Children's Knowledge of Problem Solving: A Gifted Group of Problem Solvers.

Australian Association for Research in Education

Annual Conference

Adelaide 1998

Children's Knowledge of Problem Solving: A Gifted Group of Problem Solvers.

98 Abstracts

Penny Van Deur

School of Education Flinders University of South Australia

Introduction

The Mayer report described seven Key Competencies to be introduced into Australian schools.

Solving Problems is one of the Key competencies which is being introduced into South australian schools in 1998. This competency based approach to education is a recent development in education in Australia. The Key Competencies are vocationally based skills and are argued to contribute to young people's ability to function in school, work and life.

The 7 Key competencies are:

1. Collecting, Analysing and Organising Information:

2. Communicating Ideas and Information;

3. Planning and Organising Activities;

4. Working with Others and in teams;

5. Using Mathematical Ideas and Techniques;

6. Solving Problems, and

7. Using Technology.

Cultural understanding has been discussed as a Key Competency but has not been included in current outlines of the Key Competencies.

The Key Competencies are regarded as cross-curricular skills which are able to be developed in a wide range of setting. They focus on outcomes, assume a foundation of knowledge, skills and understanding while focussing on application. It is stated in the Mayer Report that the Key Competencies are integrated in practice.

This account does not mention the characteristics of the child as important in teaching and learning the Key Competencies. The document states that the Key Competencies:

must incorporate a sense of the learner as one who builds concepts and develops understandings which inform technical applications. Competence requires both 'heads on' and 'hands on'; the capacity to think about performance and also to perform. It goes beyond pure or abstracted thinking to the skilled application of understanding (Mayer,1992 :5).

Recent curriculum documents such as South Australia's Foundations for the Future also emphasises the importance of knowing and doing in statements such as :

In our centres and schools, children and students learn to communicate, negotiate, resolve conflict, plan, solve problems, use technology, make decisions, and work collaboratively in teams. They learn skills for living and are encouraged to see learning as a self-directed activity that will benefit them throughout their lives (DECS, 1997 :Principle 4).

The emphasis on self-directed learning is widely documented in the psychological literature. Dispositional factors are regarded as important in the self-regulation of problem solving behaviour.

The Mayer Report states that :

the Key competency of Solving Problems is also about the nature of solving problems as a process, including the control that is exerted over the process (Mayer, 1992 :26).

However, dispositions in the form of attitudes and values were explicitly excluded from the Key competencies as it was felt they could not be measured.

Dispositions may be described as habits of mind which influence the way a task is approached (Resnick, 1989). Dispositions are based on belief systems, values and attitudes. Dispositions influence the attitudinal state of the learner and the strategies chosen in problem solving as well as the way the learner interprets their performance.

The Key competency of Solving Problems is represented by the Mayer Report as being about four main ideas. They are:

-Clarification and framing problems

-Achievement of appropriate completion

-Anticipation of problems, sources and contexts

- Evaluation of outcomes and processes.

It is relevant to ask if problem solving is about process, skills control or other things. The Key competencies documents contain contradictory statements about the nature of problem solving. It seems especially important to clarify the question about the nature of problem solving because this curriculum is being implemented in schools.

The Mayer Report looks at the issues of implementing the Key Competencies not at underlying research on the validity of the description of each Key Competency. The Mayer description of the Key Competencies has not been based on recent research or practical experience in school settings.

There are issues of teaching and learning involved in describing the Key Competencies so that curriculum and teaching programmes can incorporate them. This raises the question of what students know about problem solving and what they do when problem solving.

The implementation of a curriculum innovation such as the Key Competencies raises two broad questions :

1. How adequate is the Mayer Report's representation of Solving Problems?

2. What do students who are regarded as good problem solvers know about the Key competency of Solving Problems ?

It is useful to consider the problem solving knowledge of a group of students who are generally described as good problem solvers. Students identified as gifted are generally characterised as being good problem solvers (DECS,1996). The knowledge of this group can be compared with the Key Competencies description of Solving Problems. It would be expected that there would be agreement between the main ideas of the Key Competency of Solving Problems, if they are adequate, and the knowledge of the students in the gifted group.

