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145 10. Investigation of pupils’ science problem solving process using knowledge space theory at primary level 1 Ibolya Revák-Markóczi Department of Biology Teaching Methodology, University of Debrecen ABSTRACT Problem solving thinking is a part of the cognitive skills which have helped human beings become social individuals and which is even today an essential element of socialization. It is essential in life to solve social, economic and everyday problems and protect our health and environment as well. That is why the process of teaching and learning has to be planned in such a way that students become more and more success- ful in solving problems of life. To do this we need to know the micro- and macrostruc- ture, the process, the characteristics and influential factors of problem solving. The aim of this study is to investigate the characteristics of scientific problem solving strategies of 9- to 10 year- olds, which was part of the ‘Rostock Model’ didactic programme aim- ing, among others to improve the scientific thinking and positive attitude of primary school pupils. We evaluated the answers of Hungarian and German pupils, so our sam- ple involves 60 participating pupils from Hungary (39) and Germany (21) and 52 a con- trol group of pupils (26 Hungarian and 26 German). To investigate strategy elements we made two problem tasks. The assessment of answers was based on knowledge space the- ory (KST, which is also suitable for determining the most characteristic knowledge structures for student groups as well as critical learning pathway, the hierarchy of the characteristic problems and the elements of the problem solving process. According to the results the investigated pupils are in possession of the each elements of problem solving process but the levels of these are different. As provided by analysis, the skills of composing aims, planning, implementation and assessment are based on abilities of composing and understanding problems and forming hypothesis. Keywords: problem solving process, science problem solving, knowledge space theory 1 The publication is supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project. The project is co-financed by the European Union and the European Social Fund. – The study was supported by OTKA (K-105262).

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Page 1: 10. Investigation of pupils’ science problem solving …cherd.unideb.hu/old/dok/kiadvany-egyeb/3-Revak problem...145 10. Investigation of pupils’ science problem solving process

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10. Investigation of pupils’ science problem solving process using knowledge space theory at primary level1 Ibolya Revák-Markóczi Department of Biology Teaching Methodology, University of Debrecen

ABSTRACT Problem solving thinking is a part of the cognitive skills which have helped human beings become social individuals and which is even today an essential element of socialization. It is essential in life to solve social, economic and everyday problems and protect our health and environment as well. That is why the process of teaching and learning has to be planned in such a way that students become more and more success-ful in solving problems of life. To do this we need to know the micro- and macrostruc-ture, the process, the characteristics and influential factors of problem solving. The aim of this study is to investigate the characteristics of scientific problem solving strategies of 9- to 10 year- olds, which was part of the ‘Rostock Model’ didactic programme aim-ing, among others to improve the scientific thinking and positive attitude of primary school pupils. We evaluated the answers of Hungarian and German pupils, so our sam-ple involves 60 participating pupils from Hungary (39) and Germany (21) and 52 a con-trol group of pupils (26 Hungarian and 26 German). To investigate strategy elements we made two problem tasks. The assessment of answers was based on knowledge space the-ory (KST, which is also suitable for determining the most characteristic knowledge structures for student groups as well as critical learning pathway, the hierarchy of the characteristic problems and the elements of the problem solving process. According to the results the investigated pupils are in possession of the each elements of problem solving process but the levels of these are different. As provided by analysis, the skills of composing aims, planning, implementation and assessment are based on abilities of composing and understanding problems and forming hypothesis.

Keywords: problem solving process, science problem solving, knowledge space theory

1 The publication is supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project. The project is co-financed by the European Union and the European Social Fund. – The study was supported by OTKA (K-105262).

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1. Introduction

One of the most important key competencies of natural science cognitive competencies is natural scientific problem solving thinking. It is analogous to the heuristic structure of natural scientific research in terms of its logical struc-ture. This structure is the basis of those methods by the help of which we can teach our students from elementary school ages to examine, analyze and evalu-ate regularities and relations of natural phenomena.

But the learning process of natural sciences can only be effective and moti-vating if we establish and develop the cognitive abilities adapted to age needed for the application of aimed and effective methods, including the problem solv-ing thinking of students.

Problem solving thinking has research background and results that go back to decades (Molnár, 2006). The examined groups of pupils and students are at the age of 11-24 presumably with a formal way of thinking. There are only a few studies about the process of problem solving among pupils of the lower grades. These studies are predominantly qualitative in nature, they reveal various struc-tural elements, but they do not perform deep, multilevel analyses (Mulligan, 2001; Shure, 2000). The present study shows the structure of the natural scien-tific problem solving process of lower grade primary school pupils as a result of the impact assessment of an aimed experimental program by the help of Knowledge Space Theory (KST), on three superimposed, increasing levels of abstraction.

