonics”, teachers use a technique which helps their students in

memorizing important facts. When students memorize something, they can remember it and

reveal it anytime they need, without conscious mental efforts. Memorizing is useful in some

cases. When students have memorized some facts, they will easily use their minimal mental

efforts in high-order mathematical thinking. For example, if a student remembers definitions

needed in dealing with a theorem, it is likely that he or she will easily understand the theorem.

Even, when the material that should be comprehended by the students do not make sense,

mnemonics strategies still can offer a means of meaningfulness in certain level.

Mnemonic strategies have been applied to learning process since 2500 years ago, when

Simonides developed an imagery-based mnemonic system (Yates, 1966). People who used

mnemonic focused their attention into the development of useful techniques for memorizing.



Memory strategies use two distinct coding: imagery and verbal representation. This

means that there are various ways to retrieve basic facts from memory. Memory strategies also

organize new information and the existing one and forming associations between elements of

information which support the process of remembering. Therefore, mnemonic strategies assist

the learner to become more cognitively active by construct mental connections, which can put

facts firmly in memory.

The other learning strategy is structure strategy. Mayer (2002) expresses that “structure

strategies prompt active learning by encouraging learners to mentally select pieces of

information and relate them to one another within a structure”. This process can be considered

as constructing internal connections, where students may identify key ideas and put them

interrelated in a coherent organization.

Structure strategies involve mapping and outlining techniques. In organizing material

students can construct an outline by identifying and summarizing each of ideas and determine

the way they relate one to another. In knowledge mapping, developed by Dansereau

(Chmielewski & Dansereau, 1998), the students are trained to identify ideas and relations

among them. In knowledge mapping there can be spatial learning strategies or concept

mapping. In concept mapping, developed by Novak (1998), students produce maps consisting of

nodes (representing concepts) and links (lines between nodes, representing the relationship).

The other strategy for promoting meaningful learning is generative strategy, which

helps students in integrating incoming material with existing knowledge and experience. This

strategy promotes deed understanding by encouraging students to put the material into his/her

own words, and construct a link to other knowledge. Since learning is generative process in

which students actively generate the connections between ideas, or facts, then many activities

such as taking notes, underlining, answering questions, and summarizing can be viewed as

generative processing. Rothkopf (1970) called all these activities as “mathemagenic activity”,

which means any activity of students that gives birth to knowledge.

3.5. Teaching by Problem-Solving Strategies

Teachers can encourage students to become an effective problem solver by applying 4

criteria suggested by Polya. Based on Polya theory, the steps for Problem solving consist of (a)

understanding the problem (b) Devising a Plan, (c) Carrying Out the Plan, (d) Looking Back

(Reflecting).

The process of problem solving hierarchically are as follows: (a). Read the problem, (b)

Understand the problem, (c) Think of a way to solve the problem, (d) Translate the problem into

a mathematical model/sentence, (e) Do the mathematical computations, etc. (f) Arrive at a

solution, (g) Check solution for accuracy/reasonableness, etc.



In understanding the problem students should do the following activity: Look for

information given, visualize the information, organize the information – make a table, connect

the information.

In devising a plan students should do making a representation (by drawing a diagram,

making a systematic list, and using equations); making a calculated guess (by making a

calculated guess, guessing and checking, looking for a pattern); to go through the process (by

acting it out, working backwards, doing before-after); to change the problem (by restating the

problem, simplifying the problem, and solving part of the problem).

In Carrying Out the Plan students should do the following activity: Use mathematical

knowledge, use mathematical skills, and use logical thinking.

In looking back (reflecting) students should do the following steps: Check solution – Is it

reasonable?, Improve on the method used, seek alternative solutions, and Extend the method to

other problems.

There are 4 instructional strategies in PS: a. Modeling, b. Coaching (the teachers give

instructions or ask questions to help students focus on an aspect of the problem), c. Explaining,

d. Providing.

There are 3 conceptions of problem solving lessons: (a) Teaching through problem

solving, (b) Teaching about problem solving, and (c) Teaching for problem solving. In teaching

through problem solving, teachers ask students to read and understand a problem given, and

try to analyze it, and they may construct a generalization based on the available basic facts or

patterns. In teaching about problem solving, teachers grab a problem from a book, and ask the

students to work on the problem. In teaching for problem solving, teachers present preceding

mathematical concepts or skills which are needed by the students in solving the designated

problem.

3.6 Methods, Approaches, and Models of Teaching and Learning

3.6.1. Teaching for Strengthening Students’ Understanding and Reasoning

National Research Council (1989, 58-59) in Everybody Counts outlines that students

learn mathematics well only when they construct their own mathematical understanding. So,

students need to be familiar with the tasks consisting of “represent”, “find”, “transform”, “solve”,

“apply”, “prove”, “verify”, and “examine”.

In order to make mathematics meaningful to students and helping them in group

discussion sessions, teachers should focus their efforts in developing various types of questions.



In order to make students learn mathematics, teachers have to involve them in a number of

invention (guided reinvention) and constructing their own interpretation on the information,

concepts, facts, procedures they obtain in classroom teaching. This will make sense the concepts

of mathematics to students.

Enhancing teachers’ classroom environment and performance is difficult. Various efforts

should be addressed as this change will take time and needs systematic and continuous

supports from administrators, experts, supervisors, and parents.

To empower students, teachers need to move from the current practice of teaching and

seek shifts in handling classroom management, treating students, approaching concepts of

mathematics, stimulating students, presenting problems from daily life, encouraging students to

be a good problem solver.

In classroom management, teachers should treat students as individual, having unique

differences, treat all of them in a group as mathematical communities, and not as simply a

collection of individuals. This means that interactions between students and students, teacher

and students, become more important and play crucial roles. Cooperative, and if possible,

collaborative teaching should be central in practice of teaching. The students’ response to

teacher’s information and instruction will empower students in strengthening students’

reasoning. The ideas presented by other students in group discussion will enhance their

understandings and reasoning. If the climate of discussion and students’ activity is focused into

learning by doing (exploring concepts, providing mathematical evidence, verifying results,

examining open ended answers), there should be deep mathematical understandings, and

students will be ready for solving problems, conjecturing, exploring, and inventing, while

reducing an emphasis on a mechanistic answer-finding.

Building understanding through reasoning will prevent students from merely

memorizing facts, principles, and procedures. Making sense of the concepts should be put in the

first priority before asking students to do merely memorizing, as this will help students

improving their cognitive skills. Teachers can encourage students to ask by stimulating them

with some constructive questions. Stimulating questions have to help students to verify whether

a certain expression is correct logically, mathematically correct; to guide students to rely on

themselves; to work together cooperatively and collaboratively; to encourage students to think

logically and develop reasoning skills; to engage students in constructing conjectures, doing

explorations; solving real world problems by applying ideas and concepts they obtain in

teaching and learning process. Therefore, questions expressed by teachers should be in type of

scaffolding, prompting, or probing questions.



When there are two or more answers posed by the students, teachers should accept

them since the appearance of multi solutions or multi procedures given by the students are the

characteristics of open-ended approach in mathematics teaching process.

