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.

Challenges of Teaching Problem Solving

Challenges of Teaching Problem Solving - Although Polya presented the inquiry-based framework for teaching problem solving more than 50 years ago, there has yet to be widespread implementation of his ideas in U.S. classrooms. This suggests that there are a number of challenges to making this shift in mathematics teaching.

Teaching nonroutine problem solving is difficult. True problem solving is as demanding on the teacher as it is on the students. The art of teaching mathematical problem solving is best mastered over a long period of time (Thompson, 1989). Teaching problem solving is difficult, writes Schoenfeld (1992). Teachers:

• Must perceive the implications of students' different approaches, whether they may be fruitful and, if not, what might make them so.
• Must decide when to intervene, and what suggestions will help the students whill leaving the solution essentially in their hands, and carry this through for each student.
• Will at times be in the position of not knowing; to work well without knowing all the answers requires experience, confidence, and self-awareness.

Burkhardt (1988, as cited in Schoenfeld, 1992) states even more succinctly that teaching problem solving is difficult for teachers mathematically, pedagogically, and personally. Teachers must have the mathematical expertise to understand the different approaches that students might take to a problem and how promising those approaches will be. Many elementary teachers are trained as generalists and often do not have the strong mathematical background required to teach from a problem-solving approach.

Pedagogically, teachers must make complex decisions about the level of difficulty of the problems assigned, when to give help, and how to give assistance that supports students’ success while ensuring that they retain ownership of their solution strategies. Personally, teachers will sometimes find themselves in the uncomfortable position of not knowing the solution. Letting go of the “expert” role teachers have traditionally played requires experience, confidence, and self-awareness. Often, teachers are asked to teach mathematics they never encountered in school and in a way that differs from how they were taught. For these reasons, teachers may need additional training in mathematical content and theory, as well as in methods for teaching problem solving.

Nonroutine problems are difficult for students. Nonroutine, open-ended problems are often, by their nature, difficult for many students. Shannon and Zawojewski (1995) conducted a ministudy that demonstrated the difficulty presenting problem-solving tasks without providing hints and procedural steps poses to students. In the study, two groups of students were presented with similar tasks. In one task, “Supermarket Carts,” students were given a scale drawing of 12 shopping carts nested together and asked to create a rule to determine the length of storage space needed for any number of carts and the number of carts that would fit into a given space. This was essentially all the direction given.

A second group of students was assigned the task “Shopping Carts,” which included several prompts or subproblems to help guide them toward a solution. Students were asked to find the length of one shopping cart, find how much a cart sticks out when the carts are nested, find the total length of 20 carts, and find how many carts could fit into a 10-meter space. Then they were asked to find the two formulas that were asked for in the Supermarket Carts task.

The researchers reported that students attempting the Supermarket Carts task had difficulty knowing how to get started. Only a few students successfully derived the formulas required. On the other hand, none of the students working on the Shopping Carts task had any difficulty getting started, and all but one group successfully derived the requested formulas. The authors conclude that, “the sense of students’ having to struggle was greater in Supermarket Carts than in Shopping Carts.” Watching their students struggle in frustration is often very difficult for teachers. Knowing when to give hints and how much help to give requires striking a delicate balance that comes with experience and knowing students’ capabilities.

Teachers are concerned about content coverage. The TIMSS research characterized the U.S. curriculum as “a mile wide and an inch deep” compared to the mathematics curriculum in other countries (Peak, 1996, 1997; Takahira, et al, 1998). Teachers in the U.S. are generally expected to cover large areas of content each year. Yet solving challenging, nonroutine problems takes time. Often a single problem can occupy a class for a whole period or more. Therefore, it’s essential that content and skills be integrated within the context of problem solving. By selecting rich, engaging, and worthwhile tasks, teachers can ensure that time is well-spent.

Textbooks present few nonroutine problems . Although they are improving, many textbooks do not provide an adequate number of nonroutine problems from which teachers can choose. Many teachers are not comfortable straying from the scope and sequence the textbook provides, but they must develop the confidence to search out and develop other materials to supplement their texts.

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.

Why Teach Open-Ended Problem Solving?