2. Natural scientific thinking at primary level The view of Piaget (1963) about stages of cognitive development is a widely

accepted fundamental thesis about the process of learning. According to him, in-tellectual processes try to create a balance similarly to life functions (Tóth, 2000). He emphasizes the role of the appropriate level of cognitive conflicts dur-ing cognitive development. He assumes that solving intricate problems with help results in irreversible cognitive development. According to the meta-constructivism theory of Adey and Shayer (1994), in this process, during solu-tions the student reaches his own construction of the required argumentation, the construction of his own way of thinking.

Vygotsky (1987) disputes Piaget’s statements in agreement with other re-searchers, that certain sections of cognitive development can not be clearly sepa-rated. Their formation and the transitions between phases significantly depend on environmental factors at given ages. Discussions with fellow students and the teacher about the way of solving problems are such activities that contribute to the development of metacognitive abilities and through this the development of

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cognitive and general intellectual abilities (Adey, 1999). According to the theory of Vygotsky (1987) and his followers, as a result of the student’s personal psy-chological scheme and the effect of the environment, they can attain the stage of formal thinking as early as the age of six to ten.

One of the typical trends of research concerning the development of natural scientific thinking is the investigation of children’s predictions, explanations and concepts about phenomena of nature. The majority of research proves that the development of thinking is influenced by knowledge and experiences (formed as a consequence of environmental effects) of the children in connection with the given phenomena (Strunk, 1998; Stern, 2003). According to Papageor-giu at al. (2005) the concrete physical experiences of the child decisively deter-mine the duration of each stage of cognitive development. So it is an indisput-able fact that an eight years old child is capable of formal thinking, even if he can not express it suitably because of the characteristics of child’s language. During the development of thinking, beyond the existing knowledge and experi-ences, the child’s motivational and emotional attitude to the given knowledge is essential (Mahler, 1999).

According to the ‘conceptual change theory’ natural scientific thinking and learning is a move from common concepts to scientific ones (Carey, 1985; Ko-rom, 2005). The development of abilities can be realized through knowledge from special fields that can be

Investigation of pupils’ scientific problem solving process using knowledge space theory at primary level transferred to other fields too (Molnár, 2002b). In their interpretation, the development of formal thinking depends on the existing knowledge system than a determinism of a particular age of life. According to Hodson (1998) the transition towards natural scientific concepts and thinking is possible by determined instructions and the activity of students. The task of some specially qualified teachers is to create the structure of this activity system in which the social interactions play a primary role as environmental factors.

Most international surveys examining the concepts of children from different ages about living and non-living nature refer to 9- to10 year-olds (Faust, 1993; Kircher and Rohrer, 1993; Möller, 2002; Stern, 2003). The results showed that the students already have naive concepts and predictions at the beginning of the targeted activity system, and this would become scientific. During this process an essential connection has been shown between linguistic development and the way to express thoughts.

Research on cognitive abilities and the natural scientific way of thinking proved that applying targeted methods for the development of problem solving thinking is justified in primary school age as well, on condition, that during de-velopment we will take the characteristics of child’s thinking into consideration.

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3. Structure of the process of natural scientific problem solving The process of problem solving has been modelled by several researchers

from the beginning of the 20th century that are systematized by Lénárd (1984) and Rowe (1985) (Molnár, 2002a, 2006).

In every model the process of problem solving must be made up of a series of well separated steps. The emphasis of planning is typical in certain groups of processes (Pólya, 1957; Osborne, 1963). Primarily it gives the way of practical problem solving (for example phases of experimentation, research methods). Other models show the solution schemes of problems verifiable in a fictive way and the problem solving assignments appearing in theory (Newell, Show and Simon, 1962).

According to Fisher (1999) successful problem solving comprises the sys-tematic application of a given series of thoughts and actions, namely planning. The plan is a series of steps or processes which leads to the solution of the given task. The plan is versatile from the point of view that it does not contain the steps in a compulsory order. So it can make it possible to use alternative strate-gies to achieve the goal. As we approach the solution, we might need to intro-duce new ideas, to interpret new difficulties and changing conditions. The plan is nothing more than thinking over what we should do. It is the responsibility of our education to teach our students to think over thoroughly what they should do for successful problem solving.

According to Lénárd’s (1978) interpretation there are two main factors of problem solving process: a) the objective data of the task and b) the subjective features of thinking : experiences, motives and emotional factors (Balogh, 1998).

Most of the models are linear, consisting of sequential stages, taking no no-tice of the cyclical character of the problem solving process which allows feed-back for any step of the process (Molnár, 2006). Nowadays these branching models are the most accepted ones. Besides static problem solving they are the only imaginable logical ways in solving dynamic problems as well.

Schoenfeld (1985) highlights the importance of the modification of solutions. He attaches special importance to regulatory processes. The important elements of solving cycles formed by regulation are the choice of purposes, making plans, evaluation and tracking of solutions. However the ability of evaluation requires high-level abstraction by which feedback to certain phases of problem solving is completed in later stages of thinking. So during the examination of student’s so-lution strategies at the age of 11-12 we are more likely to find steps of the linear solution ways.

