“Guru Matematika Sebagai Ilmuwan dan Siswa selain sebagai
Guru Profesional”
Prof. Dr. H. Darhim, M.Si
Prof. Dr. H. Darhim, M.Si
Judging from the material structure, mathematics can be dividedinto two study materials, namely pure and applied mathematics.Pure mathematical study of mathematics emphasizes how it is structured and developed. While the studies give priority to applied mathematics in the life keterpakaiannya / field or other disciplines.
Two models of learning and study two mathematical structure of thematerial as described above, to give meaning to the arable fields of mathematics. There are four kinds of mathematics, namely: Whenthe learning done in formal mathematics and pure mathematicsstudies the structure of the material, it will come true academicmathematics. Arable fields of engineering mathematicsmathematics will form when the formal learning models, but its studyis the application of materials in the field or other sciences. If thematerial is pure mathematics study mathematics but giveninformally, then it will manifest recreational mathematics. Whileeveryday mathematics materialize, if mathematics is given in an informal but its study materials relating to the use (in life). The following table shows the study of mathematics in terms of models of learning (formal or informal) and the structure of study materials(pure or applied). Zone division of mathematics as above can be used as guidance indeveloping the field of study of mathematics, especially in order to construct a curriculum department of mathematics (pure andeducation) in colleges. Specifically for education majors to teachmathematics and mathematics teachers will result in candidates, the four kinds of mathematics that must be accommodatedproportionally.
The fourth kind of mathematics as described above, can be studiedmathematics students in the school such as: mathematicalscientists discover / construct mathematics, practitioners usemathematics, and / or studied according talents / interests / abilitiesof students.
Understand mathematics as above, there are at least three kinds ofskills that should have teachers in the teaching of mathematics,namely: as a mathematical scientist, as students learnmathematics, and as a teacher of mathematics.
Two models of learning and study two mathematical structure of thematerial as described above, to give meaning to the arable fields of mathematics. There are four kinds of mathematics, namely: Whenthe learning done in formal mathematics and pure mathematicsstudies the structure of the material, it will come true academicmathematics. Arable fields of engineering mathematicsmathematics will form when the formal learning models, but its studyis the application of materials in the field or other sciences. If thematerial is pure mathematics study mathematics but giveninformally, then it will manifest recreational mathematics. Whileeveryday mathematics materialize, if mathematics is given in an informal but its study materials relating to the use (in life). The following table shows the study of mathematics in terms of models of learning (formal or informal) and the structure of study materials(pure or applied). Zone division of mathematics as above can be used as guidance indeveloping the field of study of mathematics, especially in order to construct a curriculum department of mathematics (pure andeducation) in colleges. Specifically for education majors to teachmathematics and mathematics teachers will result in candidates, the four kinds of mathematics that must be accommodatedproportionally.
The fourth kind of mathematics as described above, can be studiedmathematics students in the school such as: mathematicalscientists discover / construct mathematics, practitioners usemathematics, and / or studied according talents / interests / abilitiesof students.
Understand mathematics as above, there are at least three kinds ofskills that should have teachers in the teaching of mathematics,namely: as a mathematical scientist, as students learnmathematics, and as a teacher of mathematics.
Mathematics
teachers should teach math skills like math when scientists discover /
construct mathematics.
There are two groups of scientists associated with the field of mathematical study. Both groups are: First, users of mathematical scientists. This group may not care about the mathematical concepts that will be used where it came from, how do I get it, and not essential to prove the truth. According to the most important group of these scientists can use it as intended.
Second, mathematical scientist (mathematician) who see mathematics as an abiding principle of structured knowledge.Mathematical concepts shall be constructed and verified according to the provisions or agreements that preceded it. Mathematical scientists, will not be quick to believe in the concept that already exist. They will explore, assess, regroup, prove, try, assume, test the concepts he learned the truth and doing math or matematisasi place is an important part in mathematics. Gravemeijer (1994) emphasize that in mathematical generalization and formalization occurs when matematisasi. Formalization includes modeling, penyimbulan, penskemaan, and definition. While generalization is the understanding in a wider sense. This is related to building a more characteristic of the application of thought in general.
Treffers (1991) matematisasi differentiate into two kinds, namely horizontal and vertical matematisasi. Lange (Darhim, 2004) termed the informal mathematics as matematisasi horizontal, while the formal mathematics as matematisasi vertical. Matematisasi horizontal as events change the contextual problem into a mathematical problem, while the vertical matematisasi is to formulate the problem in a variety of mathematical solution by using a mathematical rule accordingly.
Other views of matematisasi, associated with informal and formal learning in mathematics. In mathematics learning in the classroom, can be done with an approach taking into account aspects of informal, then look for an intermediary to bring the students' understanding of formal mathematics.
Matematisasi can also be done at first identify the purpose for transferring the problem to the problem stated mathematically.Pemvisualan penskemaan and searched through the order and its relationship to the general formula is made.
After the horizontal matematisasi used students to understand and solve a contextual problem, the next step in solving these contextual issues are matematisasi vertical. This process is done so that students' understanding of mathematical concepts learned to achieve formal aspects of mathematics. But matematisasi sequence as above is not a necessity, it may even happen matematisasi done ahead of matematisasi vertical horizontal.
Matematisasi processes in mathematics, which he passes the path may vary. Maybe someone followed the process in the horizontal and vertical lines are complicated and many. While others take the horizontal with simple lines and a short, but the vertical is reached by a complicated paths, or vice versa. Process may also be reached by horizontal and vertical lines are simple and brief.
