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DESCRIPTION OF REASONING PROCESSES USED IN SOLVING PHYSICS PROBLEMS: REASONING MAPS Sevket GUNDUZ Marmara University, Faculty of Education, Istanbul/TURKEY Prof.Dr.Mehmet Ali CORLU Marmara University,Faculty of Education, Istanbul/TURKEY ABSTRACT Problem solving requires a set of cognitive operations that is mainly reasoning. Solution can be obtained by applying these cognitive processes using necessery strategies in compliance with the desired end or goal (Arık, 1987). Reasoning can be defined as a process of deriving new information from the knowledge given explicitly guided by a rule (Kurtz,Gentre&Gunn, 1999). So, it can be said that the results or judgements of each step in solution are obtained through reasoning by using the previously acquired judgements and physics rules, and these derivation processes can be represented by graphical tools. An analysis of reasoning process in problem solving leads the educational studies in the area of teaching problem solving and diagnosing the source of students’ failures, because a student’s success in solving physics problems depends on reasoning processes that can be executed successfully. These methods of analysis can also be utilized in the field of artificial intelligence enriching the expert systems’ databases (van Someren, 1994), and developing the problem solving algorithms in softwares. The aim of our study is to answer the following research questions: “How can we describe the reasoning processes employed in problem solving explicitly?” and “How can we express the solution with graphical tools to make it easily understandable?” The main attempt is to portray the methods developed as a reasoning map model. Thus, we will have valid, reliable and fast methods that serve our needs. New analysis methods have been developed as an outcome of studies carried out with qualitative research techniques, The data was collected from the interviews conducted with five experts and five students, and from the exam documents written by 153 students attending the 11th grade of a public high school. Preparation of the reasoning maps has been carried out in three stages: first, data have been collected from interviews, written documents and observations; second, by means of the previously collected data, the knowledge elements have been determined, each of which acts as a rule in reasoning, and the judgements produced in reasoning; third, the resoning processes have been defined with input/output judgements and rules, and the relations between these processes. At the end of the third stage these descriptions have been shown in graphs. The processes of constructing reasoning maps have been refined by analyzing data cyclically in a constant comparison method of data analysis (Lodico, 2006). This procedure validates data coming from similiar sources, Besides triangulation which is comparing data from different sources. In this study, the process of preparing reasoning map model has been explained in detail, application to other fields has been left other studies. Keywords: problem solving, reasoning, reasoning maps. INTRODUCTION: Problem solving is a process of finding the unknown (Jonassen, 2000). Finding the unknown in general means production of new information for the problem solvers. The main function of problem solving is the reasoning process (Arık, 1987). Reasoning is the mental process of deriving logical consequences, which might take the form of a judgement, conclusion or prediction, from given information (Johnson-Laird, 2000; Simon, 1992; Kurtz et.al., 1999). Therefore, there is a close relation between reasoning and problem solving. It can be said that to analyze the reasoning helps the analysis of problem solving directly. Restricting the definition of problem within the physics makes the analysis easy. “Rule-based theories require that the rules are explicitly represented and activated during processing; that they be the causal force in the reasoning process” (Kurtz et.al.,1999, p:153). Reasoning by rules has been applied most often to the realm of propositional logic. As characterized by Rips, (cited by Kurtz et.al., 1999) the premises are stored in working memory, then the application of rules is coordinated by a reasoning program that searches for and produces appropriate inferences by constructing and linking steps in mental proof. This chaining process may proceed forward and backward in a constrained way and may involve building assertion trees in working memory. Human beings have explicit mental inference rules that operate on and transform propositions or judgements in working memory. The rules specify the allowable ways in which information can be put together, related, and structured. Kurtz et.al.