Describing the process of problem solving Nelson (1992) uses the term of metacognition. Schoenfeld (1985) introduces some methods that can help in the

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conscious analysis of the student’s thinking activity since students must be fully aware of the mechanism of the new ways of thinking to learn them (Leino, 1987).

In the light of previous researches we can draw the conclusion that the proc-ess of problem solving is a cyclic operation system which can be divided into definable stages and contains multiple-staged, linear connections. The output phase is always reaching the target, the solution to the problem. According to the theory of metacognition, these phases are teachable and can be brought into consciousness, as the ‘macrostructure’ of the way to the solution (Lénárd, 1984), which is a necessary but not sufficient condition for successful problem solving. So it’s forming and development similarly to the other elements of problem solving is an important task of our education.

4. The developmental program The developmental program called the ‘Rostock Model’ is the result of inter-

national cooperation. It examined the development of the natural scientific con-cept system and thinking of the pupils of the lower grades within the framework of an organized educational project with the participation of Lithuania, Hungary and Germany between 2004 and 2008. In the follow-up study the same 94 pupils of the three countries took part from first to fourth grade.

The central task of the program is the structuring, planning, execution and analysis of the learning process taking into account cognitive and constructive pedagogical and psychological as well as the neurobiological foundations that they are based on

The fundamental target of the program is to create a school environment which: 1) Takes the former knowledge and abilities of pupils into consideration; 2) Relies on the motivation and emotional factors in the process of learning; 3) Aims at competent learning. 4) Considers learning as a social and cooperative process; 5) Develops natural scientific conceptual thinking through the applica-tion of methods of natural scientific cognition; 6) Leads to performance capable knowledge as a result of the applied didactic procedures; 7) In the acquisition of natural scientific knowledge it aims at forms and develops such abilities that can be transferred to other fields as well; (8) It aims to form and develop a natural scientific teaching program which assures greater independent student activity depending on previous didactical conceptions.

Considering structure the program consisted of: 1) making detailed lesson plans of the experimental teaching that processed a unified knowledge system (‘water’ as subject matter), 2) accomplishment of the experimental lessons (8-10 lessons/year), 3) measuring the efficiency, having pre-tests written before, and

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post-tests written after the experimental teaching (the first post-test should be performed immediately after class, the

Investigation of pupils’ scientific problem solving process using knowledge space theory at primary level, second should be performed four months after teaching to measure the consolidated knowledge), 4) evaluation of the answers of students, 5) making auxiliary materials to summarize the theoretical and prac-tical relations and results of the program.

The explicit introduction of the process of problem solving and the applica-tion of certain strategic steps into consciousness was performed directly during the development program. In the lessons of the given teaching unit, the pupils discussed the purpose of the acquisition of knowledge, the main message of the lessons. These purposes and tasks were presented in written and graphic form in the schoolroom as applications during the term of the teaching units, so the pu-pils could examine them at any time. So we consciously formed the ability of goal setting. The process of learning was helped by experiments (teacher’s de-monstrative, guided by teacher’s instructions, and independent student experi-ments). On the basis of them the children wrote ‘reports’ too. The construction of the reports was consequent whereas they included the following questions: ‘What will happen? We suppose, that…’encouraging the formation of hypothe-ses; ‘What could we observe? We see that….’ recording observations and expe-riences and ...’Why did this happen? We know that…’questions concerning analysis. The report was a ready-made work sheet in all cases. It contained the descriptions concerning the planning and execution of the experiments, too.

The analysis covered four main fields: (1) The development of the natural scientific concept system at the level of knowledge; (2) Application of the scien-tific concept system during the explanation of phenomena; (3) Attributes and development of the inner structure of natural scientific concepts; (4) Appearance of attributes and changes of problem solving strategies during solving natural scientific problems

5. The study

5.1. Aims

The aims of the study concerning natural scientific problem solving strategies performed within the scope of the development program were the following: 1) To explore the strategic ways, elements and their attributes appearing in the natural scientific problem solving of pupils of the lower grades. 2) To draw con-clusions about the efficiency of the applied didactic procedure with respect to its further development. 3) To highlight the actually realized strategic elements of

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problem solving in the examined age to which the planning of the process of teaching can be accomodated. 4) Knowing the characteristics of the strategic steps, the further development of them for successful problem solving.

5.2. Sample and method We performed the research at the end of the development program in 2008.

assuming that the experimental program will show the actual impact concerning the examined ability by then. On the other hand, it was performed in the fourth class, since those pupils have more natural scientific knowledge in possession, so we were able to give more complex problem solving tasks to examine their problem solving strategies in a more detailed way. In the research we were able to consider only the answers of the Hungarian and German pupils. The

Lithuanian survey was broken off. So our sample consisted of 60 (39 Hun-garian and 21 German) fourth grade students, the 52 fourth grade students of the control group came from the two countries (26 Hungarian and 26 German).

