video 1(factoring polynomials)

The Video
Video 1 : Factoring Polynomials
To find factors of polynomials is to formed the algebraic long division. For example x-3 is a factor of x3 -7 x - 6?
When dividing x - 3 into x2 - 7 x - 6. First step the problem make a long division problem. There is you dividing x – 3 into x3+ 0 x2- 7x -6. Zero because there is no second degree term. Now, what times x give you x2? Of, course x2, so you multiply x-3 by x2, which give you x3minus 3 x square to get 3 x square. Bring it down next term negative seven x. Dividing x minus 3 into 3 x square minus seven x. Just looking at the first time 3 x square dividing x is 3x. Multiply x minus 3 by 3x. We can get 3 x square minus nine x. Subtracting you have 2x minus 6. Dividing 2 x minus 6 by x-3 which equals 2 and without a remainder. So the solution for a long division problem is x square plus 3 x plus 2. Since x minus 3 divide in x cube minus seven x minus 6. We now know x cube minus 7 x minus 6 equals ( x minus 3) times ( x square plus 3 x plus 2 ). The quadratic expression x square plus 3 plus 2 can be factored into ( x plus 1) times ( x plus 2). So, x cube minus 7 x minus 6 equals ( x minus 3 ) times ( x plus 1) times ( x plus 2 ). Substitution x cube minus 7x minus 6 to zero we get 0=(x-3)(x+1)(x+2) Thus either x-3=0 or x+1=0 or x +2=0 Solving of x we get x=3, x=-1, x=-2. The roots of x cube minus 7 x minus 6 are 3, -1, -2.
Conclusion:
*) 3 roots for this 3rd degree equation
*) 2nd degree equation always have at most 2 roots
*) 4th degree equation would have 4 or fewer roots, and so on.
*) The degree of polynomials equation always limits the number of roots.
Long division process for 3rd order Polynomial:
1. Find a partial quotient of x square, by dividing x into x cube to get x square
2. Multiply x square by the divisor and subtract the product from the dividend.
3. Repeat the process until you either “ clear it out “ or reach a remainder.

Video 2 : Solve The Problems
The next problem is question 13. The figure shows of graph of y equals g of x. If the function h is defined by h(x) = g(2x)+2, what is the value of h(1)? And we are looking for h of 1.
The information of the graph h(x)=g(2x)+2 now we are looking for h(1). We substitute h(1) into this equation. H(1)=g(2)+2 now we get g(2), see when x equals 2, y equals 1. So g(2) is 1. H(1)=g(2)+2, h(1)=1+2, so h(1)=3.

Next is the function with no function problem. This question 13 page 534. Let the function f be defined by F(x)=x+1, if 2f(p)=20, what is the value of f(3p)? We are looking f(3p) what is f when x = 3p?
First information is F(x)=x+1, 2f(p)=20 The figure of 3p start with this equation down here:
2f(p)=20, the, if we are divide by 2 so : f(p)=10. Then f(p) is just what is function f(x) of f(p) f(p)=p+1=10, then p=9. Is this right answer? No, is not. We looking for x=3p, x=27. We have an equation f(x)=x+1, f(27)=27+8=28. The last answer is 28.

Question 17.
In the x y – coordinate plane, the graph of x equals y square minus 4 intersect line l at ( 0,p ) and ( 5,t ). What is the greatest possible value of the slope of l? we’ll be looking for greatest m. The graph intersect in x = 4 line l intersect at ( 0,p ) and ( 5,t ). X is zero, y is p and when x = 5, y=t what is the possible slope for line l. What we are doing now for get the slope? m
equals (y-y1) over (x2-x1) The slope is going to be m=(t-p) over 5. Numerator is t-p we have x=y square minus 4. We can play again the point of intersect ( 0,p ) and ( 5,t ) to the equation x= x square minus 4.

