Pendekatan Pembelajaran Problem Posing

Pendekatan Pembelajaran  Problem Posing - Menurut Suryosubroto (2009:195) pendekatan pembelajaran merupakan kegiatan yang dipilih pendidik dalam proses pembelajaran yang dapat memberikan kemudahan atau fasilitas kepada peserta didik dalam menuju tercapainya tujuan yang telah ditetapkan. Sedangkan Soedjadi (dalam Harleti 2009:10) mengatakan bahwa Pendekatan pembelajaran adalah proses penyampaian atau penyajian topik pelajaran tertentu agar mempermudah siswa memahaminya.

Berkaitan dengan pendekatan pembelajaran, Herman (dalam Harleti 2009:10)  menegaskan bahwa pendekatan mengajar matematika adalah bagaimana cara penyampaian stuktur-struktur dan konsep-konsep matematika kepada siswa sedemikian rupa, sehingga mereka ikut terlibat aktif berpartisipasi dalam belajarnya.

Berdasarkan beberapa pendapat diatas dapat disimpulkan bahwa pendekatan pembelajaran matematika merupakan suatu upaya yang dilakukan pendidik untuk menyampaikan materi atau konsep-konsep matematika sehingga peserta didik ikut terlibat aktif dalam proses belajar agar dapat mencapai tujuan yang telah ditetapkan.

Salah satu pendekatan pembelajaran yang dapat memotivasi siswa untuk berpikir kritis sekaligus dialogis, kreatif dan interaktif yakni problem posing atau pengajuan masalah-masalah yang dituangkan dalam bentuk pertanyaan (Suryosubroto,2009:203). Pertanyaan-pertanyaan tersebut kemudian diupayakan untuk dicari jawabannya baik secara individu maupun bersama dengan pihak lain, misalnya dengan persera didik maupun dengan pengajar sendiri.

Pendekatan problem posing diharapkan memancing siswa untuk menemukan pengetahuan yang bukan diakibatkan dari ketidaksengajaan melainkan melalui upaya mereka untuk mencari hubungan-hubungan dalam informasi yang dipelajarinya. Semakin luas informasi yang dimiliki akan semakin mudah pula menemukan hubungan-hubungan tersebut. Pada akhirnya, penemuan pertanyaan serta jawaban yang dihasilkan terhadapnya dapat menimbulkan  rasa puas akibat keberhasilan menemukan sendiri, baik berupa pertanyaan atau masalah maupun jawaban atas permasalahan yang diajukan.

Berikut ini adalah prosedur penilaian menggunakan pendekatan problem posing

Problem Solving as an Instructional Strategy

Problem Solving as an Instructional Strategy - Polya (1945) suggests that problem solving consists of four phases: understanding the problem, devising a plan, carrying out the plan, and looking back. Lajoie (1992) defines mathematical problem solving as: “modeling the problem and formulating and verifying hypotheses by collecting and interpreting data, using pattern analysis, graphing, or computers and calculators.” This definition focuses on the processes of formulation, investigation, and verification, but it does not encompass the important elements inherent in Polya’s looking back phase, which involve evaluating and interpreting methods and results. The looking back phase includes such activities as:

• Verifying the result
• Checking for alternative methods of solution
• Determining the validity of an argument
• Applying the result or method of solution to other problems
• Interpreting the result
• Generalizing the solution
• Generating new problems to be solved

Looking back may be the most important aspect of teaching problem solving because it provides students the opportunity to learn about problem-solving processes and how a problem is related to other problems. Schoenfeld (1985) and others have shown that the principal traits that separate expert from novice problem solvers are their ability to see past the surface features of problems to their common underlying structures, and their ability to self-monitor and recognize when an approach or tactic is not being productive.

Although teachers and researchers report that it is difficult to develop a willingness in students to continue past finding the correct answer to a problem, the development of self-awareness and reflection are critical for improving problem-solving ability.