To examine the strategic elements we constructed two problem solving tasks. We considered that this was a sufficient number owing to the number of cases. We examined the possible ways of solving the two tasks before the experimental teaching at the fourth grade in April 2008; directly after the experimental teach-ing in mid-June; and four months later in

late October. We chose individual interviews as the tool of the research, dur-ing which we recorded the thoughts of the pupils in an auditive way, then we coded and evaluated it.

The problem tasks:

Task 1: ‘Rexi, the dog lives in a kennel. On a cold winter’s day his drinking water became frozen. How would you have helped?’

The aim of the problem solving is to get the dog drinking water. During the problem formulation it should be mentioned that since the water is frozen, it is solid ice, the dog can not drink it so he cannot get water, which is essential for life. The question is how we can reach the solution during which the pupils can take the opportunity to propose predictions and suggestions. This phase of the solution correspond to the stage of hypothesis-making, in which the expectations concerning the solution are about the methods and procedures that lead to melt-ing the ice. This covers the phase of the planning too, so the pupil will not have to compose it again. This is the stage of the strategic process that requires flexi-ble thinking of the part of the pupils, the application of concepts they learned in the experimental program, and the transfer of observations of the performed ex-

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periments. In this task the phase of evaluation is the step during which the pupil can give reasons based on his existing knowledge about the result of the proce-dure proposed by him. The solving of the task must be carried out in theory.

The task requires relatively little preliminary knowledge, such as: the con-cepts of solid, fluid, ice, water, melting and freezing, and the relation, that ‘there is no life without water’. So considering its nature it is semantically poor and badly defined because although there is a given starting state, the road leading to the solution is unknown, and the target state is not clearly formulated. Because in the course of the solution important knowledge and information appear only partially in the task, it is a poorly structured, incomplete task. A multitude of op-erations lead to the solution of the problem which means that we are facing a transformational problem.

Task 2: ‘This is winter and it is very cold. The lake is frozen. On the surface

of the ice, careless people are scattering more and more garbage including oily and stained papers, bottles, flasks. There will be sunnier and warmer weather af-ter the frosty times. What is the problem? How can you solve it?’

The second task requires of the pupil to solve a more complex problem. The larger amount of information given and the complicated structure of the sentence make understanding more difficult. To solve the task, the pupil needs to recog-nize, on the other hand, that the sun causes the ice to melt, so the dirt can get into the water and harm its wildlife. The other part of the Investigation of pupils’ sci-entific problem solving process using knowledge space theory at primary level solution is protecting the water from dirt. The range of former knowledge neces-sary for the solution is necessarily wider than in the first task. So the pupils need to know the concept of solid, fluid, ice, water, melting, freezing, besides regu-larities and relations that the oil floating on the water surface prevents oxygen getting into the water, which can lead to the lack of oxygen and the destruction of aquatic organisms. Since at this age the pupils does not know the exact con-cept of specific gravity, they need to be aware that the material of solid state dirt is lighter or heavier than water, therefore it will sink or float on the water surface and will do damage to the ecosystem of the lake. The exploration of ideas con-cerning water cleaning is the most flexible phase of the way to the solution, the part of the task during which the pupils have an opportunity to transfer their knowledge about water cleaning processes performed during the experimental program.

The part of the task that required the most abstraction was the recognition of the type and features of the pollution and its connection with the manner of the removal of the dirt. The exploration of water cleaning processes means the coin-cidence of the phases of forming a hypothesis and planning in this task too. Al-

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though this task contains more information compared to the first task regarding the solution, it is still semantically poor, badly defined and structured, it has a transformational problem. Compared with the other task it is more complex, re-quiring more attempts during which there is an opportunity for deeper examina-tion of problem solving strategies.

In the selection of the examined strategic elements we built on the model cre-ated by Pólya (1957) because of the previously mentioned considerations, al-though we supposed that not every step of it can be properly observed yet in the way of thinking of the examined age group. We completed the model with the phase of aim formulation because the experimental program emphasized this element well. During the research of the pupil’s hypothesis-making we exam-ined, on the one hand, whether they are capable of this operation and on the other, if they are capable, what assumptions they can make regarding the solu-tion. We took the same steps during the examination of the planning phase. We completed the examination with the question whether the pupil can explain his conception completely or only in part.

In respect of how consciously the pupil formulates the problem, the purpose of solution, we created the declared, conscious, detailed and the latent superfi-cial subcategories. In our opinion the goal setting is declared if the thoughts of the child contained the following clauses: ‘My aim is…..; With that I would like to reach…..; I want to….’ etc. This does not mean that the pupil who did not use similar terms does not know how to solve the given problem. So here the con-scious attribute mainly refers to the formulation, to the mode of expression. We thought similarly in the case of the formulation of the problem.

In order to evaluate to what extent the pupils, using already learnt natural sci-entific concepts, actually formulate their concepts in scientific or in common language, we created further subcategories.

So we examined the structure of the problem solving process of 9- to10- year-old pupils on three levels. The levels explore the features of certain steps more and more deeply (Figure 1).