(1999) have defined the reasoning process in functional form with three components, available information x and k, cognitive processes brought to bear, and generated inferences. Y = F(x,k) X represents the initial avaliable information, k refers to stored knowledge about domain and particular cases used in reasoning. F is a summary of the set of computational tools used to manipulate, recombine, or transform the input information; and y is the inferential product of reasoning process. We have defined the reasoning processes used in solving physics problems similar to the Kurtz model. This new model almost accounts for the reasonings for the physics problems: however, in other areas, this model has not been validated yet. 1012 The mental processes have a dynamic structure. Even if the same outcome has been obtained in a problem solving, the processes used in solution stages differ individually. Even if the different people use the same processes, their use orders and relations might vary. rules inputs (x,k) Operations (evoking, production, etc) output s (Y) PROCESS Figure 1. Structure of Reasoning modelProcess that is given in Figure-1 can Our be defined briefly that initial avaliable information is used as an input, this information evokes the rules from long term memory, and judgements are produced as an output by the rules. Judgements resulting from reasoning is a new information. There are two main operations in the process: the first is evoking the rules, and the second is production of actions or results. Other psychological factors were neglected in this model. Simon (1992) describes the productions in a heuristic search system corresponding to the inference rules in a system of logic. The general term “rule” that we have used, as an inference rule, represents the Simon’s production, conditional rule of modus ponens, and mental models (Gentner, 2002; Markman&Gentner, 2001)’s. Deductive, inductive and abductive thinking can be represented in this model. Output judgements may create a new appropriate situation, and they evoke the next reasoning process. The outputs of the one process are used as inputs of the next process. The chaining is accomplished by the continually changing problem situation that evokes the appropriate chain of reasoning. The reasoning chains continue in accordance with the strategy until obtaining the solution of the problem. Initial avaliable knowledge that is given in physics problems is mainly factual. It describes the situation, and it is accepted to be true. The output information that is produced by the input knowledge in reasoning process is inference or derived information. The output information is a sentence of judgement or conclusion that has a higher value than that of the input knowledge. Output judgements can be divided in some categories. It might be true, false or uncertain. The rules that are used in processes can be categorized like that principle, relation, law, model and theory. The relation is a rule that is valid only for given problem, principle is a rule that is valid for a problem group, i.e. more general than relation. The range of generalization increases as the authorized field of rules enlarges. The definitions are also used as a rule in processes, e.g. let’s consider this question: “what is the work done by force F, if it causes an object to move its own direction in the distance x?” This question can be answered with the definition of work as a rule. The sentences which express generalization such as definition and principles are called “knowledge element”. Knowledge elements are also named as field knowledge (Bloom, 1982; Özçelik, 1881). These elements are the basic factors effecting the the problem solving performance (Fisher, 2000; Kulm, 1994). In this developed model, three methods of reasoning can be graphically represented. In the first one, a judgement can be raised, if the inputs and rules are known. This process is called deduction – if there is no doubt on correctness of the premises, no suspicion on the judgement as well. In the second one, the rule can be set if the inputs and judgements are known. This is what is called induction. As the number of the processes repeated increases in deductive approach, the reliability of the knowledge produced also gets higher. This process is a type of generalization. In the third, though, the input data can be found if the rule and judgement are known. And this is called as ubduction – the recerse application of deduction. There are some cognitive events that can not be expressed in the model, e.g. The determination of a loading question, which affects the process selection after the solution starts can not be expressed in this representation. However it can be said that all questions are tried to be used which are effective on knowledge production that will be helpful in realization of the purpose related to research question leading us to final result. This procedure is, in fact, an inquiry. For this reason, the development of the inquiry abilities of the students, which are effective on problem solving will make them good problem solver. The inquiry process or fact in prblem solving is a matter that is worth to study on seperately. The representation of a knowledge takes place an important place in this model. Reasoning maps are graphical tools to represent the reasoning processes employed in solution. Jonessen (2003) argue that problem representation is the key to problem solving. He describes the tools for externalizing problem space, such as semantic networks for modeling conceptual knowledge, expert systems for representing procedural knowledge, systems modeling for strategic knowledge. Semantic networks, also known as concept maps or cognitive maps, are spatial representations of concepts and their interrelationships that are intended to represent the knowledge structuresthat humans store in their minds (Jonassen, Beissner, & Yacci, 1993). Our representation is different from those in that it represents the reasoning processes employed in problem solving. Purpose of the Study This study aims to develop new model for the analysis of reasoning employed in physics problem solving more explicitly. In order to fulfill this aim, answers to the following questions have been discussed: 1.How can we describe the reasoning processes employed in problem solving explicitly? 