To understand the knowledge of problem solving possessed by a group selected as academically gifted an estimate is needed of what these students are likely to talk about as solving problems. The behaviour of students when completing a problem solving task is examined.

The student perspective.

In this study the student perspective is used to investigate the knowledge of problem solving possessed by a group of students who were judged to be academically gifted. The same study was carried out with an average class group of students in Year 5, though this study will not be reported in this paper.

The student perspective is a little used approach in assessing the validity of new curriculum documents. Schoenfeld (1985) describes the student perspective as affecting the ways students behave and it influences what they perceive to be important in a problem and what sets of ideas or cognitive resources they use. Student belief systems determine how one chooses to approach a problem, which techniques will be avoided as well as how long and hard one will work on a problem. a change in belief systems may change control decisions and the student may use more of what they know. This may be the key to successful problem solving.

Biggs (1991) states that what students say and do while carrying out problem solving is an important source of information for teachers in planning and evaluating problem solving and in helping students to become self-sufficient and reflective in their cognitive processes.

I have reviewed the literature to develop a problem solving framework as a way of representing the events and influences that can be used in orgainsing data later in my research. I will use this framework which is developed from the research literature in examining the discussion of Solving Problems in the Key Comptetencies documents.

The following discussion relates the psychological literature to the developed problem solving framework.

A Framework to represent Problem Solving

EARLY PHASES OF PROBLEM SOLVING

Analysis.

Analyse the problem to understand what it is about.

The social environment in which children grow up influences the way they think as well as what they think about (Siegler,1991). The way students view the world may affect the ways students behave when they are confronted with a problem, influencing what they perceive to be important. The problem solvers beliefs establish the context within which a problem representation operates.

Representation

Representation to understand what could be done on the problem.

New information is interpreted in terms of what is already known and is coded according to the knowledge base possessed and in terms of the task at hand. Das,Naglieri, Kirby, (1994) argue that prior information may be used implicitly without conscious effort.

Rabinowitz and Glaser (Horowitz, O'Brien, 1985) state that understanding and insight into a problem comes from the interaction between the explicitly stated information in the problem and the student's prior experiences and knowledge. Rabinowitz and O'Brien argue that the initial representation strongly influences the choice of strategy that is used.

MIDDLE PHASES OF PROBLEM SOLVING

Planning

Plan possible solutions to try on the problem.

Planning is a domain-general strategy.

Das, Naglieri and Kirby (1994) assert that planning is a set of decisions or strategies that an individual adopts and modifies to solve a problem and reach a goal. Strategies are conscious plans for performing tasks and are responsible for deploying abilities (Das, Naglieri, Kirby, 1994).

Planning is an executive, or management procedure which is continuously applied in learning and action (Biggs,1991). The function of executive procedures is to recognise and analyse situations and tasks, form goals and sub-goals, form plans to achieve task goals and to deploy resources while monitoring progress in terms of progress toward current goals. Planning occurs during the activity as explicit or implicit decisions are made to continue, change or terminate the current activity. Students can be taught to be self-regulated by learning how to set goals, evaluate them and to use them for planning further strategies.

Selecting strategies

Selecting strategies or procedures to try on the problem.

General strategies are best learned in the context of the specific problems to be solved (Charles and Lester,1982). Across a variety of domains, strategy choices are produced by domain general processes operating on domain-specific associative knowledge ( Siegler,1991). Siegler outlines that critical information within a problem is encoded and then metacomponents are used to construct strategies.

Strategies are important in promoting self-regualted learning and perceived self-efficacy as students can see the rules and steps that improve performance. This conveys to students that they are capable of applying strategies appropriately which can instill a sense of personal control over their achievement outcomes which can raise self-efficacy (Zimmerman and Schunk, 1989). Students who attribute school successes and failure to their own effort and use of strategies are more likely to accept challenges and persist in the face of difficulties ( Watson and Lawson,1995).