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Figure 1. The examination model of the process of problem solving.

Level 1. Global existence of the steps of the process

existent non-existent

Level 2. Declaration level of the existent process step

declared, conscious, detailed latent, superficial

Level 3. The linguistic mode of expression

of declared, detailed steps scientific everyday concepts

The phases seen on Figure 1 were measured three times in the experimental

group on the basis of their attributes: a) before the experimental teaching, b) immediately afterwards (two months after the firs measurement) c) four months after finishing the experimental teaching. A similar procedure was followed in the control group. We skipped the developmental experiment of course. The re-sults presented here concern only the third measurement that shows the consoli-dated knowledge that came into being as a result of the development program. So in the case of the control group we had to consider the results of the third measurement. In the scoring we gave one point if the given category was present in the thinking of the pupil, otherwise we gave zero. In the quantitative examina-tion of hypothesis-making and planning, the pupil got as many points as many suppositions and plan suggestions he gave in the given subcategory.

development association, abstraction

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We characterized the structure of certain levels with the help of Knowledge Space Theory.

The Knowledge Space Theory is one of the methods of the examination of knowledge structure (Doignon and Falmagne, 1999). In the field of natural sci-ences – primarily in chemistry − it has been used successfully for the adaptive interrogation of knowledge (Abari and Máth, 2010), for the determination of the critical learning pathway of student groups (Taagepera et al., 1997; Tóth and Kiss, 2006; 2007), for the examination of knowledge structure as a hierarchical network (Tóth, 2005; 2007; 2012; Tóth and Sebestyén, 2009). Tóth and Ludányi combined Knowledge Space Theory and the fenomenography successfully for the examination of students’ definitions (Tóth and Ludányi, 2007a; 2007b), and extended it to all such response category fields in which the response categories are not mutually exclusive, Investigation of pupils’ scientific problem solving process using knowledge space theory at primary level and there can be some kind of precondition or connection between them (Tóth and Kiss, 2009; Tóth et al., 2008). The technical details of the examination can be found in former stud-ies (Tóth, 2005; 2007; 2012).

On the basis of Knowledge Space Theory we have an opportunity to model the characteristic knowledge structure of each learning group. The hierarchical relation between the knowledge structure and the knowledge elements in it can be displayed clearly in a Hasse diagram. You can see such a Hasse diagram for example in Figures 2 to 4. In consideration of the fundamental assumptions of Knowledge Space Theory (according to which if a student can solve a task of a higher level in the hierarchy, it can be expected that he can solve tasks at the lower levels of the hierarchy) from these hierarchies the knowledge structure (consisting of possible knowledge states) can be deduced (Tóth, 2005; 2012). For example on the basis of the Hasse diagram on Figure 2, the possible knowl-edge states are: [0], [1], [2], [1,2], [2,3], [1,2,3], [2,3,4], [1,2,3,4], [2,3,4,5] and [1,2,3,4,5].

5.3. Results and assessment

Level 1 Before comparising the structure of the experimental and the control group,

on the basis of the models created as a result of former researches concerning the process of problem solving, we constructed an expert hierarchy, using the method of Falmagne et al. (1990) (Tóth, 2012).

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Table 1. A possible relation table of Pólya’s (1986) linear model of problem solving

thinking. (1: goal setting; 2: problem definition; 3: forming a hypothesis; 4: planning, execution; 5: evaluation)

1 2 3 4 5 1 1 0 0 0 0 2 0 1 1 1 1 3 0 0 1 1 1 4 0 0 0 1 1 5 0 0 0 0 1

Total 1 1 2 3 4

Figure 2. The expert hierarchy created on the basis of Table 1.

(5)

(4)

(3)

(1) (2)

According to the data of Table 1 goal setting falls outside of the process of problem solving determined by the linear model. Nevertheless it was included into the elements to be examined, because it is incidental to the formulating of the problem. When we define a problem at the same time we make reference to the aim of the solution (for example ‘The problem is that the wildlife is damaged by to the dirt which got into the lake, so the contamination must be eliminated or prevented.’ The last clause refers to the aim of the solution). So when analyzing Table 1 it becomes understandable why 0 was given in the relation of the other elements. It is because if someone can not specify the purpose of the solution he may still be able to construct the other elements. In what follows we took Pólya’s model, its logical process and order as our basis in a strict interpretation (raising and formulation of a problem, forming a hypothesis, planning, execu-tion, evaluation) and we suppose that these elements follow one another in this order.

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The interpretation of Figure 2 is the following. The step of forming a hy-pothesis follows the formulation of the given problem and/or the purpose of solving the problem in the solving process, and it is the condition of the plan-ning, executing and evaluation elements.

Considering the course of cognitive development, these steps as the appear-ance of skill levels in formal thinking happens at the same time at different de-velopmental levels which are the essential components of hypothetic thinking. However, Figure 2 and the following analyses will prove that these process steps are in a hierarchical relation with one another. According to this interpretation there is a timing difference between the appearance of certain elements, and it is probable that the elements requiring the highest abstraction and the greatest number of cortical associative connections, like evaluation, can reach the higher developmental level.