Video 3 : Pre Calculus
Graph of a rational function which can have discontinuities because has polynomial in the denominator.
Is possible value x divide by 0
Example : f(x) equals (x plus 2) over (x minus 1) F(1)= The value become 1+2 over 1-1 equals 3 over zero. That is bad idea.
Graph f(1)=1 plus 2 over zero : Break in finction graph.
F(x) = ( x+ 2) over ( x-1) : insert 0
F(0)=0+2 over 0-1 equals -2. Insert 1 F(1)=1+2 over 1-1 equals 3 over 0 is impossible.
Rational functions don’t always work in this way! Take graph f(x) = 1 over ( x square plus 1 ). Not all rational functions will give zero in denominator because of the +1 ( never zero ).
Rational functions denominator can be zero.
Polynomial have smooth and unbroken curve and for rational function x : zero in the denominator. That impossible situation. A break can show up in two ways. A simply type break is missing point on the graph. Y = ( x square minus x minus 6 ) over ( x minus 3). The graph loose like this if x = 3 ( 3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. That is not possible, not feasible, and not allowed. So that is no way if x = 3. This is a typical example to the missing point syndrome. Y = ((3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. When you see result of 0 over 0 and also tell you direction by possible factor top and bottom of rational function and simplify. For example. Y = ( x square minus x minus 6 ) over ( x minus 3), Equals ( x minus 3 ) times ( x plus 2) over ( x minus 3) so, y = x+2.

task o(zero)

Tsukuba Journal of Educational Study in Mathematics. Vol.25, 2006

22 1

PURSUING GOOD PRACTICE OF

SECONDARY MATHEMATICS EDUCATION THROUGH

LESSON STUDIES IN INDONESIA

Marsigit

Department of Mathematics Education, Faculty of Mathematics and Science,

the State University of Yogyakarta, Indonesia

Starting in 1999 and lasting in 2005, the extending of IMSTEP_JICA Project

resulting the piloting activities through Lesson Studies for searching good practice of

secondary mathematics teaching in three cluster site West Java, Central Java and

East Java. Results of the studies significantly indicated that there are improvements

of the practice of secondary mathematics teaching learning processes in term of

teaching methodology, teacher competencies, students achievements, alternative

evaluation, teaching learning resources and syllabus. The results of the project

support the government efforts in improving secondary education by introducing the

new curriculum. However, there are still great challenges for both educationists and

practitioners to establish good practice of secondary mathematics teaching. One of

the proposed solutions is to en-culture lesson studies activities by e.g. learning them

from Japan contexts.

Key Word: good practice, lesson study, secondary mathematics teaching learning

OVERVIEW

Some philosophical backgrounds of the nature of good practice of teaching need to be

discussed as the references of the efforts to put on the grounds for the long-term

orientation of improving mathematics education. Those notions cover the questions to

mean education whether: (a) as the investment or as the need for society?, (b) as the

obligation or as the awareness of the students?, (c) as the competition or as the

collaboration?, (d) as the product or as the process, etc. Although the examples of

good practice differ in some contexts, there are some features common to the way the

children were working such as in/formality of typifies classroom, clearness of the

objectives, ethos of the school, flexibility and variety of teaching styles. The quality

of teaching was the strongest feature common to all the examples of good practice;

while the adequacy of good practice may outlined on the basis of the context of the

ideal practices.

Brown in Riley (1992) suggested that the features of good practice include fostering a

positive attitude to mathematics; emphasis on the application of mathematics; wellplanned

work; children formulating testing and revising hypotheses; work and pattern

Marsigit

22 2

and relationship; a variety of approaches to calculation used; sensible use of the

calculator; extensive experience of measurement and estimation; clear policies on

mathematics; individual, group and whole class work as appropriate; opportunities for

co-operative work; positive and well timed teacher intervention; meeting needs

through the differentiation of work; use of practical and first-hand experience; crosscurricular

work; appropriate reflection of cultural diversity; relevant exploratory work;

stimulating working environment; effective teaching.

The Third International Mathematics and Science Study (TIMSS), from the

videotaping study of classroom instruction of mathematics lessons in the U.S.,

Germany, and Japan, found that good practice of mathematics teachings were more

likely to target mathematical thinking and seeking to teach students how to solve a

particular kind of problem or carry out a specific procedure. Further, it suggested that

the concepts of mathematics were far more likely to have been developed rather than

simply presented as rules. It included multiple ways to solve mathematical problems

and asked students to perform tasks that were not “routine.” TIMSS suggested that

good practice of mathematics teaching should encourage the teachers to help students

make explicit connections between parts of the lesson to previous knowledge, and/or

to statements and problems from earlier parts of the lesson. Good teaching practice

should be supported by the research on how children learn and, particularly, on how

they learn mathematics.