Changing One's Practice: Teacher Readiness

Changing One's Practice: Teacher Readiness - A teacher’s approach to teaching mathematics reflects her beliefs about what mathematics is as a discipline (Hersh, 1986). If she characterizes mathematics as involving correct answers and infallible procedures consisting of arithmetic operations, algebraic procedures, and geometric terms and theorems, chances are, her instructional approach will likely emphasize the presentation of mathematical concepts, procedures, facts, and theorems with a focus on student practice and memorization.

The meaning and context associated with many of these theorems and procedures may be relegated to the fringes of her curricular focus. On the other hand, if she views mathematics as an active, creative endeavor involving inquiry and discovery, she will likely emphasize activities that involve students in generating and uncovering meaning and making connections. She will view her role as a facilitator, challenging students to think and to question their findings and assumptions. Ernest (1988) outlines three conceptions of mathematics, each of which prompts a
different emphasis in instruction:

First of all, there is a dynamic, problem-driven view of mathematics as a continually expanding field of human creation and invention, in which patterns are generated and then distilled into knowledge. Thus mathematics is a process of enquiry and coming to know, adding to the sum of knowledge. Mathematics is not a finished product, for its results remain open to revision (the problem-solving view).

Secondly, there is a view of mathematics as a static but unified body of knowledge, a crystalline realm of interconnecting structures and truths, bound together by filaments of logic and meaning. Thus mathematics is a monolith, a static immutable product. Mathematics is discovered, not created (the Platonic view).

Thirdly, there is the view that mathematics, like a bag of tools, is made up of an accumulation of facts, rules and skills to be used by the trained artisan skillfully in the pursuance of some external end. Thus mathematics is a set of unrelated but utilitarian rules and facts (the instrumentalist view).

Each of these views perceives the essence of mathematics differently. The instrumentalist view sees mathematics as a set of tools. Teachers with an instrumentalist view can be expected to stress rules, facts, and procedures in their classes. Their classes tend to be teacher-directed and emphasize routine drill and practice. The Platonic view sees math as a body of knowledge.

Teachers who ascribe to the Platonic view of mathematics focus on the interrelationships, underlying concepts, and internal logic of mathematical procedures. The problem-solving view focuses on the process of inquiry. Teachers with a problem-solving view tend to be more learner-focused and constructivist in their teaching style, actively involving students in exploring mathematical concepts, creating solution strategies, and constructing personal meaning in a problem-rich environment (Thompson, 1992).

Students’ beliefs about the nature of mathematics are greatly influenced by their teacher’s beliefs. Surveys of student beliefs about mathematics reveal that most students think there should be a ready method for solving problems and that that method should quickly lead to an answer (Schoenfeld 1989, 1992). Schoenfeld (1992) cites a 1983 survey conducted by the National Assessment of Educational Progress (NAEP) in which half of the students who responded agreed that “learning mathematics is mostly memorizing facts.” Three-quarters agreed that “doing mathematics requires lots of practice in following rules,” while 90 percent agreed with the statement, “There is always a rule to follow in solving mathematical problems.” Students holding such beliefs may not even attempt to solve a problem that involves too much complexity or does not appear to offer a clear-cut algorithmic approach.

Furthermore, Schoenfeld (1992) notes that most students believe that all problems have an answer; that there is only one right answer and one correct solution method; and that ordinary students cannot expect to understand mathematics but can merely memorize and apply mathematical procedures in a mechanical fashion. These beliefs largely develop out of the experiences students have in mathematics classes and from the attitudes and beliefs passed on by their teachers.

A problem-solving approach to teaching mathematics helps broaden students’ perception of mathematics from a rule- and fact-based discipline to one that involves inquiry, uncertainty, and creativity. But first, the teacher must make his own paradigm shift, and this requires him to come face-to-face with deeply held personal beliefs about teaching and learning, and to face his own propensity for risk and initiative (Dirkes, 1993). Many teachers feel unprepared to take a problem-solving approach to teaching mathematics.

Few teachers learned math themselves in this way. Even if they encountered problem solving in their college methods courses, once in the classroom, they often conform to the conventional methods that hold sway in most schools. Being an agent of change, when one is surrounded by deeply ingrained beliefs about teaching and learning, is a difficult role to perform. Teachers today are often caught between daily pressure from colleagues, parents, and others to uphold tradition in the classroom, and pressure from policymakers to employ standards-based practices (with the conflicting expectation that students will perform highly on standardized tests that measure basic skills, not performance of standards-based material).