You can see the evidence in Figure 2 that we can perform planning and evaluation only when there is a verifiable hypothesis which is however an as-sumption regarding the solution of some kind of formulated problem.

After the expert hierarchy we examined the process structure of the experi-mental and the control group.

The Hasse diagrams of the experimental group show that the level having the higher abstraction is evaluation. The condition here is that the students should be able to define the problem, the purpose of the solution, they should be able to make a hypothesis and to plan the process which means their verification. (If we assess the knowledge and the ability state of students we may realize that the evaluation is the level that is reached by the fewest students. That is considered to be in conformity with the concerning quantitative analysis too.) In the exam-ined sample the subsequent level below evaluation is goal setting. Its conditions are the formulation of the problem, the hypothesis-making and the ability of planning. This shows a difference from the expert hierarchy and demonstrates that the formulation of the solution’s purpose can cause considerable difficulties for 9- to10- year- old pupils yet. Even the aimed experimental teaching could not help in significantly.

Three different learning pathways could be observed in the experimental group that could be divided into three consecutive major blocks. The elements appearing first are the formulation of the problem, hypothesis-making, the plan-ning and evaluating. The second block (and at the same time higher level too) is goal setting, while the third, the latest learned element is evaluation. So we must emphasize the development of the two latter steps during the formulation of problem solving ability in the lower grades of primary school. The specification of the critical element proves this too. The critical element is the part of the knowledge structure the learning of which the most members of the control

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group is being prepared (Tóth, 2005; 2007; 2012). According to the data of Ta-ble 2, members of both the experimental and control groups are most generally prepared for the reception of the evaluation step at the age of 9 to 10. The effect of experimental teaching manifests itself in the case of the control group where the ability of learning the critical element (evaluating) is greater because in the experimental group students have met this phase of the solving process several times during the targeted learning methods (experimentation).

Table 2. Proportion of pupils prepared for the reception of certain elements in the ex-

perimental and control groups at the 1st level.

Group Goal set-ting

Problem definition

Forming hypothesis

Planning, execution

Evalua-tion

Experimental 11,45 % 0,12 % 0,55 % 4,96 % 44,61 % Control 14,51 % 0,11 % 2,26 % 8,78 % 79,08%

There was no typical Hasse diagram in the control group, so the pattern of the learning pathway is presumably similar to the pattern in the experimental group. This is evidence that the global existence of certain elements was less in-fluenced by the experimental teaching than by the effect of age.

Level 2 At level 2 we examined how the expert hierarchy typical for level 1 (in Fig-

ure 2) changes if we go into details, and we analyze certain steps in terms of consciousness and detailed elaboration that means a higher abstraction level (de-clared, conscious goal setting: for example sentences starting with - My aim is,….. or I do this because….; declared, conscious problem definition: for exam-ple: The problem is that….., or the formulation of the essence: ‘The water of the lake is need to be cleaned, because…’, etc.; detailed planning: if the student can formulate every detail of the plan in a proper logical order).

Table 3. Relation table of the 2nd level (1: declared, goal setting; 2: declared problem definition; 3: forming a hypothesis; 4: detailed planning, execution; 5: evaluation)

1 2 3 4 5 1 1 0 0 0 0 2 0 1 0 0 0 3 0 0 1 1 1 4 0 0 0 1 0 5 0 0 0 0 1

Total 1 1 1 2 2

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You can see in Table 3 that if someone is not able to formulate the essence of the problem straight out, he can still be able to create other elements. Con-versely, if the pupil is not able to forme a hypothesis, the following elements will not appear later. The absence of detailed planning and the ability of execu-tion mean that apart from this, the pupil can still be in possession of the lower level of planning, therefore he may be able to create other elements.

So the expert hierarchy at the 2nd level is the following (Figure 3).

Figure 3. The expert hierarchy created on the basis of Table 3.

(4) (5) (1) (2) (3) Investigation of pupils’ scientific problem solving process using knowledge

space theory at primary level According to Figure 3 if we follow the expert way, the condition of detailed

planning and evaluation is forming a hypothesis. That is to say in comparison with the hierarchy in Figure 2, planning, execution and evaluation are higher here too, while declared goal setting and problem definition show increasing ab-straction and they at the same level with the ability to form a hypothesis.

While the expert hierarchy in Figure 2 follows the element order of Pólya’s linear model, we can see in Figure 3 that the manifestation of forming a hy-pothesis is independent of the explicit or implicit manner of recognition and formulation of the problem; the condition is the presence of it in some form and level of development or another.

According to the expert hierarchy in Figure 2, there can be different abstrac-tion levels within each step, but they do not affect the movement to the next element presented at a higher level of the linear structure.