2.How can we express the solution with graphical tools to make it easily understandable?” The main attempt is to illlustrate (portray) the methods developed as a reasoning map model that the details has given in findings. Thus, we will have valid, reliable and fast methods that serve our needs. METHOD New analysis methods have been developed as an outcome of studies carried out with qualitative research techniques within the frame of interpretive paradigm (Cohen et.al.2000). The research model is the grounded theory based on the work of Glaser and Strauss (Lodico et.al. 2006). The data was collected from the interviews conducted with five experts and five students, and from the exam documents written by 153 students attending the 11th grade of a public high school. Preparation of the reasoning maps has been carried out in three stages: at first, the data from the solution of problem given in Figure-2 has been collected from interviews, written documents and observations; second, by means of the previously collected data, the knowledge elements (Merill, 1999) have been determined, each of which acts as a rule in reasoning, and the judgements produced in reasoning; third, the reasoning processes have been defined with input/output judgements and rules, and the relations between these processes. At the end of the third stage 1013 these descriptions have been shown in graphs. The processes of constructing reasoning maps have been refined by analyzing data cyclically in a constant comparison method of data analysis (Lodico et.al. 2006). This procedure validates data coming from similiar sources, beside triangulation which is comparing data from different sources. In this study, the process of preparing reasoning map model has been explained in detail, while application to other fields has been left for other studies. Table 1. The list of theme defined by content analysis of the problem solution. Codes p.def… Judgements regarding the definition the initial givens in problem case. p.gen... Judgements about determining the principles and laws of problem case. p.equ… Judgements regarding the application of first principle of equilibrium. p.dyn... Judgements regarding the application of Newton’s seconed law. p.cou... Judgements about defining the interactions of charged particles according to Coulomb’s law. p.efi... Judgements about defining the interaction of charged particle with electric field. p.tef... Judgements about finding the resultant electric field for a given point. p.sol... Judgements regarding the application of existing principles and laws in problem. FINDINGS AND INTERPRETATION Data coming from the sources have been evaluated and the reasoning map model has been developed. In this section the last version of developed model has been explained, and findings about the initial stages have been mentioned. Some definitions and rules of the model concluded by analysis of the data are given below, and application of the model evaluated through one individual’s solution. Findings About the First Stages As a subtopic of physics, “electrostatic force and field” was selected. 21 knowledge elements belonging to this topic were determined and these were grouped in categories of Coulomb force, electric field and equilibrium state. The last one was added after the preparation of problem case. The elements have been coded with the number of two digits, the first shows the category, the second shows the sequence. For example, “3.i. The force exerted on negatively charged particle placed in an external electrical field has a direction opposite to field direction”. Codes in Figure-3,4,5 that are enclosed by circles show the knowledge elements which act as a rule to direct the reasoning process. Then, the problem case and fundemental questions have been determined as given in Fig-2. The problem was solved by 5 physics teachers in interviews. 48 judgements were found from the teachers’ interviews, these are sentences giving information about the problem solution and produced in each solution step as a product of reasoning processes. Judgements have been coded like “p.equ.a”, “p.cou.b”,…symbols. The first character, “p” means of product of process. Second group of characters, “equ”, “cou”, “efi”,…etc. means the theme of judgements out of eight groups given in Table-1. For example “equ” is about equilibrium, “cou” is about Coulomb force, “efi” is about electric field. The third group of characters indicates the sequence of judgements, i.e. a is the first, b is the second, etc. A few judgements were needed to be put into fourth category, in that case a new character set was used like “x”, “y”, and “z”. For example “p.equ.b.x” from the Figure-6 means that this is the first part of the second judgement produced as an outcome of reasoning process directed by the rules of vector addition. The judgement is “The force Fe due to interaction of electric field is equal to the force Fc due to mutual interaction i.e. Coulomb Law”. The coding symbols were chosen arbitrarily. Name