Similarly, the second frequency range may occur matematisasi vary. It really depends on the materials, methods, students' abilities, environmental conditions, as well as learning objectives. Horizontal Matematisasi can occur with a frequency much the first time followed by a few vertical matetamtisasi. At different conditions, can occur matematisasi appeared first vertical, then horizontal matematisasi fever developed. There are many opportunities for learning mathematics matematisasi in terms of frequency of use of horizontal and vertical matematisasi, among others: (1) horizontal and vertical Matematisasi happen balanced and begins with matematisasi horizontal, (2) horizontal and vertical Matematisasi balanced but begins with a vertical matematisasi; ( 3) horizontal Matematisasi matematisasi more dominant than vertical and horizontal matematisasi begin with, (4) horizontal Matematisasi more dominant than vertical matematisasi but preceded by a vertical matematisasi, (5) Matematisasi more dominant than the vertical and horizontal matematisasi matematisasi begins with the horizontal, (6) more dominant than the vertical Matematisasi matematisasi horizontal but vertical matematisasi begin with.
Generalization may be regarded as the highest level in the vertical matematisasi. This means that when giving reasons for the mathematical models developed, a new mathematical model is expected to appear. The process of building a new model is needed level of thinking at a higher level. Through mechanisms such as itulan math concepts are built in full (maybe more abstract) with the presence of a new model.
As already mentioned earlier that in the view of Freudenthal (1991) mathematics as a human activity and mathematics should be taught through the rediscovery (reinvention) or through discovery (invention). In order to learn mathematics as human activity through matematisasi process, it should not be taught math in final form.The final form of mathematics is to be found students (Ruseffendi, 1988).
Approach to learning mathematics
There are two groups of scientists associated with the field of mathematical study. Both groups are: First, users of mathematical scientists. This group may not care about the mathematical concepts that will be used where it came from, how do I get it, and not essential to prove the truth. According to the most important group of these scientists can use it as intended.
Second, mathematical scientist (mathematician) who see mathematics as an abiding principle of structured knowledge.Mathematical concepts shall be constructed and verified according to the provisions or agreements that preceded it. Mathematical scientists, will not be quick to believe in the concept that already exist. They will explore, assess, regroup, prove, try, assume, test the concepts he learned the truth and doing math or matematisasi place is an important part in mathematics. Gravemeijer (1994) emphasize that in mathematical generalization and formalization occurs when matematisasi. Formalization includes modeling, penyimbulan, penskemaan, and definition. While generalization is the understanding in a wider sense. This is related to building a more characteristic of the application of thought in general.
Treffers (1991) matematisasi differentiate into two kinds, namely horizontal and vertical matematisasi. Lange (Darhim, 2004) termed the informal mathematics as matematisasi horizontal, while the formal mathematics as matematisasi vertical. Matematisasi horizontal as events change the contextual problem into a mathematical problem, while the vertical matematisasi is to formulate the problem in a variety of mathematical solution by using a mathematical rule accordingly.
Other views of matematisasi, associated with informal and formal learning in mathematics. In mathematics learning in the classroom, can be done with an approach taking into account aspects of informal, then look for an intermediary to bring the students' understanding of formal mathematics.
Matematisasi can also be done at first identify the purpose for transferring the problem to the problem stated mathematically.Pemvisualan penskemaan and searched through the order and its relationship to the general formula is made.
After the horizontal matematisasi used students to understand and solve a contextual problem, the next step in solving these contextual issues are matematisasi vertical. This process is done so that students' understanding of mathematical concepts learned to achieve formal aspects of mathematics. But matematisasi sequence as above is not a necessity, it may even happen matematisasi done ahead of matematisasi vertical horizontal.
Matematisasi processes in mathematics, which he passes the path may vary. Maybe someone followed the process in the horizontal and vertical lines are complicated and many. While others take the horizontal with simple lines and a short, but the vertical is reached by a complicated paths, or vice versa. Process may also be reached by horizontal and vertical lines are simple and brief.
Similarly, the second frequency range may occur matematisasi vary. It really depends on the materials, methods, students' abilities, environmental conditions, as well as learning objectives. Horizontal Matematisasi can occur with a frequency much the first time followed by a few vertical matetamtisasi. At different conditions, can occur matematisasi appeared first vertical, then horizontal matematisasi fever developed. There are many opportunities for learning mathematics matematisasi in terms of frequency of use of horizontal and vertical matematisasi, among others: (1) horizontal and vertical Matematisasi happen balanced and begins with matematisasi horizontal, (2) horizontal and vertical Matematisasi balanced but begins with a vertical matematisasi; ( 3) horizontal Matematisasi matematisasi more dominant than vertical and horizontal matematisasi begin with, (4) horizontal Matematisasi more dominant than vertical matematisasi but preceded by a vertical matematisasi, (5) Matematisasi more dominant than the vertical and horizontal matematisasi matematisasi begins with the horizontal, (6) more dominant than the vertical Matematisasi matematisasi horizontal but vertical matematisasi begin with.
Generalization may be regarded as the highest level in the vertical matematisasi. This means that when giving reasons for the mathematical models developed, a new mathematical model is expected to appear. The process of building a new model is needed level of thinking at a higher level. Through mechanisms such as itulan math concepts are built in full (maybe more abstract) with the presence of a new model.
As already mentioned earlier that in the view of Freudenthal (1991) mathematics as a human activity and mathematics should be taught through the rediscovery (reinvention) or through discovery (invention). In order to learn mathematics as human activity through matematisasi process, it should not be taught math in final form.The final form of mathematics is to be found students (Ruseffendi, 1988).
Approach to learning mathematics
Judging
from the use of horizontal and vertical matematisasi processes in mathematics,
the learning of mathematics can be divided into four different approaches,
namely the mechanistic approach (mechanistic), strukturalistik (structuralistic),
empiristik (empiristic), and a realistic approach (realistic). Mechanistic
approach, both horizontal and vertical matematisasi not used. This
approach is more emphasis on skills training and use of the formula. At
empiristik approach, more emphasis on process matematisasi horizontal and
vertical matematisasi process tends to ignore. Strukturalistik approach
more emphasis on process matematisasi vertical and horizontal matematisasi tend
to ignore the process. While the realistic approach, the balance of the
use of horizontal and vertical matematisasi process of learning mathematics.
Mathematics learning approach can also be distinguished by the formal and informal learning undertaken. In the mechanistic approach and strukturalistik, learning mathematics is more likely to be done formally. Empiristik approaches, learning mathematics is more likely to be done on an informal basis. While the realistic approach, there is a balance between mathematical learning is done informally and formally.
