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.

External and Extrinsic Factors of Lesson Study

Because lesson study is not an individual activity, but a collaborative activity, it depends not only on teacher's characteristics, but also on other factors come from outside teacher's personal disposition. These are highlighted as follows:

a. Students characteristics
As the lesson study focuses on the students' learning improvement, students play important roles in the success of lesson study. Students at a research lesson classroom will see "outsiders", observers, during the lesson. It can affect their learning concentration when they feel surprised with the existence of the observers.

b. Collaboration among teachers
It is clear that lesson study cannot be done individually as lesson study is a group activity. Each member of lesson study group must take participation during all steps and phases in the lesson study cycle. Collaboration and cooperation in lesson study can be made among teachers from the same school, or from different schools of the same district, among teachers teach the same subject or teach different subjects.

c. Connection to knowledgeable others, including colleagues
educational experts and facilitators (coaches) who can help the lesson study group in analyzing and setting the lesson study goals, observing the research lesson, analyzing the observation data, revising the lesson plan.

The need of knowledgeable others is urgent when the members of lesson study group are new to lesson study. Even when they already experienced lesson study, they need experts such as mathematics educators, mathematics education researchers to help them make better lesson plan preparation and lesson improvement. Knowledgeable others (outside examiner, invited advisor, or reactor to the lesson): college/university lecturers (content specialists, content educators, etc.), experienced teachers, and district supervisors have important role in the lesson study implementation and dissemination.
The following are some functions of other knowledgeable others for the lesson study group (Fernandez, 2001):

• to provide a different perspective when reacting to the lesson study work of the group
• to provide information about subject matter content, new ideas, or reforms
• to share the work of other lesson study groups, also
• to act as cheerleaders to encourage teachers to persist to act as cheerleaders to encourage teachers to persist in the process.

Effective lesson study depends on observation and subsequent discussion skills. Post-lesson discussions should focus on student thinking and should be driven by data collected during the research lesson. In turn, the data collection itself should be intentional and it is planned in advance with a particular focus. The resulting discussion should focus on students’ solution strategies, information organization, and types of errors. Formulation a set of questions in advance will be useful to guide the post-lesson discussion, rather than simply presenting data collected by each team member. Collaboration with content specialists will help teachers to get ideas about the particular aspects of student thinking to target for observation. They can also provide feedback on emerging ideas or lesson plans, participate in research lessons as data collectors, and provide comments and suggestion on post-lesson discussion.

d. Learning resources:
As lesson study concerns with teaching and learning processes, this activity requires learning resources that will be used by the teacher who teach the research lesson. This includes:

• Lesson plans that reveal and promote student thinking
• Tools that support collegial learning during lesson study
• Mathematical Tasks & Student Work
• Curriculum Materials
• Research Articles & Summaries
• Lesson Videos
• Reflection Forms & Questions

All these resources may be available readily, or need to be prepared by the lesson study group.

Sustaining Program of Lesson Study Activity

After analysing the results of monitoring and evaluation, the team uses it to design the following lesson study, revise instruments, and write a summary of a research seminar. Instruments in the monitoring and evaluation should be well-documented to support this writing and to publish in journals or to present in a seminar. The publication would greatly be advantageous for other teachers or educationists on the development of lesson. Thus one of the efforts to sustain the program of lesson study is by presenting seminar or conference for disseminating the result of doing lesson study (for example annual conference of lesson study).

 Another effort to sustain the program is hosting an open house or open class as mentioned by Jennifer Stepanek et al, that open to help to broaden the lesson study community. When lesson study teams present their research lesson in an open house, they gain additional insight into their work. Open houses are also a means for spreading professional knowledge. The teachers share what they have learned with colleagues outside the team. Open house can help to build a network of teachers and other educators to support lesson study.

The team has an opportunity to discuss the benefits and challenges of doing lesson study with other experienced practitioners. But the most important thing to sustain the lesson study activities is to make sure that the lesson study is useful for the teachers to improve their professionalism. Besides, the commitment of the teachers to do lesson study is also very important. Of course, all of this needs financial supporting and a good planning.

How Do We Implement Lesson Study ?

Lewis (2002:51-72) stated that there are six basic steps as a guide in implementing lesson study. The steps are (1) Form a Lesson Study Group, (2) Focus the Lesson Study, (3) Plan the Research Lesson, (4) Teach and Observe the Research Lesson, (5) Discuss and Analyze the Research Lesson, and (6) Reflect on Your Lesson Study and Plan the Next Steps.

Step 1. Form a Lesson Study Group

There are at least four activities that we need in forming a lesson study group. The activities are (1) recruitment of the group members, (2) commitment to provide a specific time, (3) arranging a schedule for meetings, and (4) commitment to obey the rules of the group. Members of a lesson study group can be recruited from teachers, lecturers, education personnel, and/or other people who are interested in education. The most important thing is every member should have highly commitment, interest, and willing to improve the quality of mathematics instruction. Members of the group will usually have daily, weekly, monthly, or yearly meetings and they have to present at those meetings.

