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.

BUILD A REPUTATION AS A PROBLEM SOLVER

Tonya got her first real job with a major airline as soon as she graduated from college. She didn’t realize what a good problem solver she was until her first job evaluation. Tonya’s review included these comments: Very resourceful—thinks of creative ways to solve problems. Handles obstacles conscientiously.

Generates alternative solutions when solving problems. Tonya says she learned to solve problems on high school committees and on backstage crews of community theater productions. “I was the one behind the scenes, holding things together. I never got a part in a play, but I put scenery together. And if the spotlight didn’t work, I’d figure something out. If we
needed stairs or a window for a set and didn’t have them, I’d manage to improvise.”

Tonya hadn’t been working at the airline long before coworkers discovered her problem-solving
skills. “People started coming to me with little problems. I’d fix them. But this time, they noticed. I got a reputation as a problem solver.”

DON’T TRY THIS AT WORK
Jared’s first job was with a food-service company in New Jersey. He took the position because he needed the money and liked the hours. The company had a few customer-service problems, but Jared never thought that his company’s problems had anything to do with him personally. After six months, Jared received his first employee evaluation:

Isn’t alert to problems Can’t handle complex problems or identify key issues,  Slow to take action, n Needs to be persistent in problem solving, Seldom generates more than one solution to a problem

Jared admits he deserved the poor rating. He explains, “If a customer came to me with a problem, my standard answer was, ‘That’s not my responsibility.’ Maybe I’d tell the customer to ask somebody else. If our division didn’t meet production standards, it wasn’t my fault. Not my problem, I thought.”

Jared’s evaluation woke him up to the importance of becoming a problem solver. He started paying closer attention to his company and his customers. He began to take on problems he hadn’t considered his responsibility before. And Jared’s next evaluation turned out much better.

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.

What is Open-Ended Problem Solving?

What is Open-Ended Problem Solving? In open-ended problem solving, the problem will have multiple possible answers that can be derived by multiple solution methods. The focus is not on the answer to the problem, but on the methods for arriving at an answer. Genuine problem solving requires a problem that is just beyond the student’s skill level so that she will not automatically know which solution method to use. The problem should be nonroutine, in that the student perceives the problem as challenging and unfamiliar, yet not insurmountable (Becker & Shimada, 1997).
 

In open-ended problem solving, students are responsible for making many of the decisions that, in the past, have been the responsibility of teachers and textbooks. To decide which method, or procedure, to undertake to solve an open-ended problem, a student will draw on her previous knowledge and experience with related problems. She might construct her own procedure, trying this and that, before arriving at a solution. She will then reflect on and explain to others her problem-solving experience, tracing her thinking process and reviewing the strategies she attempted, determining why some worked and others didn't. This period of reflection deepens her understanding of the problem and helps to clarify her thinking about effective solution methods, and how the problem and methods she used relate to other problems or areas of mathematics.

One of the teacher’s key responsibilities is selecting and presenting “good” problem tasks. By choosing good problems, the teacher sets up optimal conditions for her students to be engaged in meaningful problem solving. This means that the problem will:

• Be open-ended, in that it presents multiple solution methods and answers
• Address important mathematics concepts
• Challenge and interest students
• Connect to students’ previous learning

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.

Classes of Problems

There are two classes of problems: those that are considered well defined and others that are considered ill defined.Well-defined problems are those problems whose goals, path to solution, and obstacles to solution are clear based on the information given. For example, the problem of how to calculate the price of a sale item is well defined. You see the original price on the tag, calculate the discount percentage, and subtract this amount from the original price. The solution is a straightforward calculation. In contrast, ill-defined problems are characterized by their lack of a clear path to solution. Such problems often lack a clear problem statement as well, making the task of problem definition and problem representation quite challenging. For example, the problem of how to find a life partner is an ill-defined problem. How do you define “life partner”? What traits should that individual have? Where do you look to find such a person?

Only after considerable work has been done to formulate the problem can an ill-defined problem become tractable. Even at this stage, however, the path to solution may remain fuzzy. Multiple revisions of the problem representation may be necessary in order to find a path to a solution. In contrast to well-defined problems, ill-defined problems can lead to more than one “correct” solution.

The solution process for well-defined problems has been studied extensively, often using algorithms to describe how each step of a problem is solved (e.g., Newell & Simon, 1972). A well-defined problem can

be broken down into a series of smaller problems. The problem may then be solved using a set of recursive operations or algorithms. In contrast, algorithms cannot be used to solve ill-defined problems precisely because the problem cannot be easily defined as a set of smaller components. Before a path to solution is found, ill-defined problems often require a radical change in representation. For example, consider the following problem:

You have a jug full of lemonade and a jug full of iced tea. You simultaneously empty both jugs into one large vat, yet the lemonade remains separate from the iced tea. How could this happen? At first, this puzzle is difficult. You imagine two pitchers of refreshing drinks being poured into a common vessel and wonder how they could not mix. (It is safe to assume that the lemonade and iced tea have similar densities). However, if you change your mental representation of the lemonade and iced tea, you see that frozen drinks could be easily poured into the same vat without mixing. Though the problem itself does not specify the state of the drinks, most people assume that they are liquid, as is usually the case. But this constraint is simply an assumption. Of course, this puzzle is a fairly trivial one. But in life, we often make unwarranted assumptions in our everyday problem solving. Such assumptions can interfere with our ability to discover a novel solution to an ordinary problem.

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.