In conjunction with some insight into the learning of mathematics that includes mechanistic, stukturalistik, and empiristik, there are some characteristics for each view. According to the mechanistic approach, mathematics is a system of rules. These rules are given to students, then they verify and apply them to similar problems as the previous example. There was no real-world phenomena as a source, very little attention given to the application. Much attention is given to memorization (remembering) and routines. Quality as well as structure, connectedness, and insights are ignored.
According strukturalistik view, mathematics is well structured.According to this view purely mathematical axioms, definitions, and theorems. Hence the orientation of learning according to this view and the subject matter is presented in a deductive mathematics.
Empiristik view, more emphasis on environment activities. Greater attention is given to students in the hopes of cognitive maturation.Through this maturation is expected that students will be up to the expected development of cognition. However, this approach was very little in sight to arrive at a vertical level.
On a realistic approach, there is keberimbangan between environment and subject matter activity. Utilization of the environment as a learning resource is one feature of this approach.Through the utilization of contextual issues that are real world phenomena are expected to more meaningful learning of mathematics. Students must actively discover mathematical concepts in learning, through a process matematisasi (horizontal and vertical). Learning mathematics should help students rediscover the mathematical concepts that have been discovered by mathematicians, through a process matematisasi.
Mathematics learning approach can also be distinguished by the formal and informal learning undertaken. In the mechanistic approach and strukturalistik, learning mathematics is more likely to be done formally. Empiristik approaches, learning mathematics is more likely to be done on an informal basis. While the realistic approach, there is a balance between mathematical learning is done informally and formally.
In conjunction with some insight into the learning of mathematics that includes mechanistic, stukturalistik, and empiristik, there are some characteristics for each view. According to the mechanistic approach, mathematics is a system of rules. These rules are given to students, then they verify and apply them to similar problems as the previous example. There was no real-world phenomena as a source, very little attention given to the application. Much attention is given to memorization (remembering) and routines. Quality as well as structure, connectedness, and insights are ignored.
According strukturalistik view, mathematics is well structured.According to this view purely mathematical axioms, definitions, and theorems. Hence the orientation of learning according to this view and the subject matter is presented in a deductive mathematics.
Empiristik view, more emphasis on environment activities. Greater attention is given to students in the hopes of cognitive maturation.Through this maturation is expected that students will be up to the expected development of cognition. However, this approach was very little in sight to arrive at a vertical level.
On a realistic approach, there is keberimbangan between environment and subject matter activity. Utilization of the environment as a learning resource is one feature of this approach.Through the utilization of contextual issues that are real world phenomena are expected to more meaningful learning of mathematics. Students must actively discover mathematical concepts in learning, through a process matematisasi (horizontal and vertical). Learning mathematics should help students rediscover the mathematical concepts that have been discovered by mathematicians, through a process matematisasi.
mathematics
are expected to use the three principles, namely: the discovery and
bermatematika progressively guided (guided reinvention and progressive
mathematizing), the phenomenon of learning (didactical phenomenology), the
development of independent model (seff developed model).
Photo: Andri Yunardi
First, the principle of guided discovery is intended to provide opportunities to students in finding their own concept of mathematical concepts through a process matematisasi contextual issues. This fits the role of contextual issues by Treffers and Goffree (Darhim, 2004), which is to lead students to build concepts, construction of models, which have been known to apply concepts and solve problems based on rules that applies mathematical principles.
The discovery of models, concepts, and procedures to solve the problem starts with the contextual matematisasi progressively, in this context to the students formulate the problem in the form of informal and formal mathematics. This step was taken with the support of mathematical concepts and procedures that have been learned. When the support is less meaningful, the direction of the teacher has an important role. Through penskemaan, formulation, and pemvisualan problems, students are expected to try to find similarities and relationships contained in the contextual issues and turn them into informal and formal mathematical models.
Second, the phenomenon prisnsip learning emphasizes the importance of contextual issues matter to introduce the topic of mathematical topics to students. Problem issues can be derived from the real world or at least of the problems ¬ a problem that can be imagined as a matter of real students. Would certainly not be easy to create or develop an appropriate contextual issues for each student because the student experience, environment, and other social aspects can influence it.
Third, the development of independent model which means that in studying the concept of mathematical concepts and other content material, meant that students need to develop their own models or models of how to solve the problem. Model of the model act as a bridge or intermediary to develop students' thinking processes, ranging from the thinking of the best known of students who may still be intuitive or informal, to the thinking process is more formal.
In developing the model, students use the model of mathematical models that have been learned. Starting with a complete contextual issues, students find a model of the (model of) the problem is in the form of informal, followed by finding a model for the (model for) in a more formal mathematical form, until the settlement of the problem.Students learn mathematics through four stages, namely stages of the real situation, modeling stage, the stage of generalization, and pemformalan stage.
Photo: Andri Yunardi
First, the principle of guided discovery is intended to provide opportunities to students in finding their own concept of mathematical concepts through a process matematisasi contextual issues. This fits the role of contextual issues by Treffers and Goffree (Darhim, 2004), which is to lead students to build concepts, construction of models, which have been known to apply concepts and solve problems based on rules that applies mathematical principles.
The discovery of models, concepts, and procedures to solve the problem starts with the contextual matematisasi progressively, in this context to the students formulate the problem in the form of informal and formal mathematics. This step was taken with the support of mathematical concepts and procedures that have been learned. When the support is less meaningful, the direction of the teacher has an important role. Through penskemaan, formulation, and pemvisualan problems, students are expected to try to find similarities and relationships contained in the contextual issues and turn them into informal and formal mathematical models.
Second, the phenomenon prisnsip learning emphasizes the importance of contextual issues matter to introduce the topic of mathematical topics to students. Problem issues can be derived from the real world or at least of the problems ¬ a problem that can be imagined as a matter of real students. Would certainly not be easy to create or develop an appropriate contextual issues for each student because the student experience, environment, and other social aspects can influence it.