A teacher’s path to change must begin with an acknowledgment of her previous experience. She will build on her past experiences by reflecting on them in light of new ideas about effective teaching strategies (Richardson, 1990). Broadening teachers' conceptions of the nature of problem solving and its potential as an instructional tool requires that they, too, engage in solving open-ended problems. This means spending time solving a wide variety of problems and reflecting on their attempts to solve them. Changing one’s practice is further facilitated when effective teaching techniques are modeled in the classroom by a practitioner who is skilled in problem-solving instruction.

This modeling should be followed by a discussion among the teachers about the selection and use of strategies. Modeling and discussion provide concrete illustrations of the teachers' role in teaching problem solving (Richardson, 1990). Reading literature on the theory and practice of problem-solving instruction can also influence teachers to make changes in their practice (Thompson, 1989). “Examining research inquisitively and skeptically,” writes Ball (1996), “teachers can seek insights from scholarship without according undue weight to its conclusions.

They can use the broadly outlined reforms as a resource for developing inspired but locally tailored innovations.” The truth is, teachers are constantly making changes to meet the changing needs of their students and to try out ideas they've heard from other teachers. Teachers establish their own voice of authority in defining what takes place in the classroom. The notion of authority plays a critical role in conceptualizing and advancing mathematics teacher change (Wilson & Lloyd, 2000). Teachers themselves must be involved in making judgments about what change is worthwhile and significant (Richardson, 1990).

In pursuing reform goals, teachers often feel anxious about their effectiveness and knowledge. Moving in the direction of math reforms means confronting up close the uncertainties, ambiguities, and complexities of what “understanding” and “learning” might really mean. When we ask students to voice their ideas in a problem-solving context, we run the risk of discovering what they do and do not know. Those discoveries can be unsettling when students reveal that they know far less than the teacher expected or far more than the teacher is prepared to deal with (Ball, 1996). Inquiry- and problem-based teaching requires qualities beyond mathematics knowledge and skill. Personal qualities, such as patience, curiosity, generosity, confidence, trust, and imagination, matter a great deal. Interest in seeing the world from another's perspective, enjoyment of humor, empathy with confusion, and concern for the frustration and shame of others are other important qualities that can help a teacher create a learning environment that fosters students’ problem-solving abilities (Ball, 1996). “As teachers build their own understandings and relationships with math, they chart new mathematical courses with their students. And, as they move on new paths with students, their own mathematical understandings change,” Ball writes.

The Role of Problem Solving in School Mathematics

The Role of Problem Solving in School Mathematics -  Stanic and Kilpatrick (1989) identify three general themes that have historically characterized the role of problem solving in school mathematics: problem solving as context, problem solving as skill, and problem solving as art.

Problem solving as context. The authors divide problem solving as a context for doing mathematics into several subcategories. Problem solving has been used as justification for teaching mathematics. To persuade students of the value of mathematics, the content is related to real-world problem-solving experiences. Problem solving also has been used to motivate students, sparking their interest in a specific mathematical topic or algorithm by providing a contextual (real-world) example of its use. Problem solving has been used as recreation, a fun activity often used as a reward or break from routine studies. Problem solving as practice, probably the most widespread use, has been used to reinforce skills and concepts that have been taught directly.

When problem solving is used as context for mathematics, the emphasis is on finding interesting and engaging tasks or problems that help illuminate a mathematical concept or procedure. To use problem solving as context, a teacher might present the concept of fractions, for example, assigning groups of students the problem of dividing two pieces of licorice so that each gets an equal share. By providing this problem-solving context, the teacher’s goals are multiple: to create opportunities for students to make discoveries about fraction concepts using a familiar and desirable medium (motivation); to help make the concepts more concrete (practice); and to offer a rationale for learning about fractions (justification).