of theme Reasoning map given in Figure-6 has been prepared after the judgements were determined. That shows the reasoning processes used in solution and relations of each process. Each process consists of knowledge elements (enclosed by circles in figures) and input-output judgements (encloesd by squares). The direction of arrow shows knowledge process from input to output. Findings Regarding the Definitions of the Model Some definitions given here are new and some of them are common general terms. We have prefered to mention them again here to define the model as a whole. Definition-1. Process/subprocess: A process can be broken down into three basic components: the inputs, outputs, and the rules governing the operations transforming the inputs to outputs. Every process can be defined in terms of subprocess. Definition-2. Unit/fundemental/core process: is the definable smallest process. It can not be decomposed into other processes. It is usually governed by one rule. Definition-3. Process data and judgement: Input of the process is called datum, output of the process is called judgement. Definition -4. Factual Knowledge: is knowledge that is not product of infering, but given about the problem case or defining the problem case. In order to clarify the meaning of the codes used in reasoning maps, description of the categories are provided below: p.dyn.d.1 This code indicates that it is the first value of the judgement “d” belongs to “dyn” theme of “p” process. “.dyn.“ indicates theme of the application of Newton’s seconed law., “.d.“ indicates the judgement of “ if the external electric field increases, then magnitude of net force increases outward, if field decreases, net force increases inwards. “.1.“ indicates the specific judgement “if the external electric field increases, then magnitude of net force on the point of the first particle increases to the left direction”. 1014 the scientific definition or principle. These are active in problem solving, and without knowing them it is impossible to carry out the solution (Fiser, 2000; Kulm, 1994; Özçelik, 1981). Problem Case: Findings Regarding the Induced Principles of Model (Some obtained generalizations): Left T1 T2 d First particle m1, q1 right Low level generalizations, or induction of the rules, or induced principles have been drawn from the findings. These principles have been used in the analysis of the problem solution, and they were considered to be valid in the set of facts that the model covers. The induced principles about the model are explained below. Second Particle m2, q2 Two point charged particles hanging vertically by strings in a horizontal electric field are in equilibrium. Fundemental Questions: Answer the following questions by using the problem case; a) What is the ratio of charge values, q1 /q2 =? b) What is the magnitude of external electric field in terms of givens? c) What is the direction of external electric field and the signs of charges? d) How do the positions of particle change, if the magnitude of external electric field and charges of point materials, and distance of materials change? Figure 2. Problem case and related fundamental questions ask in interviews. Definition-5. Process rule: is any rule governing the operations related with inputs and outputs (judgements) in reasoning. Definition-6. Verified Judgement: If the data and rule of process are exactly known as correct or verified that they are true, the output is called verified judgement. Definition-7. Unverified Judgement: If accuracy of either the data or the rule of process has not been verified yet, the output is called unverified judgement. Definition-8. Directing question and aim of process: Directing question causes the process to start, and clarify the purpose of process. Purpose of the process are embedded in question implicitly. Operations are governed according to purpose. Definition-9.Unifold and manifold process: The process that produce single judgement is called unifold, if there are more than one judgement it is called manifold process. Definition-10. The process name: is the name of inference, and determined according to the judgement infered. Definition-11. Range of generalization (Induction, drawing a rule): If the input and the output of the process are known, then the rule of process can be drawn inductively. This is called generalization or induction. The knowledge obtained from generalization within the solution is called as byprinciples. Definition-12. Reasoning maps: The graphs showing the reasoning process with the input, the output, the rules, and the relation between them are called reasoning maps. Definition-13. Category of the process: The process is categorized according to type of judgement. Definition -14. Flowing direction of Knowledge: The direction of processing the knowledge is towards from the data to judgement Definition- 15. Evoking : is an operation that the data evokes the process rule from the long term memory (Simon, 1992). Definition- 16. Production: is an operation that the judgement is produced after evoking (Simon, 1992). Definition- 17. Knowledge Element: The knowledge elements (Merrill, 1999) can be regarded as content knowledge (Bloom, 1982). A knowledge element represents Principle-1. The general process that define the solution of the physics problem consists of the sequence of unit processes related to each