Why Teach Open-Ended Problem Solving - To help young people be better problem solvers is to prepare them not only to think mathematically but to approach life's challenges with confidence in their problem-solving ability. The thinking and skills required for mathematical problem solving transfer to other areas of life. The writers of the groundbreaking report Everybody Counts: A Report to the Nation on the Future of Mathematics Education put it this way:

Experience with mathematical modes of thought builds mathematical power—a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live (National Research Council, 1989). Learning mathematics by grappling with open-ended and challenging problems accommodates diverse learning styles. The active and varied nature of problem solving helps students with diverse learning styles to develop and demonstrate mathematical understanding (Moyer, Cai, & Grampp, 1997). Traditional teaching approaches involving rote learning and teacher-centered instructional strategies often do not meet the learning needs of many students who may be active learners or require multiple entrances into the curriculum.

Learning through open-ended problem solving helps students to develop understanding that is flexible, that can be adapted to new situations and used to learn new things (Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997).

“Things learned with understanding are the most useful things to know in a changing and unpredictable world,” explains Hiebert and colleagues. Yet, usefulness is not the only reason to learn with understanding. To learn with understanding is to also grapple intellectually with mathematics as a subject. “When we memorize rules for moving symbols around on paper we may be learning something, but we are not learning mathematics,” says Hiebert. “Knowing a subject means getting inside it and seeing how things work, how things are related to each other, and why they work like they do.”

When students encounter mathematical ideas that interest and challenge them in an openended problem solving context, they are more likely to experience the kinds of internal rewards that keep them engaged, says Hiebert (Hiebert et al., 1997). Students who must resort to memorizing will lack understanding and will likely feel little sense of satisfaction, perhaps withdrawing from learning altogether. In fact, he says, evidence suggests that if students memorize and practice procedures repeatedly in a rote fashion, it's difficult for them to go back later and gain a deeper understanding of the mathematical concepts underlying those procedures. Researchers Jerry Becker and Shigeru Shimada (1997) concur: “Lessons based on solving open-ended problems as a central theme have a rich potential for improving teaching and learning.”

Recognizing the centrality of problem solving to mathematics learning, education leaders have made it a focal point of standards reform for the past two decades. In the spring of 2000, the National Council of Teachers of Mathematics renewed its commitment to problem solving when it published Principles and Standards for School Mathematics (NCTM, 2000), an update of the council's 1989 statement about standards for teaching and learning mathematics, Curriculum and Evaluation Standards for School Mathematics. Like that seminal work, the council's updated standards identify problem solving as an essential component of math learning for all grade levels.

Furthermore, many states have adopted content and performance standards and assessments based on the NCTM standards that include an emphasis on problem solving. While the Northwest states are at different stages in the process of adopting standards and developing assessment systems, most are addressing the importance of teaching and assessing reasoning, communicating, making connections, and applying knowledge to problem situations—key tenets of problem solving.

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.

Strategies for introducing RME in teacher education in Indonesia

As reported by de Lange (1996), RME has been adopted and adapted in some projects in different countries such as USA, Portugal, England, Germany, Spain, Brazil, Denmark, South Africa, Japan, and Malaysia. For example, in the USA, through a collaboration project between the Freudenthal Institute and the National Science Foundation, RME is adopted and redesigned in the Mathematics in Context (MIC) textbooks for grade 5-8. After the books were used by students in several school districts from different states, preliminary research showed that the student achievement in the state tests increased. In one of these examples, ninth-grade students in the Ames district (Iowa state) with three years experience using these books were recently tested with the Iowa Tests of Educational Development. In this test, 25% scored in the top 1% of the nation, 47% scored in the top 10%, and 90% scored above the national median (Romberg & de Lange, 1998).

However, these positive results were achieved after facing some obstacles. As Clarke, Clarke & Sullivan (1996) mentioned that in the beginning implementation of MIC textbooks, teachers found difficulties to teach the book materials in the classrooms. Although each pupil has his/her own book, teachers were used to make their own teaching materials. They want to use the books but in some cases they want to make their own materials by adapting the materials in the books. Therefore, the project provided a professional development program for teachers learned how to teach new materials using a new approach and how to redesign the materials based on teacher needs. This experience can be used as a good example if RME materials want to be introduced then teachers are need to be trained and supported in implementing such materials.