While in Japanese context, on which Lesson Study has its culturally deep rooted,

good practice of mathematics teaching (Masami ISODA) may perceived as it is

visible, recordable in the classroom and can be showed to other people. Further, it

may be known as a good approach in an economy in which there is a teacher who is

well known by its approach. Accordingly, good practice of mathematics teaching

should be useful for the reform of mathematics education as a whole and many

teachers may have their wish to do the same approach. Good practice of mathematics

teaching also should encourage the improvement of teacher training program.

According to him, the challenges for the practitioners who develop good practice

consist of capacity to describe the nature of good practice? Why we can say it as

good practices? What kind of reform is expected by such kinds of practices? What

kind of the setting in curriculum standard for explaining why it is good. And what

kind of relation of good practice to the world of mathematics education research

movement?

One significantly reference for Lesson Study performed by Education Development

Center, Inc. (“EDC”) of Massachusetts. It organized divisions and centers of various

sizes of Lesson Studies and develops Lesson Study Communities Project to carry

research to introduce teachers to lesson study, to build a community of teachers

interested in lesson study, to enhance teacher knowledge of mathematics and

pedagogy, and to learn how the Japanese lesson study model can be adapted to

become a successful professional development model for U.S. secondary school

mathematics teachers. Indonesian educationist as well as practitioners may be able to

Pursuing Good Practice of Secondary Mathematics Education Through Lesson Studies in Indonesia

22 3

learn from Japan and other country to perform good practice of mathematics teaching

in secondary schools.

OBSERVABLE GOOD PRACTICE AND IDEAS VALUES BELIEFS OF

MATHEMATICS TEACHING LEARNING IN INDONESIA

The efforts of pursuing good practice of mathematics teaching learning in Indonesia,

starting from 1994 up to present, have its ideal values beliefs, political as well as

empirical and pragmatic ground. In the conceptual framework, it can be seen that

observable good practice components can be directly related with educational questions

'What should children learn ?' and 'How should children learn and teachers teach ?'; and

ideas values beliefs can be directly related with the questions 'Why should children be

educated in this way ?' and 'What is an educated person ?'. As they are elaborated from

socio-constructivists approach, that teacher is not only to implement the curriculum

but also to develop it. Teaching and learning in the classroom should not always be

directed teaching in which teacher dominated activities and initiations; however,

teacher needs to accommodate students’ initiatives and students’ needs. Therefore,

teacher needs to implement flexible method of teaching, in which students’

performance and achievements can be assessed during the processes of learning

activities. It implied that teaching learning process will be student centered rather

than teacher centered, in such away that students have various experiences and

opportunities to consciously uncover the nature of what they are learning.

Mixing from values beliefs and empirical evidences, there are currently demands in

Indonesia, that any educational reform should handle the issues of (a) how to promote

interactive curriculum rather than instrumental curriculum?, (b) how to promote

student centered approach rather than teacher centered approach?, (c) how to promote

students’ initiation rather than teacher’s domination?, and (d) how to promote simple

and flexible curriculum rather than crowded and tight-structured curriculum? While

in term of observable good practice, there were demands that teachers need to have a

chance to reflect their teaching in such away that they may move from older

paradigm of teaching to the new one. Teachers may move from emphasizing the

“teaching” to emphasizing the “learning”; they may move from the act of

“transferring teacher’s knowledge” to “constructing students’ knowledge”.

Marsigit

22 4

Figure: Most Indonesian teachers implement Traditional Teaching (2001)

There are also demands that teachers may move from “instructing” to “serving”; they

may move from “product oriented” to “processes oriented”; they may move from

“classical teaching” to “individual teaching”; they may move from “single method of

teaching” to “various method of teaching”; and they may move from “theoretical” to

“hands on activities”. To develop the aspects of students’ life-skill teachers need to

have a lot of time, passionate, and extra-efforts in such away that teachers are able to

motivate their students, to give them the chance to develop their skills and

experiences. In the socio-constructivist approach, teachers are expected to fully

concentrated and focused at their teachings and its preparations. It made them to be

able to promote their creativities in developing their various method of teaching such

as discussion, investigation, laboratory practice and demonstration.