other. Principle-2. The unit processes trigger each other in chain reactions. The output of one process causes the related processes. Principle-3. The output information has a higher value than the input information, because the output is produced as a consequence of cognitive operations. Principle -4. Unit process is usually unifold, and rarely twofold. Principle -5. The well defined inputs makes it easy to find the rules from long term memory. Sometimes, after the the rules have been found they help the inputs to be well defined, if they are not at the beginning. There are two way interaction between input and rules. Principle -6. This model explains the structure of inquiry. Principle-7. Deductive reasoning can be represented by this model. Principle-8. Inductive reasoning can be represented by this model. Principle-9. While the generalization level of produced knowledge becomes high, the area of validity becomes wider. Principle-10. If knowledge has been produced by generalization in one proces, it can be used as a rule that is called by-principle in other process. So, by-principles can be used in reasoning process. Principle-11. The reasoning maps of problem solvers are different even if they solve the same problem. Principle-12. If the number of processe for the problem increases, the difficulty of a problem increases. Difficulty of a problem is proportional with the number of processes. Principle-13. If the number of process for the problem increases, the need for the mathematical knowledge, that is probability of using math increases. Principle-14. The mean of difficulty index of the questions measuring the outputs is lower than the mean of questions measuring the inputs. Principle-15. There are high correlations between the scores of the questions measuring the inputs and outputs. Principle-16. Correlations between the scores of the questions measuring the inputs may not be high. Principle-17. Correlations between the scores of the questions measuring the outputs may not be high Principle-18. The success of a processes can be assesed by using the scores of question measuring the outputs Principle-19. Johnson-Laird’s mental model can be represented by reasoning maps. The elements of process may be mental models. Principle-20. Gentner(2001)’s causal mental model can be represented by reasoning maps. Principle-21. Production of the output or judgement requires the ability of application. Principle-22. The judgements may be in the types of analysis, synthesis, an evaluation. Principle-23. Heuristic reasoning can be represented by reasoning maps. 1015 Principle-24. The strategy used for the solution can be determined from the reasoning maps. Principle-25. This model doesn’t explain the seeing. Findings Regarding the Application of Model Here is an example of the application of the model in analysis of solution of one interviewee. Solver has begun the solution by searching the content area. “… since the charged particles exist, problem is related with static electric, …related with electrical force. …there are particles in equilibrium. .. related with equilibrium….” These words indicate (point) to principle-5. It means that givens in the problem evoke the Coulomb’s Law knowledge Coulomb’s Law is active. Particles are charged Rule 2.b: Two Charged particles apply a force upon each other. Coulomb force Fc are applied on each particle Outputs : 1. Coulomb force Fc is applied on first particle 2. Coulomb force Fc is applied on second particle Figure 3. Open diagram of unit process. The level of validity or influence of the process’ judgement works only for this problem, and it can be used as an input in all processes if necessery. P.cou.a P.def.a Outputs : P.cou.a. Coulomb force Fc are applied on each particle P.cou.a.1. Coulomb force Fc is applied on first particle. P.cou.a.2. Coulomb force Fc is applied on second particle. This process can not be divided into small process practically, so this can be considered as a unit process (D-2). The direction of knowledge flow in this process is bidirectional. It means, if it is known through other ways that the electrical force is exerted upon particles it may be infered that the particles are charged. Figure-4 Coded diagram of the unit process. schema in mind, and make the rules easy to be found from it. The first reasoning that the solver used has been given below: “1-… [p.gen.b] Charged particles apply a force on each other, because of Coulomb’s Law. [p.def.a] The first and second particles given in the problem are charged, so [p.cou.a] these two particles exert a force upon each other.” The first two judgements have been produced by seeing differently from the formal reasoning. The third can be used as the name of process. This reasoning process has been represented in figure-3 graphically. The components of the process have been given in Table-2 in accordance with the developed model. Table 2. Components of “P.cou.a” process Components Contents The inputs of the process are factual knowledge (definition-4) and are used as data (def-3). They are not produced in reasoning, but seeing which may be regarded as perception process. Seeing process can not be explained by this process (principle-25). The output of process is a judgement (D-3) produced by applying the process rule (D-5). This is called production (D-16). The interviewee did not do anything to verify that