Problem solving as a skill. Advocates of this view teach problem solving skills as a separate topic in the curriculum, rather than throughout as a means for developing conceptual understanding and basic skills. They teach students a set of general procedures (or rules of thumb) for solving problems—such as drawing a picture, working backwards, or making a list—and give them practice in using these procedures to solve routine problems. When problem solving is viewed as a collection of skills, however, the skills are often placed in a hierarchy in which students are expected to first master theability to solve routine problems before attempting nonroutine problems. Consequently, nonroutine problem solving is often taught only to advanced students rather than to all students. When defining the learning objectives of a problem-solving activity, teachers will want to be aware of the distinction between teaching problem solving as a separate skill and infusing problem solving throughout the curriculum to develop conceptual understanding as well as basic skills.

Problem solving as art. In his classic book, How To Solve It, George Polya (1945) introduced the idea that problem solving could be taught as a practical art, like playing the piano or swimming. Polya saw problem solving as an act of discovery and introduced the term “modern heuristics” (the art of inquiry and discovery) to describe the abilities needed to successfully investigate new problems. He encouraged presenting mathematics not as a finished set of facts and rules, but as an experimental and inductive science. The aim of teaching problem solving as art is to develop students’ abilities to become skillful and enthusiastic problem solvers; to be independent thinkers who are capable of dealing with open-ended, ill-defined problems.

Problem Recognition, Definition, and Representation

Problem recognition, definition, and representation are metalevel executive processes, called metacomponents in Sternberg’s (1985) triarchic theory of human intelligence. This theory proposes that metacomponents guide problem solving by planning, monitoring, and evaluating the problem-solving process. The metacomponents include such processes as (1) recognizing the existence of a problem, (2) defining the nature of the problem, (3) allocating mental and physical resources to solving the problem, (4) deciding how to represent information about the problem, (5) generating the set of steps needed to solve the problem, (6) combining these steps into a workable strategy for problem solution, (7) monitoring the problem-solving process while it is ongoing, and (8) evaluating the solution to the problem after problem solving is completed. In this theoretical context, the processes of problem recognition, definition, and representation correspond to the first, second, and fourth metacomponents, which are used in the planning phase of problem solving.

 

Problem recognition, also referred to as problem finding, is one of the earliest stages of problem solving. Getzels (1982) classified problems based on how they were “found.” According to Getzels, there are three kinds of problems: those that are presented, those that are discovered, and those that are created. A presented problem is one that is given to the solver directly. In this case, there is no need to recognize or find the problem; it is stated clearly and awaits solution. A discovered problem, however, is one that must be recognized. Such a problem already exists, but it has not been clearly stated to the problem solver. In this case, the problem solver must put together the pieces of the puzzle that currently exist and seek out a gap in current understanding in order to “discover” what the problem is. In contrast to presented and discovered problems, the third class of problems comprises those that are created.

Created problems are those in which the problem solver invents a problem that does not already exist in the field. For this reason, one can argue that a created problem will, in some sense, always produce a creative solution, simply because its problem statement deviated from the usual way of thinking about the problem. Getzels and Csikszentmihalyi (1976) found that artists who spent more time in the problem-finding stage while creating an artwork were judged to have more creative products than did artists who spent less time in problem finding. In fact, the artists who spent more time also remained highly creative seven years later. For the purposes of this chapter, problem recognition refers to both discovered and created problems.

Problem definition is the aspect of problem solving in which the scope and goals of the problem are clearly stated. For example, a presented problem may be easy to define if the problem statement has been prepared for the solver. However, some presented problems are not clearly stated, requiring the problem solver to clarify the precise definition of the problem. Discovered problems usually require definition because the problem solver has identified the problem in his or her field. Defining a created problem is likely to be a challenge, given that the problem solver has gone beyond the current field in inventing the need for a solution in the first place. Problem representation refers to the manner in which the information known about a problem is mentally organized. Mental representations are composed of four parts: a description of the initial state of the problem, a description of the goal state, a set of allowable operators, and a set of constraints.