The currently studies on mathematics teaching practices in Indonesia indicated that,

under the implementation of the Curriculum 1994, student’s process skill and

student’s achievement are still low; contents on Mathematics were crowded; too

many time consuming administration stipulation for teachers; there were mismatch

among the objectives education, curriculum, and National Evaluation system. Further,

National Leaving Examination assessed the children’s ability cognitively only and

considered individual differences inappropriately; while University Entrance

Examination system was considered to trigger school teachers apply much on goal

oriented rather than process oriented. Observable practices of mathematics teaching

in the period of the year 2001-2003 also indicated that many teachers still have

difficulty in elaborating the syllabus; a number of mathematics topics are considered

to be difficult for teachers to teach; a significant number of children consider some

mathematics topics as difficult to understand; teachers consider that they still need

guidelines for conducting teaching process by using science process skills approach.

PURSUING GOOD PRACTICE OF SECONDARY MATHEMATICS

TEACHING THROUGH MEDIUM-SCALE OF LESSON STUDIES

In the fiscal year 2001-2003, a medium scale of piloting of Teaching Learning Model

of secondary mathematics and sciences through Lesson Study has been carried out by

Pursuing Good Practice of Secondary Mathematics Education Through Lesson Studies in Indonesia

22 5

IMSTEP-JICA in collaboration with UPI Bandung, UNY Yogyakarta, and UM

Malang, in which Japan Government supported facilities, training as well as

Educational Experts.

.

Method

The piloting activities were carried out in three clusters i.e. West Java (Bandung),

Central Java (Yogyakarta), and East Java (Malang). Following are the sites:

West Java

(Bandung)

Central Java

(Yogyakarta

East Java

(Malang)

Year

2001/2002

Year

2002/2003

Year

2001/2002

Year

2002/2003

Year

2001/2002

Year

2002/2003

Classes

(schools)

1 2 1 3 1 1

Teacher

involved

1 2 3 3 1 1

Lecture

involved

4 4 4 3 4 4

Table: Three cluster sites of Lesson Studies of Mathematics Teaching (IMSTEPJICA

Project)

The Lesson Studies were developed in which the teachers, in collaboration with

Lecturers and Japanese Experts, tried out some teaching models at schools. The

Lecturers of Teacher Training Program and School Teachers worked collaboratively,

composes some numbers of Lesson Studies. The grounds of the Lesson Study

activities were reflecting and promoting the new paradigm of the secondary

mathematics and science education, in which learning activities are not only

perceived pragmatically and short-time oriented but also to be perceived as a longlife

time purposes.

The objectives of those Lesson Study activities were to contribute the improvement

of secondary mathematics education by pursuing good practice of mathematics

teaching. Lesson Studies for secondary mathematics were carried out by mainly

Classroom Action Research approach. They carried out to improve the teaching

learning practices and to find more appropriate methods for facilitating students

learning. Teachers’ experiences have been shared with other teachers and the lectures.

The specific objectives of Lesson Study activities are: (1) to develop instrument and

equipment for teaching learning process, (2) to develop teaching method and model

for teaching learning process, (3) to develop teaching material for teaching learning

process, and (4) to develop teaching evaluation for teaching learning process.

Lesson Study activities let the teachers to reflect and evaluate, in cooperation with

lectures or other teachers, their paradigm of teaching. Approaches of Lesson Studies

covered (a) students cooperation with others in their learning, (b) contextual teaching

and learning, (c) life-skill, (d) hands-on activities, (e) interactive process oriented

Marsigit

22 6

curriculum and syllabi development, and (f) teachers and students autonomous. From

those three sites of study, there can be produced the notions of educational

improvement, in term of teacher, student and lecture.

Result

The results of Lesson Study could be inferred from the view of students, teachers,

and of lecturers. Evidences were collected through observations, questionnaires and

interviews. It can also be noted the strengths and weaknesses of the activities. Those

are as the following:

Teacher

1. The teachers felt that the class is getting more alive. However, all the above

progresses have to be paid by spending longer time for preparation.

2. The teachers have to spend longer time for preparation, have sufficient skill to run

the experiments and use various tools.