Third, the development of independent model which means that in studying the concept of mathematical concepts and other content material, meant that students need to develop their own models or models of how to solve the problem. Model of the model act as a bridge or intermediary to develop students' thinking processes, ranging from the thinking of the best known of students who may still be intuitive or informal, to the thinking process is more formal.
In developing the model, students use the model of mathematical models that have been learned. Starting with a complete contextual issues, students find a model of the (model of) the problem is in the form of informal, followed by finding a model for the (model for) in a more formal mathematical form, until the settlement of the problem.Students learn mathematics through four stages, namely stages of the real situation, modeling stage, the stage of generalization, and pemformalan stage.
Mathematics
teachers should have the skills to teach mathematics from the standpoint of how
students should learn mathematics.
Government Regulation Number 19 Year 2005 on National Education Standards, Article 19 paragraph (1) mandates that the process of learning in the educational unit organized in an interactive, inspiring, fun, challenging, motivating learners to actively participate and provide enough space for innovation, creativity , and independence in accordance with their talents, interests, and physical and psychological development of students (PP No. 19, 2005).
Learning mathematics with a particular approach, students are required to construct a mathematical concept that is certainly with the help of the teacher when the student needs it. Help teachers be effective if the teacher knows the students and the difficulty level of maturity and thinking patterns of students.
The teacher's role in learning is as a facilitator, mentor, or learning a more experienced friend, who knows when to provide assistance (scaffolding) and how to help make the construction process can take place in the minds of students. In this context the teacher is not a light task. The most heavy duty teacher before and after the learning is taking place. Before you start learning the teacher must make plans and preparations start from defining the concept to be taught, seek and find appropriate contextual issues that concept, and plan appropriate learning strategies (not monotonous, sometimes individual or group, and so on). After learning, teacher reflection, making notes and assessments (informal and formal) to the students.
As explained above, that in the process of learning mathematics matematisasi become an important part in building a mathematical concept. There are two reasons matematisasi process is key in learning mathematics. The second reason is: First, matematisasi not only an activity of mathematicians only. It is also an activity students to understand everyday situations using mathematical approach. Here we will look at mathematical activity to determine issues related to mathematics, see possible limitations of the mathematical approach that can be used, and to know when a mathematical approach can be used when not to.
Second, matematisasi focus on math-related learning rediscovery (reinvention) of ideas. Due to obey the principles of mathematics, the ultimate goal of learning is based on the formalization of aksiomatisasi. This is not the ultimate goal should be the starting point when it will teach mathematics. In learning mathematics, students are directed as if reinventing math concepts through a process that may be different from the process conducted by experts when they found the concept.
According to Article 8 Permeneg PAN & RB No. 16 of 2009 teachers authorized to select and specify materials, strategies, methods, media teaching / coaching, and equipment assessment / evaluation in implementing the learning process / guidance to achieve quality education in accordance with professional code of ethics of teachers. The accuracy and appropriateness of the teacher when selecting materials, strategies, methods, media, and assessment tools to the point of view of students in learning mathematics, are expected to affect the ability of students.
After studying math, there are a number of skills students should possess, among other abilities: understanding, reasoning, problem solving, communication, connection, proof, representation, critical thinking, logical thinking, creative thinking and innovative thinking.
Need the right teacher to help students in this country can realize these abilities. Here are some indicators of the need for attention and help teachers be more serious. Although the results of national examinations classified as good, but based on international evaluation of students' mathematics achievement we show that less encouraging.
Since 2003, students have the ability to read primary and secondary schools in the country three times were measured and compared with the abilities of students in several other countries.'Reading' is defined not as 'recite' but 'to read and understand so that they can draw conclusions'. From surveys Literacy Progress in International Reading Study (PIRLS) conducted in 2006, the average 4th grade students in Indonesia scored 405 per 1000, so they have the competencies categorized 'low' (ie 400-474). For comparison, the average 4th grade students scored 563 in Singapore, that means they are categorized as 'high' (550-624).Singapore is ranked fourth from 45 countries, while Indonesia ranked 41 (Kemdiknas, 2009).
While students in grade 2 junior scored 382 per 1000 on the assessment of reading skills by Programme for International Student Assessment (PISA) in 2003 and scored 393 in PISA 2006, making Indonesia the average student category of 'one' (the lowest,with a score of 358 to 420). These results put Indonesia on the 39th, from 40 countries in 2003 and ranked 48th of 56 countries in 2006 (Kemdiknas, 2009).
In the field of mathematics, according to a survey conducted in the framework of Trends in International Mathematics and Science Study (TIMSS), students in grade 2 junior in Indonesia has the ability to 'low' (400-474) to obtain the value of 403 in 1999 and scored 411 in the year 2003 (Gonzales et al, 2004). In the survey conducted by TIMSS in 2007, the average Indonesian students scored 397 per 1000. Thus the ability of students in mathematics is in a category lower than the low category. This shows that the average student ability in Indonesia beyond the limits of measuring devices used by the TIMSS 2007. When compared with our neighbor, Singapore, the TIMSS 2007 survey, two junior high school students there get the mean math 593 per 1000; rate is 'high'. We obtain the value of 397 students (classified as 'low'). In Singapore, 40% of students scored 625 and above, are classified as 'cum laude', while our students are less than one percent of the gain values are 'cum laude' (Gonzales et al, 2008).
The data above is one portrait of students' skills in reading and mathematics that show is still quite alarming. From here, it turns out the great potential possessed by our students to reach almost not developed as expected capabilities. Picture of student ability is closely related to the ability of teachers to manage the learning of mathematics. Reasonably suspected that there was a correlation between the ability of the student and teacher skills. Only for a while becomes a chore that needs further scrutiny.
Government Regulation Number 19 Year 2005 on National Education Standards, Article 19 paragraph (1) mandates that the process of learning in the educational unit organized in an interactive, inspiring, fun, challenging, motivating learners to actively participate and provide enough space for innovation, creativity , and independence in accordance with their talents, interests, and physical and psychological development of students (PP No. 19, 2005).