By holding this information in memory in the form of a mental representation, the problem solver is able to remember more of the problem by chunking the information, in order to organize the conditions and rules of a problem to determine which strategies are useful, and to assess progress toward the goal state (Ellis & Siegler, 1994; Kotovsky, Hayes, & Simon, 1985; Newell&Simon, 1972).Aproblemmaybe represented in a variety of ways, for example, verbally or visually. Even a presented problem may require the generation of a new representation in order to be solved. For example, given the problem of finding your way to a new location, you may find it much easier to follow a map than to read a set of directions. If you have trouble following the map, then it may be worthwhile to write out a description of the route in words, re-representing the information in a way that makes it easier to get to your destination. It is important to note that these three aspects of problem solving are not discrete, sequential stages in the solution process, but rather are interactive and often difficult to tease apart in a real problem-solving situation. When a problem is represented in a new way, the problem solver may decide to redefine the goal accordingly. Similarly, a redefinition may lead to a new representation. It is useful to consider the roles of problem recognition, definition, and representation in the solution of well-defined versus ill-defined problems.

Recall that a well-defined problem is one whose path to solution is straightforward, whereas an ill-defined problem is one that does not lend itself to a readily apparent solution strategy. Consider the following well-defined problem, referred to as the Tower of Hanoi problem:

There are three discs of unequal sizes, positioned on the leftmost of three pegs, such that the largest disc is at the bottom, the middle-sized disc is in the middle, and the smallest disc is on the top. Your task is to transfer all three discs to the rightmost peg, using the middle peg as a stationing area, as needed. You may move only one disc at a time, and you may never move a larger disc on top of a smaller disc. (Sternberg, 1999)

The problem here is easy to recognize: One needs to move the discs onto the rightmost peg. The problem is also defined clearly; the relative sizes of the discs as well as their locations are easy to distinguish. Also, the solution path is straightforward based on this representation. Working backward, one realizes that the largest disc must be placed onto the rightmost peg, and in order to do so, the other two discs must be removed. So that the mediumsized disc does not end up on the rightmost peg, the smallest disc must first be moved to the far right. Then the medium disc is placed on the middle peg; the small disc is placed on top of the medium disc. The large disc is then free to be placed on the rightmost peg. Finally, the small disc is moved to the left so that the medium disc is free to move to the rightmost peg. The last step is then to move the small disc atop the other two and the problem is solved. Note that this well-defined problem can be expanded to include many pegs and many discs of varying sizes, but its solution will always proceed according to the algorithm described in this, the simplest case.

For the most part, well-defined problems are relatively easy to recognize, define, and represent. However, a well-defined problem may entail some degree of “problem finding,” in the sense that a problem exists but must first be discovered. For example, a scientist may struggle to identify a gap in the existing literature on a problem, but the actual process of filling that gap may come easily once the problem itself has been identified.

The solution to the discovered problem may follow a path similar to that of other problems in the field (e.g., experimental methods). For example, much early psychological research was conducted using male participants. When a researcher questioned the validity of the results for females, a new problem had been discovered. Given this new problem, the path to solution was well defined: Simply use the same experimental method but include female participants in the study. In this sense, this well-defined problem was somewhat difficult to recognize, yet once identified, it was easily defined and represented in familiar terms. The representation of well-defined problems is not necessarily easy, however. Consider another problem: Three five-handed extraterrestrial monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighborhood, both monsters and globes come in exactly three sizes, with no others permitted: small, medium, and large. The small monster was holding the large globe; the medium-sized monster was holding the small globe; and the large monster was holding the medium-sized globe. Since this situation offended their keenly developed sense of symmetry, they proceeded to transfer globes from one monster to another so that each monster would have a globe proportionate to its own size. Monster etiquette complicated the solution of the problem since it requires that: 1. only one globe may be transferred at a time; 2. if a monster is holding two globes, only the larger of the two may be transferred; and, 3. a globe may not be transferred to a monster who is holding a larger globe. By what

sequence of transfers could the monsters have solved this problem? (See Kotovsky et al., 1985)