3. Lesson Study gave positive results because it could improve teachers’

professionalism in finding variations of teaching approaches, and teaching methods.

4. It could also improve teachers’ skills in classroom management and in questioning

and in developing creative ideas.

5. Through Lesson Study, many teachers were introduced some innovations in

mathematics and science teaching and learning.

6. The new model was introduced to teachers to increase the variation of alternatives on

how to conduct classroom teaching and learning process.

7. Teachers stated that now, they have more choice to teach certain units of studies.

8. Teachers involved in these Lesson Study activities developed their competencies in

teaching mathematics and science.

9. Competencies developed for teachers are realistic approaches (RMA), authentic

assessment, and constructivist approach.

10. Teachers involved in these Lesson Study activities felt that they have to think and

develop new ways on how to let students learn and construct their own concepts.

11. They expressed their impressions that their creativity was improved.

12. There were indication that teachers’ skill to communicate, to deliver questions, to

carryout discussion method was improved e.g. in order to stimulate students to think,

teachers asked questions and by doing these, questioning skills were improved.

13. There were also evidences that teachers’ perceptions of their students’ learning in

Lesson Study activities were positive.

14. Teachers stated that Lesson Study activities needed to be continued or to be extended

in order that they could continue to develop mathematics and or science teaching.

15. Teachers stated that Lesson Study activities were useful to support the

implementation of competence-based curriculum.

Student

1. Good responds were heard during the meeting, especially the students’ enthusiasm,

students’ involvement in doing experiment and discussion.

Pursuing Good Practice of Secondary Mathematics Education Through Lesson Studies in Indonesia

22 7

2. The students respond to the contextual teaching and learning approach during Lesson

Study period may be indicated by their participation in experimentation, discussion,

and presentation.

3. More than 80% of the number of respondents were actively involved in the

experimentation, approximately 75% involved in discussion, and more than 75%

involved in presentation.

4. It can be concluded that nearly 3 out of 4 students were actively involved in the

contextual teaching and learning process.

5. Nearly 70% students were happy with the teaching learning process using a lesson

plan.

6. More than 57% of them were happy with the lesson plan and the involvement of

teacher during teaching and learning process is still high, more than 95%.

7. Students could develop responsibilities in their learning and be more active in finding

learning resources out of the classroom.

8. Students get more skillful in sharing ideas and communicating their mathematics

activities

9. Innovations practice tried out at Lesson Study schools could also improve students’

enthusiasm, motivation, activities, and performance.

10. The innovation in approaches and media of learning bring good results for students.

Most students in each class were enthusiastic in learning using the new media,

methods, or approaches.

11. It was indicated that most of students expressed their happiness during the Lesson

Study activities; the reason were: (a) the lesson was not so formal, (b) the contents

were easier to learn, (c) they were able to express their ideas, (d) they got much time

to discuss with their classmates.

12. The reasons of improving students’ motivation were: (a) they were able to detect

their competencies any time, (b) contextual teachings made them understand the

usage of learning certain subject-matter, (c) realistic teaching made them not be

boring.

13. There were more activities done by students in science laboratories, especially

activities to improve their process skills.

14. Most of the students stated that they liked to learn with hands-on activities,

discussion, demonstrations, teaching aids and worksheets.

Lecturer

1. Lecturers got experience in developing teaching materials for schools and know more

about problems faced by teachers.

2. They could also develop more creative ideas to find better methods of instruction

with teachers and better hands-on activities by utilizing local materials.

3. Through Lesson Study activities, lecturers were also benefited in knowing more

about the problems faced by teachers and schools in conducting mathematics and

science teaching and learning.

4. Lecturers’ experiences were improved due to the fact that in Lesson Study activities,

they in cooperation with teachers, should develop teaching guide, teaching materials

and assessment methods.

Marsigit

22 8

Constraint

1. Those are the tendency of teachers to keep on doing “teacher centered activities”

instead of trying new ways of teaching which need thought, energy and time to

develop, to plan, and to implement.

2. These resistances for change are caused by the “crowded curriculum” that have to be

finished and the fact that there were too many students in a classroom.

3. Lecturers and teachers should also need time to introduce and learn new innovations.

They should be very patient in developing new things; they could only develop one

thing at a time.