In the country where RME originally has been developed and implemented for about 30 years, the Netherlands, there are also positive results that can be used as indicators that RME might be promising to increase the quality of mathematics education. For instance, the results of the Third International Mathematics and Science Study (TIMSS) showed that pupils in the Netherlands gained high achievements in mathematics education which was ranked 6th out of 38 participating countries and the gap between smart pupils and weak pupils was very small (Mullis et al., 2000). The latter achievement might be interpreted that RME philosophy of mathematics as human activity was achieved or more precisely that mathematics not only for the smart pupils but also for the weak. Thus, "mathematics for all" which is only slogan for some countries including Indonesia has been achieved in the Netherlands. Still, these positive results could be achieved not in the short-term but in the long-term endeavor.

Based on the explanations in the previous sections, RME looks promising to be introduced and implemented in Indonesia because it could increase pupil's understanding and pupil's motivation toward mathematics. For instance, RME content materials are developed using the contexts that are experientially real to the pupils. This can increase pupil's motivation toward mathematics. Also, the teacher using that materials can guide the pupils learn from real level to abstract level of mathematics concepts. With the interaction and construction, this way of learning leads pupils to understand the concept of mathematics. Yet, in order to do so, there are three warnings are should be considered.

First, RME curriculum materials are not easy to be designed and learned by teachers because the mathematics materials differ from former ones in that they emphasize application problems with a loose structure and a redefinition of basic skills. For instance, the assessment materials focus on middle-level and high-level order thinking instead of the low level only. Second, teachers need to be educated how to use RME materials in their classroom. According de Lange (1993) the role of teachers in RME changes from teaching to 'un-teaching'. Finally, the implementation of RME is not a short-term program or project, but it needs many years to be institutionalized. These changes are consistent with Fullan's (2001) suggestion that the innovation of teaching is a complex undertaking for teachers, usually involving a combination of changes in the following areas:(1) new curriculum materials or changed use of existing materials; (2) new knowledge and skills required by the teacher; and (3) new values and attitudes concerning pupil learning and the new patterns of work in the classroom. Therefore, in order to introduce RME in Indonesia it is important to take in to account the obstacles that were faced either by the MIC project or Dutch experts. Related to the Fullan's suggestion, the following questions arise: How to develop or adapt new curriculum materials to the new context? How can teachers be helped in implementing these materials in the classroom practice? What advice can be given? What support can be offered? Selter (1997) pointed out that these are all mediated through the teacher, specifically through teacher's beliefs about how to organize and facilitate pupil's learning of mathematics. In this context, teacher education (pre-service as well as in-service) plays a vital role. One key strategy in this situation is to engage teachers or student teachers in their professional development using the following strategies (Loucks-Horsley, Hewson, Love & Stiles, 1998): (1) a short course (for building knowledge by teachers or student teachers); (2) curriculum development (by adapting the innovative materials into the school practice); and (3) using technology (in order to provide teachers or student teachers with a sustainable tool which support rich information about the new approach).

RME and Examples From Practice

In 1973 Indonesia government implemented modern mathematics to replace arithmetic as subject matter in primary schools. The implementation of modern mathematics in primary schools has been a problematic since its beginning. For many teachers, modern mathematics was a difficult subject. The difficulty implied to the practice on mathematics teaching and learning in which teachers were relying on the mathematics textbooks. They conducted instruction by following the textbook page by page, without considering the correctness the mathematics written in the book (Somerset, 1997).

The teaching and learning of mathematics in Indonesia schools tends to be very mechanistic, namely mathematics teachers tend to narrate mathematics formulas and procedures to their pupils. For a long time this teacher-centered approach has influenced pupils’ attitude to be passive learners. They are used to be spoon-fed by the teachers. They are not used to thinking in critical way for self-learning. The impact of the situation may be seen on the achievement in national examinations and in international comparative studies. In national examination from 1990 to 1997, junior secondary school pupils’ average score in mathematics is always below 5 out of 10 scale, making it consistently the lowest scoring of all the subjects taught in school (Manan, 1998). In international comparative studies like TIMSS and PISA, Indonesian pupils perform below most other participating countries.