3.6.2. Using Mathematical Connections in Teaching Mathematics

Mathematical connections should be emphasized in the instruction as this will prevent

students from treating mathematics as an isolated entity without any link to application in real

world. Teachers have to underline and illustrate with rich examples that mathematics concepts

come from real world problems where students might encounter in their daily life.

The idea of making connections is not new. At least, we can trace it back to mathematics

education literature in 1930s (Hiebert and Carpenter, 1992). This can be understood from the

nature of mathematics which serve the other field, yet mathematics itself has it own rigor

structure. Mathematics grows as the impact of demand in problem solving, and the solution

given by mathematics contributes to further problems, or even unsolved problems. This means,

that connections play important roles in mathematics and mathematics teaching.

In teaching mathematics, teachers have to trace back some concepts related to the topics

which are being presented. This kind of prerequisite as a requirement for teaching mathematics

cannot be avoided, as the failure of its comprehension will result in the failure of the concept

which is being explained to the students. The connections consist of three types: the

connections between mathematics concepts, between the topics in mathematics and other

fields, and between mathematics and real world (mathematics application).

The responsibility to make connection between mathematics and its application is not

merely the tasks of teachers. Curriculum developers and textbook writer have to take into

account this aspect. Textbooks prepared for students, particularly primary school students

should incorporate problems related to the concepts discussed in the description. Teachers

need to choose activities that integrate everyday uses of mathematics into the classroom

learning process as they improve students’ interest and performance in mathematics (Fong, et

al., 1986).

Students need to build meaningful connections between their informal knowledge about

mathematics and their use of number symbols, or they may end up building two distinct

mathematics systems that are unconnected – one system for the classroom and one system for

the real world (Carraher et al., 1987).

Hodgson (1995) demonstrated that the ability on the part of the student to establish

connections within mathematical ideas could help students solve other mathematical problems.

However, the mere establishment of connections does not imply that they will be used while



solving new problems. Thus, teachers must give attention to both developing connections and

the potential uses of these connections.

Vocational educators claim that the continual lack of context in mathematics courses is

one of the primary barriers to students’ learning of mathematics (Bailey, 1997; Hoachlander,

1997). Yet, no consistent research evidence exists to support their claim that students learn

mathematical skills and concepts better in contextual environments (Bjork and Druckman,

1994).

It should be noted, however, that it is not always apparent whether or not teachers can

use connection in the topics they present, as this tasks are sometimes difficult to realize.

(Lampert, 1990) indicated that the skills and concepts learned in school mathematics differ

significantly from the tasks actually confronted in the real world by either mathematicians or

users of mathematics.

3.5.3. Enhancing the Level of Understanding by Using Tools and Media

In order to facilitate an interaction in teaching that is emphasized on exploring

mathematical ideas, not just expressing correct answers, teacher must encourage the use of

various tools apart from excessive emphasis on conventional mathematical symbols. A variety

means for expressing mathematical ideas should be used, including symbols, notations,

expressions, diagrams, tables, graphs, analogies. If a range of technological tools (devices) are

already available, teachers have to encourage students to use them. Teachers also have to teach

students how to use the device, to speed up the computational operations and suggest which

devices should be used to solve particular mathematical problems.

When there are two or more tools, teachers should stimulate the students to use the

tools which might help them work on the given problems. To facilitate this interactions,

teachers are suggested to make the use of computers, calculators, LCD projector, OHP and other

technology; models consisting some local concrete material, representation consisting of

symbols, notations, diagrams, tables, graphs, and pictures; textual tools consisting analogies,

metaphors and stories; simulations, dramatization, and oral presentation.

3.6.4. Enhancing Communication Skills in Mathematical Context

Communication is the way of sharing ideas and enhancing clarity. It is used for clarifying

and developing mathematical ideas and process. Communication helps students to link their

mathematical thinking coherently and logically structured to his classmates, teachers, using

mathematical language to express mathematical ideas correctly.

Communication skills can be used for describing real objects, pictures, and diagram into

mathematical ideas; modelling problems using word, written symbols, diagrams, graphs;



expressing daily life problems in mathematical symbols and notations; listening, writing and

discussing mathematics concepts; understanding written text on mathematics; constructing

conjectures, arguments, and generalizations. Students can carry out communication by using

multiple representations.

When teachers need to focus their teaching and have to strengthen the students’

communication, teachers have to distribute tasks which are related to the importance of

mathematical concepts; solvable by various methods; contains a lot of examples; and gives

students to interpret, investigate, and formulate conjectures.

Exercises:

A. 1. Piaget states that children’s reasoning in pre operational stage is neither deductive nor

inductive. It is called transductive reasoning. One example is given by Piaget himself: His

son told him: “I have not gone to bed, so it has not come to evening”. Do you have other

examples similar to this situation in your teaching and learning process?

2. Children in this stage cannot think reversely. Can you give one example in mathematics

operation?

3. The following is a conversation involving a child:

A: “Do you have a brother?”

Y: “Yes”.

A: “What is his name?”

Y: “Hardi”.

A: “Does Hardi have a brother?”

Y: “No”.

4. Suppose there are 2 strings, each of which is 2 meter long. Each string is converted into the

following pattern:

What do you predict about children response when they are asked “Which one is now

shorter?”



B. 1. Do you think that a seventh grade teacher should not explore the nature of intuitive and

formal mathematical proofs with students? Discuss it in your group.

2. Do you think that a seventh grade teacher should realize that a twelve year old adolescent

has a different mental structure (as well as an obviously different physical structure) than

a twenty-two year old teacher?

3. What do you think about the following statement: It is appropriate for us to examine the

intellectual attributes which some secondary school students do not have, but which are

required to carry out many standard school mathematics learning activities?

4. Do you agree that concrete thinkers will not be able to solve logical puzzles or to resolve

mathematical paradoxes?

5. Mathematical symbols and manipulations involve formal operations, and many students

learn algebra by memorizing rules for combining and manipulating symbol with little

understanding of the meaning of algebraic techniques. Give 3 examples of this kind wrong

manipulation that students experience in mathematics.

C.1. Mention some examples of figural content and symbolic content, semantic content and

behavioural content in mathematics topics.

2. Construct a table depicting how the information is identified and organized in the mind by

writing a mathematics concept and related products of learning based on Guilford’s

factors of Intellectual Ability.

D. 1. Write 2 examples of concept learning in mathematics.

2. Write 2 examples of rule learning in mathematics.

3. Write step by step how to derive the quadratic formula for solving a quadratic equation.

Write also what kind of ability students should have in each step of the process.

E. 1. Do you agree with the statement “Pure concepts should be learned by students before

notational concepts are presented”.

2. Do you think that if students learn notational concepts followed by pure concepts will

make students merely memorize patterns for manipulating symbols without

understanding the underlying pure mathematical concepts?

3. Some students write the following symbol manipulations: 392+=+ xx , .1553

aaa =⋅

Discuss why they have this kind of errors.



F. 1. Some critics of reception learning and some proponents of discovery learning claim that

reception learning usually is rote learning and discovery learning usually is meaningful

for students. What do you think about this claim? Do you agree to what Ausubel said that

“both reception and discovery learning can each be rote or meaningful depending on the

conditions under which learning occurs”?