4. Based on their experience in conducting processes, there were different perceptions

of those lecturers and teachers on new paradigm.

5. Some teachers tend to skill use the “old” orientation in teaching and learning

processes, which is to try to achieve the highest possible score in final examination.

6. In order to do that they usually practice “teacher centered” and “product oriented”

teaching and learning processes and forget to develop students’ thinking processes

skills.

7. Based on the implementation of Lesson Study activities, it was found that some

aspects of Lesson Study activities have difficulties to implement in term of teachers,

students, class management, equipment or facilities, and schooling system.

8. Number of students in each Lesson Study class was relatively high i.e. 35 or more;

this made the teachers have difficulties to facilitate their needs in learning activities.

With a large number of students in the class, it was difficult for the teachers to

monitor and supervise their activities.

9. Due to the fact that we do not implement moving class, it needs more time for the

teachers to prepare equipments and other facilities.

10. Another difficulties faced by the teachers related to the development of students’

worksheets was that teachers have some problems on how to prepare many kinds of

worksheets, to decide the topics to be piloted and to prepare teaching guides.

Figure: Group discussion in piloted teaching practice (2003)

DISCUSSION AND CONCLUSION

There were strong evidences that Lesson Studies activities improved students’

enthusiasm, motivation, activities, and performance. It also improved teachers’

professionalism in terms of teaching performance, variation of teaching

methods/approaches, collaboration. Lecturers got knowing more about the problems

Pursuing Good Practice of Secondary Mathematics Education Through Lesson Studies in Indonesia

22 9

faced by teachers. It was take time for teachers to shift from teacher-centered to

student-centered. Teachers developed teaching methods based upon more hands-on

activities and daily life utilizing local materials. Students were active learning and

involved in discussion to share ideas among classmates. Students enjoyed learning

science and math during Lesson Study activity due to some reasons. According to

students’ respond, the lesson was not so formal, the contents were easier to learn,

students able to express their ideas, students got much time for discussion with their

classmates, more experiment science and math. Teachers got alternative method to let

students learn and construct their own concepts. However, teachers took time to get

used to develop teaching model by their own.

The Lesson Study project was proven to be very effective in lifting students’

enthuciastic in learning science, helping students to develop their experimental and

discussion skill, giving opportunities to students in developing their own scientific

concept by themselves. It was also reported that by using constructivism approach,

the students may find out their best style of learning. Competition rises among groups

of students in presenting the results of their work and defending their presentations.

This forces students to learn theory more on their own. As a result of Lesson Study

activities there were many teaching material developed either by lecturers and

teaching together or by lecturers or teachers themselves. Those materials were either

developed by lecturers or teachers in their own classroom or by lecturers and teachers

together during Lesson Study activities. In general lecturers and/or teachers

developed the teaching materials after thinking extensively what and how to develop

teaching materials for a certain topic, and then develop the materials. Further they try

out the teaching materials in their classroom and revise those according to the result

of the try out.

The results of Lesson Study activities and exchange experiences come to a suggestion

that to improve mathematics and science teaching in Indonesia; it needs to deliver

obvious messages to the government, teachers and head-teachers or schools.

Learning from study, it was also suggested that to promote good practice of

mathematics and sciences teaching, the teachers need to en-culture their efforts in

inovating teaching learning processes which meet to academic students needs,

encouraging students to be active learners, developing various strategic of teaching,

developing various teaching materials, and in developing teaching evaluation. In

developing teaching learning methods, the teachers need to: plan the scenario of

teaching, plan students activities, plan teachers’ roles, distribute the assignments,

develop assesment methods, and monitor the progress of students achievements.

To develop their experiences, the teachers also need to participate frequently in such

kinds of workshops or seminars. By using those teaching materials teachers could

conduct the teaching and learning process more efficiently. Students enjoyed their

learning process because they were involved in observing and doing things. Those

teaching materials also improve students’ motivation and interest in learning the

materials. Although there were may kinds of teaching materials that have been

Marsigit

23 0

developed through those Lesson Study activities, there still more topics that need to

have or to have better teaching materials. Therefore lecturers from three universities

need to have further collaborative work to develop more teaching materials in the

future.