Learning mathematics with a particular approach, students are required to construct a mathematical concept that is certainly with the help of the teacher when the student needs it. Help teachers be effective if the teacher knows the students and the difficulty level of maturity and thinking patterns of students.
The teacher's role in learning is as a facilitator, mentor, or learning a more experienced friend, who knows when to provide assistance (scaffolding) and how to help make the construction process can take place in the minds of students. In this context the teacher is not a light task. The most heavy duty teacher before and after the learning is taking place. Before you start learning the teacher must make plans and preparations start from defining the concept to be taught, seek and find appropriate contextual issues that concept, and plan appropriate learning strategies (not monotonous, sometimes individual or group, and so on). After learning, teacher reflection, making notes and assessments (informal and formal) to the students.
As explained above, that in the process of learning mathematics matematisasi become an important part in building a mathematical concept. There are two reasons matematisasi process is key in learning mathematics. The second reason is: First, matematisasi not only an activity of mathematicians only. It is also an activity students to understand everyday situations using mathematical approach. Here we will look at mathematical activity to determine issues related to mathematics, see possible limitations of the mathematical approach that can be used, and to know when a mathematical approach can be used when not to.
Second, matematisasi focus on math-related learning rediscovery (reinvention) of ideas. Due to obey the principles of mathematics, the ultimate goal of learning is based on the formalization of aksiomatisasi. This is not the ultimate goal should be the starting point when it will teach mathematics. In learning mathematics, students are directed as if reinventing math concepts through a process that may be different from the process conducted by experts when they found the concept.
According to Article 8 Permeneg PAN & RB No. 16 of 2009 teachers authorized to select and specify materials, strategies, methods, media teaching / coaching, and equipment assessment / evaluation in implementing the learning process / guidance to achieve quality education in accordance with professional code of ethics of teachers. The accuracy and appropriateness of the teacher when selecting materials, strategies, methods, media, and assessment tools to the point of view of students in learning mathematics, are expected to affect the ability of students.
After studying math, there are a number of skills students should possess, among other abilities: understanding, reasoning, problem solving, communication, connection, proof, representation, critical thinking, logical thinking, creative thinking and innovative thinking.
Need the right teacher to help students in this country can realize these abilities. Here are some indicators of the need for attention and help teachers be more serious. Although the results of national examinations classified as good, but based on international evaluation of students' mathematics achievement we show that less encouraging.
Since 2003, students have the ability to read primary and secondary schools in the country three times were measured and compared with the abilities of students in several other countries.'Reading' is defined not as 'recite' but 'to read and understand so that they can draw conclusions'. From surveys Literacy Progress in International Reading Study (PIRLS) conducted in 2006, the average 4th grade students in Indonesia scored 405 per 1000, so they have the competencies categorized 'low' (ie 400-474). For comparison, the average 4th grade students scored 563 in Singapore, that means they are categorized as 'high' (550-624).Singapore is ranked fourth from 45 countries, while Indonesia ranked 41 (Kemdiknas, 2009).
While students in grade 2 junior scored 382 per 1000 on the assessment of reading skills by Programme for International Student Assessment (PISA) in 2003 and scored 393 in PISA 2006, making Indonesia the average student category of 'one' (the lowest,with a score of 358 to 420). These results put Indonesia on the 39th, from 40 countries in 2003 and ranked 48th of 56 countries in 2006 (Kemdiknas, 2009).
In the field of mathematics, according to a survey conducted in the framework of Trends in International Mathematics and Science Study (TIMSS), students in grade 2 junior in Indonesia has the ability to 'low' (400-474) to obtain the value of 403 in 1999 and scored 411 in the year 2003 (Gonzales et al, 2004). In the survey conducted by TIMSS in 2007, the average Indonesian students scored 397 per 1000. Thus the ability of students in mathematics is in a category lower than the low category. This shows that the average student ability in Indonesia beyond the limits of measuring devices used by the TIMSS 2007. When compared with our neighbor, Singapore, the TIMSS 2007 survey, two junior high school students there get the mean math 593 per 1000; rate is 'high'. We obtain the value of 397 students (classified as 'low'). In Singapore, 40% of students scored 625 and above, are classified as 'cum laude', while our students are less than one percent of the gain values are 'cum laude' (Gonzales et al, 2008).
The data above is one portrait of students' skills in reading and mathematics that show is still quite alarming. From here, it turns out the great potential possessed by our students to reach almost not developed as expected capabilities. Picture of student ability is closely related to the ability of teachers to manage the learning of mathematics. Reasonably suspected that there was a correlation between the ability of the student and teacher skills. Only for a while becomes a chore that needs further scrutiny.
Mathematics
teachers should have the skills to teach mathematics from the standpoint of
authority and responsibility of teachers as professionals.
Teachers as professionals have the functions, roles, and position are very important in achieving the vision of education is to create intelligent beings Indonesia and competitive. More importantly the presence of a professional teacher in the management of learning.This matches the findings of the World Bank (Kemdiknas, 2009) through a comparative study of teachers 'performance results that are categorized as high and low categories of students' low group.The results showed, after a year of the conclusion that there is a difference, although not very significant, the results of student learning that lesson by the teacher of high and low categories. But, after learning last three years there was a difference of student learning outcomes is significant.
What happens if a teacher is not master the material?
Some idea of the difficulty finding math teachers Sadiq (2008) that in general it can be concluded that the vocational school mathematics teachers have difficulty in implementing Permendiknas No. 22 of 2006 in class, especially with regard to the application of contextual learning and attainment of learning objectives related to aspects of reasoning, communication, and problem solving.
Chick (Kemdiknas, 2009) investigated the language used by elementary school teachers in South Africa when they are teaching math. Apparently teachers are less mastered the art so that they worry if they lack understanding will appear. Therefore, they tend to use certain types of activities they considered 'safe', ie the activities in which students made 'active' and 'busy' when in fact they are not challenged to do something new. Thus as long as students do the activity risk of 'false' is very low, so teachers will not lose face in front of students.