2. Discuss the following arguments, and write your own opinion regarding this matter:

a. The distinction between reception and discovery learning is not difficult to

understand. In reception learning the principal content of what is to be learned is

presented to the learner in more or less final form. The learning does not involve any

discovery on his part.

b. In reception learning students are required to internalize the material or incorporate it

into their cognitive structure so that it is available for reproduction or other use at

some future date.

c. The essential feature of discovery learning is that the principal content of what is to be

learned is not given but must be discovered by the students before they can internalize

it.

G. 1. Seventh and eight graders are not yet ready to use the notation of y = f(x) to represent the

concept of a mathematical function. What kind of representation should be given to them?

2. Do you agree if a high school sophomore is given examples of functions such as sets of

ordered pairs of objects or linear relations such as y= -x ? Why?

H. 1. If a student turns red with embarrassment after wrongly answering teacher’s question,

what kind of conditioning and learning the teacher might be done?

2. A student had learned a number of incorrect algebra techniques, such as 222)( baba +=+ and he continued to do the same error, even after being corrected

many times. How do you suggest to “unlearn” this misconception, something which has

been learned incorrectly the first time and has been reinforced through repeated use?

3. The following are some examples of considerable abilities which students can show:

Determine whether the abilities can be categorized into memory, cognition, evaluation,

convergent production, or divergent production.

a. A student answers “1” when a question of “sin 90o” is given.

b. A student can separate a set of polygons consisting of quadrilaterals and triangles into a

set of quadrilaterals and a set of triangles.

c. A student comes to a conclusion that a figure of triangle he has is an isosceles triangle.



d. A student manages to solve a system of linear equation in two variables.

e. A mathematician discovers and proves a new important mathematical theorem.

I. 1. Suppose that a teacher asks his students to recognise what kind of geometric shape he has by

asking whether the geometric shape is a square, and the figure he has one at a time, are

rhombus, square, trapezoid, and parallelogram. If there is a student who answers that the

figure is a square when a figure of rectangle is given. Do you think that this kind of teacher’s

efforts can reveal the real concept possessed by his students? Do you think that this activity

can be viewed as a kind of concept learning task?

2. What do you think the behaviourist and the cognitive theorist will react to the above

problems? Will they think that it is in line with behaviour principle?

J. 1. Do you agree that memorizing skills are still relevant in current mathematics learning? If it

is still relevant, in what context it should be used?

2. Give 3 examples of “mathemagenic activities” in mathematics. Explain your description.

3. Give an example of concept mapping for sets of numbers, sets of geometric shapes, and sets

of solids.

K. 1. (Teaching through problem solving). Draw a number of different polygons and ask

students how many triangles there are in each polygon. Ask them to find the general

methods for computing the number of triangles in a polygon.

2. (Teaching about problem solving). Suppose a teacher have a problem on mathematics:

What are the four consecutive numbers whose sum is 50? Ask the students, and analyse

their different types of answer. Who comes with variables?

3. (Teaching for problem solving). In teaching for a problem solving, problem solving is

preceded by teaching of mathematics concepts or skills needed to solve the problem.

What concepts do you think should be given to students when you will use problem

solving in the solution to quadratic equations?

L. 1. A student constructs their own mathematical understanding through a number of tasks.

For example, a student find the method how to prove a theorem by himself/herself

rather than copying or and reading the proof given by his/her teacher. Can you explain

give an illustration how this student learn mathematics by (a) “represent” a particular

notation in any other form, and (b) “transform” an algebraic expressions into another

one?



2. Developing a number of good questions is central to mathematics teaching. Give your

reason and analysis why these efforts can make mathematics meaningful to students

and helping them in group discussion sessions.

3. In order to make students learn mathematics, teachers have to involve them in a number

of invention (guided reinvention) and constructing their own interpretation on the

information, concepts, facts, procedures they obtain in classroom teaching. Can you

explain why this will make sense the concepts of mathematics to students.



Chapter 4

4.1. Current Trends of Mathematics Teaching Practice

4.1.1. Current Methods and Approaches in Teaching Mathematics

Students’ abilities to learn mathematics can be largely an impact of how teachers

conduct teaching and learning process. Teachers should positions themselves as a charismatic

figure and a persuasive presenter. Teachers need to engage students in cognitive and social

tasks, and to use them productively. Most of students are keen on listening to teachers’

presentation in one way traffic communication. However, in most of the cases students should

do their own learning, take efforts to understand the information and make it their own.

Effective students draw facts, procedures, principles, ideas, and information from their teachers

and take the advantages of learning resources and facilities effectively. Teachers have a major

role: how to create powerful learners.

In teaching it is important to teach the students to learn, as effective teaching means that

the students have been taught to be stronger learners. Various models of teaching can be

effective, when it can achieve a number of specific objectives, including cognitive, affective,

psychomotor skills, as well as self esteem, social skills, information, ideas, and creativity. This

also includes the students’ ability to learn.

(a) Partners in Learning

A number of developments work on cooperative learning are in progress, and great progress

has been made in developing strategies that encourage students to work together effectively.

There have been a lot of effective means for organizing students to work together in classroom.

This includes teaching students to do some simple learning tasks in pairs and complex models

for organizing classes in learning communities that strive to educate themselves.

In cooperative learning students are facilitated across cognitive, affective and

psychomotor skill, self-esteem improvement, and social skills. In academic learning they obtain

the acquisition of information and skills, and how to achieve them through various inquiry.



(b) Group Investigation

Group investigation can be considered as a direct path to students’ development

learning community. Group investigation is a complex type of cooperative learning, which

combines preparation for life and academic study.

Group investigation has been used in all subject areas, including children in all ages. The

model leads students to define problems, explore a number of perspectives on the problems,

and study with their partners to gain and understand information, ideas, and skills, where they

also learn to develop their social competence. The roles of teachers are organizing the process

of group work, helping students in finding and organizing facts, procedure, principles.

(c) Role Playing

In role playing, students learn to solve various problems, understand social behaviour, social

interactions and their role in those interactions. Role playing is good for helping students in

collecting and organizing information about some issues, develop empathy with other through

interactions within the group, and to enhance their social skills.

4.1.2. Current Models of Teaching

a. Information-Processing Models

This model emphasized the ways of enhancing the human being’s innate drive to make

sense of the world by acquiring and organizing data, sensing problems and generating solutions

to them, and developing concepts and language for conveying them (Joyce and Weil, 1996).

In information-processing models, students are given with information, concepts, facts,

procedures, concept formation, Some of these models also generate creative thinking, and the

others gives students with general intellectual ability. The following models are classified as

information-processing models: inductive thinking (classification-oriented), concept

attainment, mnemonics (memory assists), advance organizers, scientific inquiry, inquiry

training, and synectics.

The ability to analyze information and create concepts is generally regarded as the

fundamental thinking skills (Joyce and Weil, 1996). Inductive thinking is not only for students

who study sciences, but also for those who study social sciences.

Some concepts require connection between the exemplars and some other entity. To

attain a concept, one need to (1) construct the concept attainment exercise so that we can study

how our students think; (2) the students can not only describe how they attain concepts, but

they can learn to be more efficient by altering their strategies and learning to use new ones; (3)



by changing the way we present information and by modifying the model slightly, we can affect

how student will process information.