Applying strategies

Applying strategies to work out a problem

Self-efficacy beliefs influence the execution of behaviours relevant to a specific task or situation (Bandura,1992). Bransford and Vye (in Resnick and klopfer, 1989) argue that it is important that knowledge is conditionalised so that students know when to apply principles, concepts and strategies.

LATE PHASES OF PROBLEM SOLVING

Evaluation of progress on the solution

Evaluating to see progress being made and to see if new sub-goals need to be planned.

Biggs(1991) states that increased awareness of learning changes attitudes and procedures and that self-evaluation was the key to progress. The quality of the learning that occurs is determined by the decisions made by the learner and those decisions depend on the learner's perceptions and interpretations.

Garafalo and Lester (1982) argue that the managerial skills necessary to regulate problem solving activity need to be taught. Evaluating one's plans and strategies is an important part of the problem solving process.

Review

Review of problem solving activity in order to revise or abandon non-productive strategies and plans.

Students can be trained to be reflective in their problem solving by looking back to find errors and to check that a problem has been solved. Prawat (1989) describes a mastery disposition where the goal is to increase competence. Errors may be interpreted as providing useful information in problem solving. Teachers are seen as guides in learning. This orientation favours complex learning strategies such as planning, monitoring and checking work.

CONTROL OR DISPOSITIONAL PHASES OF PROBLEM SOLVING

Monitor

Monitor progress on the solution attempt.

It has been found (Hong,1993) that self-monitoring of performance increases students learning and subsequent performance.

Self-efficacy for problem solving is enhanced when students believe that they have the capabilities to solve problems and reach desired outcomes by effectively employing their skills and knowledge. Schunk (1991) argues that motivation is enhanced when students perceive that they are making progress in their learning.

Students who believe that they are capable of learning will attribute successes to their own abilities and efforts rather than to outside assistance. This belief contributes to self-regulation (Zimmerman and Schunk, 1989). Self-monitoring is a general skill.

Persist

Persistence or self-control during the problem solving attempt.

Problem solving involves making efforts to solve problems and coping with difficulties. Gilhooly ( 1989) argues that a problem solver's beliefs about problem solving can affect the process of problem solving by affecting persistence.The way the student views difficulties in problem solving is affected by their beliefs about problem solving (Winne,1991). Students who have high self-efficacy are likely to accept greater challenges, expend more effort and may be more successful in reaching their goals as a result (Bandura in Hjelle, Ziegler, 1992).

Ask the Teacher (interaction ).

The instructional activities of planning, organising practical and cognitive activity can be learnt in co-operation with more knowledgeable others such as peers, or adults including the teacher (Wood,1988). Wood asserts that it is through social and instructional interactions that self-regulation is discovered and developed.

The phases of late and control or dispositions are classified together as Self-management aspects of problem solving.

Problem Solving Framework

Tittle (1994) discusses a framework for the assessment of teaching and learning and says that it is important to view the interpreter (teacher) and learner (student) as central to the development and evaluation of assessments.

A study of problem solving by Fortunato, Hecht, Tittle, Alvarez (1991) found a discrepancy between students actual and reported behaviour which suggests the need to clarify the meaning of problem solving strategies.

The problem solving framework is made up of four phases which are also comprised of events which describe different parts of the problem solving process.

Early Analysis to understand what a problem is about

Representation to understand what could be done on the problem

Middle Planning possible solutions to the problem

Selecting strategies or procedures to try on the problem

Applying strategies to work out a problem

Late Evaluation of progress on the problem

Review of problem solving activity

Control Monitor progress on the solution attempt

Persistence or self-control during the problem solving attempt

Ask the teacher for clarification

The phases of Late and Control (disposition ) are classified as self-management.

The problem solving framework is compared with the main ideas of the Key Competency of Solving Problems in Table 1.