Hanoi problem (Newell & Simon, 1972). However, it is actually directly isomorphic to (i.e., its structure is exactly the same as that of) the Tower of Hanoi problem. In this case, it is the difficulty of representing the problem correctly that increases the level of difficulty of the problem as a whole. After you are told of the isomorphism between the two problems, the solution is simply a matter of mapping relationships from one problem to the other. In summary, problem definition is usually easy for the class of well-defined problems; however, accurate problem recognition and representation are not necessarily straightforward, even when the scope and goals of the problem are clear. In the case of ill-defined problems, however, it is often the case that all aspects of problem formulation are relatively challenging. Perhaps the easiest stage in attempting to solve an ill-defined problem is that of problem recognition. It is often relatively simple to identify a fuzzy problem. For example, it is easy to identify the problem of developing a test of creativity. It is hard, however, to define the exact contents of such a measure. The real difficulty in solving an ill-defined problem is in clarifying the nature of the problem:howbroad it is, what the goal is, and so on. Although well-defined problems have a clear path to solution, the solution strategy for an ill-defined problem must be determined by the problem solver. To develop a problem-solving strategy, it is first necessary to specify the goals of the task. For example, if we take on the task of designing a creativity test, we must decide whether the goal is (a) to estimate the creativity of undergraduate psychology majors or (b) to measure creative potential among people of all ages and educational and cultural backgrounds. Before the path to solution can be constructed, the goal must be clear.

Errors in Problem Solving Story

The errors of students in solving problems closely related stories difficulties experienced by students in solving story problems. Difficulties in resolving the matter of the story students, according to Ahmad Syafri in Rahardjo and Astuti (2011: 14) can be broadly classified as follows.

• Difficulty in understanding the issues (problems), the difficulty in determining what is known and what is being asked in the matter. 
• Difficulties in the settlement plan, which is about the difficulty in translating the story into the model (phrase) mathematics. 
• Difficulty in completing the plan, the difficulty in resolving models (phrase) mathematics.
• Difficulty in seeing (check) returns results that have been obtained. 
• The difficulty in interpreting the answers to the situation of the problem contained in the matter.

Sudjono in Supin (2001:8) describes some of the difficulties in solving the problem in terms of a psychological story is:

• Students do not understand what is read, due to the lack of knowledge of some terms that do not know, whether from foreign languages ​​or from traditional languages.
• Students are not able to live up to what is described in the matter, due to the events in question are not in accordance with the student experience. 
• Students are not able to choose the mathematical concepts that are relevant to the demands of matter in developing a mathematical model. 
• Students are not capable of computing (the math). In this case also means that the students already know conceptually counting regulations, but many can count finished insufficient servings specified time, or there are many wrong answers students because they lack accuracy.

Based on the difficulties of solving the problem raised by the story and Soedjono Syafri Ahmad, the general errors that may occur in solving story problems are (1) an error in determining the things that are known to the matter, (2) difficulty in determining things that are asked by the question, (3) errors in the mathematical model, (4) errors in computing, and (5) errors in interpreting the results obtained.
According to Allan L.White (2005:17) errors in solving math problems include:

a.       Reading Errrors (R)

Errors will be classified as a reading if students can not read key words or symbols written on the issue. This discourages students perform the next procedure in the proper groove problem solving.

b.      Comprehension Errors (C)

Students have been able to read all the words in question, but do not understand the whole meaning of the words, so that students are not able to move further along the grooves solving the right problem.

c.       Transformation Errors (T)

Students have been able to understand what the question is to be found but was unable to identify the operation or sequence of operations, which are necessary to solve the problem.

d.      Process Skills Errors (P)

Students recognize the corresponding operation or sequence of operations but did not know the procedures necessary to carry out operations accurately.

e.      Encoding Errors (E)

Students correctly solve the solution of a problem, but the solution cannot be expressed in the form of proper notation.

From the description of the types of errors completing a math problem above in relation to the type of errors in solving math story problems. Types of mistakes made by students in solving math story problems can be determined based on step-by-step story about the settlement. Reading and comprehension errors are misunderstanding about the story. In the aspect of the student should be able to read the entire sentence and look for important information when reading and understanding about the story. In the aspect of mathematical modeling, students can perform transformation errors, it is because students are not able to change the sentence into a sentence about math. Process skill errors are errors in the process of completing math story problems. In this case, students make the mistake of computing the student made a mistake in arithmetic operations or procedural errors. While encoding errors is error infer that students cannot write about the conclusions in the context of the question.