(1) Concept Attainment

Concept Attainment is a model of teaching which study of thinking developed by Bruner,

Goodnow, and Austin. It is a close relative of the inductive model. In concept attainment there

are some objects. One of them is attributes, which can be considered as features that the items

of data has; essential attributes, attributes which are critical to the domain under consideration.

Exemplars of a category have many other attributes that may not be relevant to the category

itself. Attribute value, which refers to the degree to which an attribute is present in any

particular example.

Concepts are defined by the presence of one or more attributes conjunctive concepts.

The Exemplars are joined by the presence of one or more characteristics. Disjunctive concepts

are defined by the presence of some attributes and the absence of others.

Some concepts require connection between the exemplars and some other entity. To

attain a concept, one need to (1) construct the concept attainment exercise so that we can study

how our students think; (2) the students can not only describe how they attain concepts, but

they can learn to be more efficient by altering their strategies and learning to use new ones; (3)

by changing the way we present information and by modifying the model slightly, we can affect

how student will process information.

The key to understanding the strategies the students use to attain concepts is to analyze

how they approach the information available in the exemplars. They can concentrate on just

certain aspects of the information (patristic strategies), or do they keep all or most of the

information in mind (holistic strategies).

In model of concept attainment there are 3 phases: (1) Presentation of Data and

Identification of Concept; (2) Testing Attainment of the Concept; (3) Analysis of Thinking

Strategies.

In the first phase, teacher presents labelled examples, students compare attribute in

positive and negative examples, generate and test hypotheses, state a definition according to the

essential attributes. In the second phase, students identify additional unlabeled examples as yes

or no, generate examples, teacher confirm hypotheses, names concept, and restates definitions

according to essential attributes. In the third phase, students describe thought, discuss role of

hypotheses and attributes, and discuss type and number of hypotheses.



(2) Scientific Inquiry

Jurisprudential inquiry is designed to study of social issues at community state. Initially,

this model was created especially in social studies. In its development, however, this model can

be used in many areas, as well as in mathematics and science.

The main idea of the model is to involve students in a genuine problem, of inquiry by

confronting them with an area of investigation, helping them identify a conceptual or

methodological problem within this area of investigation, and inviting them to design ways of

overcoming that problem. The students will gain respect for knowledge and will probably learn

both the limitations of current knowledge and its dependability (Scahubel, Klopfer, and

Aragheven, 1991)

Phase 1: an area of investigation is posed to student, including the methodologies used in

investigation.

Phase 2: students structure the problem, and identify a difficulty in the investigation. (The

difficulty may be one of data interpretation, data generation, the control of

experiments, or the making of inferences)

Phase 3: Students identify the problem in the investigation, asked by teachers to speculate

about the problem.

Phase 4: Students speculate on ways to clear up the difficulty, by redesigning the experiment,

organizing data in different ways, generating data, and developing constructs.

(3) Inquiry Training

Inquiry training is designed to bring students directly into scientific process through

exercise that compress the scientific process into small periods of time. Inquiry training

resulted in increased understanding of science, productivity in creative thinking, and skills for

obtaining and analyzing information. In term of the acquisition of information, it is not more

effective than the conventional methods of teaching, but it was as efficient as recitation or

lectures accompanied by a laboratory experiences (Schlenker, 1976).

Ivany (1969) and Collins (1969) reported that the method works best when the

confrontations are strong, arousing genuine puzzlement, and when the materials the students

use to explore the topics under consideration are especially instructional.

Voss (1982) states that students in junior and senior high school can have benefit from

the model. Elefant (1980) suggests that the method can be powerful with students who have

severe sensory handicaps.



Inquiry training is originated in a belief in the development of independent learners. Its

method requires active participation in scientific inquiry. The general goal of inquiry training is

to help students develop the intellectual discipline and skills necessary to raise questions and

search out answer stemming from their curiosity.

Inquiry training begins by presenting students with a puzzling event. Individuals faced

with such a situation are naturally motivated to solve the puzzle.

Suchman’s theory:

1. Students inquire naturally when they are puzzled.

2. They can become conscious of and learn to analyze their thinking strategies.

3. New strategies can be taught directly and added to the students’ existing ones.

4. Cooperative inquiry enriches thinking and helps students to learn about the tentative,

emergent nature of knowledge and to appreciate alternative explanations.

There are 5 phases in inquiry training:

Phase 1: Students’ confrontation with the puzzling situation.

In this phase, the teacher presents the problem situation and explains the inquiry

procedures to the students.

Phase 2: Data-gathering operations of verification.

The students verify the nature of objects and conditions, and verify the occurrence of

the problem situation.

Phase 3: Data-gathering operations of experimentation.

The students introduce new elements into the situation to see if the event happens

differently.

Phase 4: Students organize the information they obtained during the data gathering and try to

explain the discrepancy.

Phase 5: Students analyze the problem-solving strategies they used during the inquiry.

Experiments serve two functions: exploration and direct testing. Exploration – changing

things to see what will happen- is not necessarily guided by a theory or hypothesis, but it may

suggest ideas for a theory. The process of converting a hypothesis into an experiment is not easy

and takes practice. Many verification and experimentation questions are required just to

investigate one theory.



Teachers should broaden the students’ inquiry by expanding the type of information

they obtain. During verification they may ask questions about objects, properties, conditions,

and events. Object questions are intended to determine the nature or identity of objects. Event

questions attempts to verify the occurrences or nature of an action. Condition questions relate

to the state of objects or systems at a particular time. Property questions aim to verify the

behaviour of objects under certain conditions as a way of gaining new information to help build

a theory.

The main idea of the model is to involve students in a genuine problem, of inquiry by

confronting them with an area of investigation, helping them identify a conceptual or

methodological problem within this area of investigation, and inviting them to design ways of

overcoming that problem. The students will gain respect for knowledge and will probably learn

both the limitations of current knowledge and its dependability (Scahubel, Klopfer, and

Aragheven, 1991)

b. Approaches for Enhancing Creative Thought

(1) Synectics

Synectics is a very interesting and delightful approach to the development of

innovations. The initial works with synectics procedures was to develop creativity groups, who

work together to function as problem solver or product developers. In synectics exercises,

students “play” with analogies until they relax and begin to enjoy making more and more

metaphoric comparisons. They use analogies, to attack the problems or ideas.

Ordinarily, when one is confronted with a task, he or she consciously become logical,

and try to analyze the element of the problems and try to think it through. When we use

synectics, it leads us into a slightly illogical world, to give us the opportunity to invent new ways

of seeing things, expressing ourselves, and approaching problems.

Synectics is used to help us develop fresh ways of thinking about the students, their

motives, our goals, and the nature of the problem. If we can relax the premises that have

blocked us, we can begin to generate new solutions. We can consider that we have been taking

responsibility for the students in areas where they may need to be responsible for themselves.

The specific processes of synectics are developed from a set of assumptions about the

psychology of creativity: by bringing the creative process to consciousness and by developing

explicit aids to creativity, we can directly increase the creative capacity of both individuals and

groups.