At the meeting, members of the group will prepare everything related to lesson study activities that will be conducted. A member of the group sometimes will be assigned to be a model that will teach a research lesson. As noted above, the group members will arrange an agenda and a schedule. The agenda and schedule will be very useful for every member of the group. Tasks as well as job description of each member should be clearly defined.

Step 2. Focus the Lesson StudyThis step consists of three main elements. The elements are (1) to determine the research theme, the main aim, the research focus, or the main objective of lesson study; (2) to choose subject-matter; and (3) to select a topic or a unit and lessons. There are three things that should be considered in determining the research theme of lesson study. First, what is the actual quality of our students right now? Second, what ideal quality of students will we expect at the end of lesson study activities? Finally, is there any gap between the ideal and the actual quality of our students? The gap (if any) will be used to choose the research theme of lesson study. Subject-matter area can be selected from a series of subject-matters offered at a school. The subject-matters can be science, chemistry, physics, biology, mathematics, language, or social studies. In selecting the subjectmatter, we have to consider the following three main questions. What subject-matter is the most difficult for students? What subject-matter do the teachers find as the most difficult to be taught? What subject-matters in a new curriculum do teachers want to understand and master? In our case,

of course, the subject-matter will be mathematics.

After selecting research theme and subject-matter, the next activity will be choosing a topic and lesson. We choose a topic that is (1) a prerequisite for the subsequent topics, (2) very difficult for students or the one that students hate, (3) difficult to teach or the one that teachers dislike, (4) new in a new curriculum. After choosing a topic, we formulate the objectives of the topic and then determine the number of lessons related to the topic and its objectives. Suppose, for example, we choose inequality as our topic. The lessons may consist of the concept of inequality, linear inequality, linear inequality that involve absolute values, quadratic inequality, the relationship between quadratic inequality and quadratic function, fractional inequality, graphical method for solving linear inequility, and inequality in two variables.

Step 3. Plan the Research Lesson

Planning has a very important role in every activity, especially in educational research. A plan can be a guide for researcher. A good plan will help researcher to solve the research problem or the research question. In planning a research lesson, we need to develop a plan to guide learning. This plan will be very useful to guide us in teaching, observing, and discussing the research lesson. There are some versions of format that can be used. It depends on the characteristics of subject-matter and the agreement of the users.

Each country has its own format. Every expert has his or her own format. Which format shall we choose and use in our research lesson? This is a problem of choice. So, we can choose a format that is fit to our need. The most important thing to be kept in mind is that the plan should be a complete document. In other word, the plan should be a stand alone document or a self-contained document. In research lesson, there are three kinds of plan, namely (1) general or comprehensive plan, (2) unit or topic plan, and (3) (research) lesson plan. Comprehensive plan is related to the research theme and the longterm goal for students. This plan may relate to one semester planning and it will consist of several topics. Unit or topic plan will cover several research lessons. In other words, unit plan will consist of some lessons. In our example above, the topic will be inequality and the lessons will be the concept of inequality, linear inequality, linear inequality that involve absolute values, quadratic inequality, the relationship between quadratic absolute values, quadratic inequality, the relationship between quadratic inequality and quadratic function, fractional inequality, graphical method for solving linear inequility, and inequality in two variables. Research lesson plan is the core of plan related to research lesson we propose. This plan will answer the question ”What changes in student thinking will occur during the lesson, and what will provoke them.” (Lewis, 2002:64).

Template of plan to guide learning propose by Lewis (2002:127- 130) can be examined in the Appendix 5 below. Of course, we do not have to use this template. We can compare this template to our format that we usually use in each country. For Indonesian people, we have a lesson plan called ”Rencana Pelaksanaan Pembelajaran” (RPP) or ”Instructional Implementation Planning.” For the time being we can use the format of RPP. Please ask your colleagues from Indonesia to show their RPP. Collecting data is an important activity to be considered and included in a plan to guide learning. We need an instrument that fits our objectives. Instrument that we usually use in implementing lesson study is observation guideline, observation sheet, or observation form. Observation form contains three main questions, namely (1) when does a student begin to concentrate on the lesson?, (2) when does a student end their concentration on the lesson?, and (3) what lesson learned can be taken from the lesson implemented?

Step 4. Teach and Observe the Research Lesson

The research lesson that has been prepared can be implemented and observed. One of the group members that has been assigned by the group will teach the research lesson. During the implementation of the research lesson, other members of the group observe the lesson. Other participants, teachers, lecturers, or even anybody who is interested in mathematics education can also observe the lesson. Observer will collect data related to students’ thinking, learning, motivation, and/or behavior. This activity is usually called open lesson. Recording and documenting the research lesson can be done by using audiotape, videotape, handycam, camera, student work, worksheet, and/or narative observation notes. The main thing to be remembered is that the record should contain facts and real events that found during the lesson and should not contain the opinions of the observer. The facts documented will be very useful and needed at discussion and reflection step.