On the basis of the analysis of the Hasse diagram of the experimental group the declared goal setting, evaluation and detailed planning got to the highest level. The condition is that the pupil must be able to formulate the problem con-sciously. Namely, if the consciousness of any element can be observed, probably the other elements will find similar expression too. This refer to a higher degree of metacognition.

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Compared to the expert hierarchy (Figure 3) in the experimental group, form-ing a hypothesis is always a precondition for the declared formulation of the problem, while if we take no notice of the abstraction level of the problem’s formulation (Figure 2, besides the Hasse diagrams regarding the first level in the experimental group and the learning pathway showed by the control group at the second level), it is the other way around. At first it is needed to see the problem clearly, which is followed by forming a hypothesis. The result of the experimen-tal group can be explained by the fact that during experimental teaching the forming of a hypothesis was emphasized, while the problem definition some-times appears as a secondary factor.

Making children conscious of the learning process was the primary purpose of the program of the Rostock Model. It emphasized the experimentation among its methods by means of which the children learnt problem formulation, hy-pothesis-making, planning and evaluation (although not with the same impor-tance). After the specification of learning pathways and Hasse diagrams we can say that there was already a learning pathway where the ability of evaluation and conscious goal setting developed before detailed planning. This is probably the effect of experimental teaching. This is so much the truer because in the control group conscious goal setting and evaluation got to the very end of the learning pathway.

Table 4.

Proportion of students prepared for the reception of certain elements in the experimental and control groups at the 2nd level

Group Goal set-

ting Problem definition

Forming hypothesis

Detailed Planning, execution

Evalua-tion

Experi-mental 49,27 % 13,55 % 2,96 % 40,22 % 38,53 %

Control 88,48 % 10,00 % 7,83 % 66,44 % 69,80 %

The specifications of critical elements of the 2nd level show an analogy with

the lessons of Hesse diagrams and learning pathways (Table 4). In the case of both the experimental and control groups the next step is the learning of de-clared goal setting, close to it detailed planning, execution and evaluation. The values of the experimental group are lower owing to the effect of the program.

So it is important information for the teacher that in the fourth grade of pri-mary school (9-to 10- year-old pupils) the highlighted task is not hypothesis-

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making or problem formulation, but the effort to be conscious in the specifica-tion of the purpose of the solution, planning and evaluation. But this can only happen in this way if we develop the problem solving thinking of the a given student group gradually in a planned way, taking into account the age character-istics from the first class of primary school.

Level 3 The third level is the highest attainable level on the basis of the criteria of our

examination. This means that in addition to the given element being consciously formulated and entirely elaborated, the child applies natural scientific phraseol-ogy in his language which means a higher cognitive level.

Table 5.

Relation table of the 3rd level (1: formulated goal setting; 2: declared, scientifically formulated problem definition; 3: hypothesis, scientifically formu-lated; 4: detailed planning, execution, scientifically formulated; 5: evaluation,

scientifically formulated)

1 2 3 4 5 1 1 0 0 0 0 2 0 1 0 0 0 3 0 0 1 0 0 4 0 0 0 1 0 5 0 0 0 0 1

Total 1 1 1 2 2

Investigation of pupils’ scientific problem solving process using knowledge space theory at primary level

Figure 4.

The expert hierarchy created on the basis of Table 5.

(1) (2) (3) (4) (5) According to the data of Table 5 if the student is not able to formulate a given

element in scientific language, he can still be able to formulate the other ele-ments scientifically or in everyday terms. So here the expert hierarchy (Figure 4) shows an equal level of the steps, there is no relation between them. So scientific

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phraseology does not have an influence on the hierarchy of the process steps, it means only a higher abstraction level of the given element.

According to the examination, the experimental group shows an absolutely different picture in respect of expert hierarchy. According to Hasse diagrams, the formulation of hypothesis-making is the condition for the similar formulation of other elements. Whoever is able to do this is probably also able to represent other elements in scientific language as well.

The learning pathway shows an unidirectional hierarchy in the direction of scientifically formulated hypotheses, problem formulation, evaluation, goal set-ting and planning. So the key element in this row is the formulation of hypothe-sis-making in scientific language. Anyone who has achieved this level can count on the appearance of the other steps typical for the third level. This process can be accelerated and stabilized by the help of the proper method.

The end of the experimental group’s learning pathway shows a great similar-ity with the learning pathway typical of the second level whereas the evaluating and goal setting in scientific terms are placed before detailed planning. One rea-son for this is the characteristic of experimental learning that it strengthens goal setting and evaluation. The other reason can be that the detailed planning (which serves for the verification of the hypothesis) requires the recollection and appli-cation of a complex preliminary knowledge system (and what is more, in scien-tific language). This can happen only by means of multiple abstractions, which is one of the most difficult tasks for students at the examined age. However ac-cording to quantitative assessment many pupils reached this level as well and produced significantly better average compared with the scientifically formu-lated evaluation (any evaluation at all) and goal setting.