Creativity is the development of new mental patterns. Non rational interplays leave for

open-ended thoughts that can lead to a mental state in which new ideas are possible. The basis



for decisions, however, is rational. The irrational state id the best mental environment for

exploring and expanding ideas, but it is not a decision-making stage.

Logic is used in decision making and that technical competence is necessary to the

formation of ideas in many areas. The creativity is essentially an emotional process, one that

requires elements of irrationality and emotion to enhance intellectual processes. Much problem

solving is rational and intellectual, but by adding the irrational we increase the likelihood that

we will generate fresh ideas.

The other assumption is that the emotional, irrational elements must be understood in

order to increase the probability of success in a problem solving situation. In other words, the

analyses of certain irrational and emotional processes can help the individual and the group

increase their creativity by using irrationality constructively. Aspects of the irrational can be

understood and consciously controlled. Achievement of this control, through the deliberate use

of metaphor and analogy, is the object of synectics.

Through the metaphoric activity of the synectics model, creativity becomes a conscious

process. Metaphors establish a relationship of likeness, the comparison of one object or idea

with another object or idea by using one in place of the other. Through these substitutions the

creative process occurs, connecting the familiar with the unfamiliar or creating a new idea from

familiar ideas.

Metaphor introduces conceptual distance between the student and the object or subject

matter and prompts original thoughts. By metaphor, teachers provide a structure, a metaphor,

with which the students can think about something familiar in a new way. Conversely teachers

can ask students to think about a new topic, in an old way by asking them to compare it to the

transportation system. Metaphoric activity thus depends on and draws from the students’

knowledge, helping them connect ideas from familiar content to those from new content, or

view familiar content from a new perspective. Synectics strategies using metaphoric activity are

designed, then, to provide a structure through which people can free themselves to develop

imagination and insight into everyday activities. Three types of analogies are used as the basis

of synectics exercises: personal analogy, direct analogy, and compressed conflict.

D. Explain some advantages of scientific inquiry and inquiry model? Give some examples in

mathematics.



(2) Personal Analogy

In constructing personal analogy, the students require to empathize with the ideas or

objects to be compared. Students must feel they have become part of the physical elements of

the problem.

The emphasis in personal analogy is on empathetic involvement. Personal analogy

requires loss of self as one transports oneself into another space or objects. The greater the

conceptual distance created by loss of self, the more likely it is that the analogy is new and that

the students have been creative or innovative. Gordon (..) identifies 4 levels of involvement in

personal analogy: first-person description of facts; first-person identification with emotion;

empathetic identification with a living thing; and empathetic identification with a nonliving

object.

(3) Direct Analogy

Direct analogy is a simple comparison of two objects or concepts. The comparison does

not have to be identical in all respects. Its function is simply to transpose the conditions of the

real topic or problem situation to another situation in order to present a new view of an idea or

problem.

(4) Compressed Conflict

The third metaphorical form is compressed conflict, generally a two-word description of

an object in which the words seem to be opposites or to contradict each other. Compressed

conflicts, according to Gordon, provide the broadest insight into a new subject. They reflect the

students’ ability to incorporate two frames of reference with respect to a single object. The

greater the distance between frames of reference, the greater the mental flexibility.

c. Enhancing Students’ Metacognition

Metacognition, or “thinking about thinking”, refers to the awareness of, and the ability to

control one’ thinking processes, in particular the selection and the use of problem-solving

strategies. It includes monitoring of one’s own thinking, and self-regulation of learning.

The provision of metacognition experience is necessary to help students develop their

problem-solving abilities. The following activities may be used to develop the metacognitive

awareness of students and to enrich their metacognitives experience:

a. Expose students to general problem-solving skills, thinking skills and heuristics, and

how these skills can be applied to solve the problem.

b. Encourage students to think aloud the strategies and method they use to solve particular

problems.



c. Provide students with problems that require planning (before solving) and evaluation

(after solving)

d. Encourage students to seek alternative ways of solving the same problems and to check

the appropriateness and reasonableness of answer.

e. Allow students to discuss how to solve a particular problem and to explain the different

methods that they use for solving the problem

4.2. Teaching Mathematics by Using Information Technology (IT-Based Teaching)

Nowadays the main role of the computer is that it can assist teachers in their teaching

and learning activities. Kieren (1973) revealed that evidence reported from a number of studies

designed to examine the effectiveness of computer-assisted teaching and learning indicates that

computers can be used very effectively to enhance the learning of mathematics.

Much information consisting of questions, problems, proofs, facts, procedures,

principles, can be stored in the computer memory as a programmed text of an integrated

course. All this information can be called back and communicated to students, where the

computer can acts as a tutor by presenting information, and asking some challenging questions.

These processes may give students a series of discovery experiences, a series of practice

exercises, process of deduction in presenting a proof, or thought-challenging problems.

There are a number of responses which can be posed by students in reacting to the

computer stimulus: solving the problems, answering the questions, drawing on the computer

screen, selecting a multiple-choice answer. The computer can give requested pictorial

information, repeating explanation, pronouncing new words, recording and presenting

students’ test scores, and even telling the students’ the right answer when they have wrong

answers.

The teachers with the assistance from the computer could have extra time to do creative

and more professional work. They can inspire students, developing students’ interests,

creativities, attitudes, and values, teaching a basic concept and its application. So, they can avoid

a lot of routine works including giving tests, keeping students’ records, and selecting

individualized assignments.

By the assistance of the computer (in a computer-tutor classroom), a teacher can focus

to the following aspects: selecting or developing a new topic, new programs, new sequences,

and new materials for his students. The teacher can also assist students individually, based on

their interest, needs, and difficulties, as information is already provided by the computer. The

teacher can also analyze students’ response from their answer on the screen, focusing their

attention to what they should teach and enhance students’ achievement, more than ever before.



With the assistance of the computer the teacher could carry out their usual activities. However,

they now can do much wider work in extra time: selecting proper materials and tailoring all

programs and activities suitable to students needs.

Computer programming should be a part of the models of teaching implemented in

classroom practice, if the attaining goals cover the following aspects: (1) Learning how to solve

problems; (2) communicate mathematical ideas; (3) developing desirable attitude, interests,

and appreciations; (4) teaching mathematical concepts; (5) developing skills in computation;

(6) adapting instruction to individual abilities and interests.

Computer can be used to analyze functions. Equations or ordered pairs can be entered

into a computer where the results in graphical form of a function can be observed on the screen.

The change of its slope, the change of its positions as the function representations can add to

students’ comprehension. When an equation as an input is entered into the program, students

can observe how a graph of the function moves to certain directions, or change its form. The

students could be given a general form of a linear equation, they could be asked to change its

constants (or its slope), the computer can present the equations, and the graph would be

displayed instantly. The students then could have an opportunity to analyze the characteristics

of the functions, including the regularities that are apparent, and generalization that can be

drawn. By this process, students may have a good experience in experimentation which can

encourage their discovery on the other concepts.

Computers can be worthwhile when a teacher presents simulation of a topic, as

simulation can describe the situations of the real world which might occur. Through simulation

students can understand the process of certain activities or systems, they can see how the

system reacts to the input they feed, and when there are a number of various computer

responses to stimulus given by students.