process rule is correct or not. He was certain that it was a true and well-known principle. This reasoning process is deductive. Since major premise is not given in the problem, and it is used from the cognitive schema of physics knowledge, so it may be regarded as heuristic reasoning (p-23). There are cause and effect in this process, so it may be named as causal reasoning. P.gen.b inputs: P.def.a. Particles are charged, q1 and q2 p.def.a.1. First particle is charged q1. p.def.a.2. Second particle is charged q2. P.gen.b. Coulomb’s Law is active in problem, Rule : 2.b: Electrical charged particles exert a force on each other. This process may be used in other problems where there are charged particles. The detailed analysis of process in terms of model is given below. The process’ aim and the directing question aren’t expressed in the diagram. Since the aim of the process is to produce the output, the output implies the aim and the directing question implicitly. Inputs: 1. First particle is charged q1. 2. Second particle is charged q2. 3. Coulomb’s Law is active. 2.b This is the process that it has produced p.cou.a output from p.gen.b and p.def.a inputs by using the rule 2.b The meaning of codes are given in table-2. This process is given in coded schema in figure-4, where inputs, outputs and rule are given with code numbers. This technique is prefered in order to show more processes in per page. In figure-4 the input and output are expressed the most specific form like p.def.a.1, p.def.a.2, p.cou.a.1, p.cou.a.2. These are the particle specific sentences. P.def.a, p.cou.a are more general judgements than the particle-focused sentences. Finding the rules (principles) from the knowledge schema affects the problem solving. The finding algorithms or evoking ability which can be named remembering or application skill varies from person to person. Problem solution has been continued; “2-… [p.def.f] The line connecting the particles is horizontal, so [p.cou.d] the direcetion of force exerted on particles is horizontal. ….” The components of this infering are given in Table 3. Tablo 3. Components of “P.cou.d” process Process name P.cou.a.Infering process that particles exert a force on each other because of their charges. Directing How does an effect appear due to particles question to be charged? İnputsp.def.a.1. First particle is charged q1. (data) p.def.a.2. Second particle is charged q2. p.gen.b. Coulomb’s Law is active in problem. Process 2.b. Two charged particles apply a force on rule each other. Outputs P.cou.a.1. Coulomb force Fc is applied on first particle. (judgement P.cou.a.2. Coulomb force Fc is applied on second s) particle. Components Process name Contents P.cou.d. Infering process that the dicetion of force exerted on particles due to their charges is horizontal. Directing What is the direction of electrical forces question exerted on particles? İnputs (data) p.def.f. The line connecting the particles is horizontal. P.cou.a.1. Coulomb force Fc is applied on first particle. P.cou.a.2. Coulomb force Fc is applied on second particle. 1016 particles. The questionthat was asked to the interviewee was: “what did you think between the third and fourth sencences?” He replied that: Process rule 2.e. The direction of the Coulomb Force exerted on each other is on the line connecting particles charge q1 and q2. Outputsp.cou.d.1. The force Fc applied on the first particle (judgements) is horizontal. p.cou.d.2. The force Fc applied on the first particle is horizontal. The output of the process “p.cou.a” was used here as the input of the process “p.cou.d” (D-3). If the sequence is considered, p.cou.a must precede p.cou.d. The first process has triggered the second. These two processes and their relations have been shown in figure-5. This model briefly represents the scientific method. In scientific method, if the factual data (inputs) are known, the principles (rules) are constructed theoretically from them, this is called invention. The hypothesis can be infered by using the constructed frame of theory. If the validity of hypothesis are verified either theoretically or experimentally, the validity of the principles are also verified. These procedures can be represented by reasoning maps. As the analysis of problem solving is continued: “3-… Since the objects are in equilibrium, the net forces exerted upon the objects are equal to zero. … [p.equ.a]” The components of this process are as follows: Input: P.def.g. Particles are in equilibrium. P.def.g.1. The first particle is in equilibrium. P.def.g.2. The second particle is in equilibrium. Rule: 1.a.If an object is in equilibrium, then the resultant force exerted upon that particle is equal to zero Output: P.equ.a. the resultant forces exerted upon the particles are equal to zero. P.equ.a.1. the resultant force exerted upon the first particle is equal to zero. P.equ.a.2. the resultant force exerted upon the second particle is equal to zero. This inference has been made according to the first principle of equilibrium, that is special case of Newton’s First Law. This process has been triggerd by the electrical force obtained previously and initial information of being equilibrium given in the probem case. “4-… since the objects are in the horizontal homogenous external electric field, a secondory electrical force is exerted upon the objects due to that field. And that force