Interaction between teachers and students for their involvement in learning activities using the language of the 'safe'. Teachers choose activities that secure communications or conversations from any risk, so that all parties should not feel ashamed. Of course by way of teaching such as this will result in students are not encouraged to ask questions or to innovate. Such conditions would hamper the development of student competence.
Based on research we can conclude that Chick is a master teacher is a teacher of learning materials which tend to be confident and brave to innovate. Conversely, teachers who lack confidence to master the art would be less when teaching, as a further consequence will tend to avoid all forms of innovation and risk.Though the position of teachers as professionals working to enhance the dignity of the teacher's own learning as well as its role as an agent to improve the quality of national education. While the goal to implement a national education system and achieve national education goals, namely the development of potential learners in order to become a man of faith and fear of God Almighty, noble, healthy, knowledgeable, skilled, creative, independent, and democratic citizenship and responsible (Law No. 14, 2005).
Facing the education system innovation
The presence of Act No. 14 of 2005, Government Regulation No. 74 of 2008, and Permeneg PAN & RB No. 16 of 2009 had a direct impact on teachers as professionals.
Law Number 14 Year 2005 on Teachers and Lecturers, Article 10 Paragraph 1 requires teachers to have four competencies, as follows:
Pedagogical competence: the ability of teachers in managing student learning
Competence of personality: a steady personality, noble, wise and dignified and exemplary students
Social competence: the ability of teachers as part of the community to communicate and interact effectively and efficiently with students, fellow teachers, parents / guardians of students, and surrounding communities
Professional competence: the ability to master the knowledge of teachers in science, technology, and / or cultural arts diampunya.
Government Regulation Number 74 Year 2008 on the Guru, Article 3, Paragraph 4-7, giving details of the four competencies. Teachers need support so they can quickly deal with and adjust to the demands given by the Act and the Regulation.
One of the 'social competence' which require special attention are the 'use of information and communication technologies are functionally'. It is inevitable that all teachers must have adequate information and communication technology (ICT). However, based on research conducted by the Economist Intelligence Unit in Britain, in 2006 the people of Indonesia derives score 3.39 per 10 in readiness to function in cyberspace (e-readiness), thus Indonesia is ranked 62 of 68 countries (Economist Intelligence Unit, 2007).
One year later, in 2007, Indonesia still get a score of 3.39, but its ranking has dropped to 67 from 69 countries. In other words, the people of Indonesia's readiness to use the internet does not grow from 2006 to 2007, but some other countries - including Algeria, Kazakhstan, Pakistan and Vietnam - managed to catch up and surpass the position of Indonesia. Only Azerbaijan and Iran are ranked below Indonesia. For comparison, the Singapore reached a score of 8.6 per 10 in 2007 and occupied the 6 from 69 countries.Whether conditions had changed for 2012?
From this it can be seen just how heavy a task that must be endured to help teachers achieve competence 'use of information and communication technologies are functionally' as required by Regulation 74 of 2008.
Referring to the Minister of Administrative Reform and Reforms No. 16 of 2009, teachers who have been carrying out duties in accordance duties, authority, and the workload, the teacher still had to carry out activities related to sustainable development of professionalism. According to this conception is essentially a teacher has the authority, responsibilities, and duties are graded according to rank and position. The higher the rank and position the area of authority, responsibility, and the demands of his job.
Continuing professional development is intended that the teacher be able to reduce the distance between the knowledge, skills, and personalities that have been owned by the demands of his profession in the future. Teachers who perform their duties effectively supported by an increase in professionalism in a sustainable manner, can have an impact on career development for teachers is concerned.
Career development of teachers interpreted as an increase in the rank / position, followed by an increase in competence, responsibility, and authority to carry out the duties pembelajaan and guidance and / or additional tasks relevant to the functions of the school / madrasah which is accompanied by the acquisition of additional benefits in carrying out these tasks.
Career development of teachers when grouped by level and kewenangnnya divided into 2 (two) categories, namely: (1) the career development of vertical lines, in the form of a promotion / job as a teacher (progression) and (2) horizontal career development path, in the form of increased kewengan and responsibility to carry out additional duties and / or other duties relevant to the functions of the school / madrasah (promotion).
Signs teacher coaching career in general are described on the Government Regulation No. 17 of 2010 Section 176, namely: (1) The Government to develop and establish patterns of career guidance teachers and in accordance with the provisions of legislation, (2) Government and / or local government shall conduct career guidance teachers and career coaching in accordance with the pattern referred to in paragraph 1, (3) Operator established public education shall conduct career guidance teachers and the educational unit to be held in accordance with the pattern of career coaching as intended in paragraph 1, (4) Coaching career educators implemented in the form of increased academic qualifications and / or competence as a learning agent with reference to the National Education Standards, and (5) Coaching career education personnel carried out in the form of increased academic qualifications and / or managerial competencies and / or technical education personnel with reference to the National Education Standards.
Promotion of teachers under Government Regulation No. 74 of 2008 Section 36 that the task professionalism, teachers eligible for promotion in accordance with his duties and accomplishments.Furthermore, the promotion would include promotion and / or increase the functional hierarchy.
Promotions and awards for teachers, both civil servants and teachers are not civil servants in the educational unit organized by the society, is also a message that mandated the Government Regulation No. 17 of 2010 Section 177 and Section 178. Article 177 of the Regulation states that promotions and rewards for teachers and is based on educational background, experience, ability, and job performance in the field of education. While Article 178 states that: (1) Promotion for educators and education personnel referred to in Article 177 provided in the form of a promotion / classes, promotions, and / or other forms of promotion are carried out in accordance with the provisions of legislation, and (2) Promotion for teachers and civil servants rather than the educational unit organized by the society carried out in accordance with the statutes and bylaws and the provisions of the education legislation.
Along with the increasing rank / position reflects increased professionalism for teachers, in addition to his salary scale followed by appropriate levels of rank / position, if it meets the other requirements according to current regulations, a teacher can be promoted to carry out additional duties and / or other tasks that are relevant to function of the school / madrasah.