When simple computer programming is presented to students, they will have some skill

in logical and systematic thinking as it contains flowcharts and algorithms, where students learn

how the process is carried out step by step from the beginning of the program.

Computer-based teaching is an execution of programs based on instructional objectives

(Hatfield, 1985: 1-4). This includes Computer-Aided Instruction (CAI), Computer-Assisted

Learning (CAL), and Computer-based Training (CBT), computer conference, e-mail, web site,

and multimedia computer. There are various types of interaction in computer-based teaching,

some of them are drill and practice, tutorial, game, simulation, discovery, and problem solving,

presentation or demonstration, communication, test, and information resources.

Other mode of computer use in mathematics education is computer-managed

instruction (CMI). It can be considered as an indirect mode of classroom use of computers,



beacuse students usually do not have much control over a computer operating in a CMI mode

and may not even have direct contact with computer. In the CMI mode the computer is used as a

manager of the learning environment. It may be used to perform the following tasks: (1)

Administering drill-and-practice skill exercises to individual students; (2) Evaluating, scoring,

and providing feedback from students’ answer to drill-and-practice exercises; (3) Administering

pretest and posttest to individual students as well as evaluating and monitoring each students’

work; (4) Keeping academic, personal, medical, and counseling records of students; (5) Setting

cognitive (and occasionally affective) learning objectives for each students; (6) Prescribing

learning activities for individual students by analyzing and evaluating each student’s progress

toward specific learning objectives (Bell, 1978).

CAI which is also computer software for education is the initial educational applications

of computer in mathematics education, which was used experimentally to provide drill and

practice in arithmetic skills since 1965. CAI, in its current development, has become a very

sophisticated instructional mode which provides for complex evaluation of student responses.

Students working in the CAI mode are ordinarily involved in drill, practice, and testing of skills,

exploration concepts, or demonstration of principles.

Courseware used in the interaction of computer-based teaching or CAI, as a set of

techniques, software and supplement, has to assist teachers in motivating students and

enhancing their interest in understanding and exploring concepts. The software should be

designed so that it is appropriate for students in their level of thinking. Kaput (1992) indicates

that programs (software) for computer-based teaching is not only based on assumptions as a

guidance, but it should also based on the experts’ competence in implementing those

assumptions in presenting the software.

By understanding the characteristics of software for mathematics teaching together

with their advantages, we can start developing e-learning for teaching concepts of mathematics.

Based on the current development of technology, we can see that the computers nowadays as

devices for enhancing students’ learning concepts, skills, and problem solving are already

available. The problem is: have we provided teachers with all skills in using computers,

especially in preparing materials, to be established in a computer-based teaching?

4.3. Promoting Teachers’ Professional Development

Teacher professional development, particularly mathematics teacher, is a complex

problem. Mathematics teachers, similar to non mathematics teachers, develop their own

profession through the implementation of professional knowledge within relevant and

meaningful mathematical context devoted to the mathematical understanding of students.

Teachers have to develop their own vision for professional development. This requires a level of



teacher autonomy which depicts an internal drive toward professionalism. To be more

professional, teachers can use some various efforts, including further study, attending

workshops/seminars/conferences, self study, conducting classroom action research, and being

involved in curriculum analyses and its implementation. This professional development needs

special efforts which can affect teachers’ belief, attitudes, expectations, and classroom practices.

Professional development can come in various forms. True professional development,

however, with meaningful and long lasting change is autonomous activity chosen by the

teachers themselves, when they seek the way of understanding and teaching mathematics. On

the other hand, professional development which is forced from outside of the teachers will not

foster the long term impact, as it is not a kind of developmental qualitative change. Further,

Castle dan Aichele (1994) stated that “Externally imposed professional development activities,

although well intentioned, are doomed to failure, like other passing educational fads on the junk

heap of discarded simplistic solutions to complex problems”.

Professional competency is not transferable from other people. It has to be constructed

through direct experience and exploration. It is the teachers themselves who should learn and

interact with their environment. They have to find the connection between prior knowledge and

new knowledge to interpret the meaning of their new finding properly. Professional knowledge

should be actively constructed by the teachers themselves to be active as a decision maker for

their students’ interest and benefit. Professional teachers work and act based on their

professional knowledge and not because of the commands or some rewards given by the others.

Jones, et al. (1994) indicates that professional development can be enhanced by attending

training sessions, seminars, workshops, which are organized for fostering new understanding

towards mathematics teaching and for formulating new perspective in teaching mathematics.

These organized programs should be devoted to develop teachers’ teaching plan together with

its assessment under the supervision of relevant experts. Teachers also should observe the

interactions which occur in other teacher’s teaching and learning process. They have to be

involved in some efforts for improving the quality of teaching and learning process, and to be

involved in an inquiry about student’s thinking,

Learning is an integrated process, and so is learning about teaching. Although the

professional development standard for mathematics teacher indicates that the component of

teachers’ knowledge is separated from its practice, the success of teachers’ work depends on the

integration of theory and practice. Ideally, teachers are involved simultaneously in the study of

mathematics and its pedagogy. Although this competency is not easily accomplished, for the

sake of the innovation of education, teachers need to struggle in achieving this target.



Teaching mathematics is a highly complex activity. Teachers who teach mathematics need a

broad horizon of mathematical knowledge, characteristics of students, the theory of learning,

the use of media and manipulative. All this competencies require understanding in some aspects

related to socio-economical background, attitude, culture, and belief.

Most of the teachers teach mathematics using their own experience based on the lesson

they attended in primary school, junior high school, or senior high school. This experience

influences their way of thinking, teaching and learning process, as well as their professional

development (Ball, 1988).

In the earliest time of their career, prospective teachers engaged in knowledge,

understanding, skills, and disposition enhancement. This development consists of mathematical

knowledge, pedagogy of mathematics and its application in schools. During this period of time,

they obtain some experiences which made them interacting with supervisor teachers, who act

as their mentors. It is in this period they try out their own ideas, and then analyze it under

guidance and motivation from their mentor (Ball, 1988).

Standard for professional development of mathematics teachers underlines the importance

of pre-service and in-service program for mathematics teachers. This standard is not only

devoted for programs in the beginning level, but also for further programs which provide

advance mathematics, advance seminars and workshops, or programs for sharing ideas and

experiences, during the teachers’ career.

There are a number of programs which can enhance teachers’ professionalism. Attending

some courses in a university (as a pre-service program) and teaching practice in some schools

(as open lessons) are some examples of professional development program. The other program

that can be realized for professional development is the involvement of teachers in some

training sessions (as in-service trainings) to update their knowledge and competency, and to

anticipate the changing global world.

Based on NCTM standard, i.e. Standards for the Professional Development of Teachers of

Mathematics (NCTM, 1991), there are at least 6 standards in professional development for

mathematics teachers, which focus on what a teacher needs to know about mathematics,

mathematics education, and pedagogy for mathematics, in order to be able to carry out the

vision of teaching: (1) Modelling good mathematics teaching; (2) Knowing mathematics and

school mathematics; (3) Knowing students as learners of mathematics; (4) Knowing

mathematical pedagogy; (5) Developing as a teacher of mathematics; and (6) Teachers’ roles in

professional development.