A transition from a more traditional, skill-oriented approach towards a problem-based, reform mathematics approach constitutes a complex innovation, because it does not only involve the introduction of new instructional sequences and new instructional activities, but also asks for new roles for the teacher and new social and sociomathematical norms. Teachers will have to foster a problem-solving classroom culture, they will have to bring students to changing their current more passive receptive role into one of more active participation, in which they take initiative and responsibility, and learn to think and reason for themselves. In addition, teachers will have to learn to guide the new learning process by choosing or designing those instructional tasks that may generate productive mathematical thinking at any given moment in time, and to organize and orchestrate whole-class discussions that help students in making mathematical process. The role of the teachers as a consequence will change from authoritarian, instructional oriented, and teacher centered, toward more supportive, more student centered, more learning condition oriented.

Consequently, key component of successful mathematics education reform will consist of in-service and pre-service teacher education, co-teaching in classrooms, and the production of supportive textbooks and teacher manuals.. Another, equally important element concerns the requirement to make sure that the intended innovation fits the Indonesian context. In this respect, an important prerequisite for success will be a sense of ownership of the teachers and teacher educators who are involved. Thus a bottom-up approach is called for, in which the Indonesian teachers and teacher educators ‘re’-invent a form of realistic mathematics education that fits the Indonesian situation and priorities. The theory of RME is useful in several countries, such as in the Netherlands and the US. However, much more important than this evident is that the concept of RME itself is in line with the current thinking in Indonesia about mathematics learning which emphasize on pupil-active learning, problem solving and application of mathematics. It is a common phenomenon currently in Indonesia that the objective of teaching and learning mathematics is to develop pupils’ reasoning and logical ability. If we carefully listen to the messages from mathematics teachers in Indonesia, then one of their concerns is how to make mathematics lesson relevant for pupils in dealing with the daily life problems. It is also argue that mathematics should be mastered as a systematic pattern of reasoning (Nasution, 1996). The (re)construction of mathematics ideas and concepts goes hand in hand with with the process of the development of pupil’s reasoning ability. This can be achieved in RME through pupils’ exposure to contextual problems within the framework of an interactive teaching and learning
process.

Examples From Practice
Primary school teachers from several cities in Indonesia use RME in their mathematics lesson. In the following two examples are given. The teaching and learning of fractions

 “Students, tomorrow we will have lunch together with bread. You will be devided into groups and each of groups bring your own bread, bread-knife, and jam. Wow, they imagine that how cheerful it will be. “I don’t like jam, Mom, may I bring such butter or sugar? “Alright, You can bring cheese even everything what you like”. It is a situation rarely found in mathematics teaching session. With RME mathematics is so real and exciting. At a glance they seem to gladly play but actually they are learning mathematics since first time. It is not only easy to digest mathematics concept but also it will be strengthen their concept because of their own experiences. Learning using RME is exciting but it isn’t without risks. What are the risks?? They are two important things in learning using RME, realistic and re-invention. We can imagine if in every topic or sub topic we have to find activity that is able to bring student to the relevant material, how it will be spend time so much? Furthermore, it exclude time we need to have them build their conception until re-invention.

Obtaining the difficulties, it seems to be hard to realize RME. Nevertheless, it can believed there is problem without solution. What is that? The appropriate solution is intertwining. What is that and what about the application? Reality in life shows that a problem is actually a link of things linked each other. It is called intertwine. Link among units will make the all aspects of mathematics learning effective because a teacher doesn’t have to explain the material page to page . 

The following example is one of in intertwine implementation in fraction

By simple activities teacher gives stimulant and direction to the students. They are able to re-invent and learn several fraction materials in intertwine all at once as follows:

• Fraction as division (in activity-1)
•  Fraction as part of a whole. (in activity -1&2)
• Comparing fractions with the same denominator(in activity-2)
• Putting fraction with the same denominator in the right order. (in activity-2)
• equivalent fraction and simplifying fraction (in activity-3)

They have sliced bread and spread it with jam and the end of the activity they have
eaten it together. Unconsciously,they have learnt mathematics joyfully.

Five Tenets of RME

According to van Hiele (de Lange, 1996) learning process is a process through three
levels:

• A student reaches the first level of thinking if she/he could manipulate familiar characteristic from recognized patterns.
• A student reaches the second level of thinking if she/he could manipulate the relation of those characteristic.
• A student reaches the third level of thinking if she/he could manipulate intrinsic characteristics of relations.