The study also recommended that to encourage educational innovations, the headteachers

need: (1) to make good atmosphere for teaching and learning, (2) to promote

to implement various teaching methods and teaching learning resources, (3) to give

the chances for the teachers and their students to perform their initiatives, (4) to

promote cooperative learning, (5) to promote research class as a model for

educational innovations (as Japanese teachers do), (6) to support the teachers to be

the developer/maker of the curriculum, (7) to promote teachers’ autonomy in

developing model of teaching learning activities, (8) to implement school-based

management, (9) to encourage students’ parents participations, and (10) to promote

cooperation with other educational institutions.

Further, the study also recommended that to improve the quality of mathematics and

sciences education, the central government needs to: (1) implement more suitable

curriculum i.e. more simple and flexible one, (2) redefine the role of the teachers i.e.

teachers should facilitate students' need to learn, (3) redefine of the role of principals;

principals should support the professional development of teachers by allowing them

to attend and participate in scientific, meetings and trainings, (4) redefine the role of

schools; schools should promote school-based management, (5) redefine the role of

supervisor; the supervisors need to have similar background with the teachers they

supervise in order to be able to do academic supervision, (6) improve teachers’

autonomy to innovate mathematics and science teaching and learning, and (7)

promote better collaboration between school and university; communication among

lecturers and teachers should be improved; these could be done through collaborative

action researches and exchange experiences through seminars and workshops, (8)

redefine evaluation system, and (9) to extend project for promoting new paradigms

and educational innovations.

REFERENCE:

Bourne J. (1994). Thinking Through Primary Practice. Routledge: London

Cohen, L. and Manion, L. (1994). Research Methods in Education : Fourth Edition.

London : Routledge.

Ebbutt, S. and Straker, A. (1995). Children and Mathematics', Mathematics in Primary School,

Part I. London: Collins Educational.

Ernest, P. (1989). Mathematics Teaching : The State of the Art. London : The Falmer Press.

Gorman, J. (1994-2005). Lesson Study Communities Project. Education Development

Center, Inc., Center for Mathematics Education, 55 Chapel Street, Newton, MA 02458.

Grouws, A. D. and Cooney, J. T. (1988) Effective Mathematics Teaching : Volume 1,

Virginia : The NCTM, Inc.

Herawati Susilo. (2003). FINAL REPORT: Improvement of Secondary School Education.

IMSTEP-JICA Project

Pursuing Good Practice of Secondary Mathematics Education Through Lesson Studies in Indonesia

23 1

Isoda, M. (2005). Information for Workshop in APEC specialist session from January 17:

APEC-Tsukuba meeting focus on Innovation of mathematics education through the

lesson study. Retrieved 

Marsigit. (2003). The Implementation Of Project Ctivities October To 20. 1998 -

September 2003, IMSTEP-JICA Project

TIMSS. (2005) Lessons from the World : What TIMSS Tells Us About Mathematics

Achievement, Curriculum, and Instruction. National Center for Education Statistics

(NCES), Retrieved 2005

SENTENCES

STIPULATION= Ketentuan, syarat; Certainty (rule, direction) that must we do
example :if we want to enter a college, several STIPULATION is waiting us.

ELABORATED= Merinci atau menekuni; an activity(learn,etc) with diligent
example :the student of mathematics should ELABORATED math.


JURING= Section, Segment of the cyrcle
example :in a circle, we can make a SEGMENT, with just one line.

BERSISIAN= Adjacent; from side to side.
example: we can make several adjacent in two form.

Representing video of learning mathematics (video 3)

Pre-Calculus
• Graps of a rational function
Can have discontinuities
Has a polynomial in the denominator
It’s possible that some value of x we’ll need to division by zero, if so that value is straight up OFF LIMITS
Example: f(x)=(x+2)/(x-1), and take x=1, then we get f(x)=3/0, with zero in the denominator(that’s no good), so choosing x=1 is a bad idea(baaad).
3/0 is break in Function Graph
Not all rational functions will give zero in denominator, for example is f(x)=1/(x2 )+1, the denominator is never zero because of the +1, so the graph is no break, and don’t forget as general rule, when you deal with Rational function, you must expect the possibilities that the denominator is can be zero.
Break can show up in 2 ways:
1. Missing point on the graph,
Example: y=(x2 –x-3)/(x-3), when we take x-3 the result is 0/0, this is Tyical Example of Missing Point Syndrome.
Missing point is a kind of a loophole,
y=(x2 –x-3)/(x-3), when we take x-3 the result is 0/0, but if we simplify first, we got
y=(x2 –x-3)/(x-3)>>>y=(x-3)(x+2)/(x-3)>>>y=x+2.