Density of Decimals

Density of decimals : One of the features distinguishing decimals from whole numbers is the density of decimals. Hiebert et al., (1991) found improving the continuity aspects of decimals’ density was particularly difficult. Working with problems involving continuous models in the written tests and the interviews, such as marking a representation of a decimal number on a number line, or finding a number in between two given decimals such as 0.3 and 0.4 were found to be more challenging than working with discreterepresentation task utilizing MAB models. Analysis for this finding suggested that an extra step in finding the unit of the continuous models explained the lower performance on continuous-representation tasks. In Table 2.1, A2 (money thinking) describes students thinking like this.

Likewise, Merenlouto (2003) found that only a small portion of Finnish students aged 16-17 years old in her study changed their concept of density. She attributed difficulties with grasping density to students’ reference to natural numbers and difficulties in extending their frame of reference to rational or real numbers. Some students relied on the possibility to add decimals in their explanations for recognizing density of decimals. Furthermore, she contended that this kind of explanation was based on an abstraction from natural numbers properties rather than a radical conceptual change from natural to real numbers.

Difficulties with density were also evident in studies involving pre-service teachers. Menon (2004) found only 59% of 142 pre-service teachers recognized the density of decimals. A similar trend was noted by Tsao (2005) who found that of 12 pre-service teachers involved in her study, only the six high ability students demonstrated an understanding of density.

The nature of incorrect responses with regard to the density of decimals is reflected in common misconceptions drawing on analogies between decimals and whole numbers. Clearly density will not make sense to students holding misconceptions identified in Table 2.1 such as money thinking, denominator focussed thinking, reciprocal thinking, and place value number line thinking. In general incorrect answers in recognizing the density of decimals could be classified in two categories. The first category of incorrect answers is identifying no decimal existing in between pairs of decimals. Fuglestad (1996) found that most students in her study of Norwegian students claimed there were no decimals in between two given decimals such as between 3.9 and 4 or between 0.63 and 0.64. Similarly, Bana, Farrell, and McIntosh (1997) reported that the majority of 12 year olds and 14 year olds from Australia, US, Taiwan and Sweden displayed the same problem. Only 62% of 14 year olds from Australia and 78% of 14 year olds from Taiwan showed understanding of decimal density. This evidence reflected incorrect extension of whole number knowledge that there is no whole number in between two consecutive whole numbers such as 63 and 64. Note that students holding money thinking (allocated to A2 code in Table 2.1) also will have difficulty in grasping the density notion of decimals and identify no decimals in between decimals such as 0.63 and 0.64. However, these students might identify 9 decimals in between 0.6 and 0.7 for instance, if they interpret decimals only as a number system for dollar and cents.

The second category of incorrect answer translates knowledge of multiplicative relations between subsequent decimal fractions. For instance, Hart (1981) reported that 22 to 39% students age 12 to 15 year-old thought there were 8, 9, or 10 decimals in between 0.41 and 0.42. Similarly, Tsao (2005) observed the same phenomenon in her study with pre-service teachers. She found that three pre-service teachers from a low ability group believed there were nine decimals in between 1.42 and 1.43 by sequencing only the thousandths: 1.421, 1.422,…, and 1.429. Along with most of the students in L and S groups (see Table 2.1), some students holding A thinking, such as A2 thinking with reference to metric measures (m, cm, mm) might possibly respond in this way.

Education System of Indonesia

The character of Indonesia's educational system reflects its diverse ethnic and religious heritage, its struggle for a national identity, and the challenge of resource allocation in a developing archipelago nation with a young and rapidly growing population. Although a key government goal is to provide every Indonesian with at least nine years of basic education, the aim of universal education has not been reached, particularly among females – although great improvements had been made. Obstacles to meeting the government’s goal include a high birth rate, a decline in infant mortality, and the shortage of schools and qualified teachers. In 1973 the government issued an order to set aside portions of oil revenues for the construction of new primary schools. This act resulted in the construction or repair of nearly 40,000 primary schools by the late 1980s, a move that greatly facilitated the goal of universal education. 