Traditional instruction has been modeled from the second to the third level. According to de Lange (1996) this never happen if we start from real world. Researchers in the Freudenthal Institute notice that van Hiele theory is important not because of its theoretical application but its practical implication. First, mathematics might be started from the level in which concepts being used have a high degree of familiarity for students, and second the goals of mathematics instruction is to create a relational framework (Gravemeijer, 1994).

So, mathematics learning process should be emphasized to concepts in which students familiar with. Each student has a set of knowledge that she/he has as a result of her/his interaction with environment or from previous learning process. After students are involve in meaningful learning process, they develop their knowledge to the higher level. In this process students are actively involve in gaining new knowledge. The construction of knowledge is a modification process that move slowly from first level to second level and third level. In this process students are responsible to their own learning activities.

The thinking framework of mathematical ideas go through three phases, namely exploration, identification, and application phases which stimulate new exploration phase that could be used to develop new mathematical concept. Those phases are cyclic processes without objective that was determined beforehand. There is no intention to come to certain goal. Learning progress highly depends on students’ ability to acquire new knowledge. Gravemeijer (1994) argues that there is no place for pre-program teaching and learning process, since all processes depend on students individual contribution and should be approved by teacher and students. Based on this argument Clarke, Clarke, and Sullivan (1996) contend that RME put its theory as process of inquiry as well as socio-constructivism of learning.

Cobb (1994) contends that RME theory is compatible to domain specific instructional theory which is rely on real world application and modeling. Alignment between constructivism and RME is mainly on similar characteristics on mathematics and mathematics learning. Both theories use premise that mathematics is human creative activity and mathematics learning took place when students are able to develop effective way of solving problems (de Lange, 1996; Steefland, 1991; Treffers, 1987). Using Cobb’s description, de Lange (1996) states RME tenets:

1. Use of contextual problems (contextual problems as applications and as starting points from which the intended mathematics can come out).
2. Use of models or bridging by vertical instruments (broad attention is paid to development models, schemas and symbolization rather than being offered the rule or formal mathematics right away).
3. Use of students' contribution (large contributions to the course are coming from student's own constructions, which lead them from their own informal to the more standard formal methods).
4. Interactivity (explicit negotiation, intervention, discussion, cooperation and evaluation among pupils and teachers are essential elements in a constructive learning process in which the student's informal strategies are used as a lever to attain the formal ones).
5. Intertwining of learning strands (the holistic approach implies that learning strands can not be dealt with as separate entities; instead an intertwining of learning strands is exploited in problem solving).

Conception of Learner and Learning

According to van Hiele (cf. de Lange, 1996) the process of learning proceeds through  three levels:

• A student reaches the first level of thinking as soon as he or she can manipulate the known characteristics of a pattern that are familiar to him or her.
• As soon as he or she learns to manipulate the interrelatedness of the characteristics he or she will have reached the second level.
• He or she will reach the third level of thinking when he or she starts manipulating the intrinsic characteristics of relations. Traditional instruction is inclined to start at the second or third level. According to de

Lange (1996) this should not be the case if we start in the real world. The researchers at the Freudenthal Institute notice that the significance of the level theory of van Hiele does not exist in its theoretical use, but it does in its practical implications. Firstly, mathematics has to start on a level at which the concepts used have a high degree of familiarity for the students, and secondly its aim has to be the creation of a relational framework (Gravemeijer, 1994). Although RME and constructivism are not the same, to some extent they have some compatible characteristics, one of which is the similarity of the conceptions of learning and learners in both theories. As are the case to constructivism, the following conceptions are relevant to RME (Anderson et al., 1994; Louck-Horsley, et al., 1998; van den Berg, 1996):

• Each learner brings his or her preconceptions to the educational experience.These preconceptions are highly influential on subsequent learning. Learnerspossess a diverse set of alternative conceptions about mathematical ideas thatinfluence their future learning;
• Each learner actively constructs meaning. Learners acquire new knowledge by constructing it for themselves;
• Each learner is ready to share his or her personal meaning with others, and based on this negotiation process, reconceptualizes the initial knowledge structures. The construction of knowledge is a process of change that includes addition, creation, modification, refinement, restructuring, and rejection;
• Each learner takes responsibility for his or her learning. The new knowledge learners construct for themselves has its origin in a diverse set of experiences;
• Each learner is convinced that success in learning with understanding is possible. In other words, all students regardless of race, culture, and gender are capable of understanding and doing mathematics.