is horizontal. …” [p.efi.a] The components of this process are as follows: Inputs: P.def.e. There exists a homogeneous external electric field. P.def.c. The direction of external electrical field is horizontal. P.gen.c. Charged particles interact with electrical field. Rule: 3.a.The electrical force is exerted upon charged objects placed in an external electrical field in the direction of field. Output: P.efi.a. The horizontal electrical force Fe is exerted upon the particles due to external electrical field. P.efi.a.x.1. Fe1 is exerted upon first particle. P. efi.a.x.2. Fe2 is exerted upon second particle. P. efi.a.y.1. Fe1 is horizontal. P. efi.a.y.2. Fe2 is horizontal. The judgement that the net forces acted on particles are zero implies that there must be a second force applied on “5-… in fact I found that the net force was zero, but, because of vector addition if the net force is zero there must be more than one force. I have looked to weights and tensions, these are perpendicular to the electrical forces [existing due to the charges of particles], I couldn’t use them. And again I looked at the problem and I realized the external electrical force. I thought to consider the force due to that field….” Interviewee executed the process of finding the resultant vector (p.equ.b) after the process of p.equ.a. Since the data was insufficient, he couldn’t succeed (principle-5). The evoked rule has been implied that there must be an additional data, this implication caused the interviewee to ask new questions and to produce the process “p.efi.a” explained above. This is in fact the process of inquiry. Interviewee executed the process p.equ.b: “6-… The [horizontal] forces acted on objects equal each other and oppsite direction. ..”(p.equ.b) The analysis of individual’s solution can be proceeded by using this method as explained in model. Our purpose is to explain the application of our model in the analysis of the physics problem solution, so we didn’t mentioned any more details. The reasoning map of the first question’s solution is given in the Figure-6. In short, it can be said from the map that this map varies individually, and gives the information about the solution strategies of the persons. At the stage of physical description thare are 5 processes, at the stage of converting the physical representation to mathematical representation there are 4 processes, and finall there is one process at the stage of equation solution. CONCLUSION AND SUGGESTIONS The main purpose of this study is to describe the model explaining explicitly the resoning processes employed in problem solving. First of all, the definitions and principles of the model that was constructed by the qualitative analysis of the solutions of the participants were given, then the individual’s solution was explained by using the model. This model has of course some limitations. It is hard to represent the higher order thinking process. Having simple explanations of processes facilitates the assesment of problem solving. Application of the model to the assesment of problem solving has been made already in another study (Gunduz, 2007). Gunduz has shown how the resoning maps can be used in preparing the diagnostic test of physics. Judgements given with a reasoning map indicate the objectives that the items measure. Questions measuring the judgements have been prepared accordingly for the diagnostic test. In addition to these studies, it is recommended to search how the reasoning maps can be used in teaching problem solving. Jonessen (2003) argue that problem representation is the key to problem solving. He mentions in his article that Mayer reports that diagrams or flowcharts produces better performance than verbal representations, especially for more complex problems. He also cites that the spatial reorganization of information facilitates some of the cognitive activities that are required to solve problems. ACKNOWLEDGEMENT: This study was supported by the Scientific Research Projects Department of Marmara University with the project numbered EĞT-117/081004 and dated 08.10.2004. 1017 Question:what are the relations between the magnitude of charges, external electrical field and distance between particles? Particles are in equilbrium The line connecting particles is horizontal Particles are in the uniform electrical field Ed. Paricles are charged Ed is horizontal Level of determining the givens 1th rule of equilibrium (1.a) E Field (3.a) Coulomb’ s Law (2.b) Fnet=0 Stages of physical description: Determination of definitions/principles/laws. Fc is horizontal 2.e 2.d Fc is acted Fc1=Fc2 Horizontal Fe is acted 2.c 4.a 3.g FC Fc and Fe are horizontal and opposite. Fe=Fc kq1q2 d2 the stage of converting the physical representation to mathematical representation: Aplication of rules and writing the equations Math rule (5.a) If an object is in equilibrium, then the resultant force exerted upon that particle is equal to zero. Particles are in equilibrium. So, the resultant forces exerted upon the particles are equal to zero. q1Edis kq1q2 q2 Edis d2 particle-1 Fe = q.E kq1q2 d2 particle -2 Math rule (5.a) Edis kq2 d 2 kq1 q1 = q2 d2 Figure 6. Reasoning Map of solution given to the first question of problem case. 1018 The stage of the equations’ solution REFERENCES Arık, I.A. (1987). Yaratıcılık (Üç Derleme). 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