These 6 standards underline essential guidance to teachers, lecturers, and people who are

involved in education and professional development of teachers. These standards give direction



toward excellence in teaching mathematics and guidance to all people interested in the

improvement of teaching,

It is the responsibilities of teachers to develop their own professional development and

realize that good mathematics teaching should be modelled in teachers’ professional

development experiences. Of course, teachers should be encouraged to attend some open

lessons in which they have an opportunity to observe other teacher’s performance. At the same

time they can develop their own competency in mathematics and its instruction. Through this

lesson some teachers can share their experiences, ideas, and recommendation. Teacher who

acts as the model will cultivate some benefits coming from his or her colleagues who share

information, ideas, or comments on his/her performance.

All educators, including teachers and lecturers should have deep understanding regarding

the theory and implementation of teaching and learning process for enhancing their

professional development. This effort can be carried out through pre-service training program,

in-service training program, as well as on-service training program.

One example of on-service teacher training program is Lesson Study. Lesson study is a

direct (translation for the Japanese term jugyokenkyu, which is composed of two words: jugyo

means lesson, and kenkyu, which means study or research. Therefore, based on the meaning of

these two words, Lesson Study is study or examination of teaching practice. It should be noted,

however, that this word is so much different from the word kenkyujugyo which is simply a

reversal of the term jugyokenkyu, and thus literally means study or research lessons, that is

lessons that are the object of one’s study. Study lessons are “studied” by carrying out the steps

described next in an attempt to explore a research goal that the teachers have chosen to work

on (e.g., understanding how to encourage students to be autonomous learners) (Fernandez and

Yoshida, 2004).

Lesson Study is an innovation of mathematics teaching and learning process developed in

Japan. This activity, implemented under the principle of Plan-Do-See, is now recognized

worldwide (Fernandez and Yoshida, 2004). The aim of Lesson Study is to improve the quality of

teachers’ teaching and their professional capacity through daily teaching activities (Baba and

Kajima, 2003; NRC, 1995). Basically, Lesson Study can be considered as a real collaboration

between teachers (under collegiality principle) for improving the quality of their teaching and

learning process. This collaboration is carried out by reviewing teachers’ daily teaching and

learning. Lesson Study can be used for achieving the goals of teaching and for enhancing the

quality of teaching (Stigler & Herbert, 2000; IFIC & JICA, 2004).

In Lesson Study, a number of teachers, with the help and guidance of faculty members from

relevant area in universities, design teaching materials, present their teaching in the classroom,



and then discuss the result of teaching activities with the other teachers. The result of the

discussion with different peer group can be used as feedback for future improvement (Kumano

2004; IFIC & JICA, 2004).

There are 3 main steps for refining the quality of teaching in Lesson Study: Plan-Do-See,

where (a) Plan the teaching of certain topics and concepts via academic exploration (Plan); (b)

Conducting teaching activities based on the prepared plan and material, and inviting

counterparts from other schools for observation purposes (Do); (c) Conducting reflection

session (post-class discussion) for exchanging ideas, opinion, comments, and discussions with

the observer (See).

The first step in Lesson Study is study of teaching materials, started by selecting topics and

constructing teaching plan. In analyzing the topic, teachers together with some experts on

relevant area, analyze some resources deeply and make an attempt to see the relationships

between mathematics concepts and its application in daily life, in order that the concepts are

more contextual, understandable, and interesting.

Soon after the topics have been selected and the teaching plans have been designed, the

next step is the implementation of the teaching, observed by principals, teachers, lecturers,

supervisors, and other practitioners. After implementing the teaching process, a forum of Post-

Class Discussion for discussing the process of teaching, its strength and weaknesses, the depth

of mathematics concepts, misconceptions (if any), and the pedagogy or learning theory related

to the teaching. All recommendation and information from other colleagues will enrich the

knowledge of the model teachers.

From some discussions with some teacher models, concerning the lesson plan they created,

some observations on their performance during the classroom activity, and some reflections

(post-class discussion), there are some indications that this kind of teacher training results in

some new ideas and recommendation for teachers which can enrich teacher’s horizon,

competence, and performance. Through discussion phase in reflection session, there were many

inputs given by the observers to the teachers (who acted as a model in teaching and learning

process). This can influence teachers’ professional development since by being involved in

lesson study they obtain some comments, recommendations, deep explanation of mathematics

concepts, and the most recent models of teaching which are introduced by experts on relevant

area. It is clear that teachers can develop their competency through lesson study in their own

fields.



Exercises:

A. 1. What kind of ability students should have when they come to the end of their lesson?

2. How do you explain that working with partners is a strategy that encourage students to

work together effectively. What kind of methods can be applied for organizing students to

work together in a classroom?

3. In group investigation, a teacher should lead students in exploring a number of

perspectives on problems. Mention 2 problems for explorations.

B. 1. Explain 2 information-processing models and give an illustration of their implementation

in teaching mathematics topics.

C. 1. How do you explain that some concepts require connection between the exemplars and

some other entity? Give examples of the connection between mathematics topics and

other fields or real world.

2. What do students need when they attain a concept? Give illustrations and elaborations.

D. 1. In synectics procedures a teacher develops creativity groups, who work together to

function as problem solver or product developers, by asking students to use analogies and

metaphoric comparisons. Give 2 examples of topics on mathematics and explain how you

teach these topics using synectics procedures.

E. 1. How do you develop students’ metacognitive awareness in mathematics?

2. In developing students’ problem-solving abilities, students should be given particular

experience. Do you agree if the students are enriched by various worded problems? Why?

3. To develop students’ metacognitive awareness, their metacognitive experience has to be

enriched. How do you do this effort when you want your students possess this

competence through geometry concepts?

F. 1. To react to a computer stimulus, a student may pose a number of responses. Mention some

of these possible answers.

2. Explain some responses given by a computer, when a computer gets stimulus or resposes

from students.

3. Do you agree that in IT-based teaching, a teacher need not to work harder than in

traditional teaching? By the assistance from the computer, what activity the teacher can

develop?



4. In what situation, computer programming is a part of model implemented in your teaching

and learning activities?

5. Give a number of concepts of mathematics, and explain how these concepts are delivered

by using a computer.

6. How do you explain matrices by using IT-based teaching with the types of interaction of

tutorial or simulation?

G. 1. To develop teachers’ competences toward professionalism, some various efforts can be

taken. What kind of efforts you should take based on the situation and conditions of your

school?

2. Explain why attending workshops, seminars, or conferences can enhance and broaden

your horizon and strengthen your skills in teaching.

3. Can you explain what is meant by “Professional competency is not transferable from other

people”.

4. To be more professional, a teacher should focus on what he/she needs to know about

mathematics, mathematics education, and pedagogy for mathematics. What should

he/she master when he/she will carry out the vision of teaching?

5. Lesson Study is an innovation of mathematics teaching and learning process and a real

collaboration between teachers under collegiality principle. Explain the main three

principles of Lesson Study. Explain why Lesson study is a good improvement procedure

for strengthening teachers’ competence. Can Lesson Study be applied into all subjects in

all different levels of education?