TASK OF ENGLISH

STIPULATION= Ketentuan, syarat; Certainty (rule, direction) that must we do
example :if we want to enter a college, several STIPULATION is waiting us.

ELABORATED= Merinci atau menekuni; an activity(learn,etc) with diligent
example :the student of mathematics should ELABORATED math.


JURING= Section, Segment of the cyrcle
example :in a circle, we can make a SEGMENT, with just one line.

BERSISIAN= Adjacent; from side to side.
example: we can make several adjacent in two form.

From: Bibid Bagasworo
NIM: 07305141013
class: Math R '07
Yogyakarta Math University

Representing video of learning mathematics(video 2)

We will describe the video that I’ve been looked, the video content some question, these are:
13.the figure shows the graph of y= g(x). if the function h is defined by h(x) = g(2x)+2, what is the value of h(1)?
First piece of information is h(1) then y=g(x) and h(x) = g(2x)+2 then we count h(1)=g(2)+2, refer to the graph g(2)=1 so h(1)=1+2=3 is the answer.



In another question:
13. Let the function f be defined by f(x)=x+1. If 2f(p)=20, what is the value of f(3p)?
Answer: first information is f(x)=x+1
second information is 2f(p)=20
2f(p)=20f(p)=10, because f(p)=p+1=10, so p equals to 9, we looking for x, so x=3px=27,is this the answer? No, we looking for f(3p)f(27)=27+1=28 is the answer

Representing video of learning mathematics (video 1)

F(x,y)=0,
Function=f(x) : VLT
Function x=g(y) : HLT :INVERTIBLE



===>y=2x-1 and y=x ;
We can find the intersection by ===>
x=2x-1 ;
1+x=2x ;
1=x ;
so the intersect is 1
2x-1=y
2x=y+1
x=(1/2)(y+1)
x=(1/2)y+(1/2, excange x become y, we get
y=(1/2)x+(1/2)
example;


f(x)=2x+1
g(x)=(1/2)x+(1/2)
f(g(x))=2[(1/2)x+(1/2)]-1
=x+1-1=x, In the other form we have g(f(x))=(1/2)[2x+1]+( 1/2)===>
x-(1/2)+(1/2)=x
g=f-1
f(g(x))=f(f-1)
=x
g(f(x))= f-1(f(x))=x
let’s do one more example, we take y=(x-1)/(x+2),in vertical asymtot in x=-2, and horizontal asymtot in y= 1, the x intercep is going to (1,0), the y intercept is (0,1/2)
we make the equation simple to be calculate:
y(x+2)=x-1
yx+2x=-1-2y
(y-1)x=-1-2y
X=(-1-2y)/(y-1)
Y=(-1-2x)/(x-1)
When x=0 ,y=-1, then when y=0,x=-1/2, so we get graphic that have vertical asymtot x=1,and the horizontal asymtot y=-2,
So f-1 (x)=(-1-2x)/(x-1)
f(f-1 (x))= ([-1-2x/x-1]-1)/([-1-2x/x-1]+2)
=[-1-2x-(x-1)/(x-1)][(x-1)/(-1-2x+2x-2)]
=(-1-2x-x-1)/( -1-2x+2x-2)
=-3x/-3
=x
Back to the front statement, is
f-1 (x)= (-1-2x)/(x-1) and
f(x)=(x-1)/(x+2), so
f-1 (f(x))= (-1-2{(x-1)/(x+2)})/({ (x-1)/(x+2)})-1
=-x-2-2x+2/x-1-x-2
=-3x/-3
=x
F(g(x))=x,wich the range of g is domain of f
G(f(x))=x,wich is the range of f is domain of g
G= f-1 it’s important thought, to remember, the (-1) it’s mean the reciprocal or it’s just mean invers function, when we doubt line we found out the slope of the invers function with the reciprocal of the other slope,that’s true general.