Teachers are promoted to carry out additional duties and / or other duties relevant to the functions of the school / madrasah requires additional knowledge and skills in accordance with the additional duties and / or such other relevant duties. Additional knowledge and skills is evidenced by a certificate from an educational institution that is authorized for it. Teachers who perform additional duties and / or other relevant tasks that have the authority and responsibility are more than teachers who do not obtain additional duties and / or other duties relevant to the functions of the school / madrasah.
In addition to the ownership of knowledge and additional skills in accordance with the additional duties and / or other duties relevant to the functions of the school / madrasah, the promotion will also affect the provision of incentives or allowances from the Government, local governments, and / or community. Promotion (promotion) can also affect the provision of credit points according to their performance when carrying out additional duties and / or other duties relevant to the functions of the school / madrasah was.These credit points can be used for promotion / the teacher in question functional (progression). In other words, there is a very close relationship between promotion and progression. The two are linked causally causation.
Facing the demands of current theories
Research conducted by Malderez & Wedell (Kemdiknas, 2009) about the competency of the teachers are skilled at finding some of the same competencies as set out in Regulation No. 74 of 2008.But there are still some other competence according to Malderez & Wedell possessed by a competent teacher but does not appear in PP No. 74 of 2008. The competencies of 'extras' are: (1) Knowing about, namely the ability of teachers to understand their duties and how the school where students learn. (2) Knowing how, the ability of teachers to make use of models / methods / approaches / teaching techniques, analyzing that happens in the classroom, and develop a situation that supports the learning process. (3) Knowing to, the ability of teachers to utilize his knowledge in a way and at the right time so that the learning process can be fully supported.
The ability of teachers to 'observe and analyze the things that happened in class', according to Malderez & Wedell, is one of the characteristics of skilled and experienced teachers. The sensitivity of the learning environment can not be taught and not necessarily going to grow by itself. Therefore, most teachers need guidance from a mentor or teacher coaches to improve their level of sensitivity. This is nothing but the ability to 'investigate and understand the context in which teachers teach'.
Conclusion
Assessed based on four different mathematical models of learning and study of the structure, namely mathematics: academic, technical, recreational, and life.
Understand mathematics as above, there are at least three kinds of skills that should have teachers in the teaching of mathematics, namely: as a mathematical scientist, as students learn mathematics, and as a teacher of mathematics.
Teachers as professionals have the functions, roles, and position are very important in achieving the vision of education is to create intelligent beings Indonesia and competitive. More importantly the presence of a professional teacher in the management of learning.This matches the findings of the World Bank (Kemdiknas, 2009) through a comparative study of teachers 'performance results that are categorized as high and low categories of students' low group.The results showed, after a year of the conclusion that there is a difference, although not very significant, the results of student learning that lesson by the teacher of high and low categories. But, after learning last three years there was a difference of student learning outcomes is significant.
What happens if a teacher is not master the material?
Some idea of the difficulty finding math teachers Sadiq (2008) that in general it can be concluded that the vocational school mathematics teachers have difficulty in implementing Permendiknas No. 22 of 2006 in class, especially with regard to the application of contextual learning and attainment of learning objectives related to aspects of reasoning, communication, and problem solving.
Chick (Kemdiknas, 2009) investigated the language used by elementary school teachers in South Africa when they are teaching math. Apparently teachers are less mastered the art so that they worry if they lack understanding will appear. Therefore, they tend to use certain types of activities they considered 'safe', ie the activities in which students made 'active' and 'busy' when in fact they are not challenged to do something new. Thus as long as students do the activity risk of 'false' is very low, so teachers will not lose face in front of students.
Interaction between teachers and students for their involvement in learning activities using the language of the 'safe'. Teachers choose activities that secure communications or conversations from any risk, so that all parties should not feel ashamed. Of course by way of teaching such as this will result in students are not encouraged to ask questions or to innovate. Such conditions would hamper the development of student competence.
Based on research we can conclude that Chick is a master teacher is a teacher of learning materials which tend to be confident and brave to innovate. Conversely, teachers who lack confidence to master the art would be less when teaching, as a further consequence will tend to avoid all forms of innovation and risk.Though the position of teachers as professionals working to enhance the dignity of the teacher's own learning as well as its role as an agent to improve the quality of national education. While the goal to implement a national education system and achieve national education goals, namely the development of potential learners in order to become a man of faith and fear of God Almighty, noble, healthy, knowledgeable, skilled, creative, independent, and democratic citizenship and responsible (Law No. 14, 2005).
Facing the education system innovation
The presence of Act No. 14 of 2005, Government Regulation No. 74 of 2008, and Permeneg PAN & RB No. 16 of 2009 had a direct impact on teachers as professionals.
Law Number 14 Year 2005 on Teachers and Lecturers, Article 10 Paragraph 1 requires teachers to have four competencies, as follows:
Pedagogical competence: the ability of teachers in managing student learning
Competence of personality: a steady personality, noble, wise and dignified and exemplary students
Social competence: the ability of teachers as part of the community to communicate and interact effectively and efficiently with students, fellow teachers, parents / guardians of students, and surrounding communities
Professional competence: the ability to master the knowledge of teachers in science, technology, and / or cultural arts diampunya.
Government Regulation Number 74 Year 2008 on the Guru, Article 3, Paragraph 4-7, giving details of the four competencies. Teachers need support so they can quickly deal with and adjust to the demands given by the Act and the Regulation.
One of the 'social competence' which require special attention are the 'use of information and communication technologies are functionally'. It is inevitable that all teachers must have adequate information and communication technology (ICT). However, based on research conducted by the Economist Intelligence Unit in Britain, in 2006 the people of Indonesia derives score 3.39 per 10 in readiness to function in cyberspace (e-readiness), thus Indonesia is ranked 62 of 68 countries (Economist Intelligence Unit, 2007).