The education system has its roots in the Indonesian culture based on Pancasila (five principles of the national ideology) and the 1945 Constitution. In the National Education Act No. 2 of 1989, the system aims at generating abilities and increasing the standard of living and dignity of the Indonesian people in order to achieve the national development goal.

In Indonesia the schools are operating under the auspices of several ministries. Public secular schools and non-Muslim private schools are under the responsibility of the Ministry of National Education, while Islamic schools are administrated under the Ministry of Religious Affairs. A small number of institutions designed for training specialized personnel are managed by several other ministries, such as the Ministry of Home Affairs and the Ministry of Defense and Security.

The system consists of seven types of education (see Figure 2.2), they are: (1) General education emphasizes the development of general knowledge and improvement of skills for the students. Specialization is also needed in the 12th grade; (2) Vocational education prepares students in mastering a number of specific vocational skills needed for employment; (3) Special education provides important skills and abilities for student with physical and/or mental disabilities; (4) Servicerelated education aims at increasing abilities required for certain government officials or prospective officials to implement a certain task; (5) Religious education prepares students to play a role which demands the mastery of specific knowledge about religion and related subjects; (6) Academic-oriented education focuses primarily on improving the mastery of science; and (7) Professional education prepares students primarily on mastering specialized or job-related knowledge and skills.

The level of education that includes formal school system consists of: basic education, secondary education, and higher education. Apart from the levels of education mentioned above, pre-school education is also provided. Out-of-school education can be held at the outside schools and provided by governmental and non-governmental agencies of private sector and the community.

Following kindergarten, Indonesians of between six and fifteen years of age have to attend a six-year primary school and a three-year junior high school. They could choose between state-run, nonsectarian public schools supervised by the Ministry of National Education or private or semiprivate religious (usually Islamic and called Madrasah Ibtidaiyah and Madrasah Tsanawiyah) schools supervised and financed by  the Ministry of Religious Affairs. Although 85 percents of the Indonesian population are registered as Muslims, less than 15 percents attend the religious schools. Enrolment was slightly higher for girls than boys and much higher in Java than in the rest of Indonesia.

A central goal of the national education system in the early 1990s was not merely to convey secular wisdom about the world, but also to instruct children in the principles of participation in the modern national-state, its bureaucracies, and its moral and ideological foundations. Since 1975, a key feature of the national curriculum – as in other parts of society – had been instruction in the Pancasila (The five principles of national identity of Indonesia). Children aged six and above learned the five principles – belief in one God, humanitarianism, national unity, democracy, and social justice – by rote and were instructed daily to apply the meanings of this key national symbol to their lives. Inside the public school classroom of the early 1990s, a style of pedagogy prevailed that emphasized rote learning and respect to the authority of the teacher. Although the youngest children were sometimes allowed to use the local language, by the third year of primary school nearly all instruction was conducted in formal Indonesian. Instead of asking questions of the students, a standard teaching technique was to narrate a historical event or to describe a mathematical problem, pausing at key junctures to allow the students to fill in the blanks.

After completion of the nine-year basic education, senior secondary education is available. The paths of senior secondary education include general high school, vocational high school, religious high school, service related high school, special education, and professional secondary education. The length of study in secondary education is three years for general high school and three or four years for other senior high schools. In addition to the senior secondary education, there is also an Islamic general senior high school called madrasah aliyah, which is equivalent to general senior high school, but it is managed and run by the Ministry of Religious Affairs.

Higher education is an extension of secondary education consisting of academic and professional education. Academic education is mainly aimed at mastering science, technology, and research, whereas professional education is aimed more at developing practical skills. Institutions involved in higher education are of the following types: academies, polytechnics, schools of higher learning, institutes, and universities. The length of study in higher education is three years for diploma program and four years for graduate program. Upon completing a graduate program, students can continue to master program for two years, and finally to doctorate program for three years 

In 1998, school enrolment for population aged 7-12 years and 13-15 years was 95.07 percents and 77.31 percents respectively, and only 49.52 percents of those aged 16-18 years went to schools. The national adult literacy rate was 89.42 percents in 1998 (93.41 percents for males and 85.54 percents for females) (Table 2.1).