References

........., 2009, Nature of the Students, Going to a public school, New South Wales, Department of

Education and Training. Retrieved 2009 <http://www.schools. nsw.edu.au/gotoschool/

highschool/transitions/natureofstud/index.php>

Baba, T., dan Kajima, M. (2003). Lesson Study. Tokyo: Japan International Cooperation Agency

(JICA).

Ball, D.L. (1990). Halves, Pieces and Twoths: Constructing and Using Representational Contexts in

Teaching Fractions. East Lansing, Mich.: National Center for Research on Teacher Education.

Bell, F.H. (1978). Teaching and Learning Mathematics (In Secondary Schools). New York: Wm. C.

Brown Company Publisher.

Banchoff, T. (1986). Computer Graphics Applications in Geometry: Because the Light is Better over

Here. In The Merging of Disciplines: New Directions in Pure, Applied, and Computational

Mathematics, edited by R.E. Ewing, K.I. Gross, and C.F. Martin. New York: Springer Verlag.

Bloom, L.M., Comber, G.A., and Cross, J.M., (1987). Graphing through Transformation via the

Microcomputer. Bandung: PPPG.

Brown, T., 1997, Mathematics Education and Language: Interpreting Hermeneutics and Post-

Structuralism. Dordrecht: Kluwer Academic Publisher.

Castle, K. dan Aichele, D.B. (1986). Professional Development and Teacher Autonomy in

Professional Development for Teachers of Mathematics (1994 Yearbook). Reston, Virginia:

National Council of Teachers of Mathematics, Inc.

Cockroft, H. W., 1982, Mathematics counts: Report of the Committee of Inquiry into the Teaching of

Mathematics in School under the Chairmanship of Dr.W.H. Cockroft. London: Her Majesty's

Stationery Office.

Ebbutt, S. and Straker, A., 1995, Mathematics in Primary Schools Part I: Children and

Mathematics. Collins Educational Publisher Ltd.: London

Ernest, P., 1994, Mathematics, Education and Philosophy: An International Perspective. The

Falmer Press: London.

Ernest, P., 2002, What Is The Philosophy Of Mathematics Education? Paul Ernest University of

Exeter, United Kingdom. Retrieved <http://www.icme-organisers.dk/dg04/contributions/

ernest.pdf>



Ernest, P., 2007, Mathematics Education Ideologies And Globalization. Retrieved<http://people. exeter.ac.uk>

Fernandez, C., and Yoshida, M. (2004). Lesson Study, A Japanese Approach to Improving

Mathematics Teaching and Learning. London: Lawrence Erlbaum Associates, Publishers.

Fey, J. dan Heid, M.K. (1984). Imperatives and Possibilities for New Curricula in Secondary

School Mathematics in Computers in Mathematics Education (Year Book). Hansen, V.P. and

Zweng, M.J. (Editors). Reston, Virginia: National Council of Teachers of Mathematics, Inc.

Fullan, M., 2002, Leading and Learning for the 21st Century, Vol 1 No. 3 - January 2002.

Furlong, J., 2002, Ideology and Reform in Teacher Education in England:

Fletcher, T.J. (1988). Microcomputers and Mathematics in Schools, United Kingdom: Department

of Education and Science.

Glenn, A., 2009, Philosophy of Teaching and Learning "Your job as a teacher is to make every

single student feel like a winner”. Retrieved <http://depts.washington.edu

/ctltstaf/example_portfolios/munchak/pages/87361.html>

Grouws, A. D. and Cooney, J. T., 1988, Effective Mathematics Teaching: Volume 1, Virginia : The

NCTM, Inc.

Glass, E.M. (1984). Computers: Challenge and Opportunity in Computers in Mathematics

Education (Year Book). Hansen, V.P. and Zweng, M.J. (Editors). Reston, Virginia: National

Council of Teachers of Mathematics, Inc.

Hartman, 1945, Validity Studies Of The Hartman Profile Model, Ai, Axiometrics International, Inc.

Retrieved 2007< http://www.google.com>

Heinich, R. (1996). Instructional Media and Technology for Learning. New Jersey: Prentice Hall,

Inc.

IFIC dan JICA (2004). The history of Japan’s Educational Development. Boston: Allyn and Bacon.

Jaworski, B., 1994, Investigating Mathematics Teaching: A Constructivist Enquiry. London: The

Falmer Press.

Jones, A.G., Lubinski, C.A., Swafford, J.O., dan Thornton, C.A., A Framework for the Professional

Development of K-12 Mathematics Teachers in Professional Development for Teachers of

Mathematics. Reston, Virginia: National Council of Teachers of Mathematics, Inc.



Jonhnson, D.A., and Rising, G.R. Guidelines for Teaching Mathematics (2nd Edition). Belmont:

Wadsworth Publishing Company, Inc.

Joyce, B. and Weil, M. (1996). Models of Teaching (5th edition) Boston: Allyn and Bacon.

Kaput, J. J. (1992). Technology and Mathematics Education in Grouws, D. A. (Ed.), Handbook of

Mathematics Teaching and Learning. Reston, Virginia: National Council of Teachers of

Mathematics: Hillsdale , H. J.: Lawrence Erlbaum.

Kulik, J.A., Kulik, C. C., dan Bangert-Drowns, R. L. (1985). Effectiveness of Computer-Based

Education in Elementary Schools [Online]. Tersedia: http://www.nwrel.org/

scpd/sirs/5/cu10.html [29 January 2004].

Kumano, Y. (2004). Lesson Study. Makalah Pelatihan Dosen LPTK dalam Pengembangan

Kemitraan LPTK dengan Sekolah dalam Rangka Peningkatan Mutu Pembelajaran MIPA.

Bandung: Universitas Pendidikan Indonesia.

Mayer, R.E. (1999). The Promise of Educational Psychology, Vol. II: Teaching for Meaningful

Learning. New Jersey: Pearson Education, Inc.

NCTM. (1991). Professional Standards for Teaching Mathematics. Reston, Virginia: The National

Council of Teachers of Mathematics, Inc.

Orton, A. and Wain, G., (1994). Issues in Teaching Mathematics. London: Cassell.

Runes, D.D., (1942). Dictionary of Philosophy. Retrieved 2007 <http://www.ditext.

com/index.html>

Saito, E. (2004). Indonesian Lesson Study in Practice: Case Study in IMSTEP. Paper presented to

Workshop for Mathematics and Science (MGMP Mathematics Teachers). Bandung:

Indonesian University of Education.

Swanson, R.A. and Holton III, E.F. (2009). Foundation of Human Resources Development: Second

Edition, Berrett-Kohler Publisher Inc.

Stigler, J. dan Hiebert, J. (1999). The Teaching Gap: Best ideas from the world’s teachers for

improving education in the classroom. New York: Free Press.

Wilson, B. (1988).Making Sense of the Future. A Position paper on the Role of Technology in

Science, Mathematics and Computing Education. [Online]. Tersedia: http://www.

hometown.aol.com [29 January, 2004]

philosophical theoretical ground mathematics teaching

Documents