One year later, in 2007, Indonesia still get a score of 3.39, but its ranking has dropped to 67 from 69 countries. In other words, the people of Indonesia's readiness to use the internet does not grow from 2006 to 2007, but some other countries - including Algeria, Kazakhstan, Pakistan and Vietnam - managed to catch up and surpass the position of Indonesia. Only Azerbaijan and Iran are ranked below Indonesia. For comparison, the Singapore reached a score of 8.6 per 10 in 2007 and occupied the 6 from 69 countries.Whether conditions had changed for 2012?
From this it can be seen just how heavy a task that must be endured to help teachers achieve competence 'use of information and communication technologies are functionally' as required by Regulation 74 of 2008.
Referring to the Minister of Administrative Reform and Reforms No. 16 of 2009, teachers who have been carrying out duties in accordance duties, authority, and the workload, the teacher still had to carry out activities related to sustainable development of professionalism. According to this conception is essentially a teacher has the authority, responsibilities, and duties are graded according to rank and position. The higher the rank and position the area of authority, responsibility, and the demands of his job.
Continuing professional development is intended that the teacher be able to reduce the distance between the knowledge, skills, and personalities that have been owned by the demands of his profession in the future. Teachers who perform their duties effectively supported by an increase in professionalism in a sustainable manner, can have an impact on career development for teachers is concerned.
Career development of teachers interpreted as an increase in the rank / position, followed by an increase in competence, responsibility, and authority to carry out the duties pembelajaan and guidance and / or additional tasks relevant to the functions of the school / madrasah which is accompanied by the acquisition of additional benefits in carrying out these tasks.
Career development of teachers when grouped by level and kewenangnnya divided into 2 (two) categories, namely: (1) the career development of vertical lines, in the form of a promotion / job as a teacher (progression) and (2) horizontal career development path, in the form of increased kewengan and responsibility to carry out additional duties and / or other duties relevant to the functions of the school / madrasah (promotion).
Signs teacher coaching career in general are described on the Government Regulation No. 17 of 2010 Section 176, namely: (1) The Government to develop and establish patterns of career guidance teachers and in accordance with the provisions of legislation, (2) Government and / or local government shall conduct career guidance teachers and career coaching in accordance with the pattern referred to in paragraph 1, (3) Operator established public education shall conduct career guidance teachers and the educational unit to be held in accordance with the pattern of career coaching as intended in paragraph 1, (4) Coaching career educators implemented in the form of increased academic qualifications and / or competence as a learning agent with reference to the National Education Standards, and (5) Coaching career education personnel carried out in the form of increased academic qualifications and / or managerial competencies and / or technical education personnel with reference to the National Education Standards.
Promotion of teachers under Government Regulation No. 74 of 2008 Section 36 that the task professionalism, teachers eligible for promotion in accordance with his duties and accomplishments.Furthermore, the promotion would include promotion and / or increase the functional hierarchy.
Promotions and awards for teachers, both civil servants and teachers are not civil servants in the educational unit organized by the society, is also a message that mandated the Government Regulation No. 17 of 2010 Section 177 and Section 178. Article 177 of the Regulation states that promotions and rewards for teachers and is based on educational background, experience, ability, and job performance in the field of education. While Article 178 states that: (1) Promotion for educators and education personnel referred to in Article 177 provided in the form of a promotion / classes, promotions, and / or other forms of promotion are carried out in accordance with the provisions of legislation, and (2) Promotion for teachers and civil servants rather than the educational unit organized by the society carried out in accordance with the statutes and bylaws and the provisions of the education legislation.
Along with the increasing rank / position reflects increased professionalism for teachers, in addition to his salary scale followed by appropriate levels of rank / position, if it meets the other requirements according to current regulations, a teacher can be promoted to carry out additional duties and / or other tasks that are relevant to function of the school / madrasah.
Teachers are promoted to carry out additional duties and / or other duties relevant to the functions of the school / madrasah requires additional knowledge and skills in accordance with the additional duties and / or such other relevant duties. Additional knowledge and skills is evidenced by a certificate from an educational institution that is authorized for it. Teachers who perform additional duties and / or other relevant tasks that have the authority and responsibility are more than teachers who do not obtain additional duties and / or other duties relevant to the functions of the school / madrasah.
In addition to the ownership of knowledge and additional skills in accordance with the additional duties and / or other duties relevant to the functions of the school / madrasah, the promotion will also affect the provision of incentives or allowances from the Government, local governments, and / or community. Promotion (promotion) can also affect the provision of credit points according to their performance when carrying out additional duties and / or other duties relevant to the functions of the school / madrasah was.These credit points can be used for promotion / the teacher in question functional (progression). In other words, there is a very close relationship between promotion and progression. The two are linked causally causation.
Facing the demands of current theories
Research conducted by Malderez & Wedell (Kemdiknas, 2009) about the competency of the teachers are skilled at finding some of the same competencies as set out in Regulation No. 74 of 2008.But there are still some other competence according to Malderez & Wedell possessed by a competent teacher but does not appear in PP No. 74 of 2008. The competencies of 'extras' are: (1) Knowing about, namely the ability of teachers to understand their duties and how the school where students learn. (2) Knowing how, the ability of teachers to make use of models / methods / approaches / teaching techniques, analyzing that happens in the classroom, and develop a situation that supports the learning process. (3) Knowing to, the ability of teachers to utilize his knowledge in a way and at the right time so that the learning process can be fully supported.
The ability of teachers to 'observe and analyze the things that happened in class', according to Malderez & Wedell, is one of the characteristics of skilled and experienced teachers. The sensitivity of the learning environment can not be taught and not necessarily going to grow by itself. Therefore, most teachers need guidance from a mentor or teacher coaches to improve their level of sensitivity. This is nothing but the ability to 'investigate and understand the context in which teachers teach'.
Conclusion
Assessed based on four different mathematical models of learning and study of the structure, namely mathematics: academic, technical, recreational, and life.
Understand mathematics as above, there are at least three kinds of skills that should have teachers in the teaching of mathematics, namely: as a mathematical scientist, as students learn mathematics, and as a teacher of mathematics.
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