Observer’s role during the lesson is to collect data and it is not to help students. Students should be reminded that if they need some help, they can ask their teacher to help them. They are prohibited to ask some help from other people in their classroom.

Step 5. Discuss and Analyze the Research Lesson

Research lesson that has been implemented need to be discussed and analyzed. This is necesary to be done because the results of analysis and discussion can be used to revise or refine the research lesson. We hope that the revising research lesson will be more complete, effective, and effcient.

Before conducting discussion and analysis, it will be better to prepare a discussion manual and agenda. The contents of the manual may cover the following items. First, the member of the group who teaches the lesson will be the first speaker in the discussion activity. He or she will have chances to deliver or express all of his or her difficulties in teaching the lesson. Second, as a rule of the game, the lesson belongs to the members of the group. Consequently, the lesson is ”our lesson” and it is not ”my lesson.” The members of the group have responsibility to explain everything related to the lesson. Third, discussion should be focused on the data collected during the lesson by the observers. Fourth, discussion and analysis of the research lesson should be done directly at the same day and right after the lesson implemented. This is reasonable, because the results of the

discussion and analysis will be used to revise the lesson. Finally, the role of a moderator should be clearly stated and implemented in order to get an effective and efficient discussion.

Step 6. Reflect on Your Lesson Study and Plan the Next Steps

In reflection session, there are some questions that should be answered. What parts of the lesson have been practiced well? Is there any progress by doing the research lesson? What parts of the lesson need to be improved? What are the benefits of the lesson study efforts? What we would like to change in the next cycle? Is lesson study enriching our knowledge? Is lesson study helping student learning? Is lesson study helping student development? Who will re-teach the revised lesson. Of course, there must be many more questions that can be raised.

Reasons For Applying Study Lesson

Lesson study has been chosen and implemented because of the following reasons. Firstly, lesson study can improve the quality of teaching and learning process as well as lesson in a classroom. This is understandable because (1) lesson study is an effective way to improve teaching and learning process through development of a shared professional knowledge among lesson study group members that is based on a real teaching practice, (2) the basic idea of lesson study is every student achieves his or her highest quality of learning, (3) the objectives of a lesson become a main focus of a classroom instruction, (4) based on the real experience in classroom, lesson study can be able to become foundation of instructional development, and (5) lesson study makesteachers’ role as instructional researchers.

Secondly, a well-designed lesson study will produce professional and innovative teachers. By doing lesson study, teachers can (1) effectively determine instructional objectives of a particular lesson, unit, and subjectmatter area; (2) study and improve lessons that are useful and suitable for students; (3) deepen their subject-matter knowledge they deliver; (4) determine the long-term objectives that will be achieved by their students; (5) plan collaboratively their lessons; (6) study accurately student learning and behavior; (7) develop their best instructional knowledge; and (8) reflect their teaching process based on students and colleagues perspective or opinion. Thirdly, Wang-Iverson and Yoshida (2005:13) state that lesson study has some advantages or benefits as follow:

• to reduce teacher isolation.
• to help teachers learning to observe and criticize.
• to deepen teachers’ understanding of content and curricular scope and sequence.
• to allow teachers to focus on helping all students’ learning.
• to create shared expectations for and understand of student thinking and learning.
• to increase collaboration and respect for each other.

Finally, Wang-Iverson and Yoshida (2005:16) stated also that lesson study is unique because of the following reasons:

• It is teacher-led, long-term professional learning.
• It is plan collaboratively over a period of time through intensive study of materials, standards, and students (in a practical called kyozaikenkyu).
• It supports a collaborative focus on students’ thinking through observation of classroom practice in real time with outside guests.
• It offers a process that makes concrete in an actual lesson and it is good for learning (e.g., enhancing student motivation for learning).
• It provides new and outsiders’ perspectives of teaching and learning.
• It fosters shared reflection based on classroom evidence.
• It makes concrete what reflection means, what problem solving looks like, and what thinking entails.
• It involves long-term participation of knowledgeable others.

Understand The Theory of Lesson Study

There are some definitions of lesson study. Here are some of them. Fernandez and Yoshida (2004:7) state, ”Lesson study is a direct translation for the Japanese term jugyokenkyu, which is composed of two words: jugyo, which means lesson, and kenkyu, which means study or research. As denoted by this term, lesson study consists of the study or the examination of teaching practice. How do Japanese teachers examine their teaching through lesson study? They engage in a well-defined process that involves discussing lessons that they have first planned and observed together. These lessons are called kenkyujugyo, which is simply a reversal of the term jugyokenkyu and thus literally means study or research lessons, or more specifically lessons that are the object of one’s study. Study lessons are “studied” by carrying out the steps described next in an attempt to explore a research goal that the teachers have chosen to work on (e.g., understanding how to encourage students to be autonomous learners).”