Hasse diagrams of the control group shows a more varied picture than in the experimental group. One of their characteristic points is the detailed scientific planning which has preconditions as problem formulation and hypothesis-making by using the similar language. On the other hand planning is the precon-dition of evaluation and goal setting. We can find linear structures among dia-grams which are similar to the learning pathways of the experimental group with the difference that here the goal setting and evaluation never precede the step of planning.

It is a fundamental difference compared with the experimental group that the determining element is not hypothesis-making but the formulation of a scientifi-cally determined problem. This stands at the very front of the learning pathway, every other elements just come after it.

The reason why the members of the experimental group learn the formulation of a hypothesis in scientific language first is the increased development effort di-

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rected to this element by the experimental teaching. In this respect the control group shows the pathway typical to age, the formulation of the less abstract pro-gram appears first.

Table 6. Proportion of students prepared for the reception of certain elements

in experimental and control groups at the 3rd level.

Group Declared goal setting in scientific

language

Declared problem definition

in scientific language

Forming hypothesis- in scientific

language

Detailed planning, execution

in scientific language

Evaluation in scientific

language

Experi-mental 52,49 % 29,99 % 14,99 % 67,25 % 61,25 %

Control 88,57 % 50,00 % 62,82 % 84,22 % 89,91%

In the examination of critical elements of the third level we can come to si-

milar conclusions as in the second level (Table 6). The development of goal set-ting and evaluation must be emphasized with 9- to10 year- old pupils because of their age in the future too. On the other hand we can see that in the explanation of the course of problem solving we must emphatically keep close watch on the verbal application of scientific knowledge during the entire course of the prob-lem solving process. The effects of experimental teaching manifest themselves here too and at the same time it proves that if we know that the development of which step of problem solving process is actual at the given age and time inter-val, we can achieve good results by the exploration, strengthening and practice of it.

6. Conclusion Knowledge Space Theory as a method provides an opportunity to character-

ize the structure of the problem solving process (the sequence of certain steps) of a given group. It is suitable for the demonstration of the structural differences in problem solving between the experimental and the control group after experi-mental teaching, which can help in pointing out what is the result of age effect and what can be influenced by development methods.

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In our examination we tested the effect of a didactic program called the Ros-tock Model relating to the structure of Pólya’s linear model among 9- to 10- year- old primary school students. We performed the analysis at three different levels. One of the conclusions is that the linear model’s learning order can be temporarily influenced by targeted teaching-learning methods. This does not mean that the classic and inherently evolved process of problem solving is turn-ing over, only that with the help of the given method the child learns earlier to formulate and declare certain elements compared with other elements. So in our examination, Investigation of pupils’ scientific problem solving process using knowledge space theory at primary level in the higher and higher levels of the analysis of the given element in the experimental group, we saw, that they learned the formulation of a hypothesis earlier than the formulation of the prob-lem, furthermore that the most difficult task for them is goal setting and evalua-tion.

The applied didactic program set out from the assumption that among these pupils there are some who have reached the level of formal thinking so the methods (aiming the raising of awareness of the problem solving process) can be applied during the learning process. On the other hand the program also assumed that the structure of this problem solving is incomplete yet, the elements requir-ing higher level abstraction are have not reached the appropriate level. So the programme tried to attempted the development of the presumably existing, more and less stable process elements in a complex manner. Of the elements of the linear model the programme did not put much emphasis on problem formulation, while it systematically developed the ability of goal setting, forming a hypothe-sis, planning and evaluation.

This is the reason why evaluation and goal setting elements are more devel-oped in the learning pathways in the experimental group compared to the control group, at higher levels (2nd and 3rd levels) and that the members of the experi-mental group learned the higher level of hypothesis making earlier than the for-mulation of problem which generally precedes it. The order can be influenced temporarily which means that the problem solving ability of a 9- to 10- year- old child is very plastic. The elements and the order of the solving process already exist (and this is probably a hereditary characteristic, since it is also able to solve problems intuitively with those, who were never instructed in the process), but certain steps do not have the same strength and stability in certain students.

Starting from the fact, that the formulation of the problem and the ability of forming a hypothesis was the condition of the appearance of other elements in almost every case, we can draw the conclusion that similarly to the classic order we must keep a close watch on the targeted development of them already from the first class of primary school. The strengthening of these two elements is the

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condition for planning, goal setting and evaluation. Most of the 9- to 10- year-old pupils are prepared to learn them and to bring the three elements to con-sciousness, so emphatic development can take place already in the fourth grade. Nevertheless it also need to be kept in mind, that the process of problem solving is an integral whole, there are no more important or less important steps. It can lead to the failure of a solution if whatever is missing or has an inadequate de-velopment level. So the more focused development of one or another element does not mean that we can afford not to pay attention to the others. We can only harmonize the cognitive developmental level of the given age with the abstrac-tion levels of problem solving.

The question is whether there is any need for such explicit intervention, will the problem solving of pupils be improve by that? The longitudinal examination of students from the sample can answer this question, but it will presumably make the process of problem solving faster and more predictable, which is a great necessity today.

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