The conception of learning in RME is in line with the conception of learners. The starting point in the learning process of the realistic approach is emphasized on the conception that the students are familiar with. Each learner has a preconception or a set of alternative conceptions about mathematical ideas. After a student is involved meaningfully in a learning process, the student develops the conceptions to a higher level. In this step, the student actively acquires new knowledge. The construction of knowledge is a process of change that proceeds slowly from the first to second and then to the third. In this process the student is responsible for his own learning.

Realistic Mathematics Education (RME)

Realistic Mathematics Education (RME) can not be separated with Freudenthal Institute. The institute was established in 1971 under the University of Utrecht, the Netherlands. The name of the institute is after Professor Hans Freudenthal (1905 – 1990), a Germany/Dutch author, educator and mathematician.

Since 1971, Freudenthal Institute has been developing a theoretical approach toward mathematics teaching and learning called RME (Realistic Mathematics Education). RME combines the ideas of what is mathematics, how students learn mathematics, and how mathematics should be taught. Freudenthal believed that students should not be considered as passive receivers of ready-made mathematics. According to Freudenthal education might be arranged in favor to students to use every single situation and opportunity to reinvent mathematics themselves. Many problems could be developed from various contexts which are considered meaningful as learning resources. Mathematical concepts are developed by mathematization process that is starting from context-link solution students gradually develop tools for mathematical understanding to formal level. Models which are emerged in students’ mathematical activities might prompt interactivities that lead to a higher level of mathematical thinking.

RME, a promising theory
RME is a promising theory in mathematics teaching and learning. Many literatures mention that RME is potential to improve students understanding of mathematics (Streefland, 1991; Gravemeijer, 1994, 1997). The Netherlands is a pioneer in RME. Thanks to research and development conducted by Freudenthal Institute. Later on, in the United States of America (USA), several schools use RME curriculum materials developed by University of Wisconsin in collaboration with Freudenthal Institute through MiC (Mathematics in Context) project. The Netherlands’s experiences in implementing RME in schools have been used as starting point for curriculum development in the USA (Clarke, 1993; Clarke, Clarke, & Sullivan, 1996; de Lange,
1994). MiC Projects resulted in curriculum materials for grades 5 to 9. After several schools in several districts use the materials, research showed that students’ achievement in national examination highly improved (Romberg & de Lange, 1998). Also, in the Netherlands there has been a positive result which can be used as indicator of successful implementation of RME in schools. In international comparative studies like TIMSS the Netherlands students gained high position (Mullis, et al., 2000).

Freudenthal Institute and University of Western Cape in South Africa worked together in a project called Remesa (Realistic Mathematics Education in South Africa). Project’s goal was developing and researching the impact of innovative mathematics learning materials with the premise reality is the basis of and the domain of application of mathematics. Learning materials developed by Remesa were expected to be used as learning resources by mathematics teachers, book writers and others in developing mathematics instruction program relevance to South Africa context. Beside USA and South Africa, RME theory seems to be in line with mathematics curriculum development in other countries such as Portugal and British (de Lange, 1996).

One of the reasons why RME accepted in many countries is its own concept. According to Freudenthal mathematics should be seen as human activity and linked to reality. In the concept of RME students should be given opportunity to reinvent mathematics under guidance of adults (Gravemeijer, 1994). The reinvention of mathematical ideas and concepts might be started from exploration various problems and ‘real world’ situation (de Lange, 1995).

Moreover, in RME the learning process plays important role. Learning route as a result of students own process of thinking should be mapped out (Gravemeijer, 1997). In this process teacher should develop interactive teaching and learning process, namely give students opportunity to actively involve in their own process of learning.

In RME, real world is utilized as starting point in developing mathematical ideas and concepts. Real world is the world outside mathematics, such as subject matter other than mathematics, or our daily life and environment (Blum & Niss, 1989). De Lange (1996) defined real world as a concrete real world which is transferred to students through mathematical application. That is the way we see mathematics learning process on students, that occurs in real situation. The process of developing mathematical ideas and concepts which start from real world is called conceptual mathematization (de Lange, 1996). A schematic model for this learning process is illustrated as cyclic process, and the process is more important than the product (Figure 1). We assume that knowledge is a transformation process which is continuously created and recreated, not a free entity to be mastered or communicated. Real world is always adjusted (de Lange, 1996).