Phase	Event	Key Competencyof Solving Problems
Context	Context General Specific	Context General
EARLY	Analysis	Clarification
	Represent	Framing

MIDDLE	Planning	Anticipation
	Selection	xxxxxxxxx
	Application	xxxxxxxxx

LATE	Evaluation	Evaluation
	Review	Evaluation

CONTROL/ DISPOSITION	Monitor	xxxxxxxxx
	Persist.	xxxxxxxxx
	Ask T.	xxxxxxxxx

xxxxxxxxxxdenotes that the event is

not stated explicitly in the Key Competencies document.

Table 1 The Problem Solving Framework compared with the main ideas of Solving Problems.

The table shows that there are events which are omitted from the Key Competencies description but are cited in the literature as being part of problem solving. These events describe selecting and applying strategies, monitoring the course of problem solving activity, persisting when striking difficulty and asking the teacher for clarification.

This study looks at the knowledge (views and behaviours ) of a group of Year 5 students selected as being academically gifted. The knowledge of the students is compared with the Key Competencies outline of Solving problems to assess the adequacy of the description presented by the Mayer report.

Research Questions

1. What knowledge do students in the gifted group have of problem solving ?

2 . What is the knowledge state of students in relation to the influence of dispositional factors in problem solving ?

3. To what extent do students indicate that they can exert control over their problem solving performance ?

4. Do the students describe problem solving in a domain general way or a domain specific way ?

5. What is the relation between knowledge of students and their actions as exhibited on a problem solving task ?

6. In what way does the problem solving knowledge of the group of students identified as academically gifted compare with the Key Competencies description of Solving Problems?

The Studies

147 students in Year 5 (from 4 schools) were tested on the screening test Raven's Standard Progressive Matrices. Twenty-one students who scored above the 90th. percentile were selected to participate in the problem solving session.

The students were trained in think aloud procedures before the problem solving task was undertaken.

Each student was interviewed and observed individually. Each session was audio-taped and video taped. Later the sessions were transcribed and the protocols were analysed and coded.

Tasks and Data Taken to assess PROBLEM SOLVING Knowledge

TASK	DATA TAKEN

Students
Individual student interview to ascertain views of problem solving behaviour.	Student descriptions of the phases of problem solving.
Attitudes to a similar problem..9 dots problem	Dispositional attitudes to problem solving.
Attitudes to a dis-similar problem.Social problem involving bullying.
An invention problem to create a robotic shopping trolley which satisfies 4 constraints.	Observation of problem solving actions
Student self-report questionnaire on the robotic trolley problem solving task	Attributions for success on the problem solving task
Student ratings on problem solving ability in 6 curriculum areas.	Student ratings.

RESULTS

1.What knowledge do students in the gifted group have of problem solving ?

In the interview students were asked a general question about their knowledge of the nature of problems:

Q.1 What do you think problems are?

The responses to the questions were tabulated and percentages of the total number of responses for the group were calculated. The results are shown in Table 2.

Gifted Group	%
Something you have to solve	27
Think over	9
Difficulties to sort out	9
Get answers	9
Exercises to teach you	5
Having trouble	4.5
Mathematics	4.5
Answer questions	9
Don't Know	23

Table 2.The knowledge of the nature of problems of students in the gifted group.

The focus of interest with these questions was interpretations made by the students of these terms at a general level, interpretations that would be influential in situations such as when students would engage in problem solving activity in classrooms.

Most of the responses indicated that the students were able to provide some interpretation of problem. 54% of the responses included recognition that the problem required effortful activity and that the situation needed some resolution.Recognising that problem solving requires effort may have dispositional implications for the way students approach problem solving.

For a small number of students the term problem was associated most strongly with mathematical tasks. These students did not associate problems with other areas of the curriculum. This may reflect the fact that the students' work in mathematics is frequently discussed as involving problems and problem solving.

23% of the responses were 'Don't know' which seems puzzling though this response may indicate a view of problem solving as specific to a particular task.

The children did not clearly specify that the process in problem solving is important.

During the problem solving interview the students were asked for their views about their problem solving ability, the possibility of improving their problem solving and whether their problem solving ability varies in relation to different types of problems.

The responses were assigned to four categories. The categories are:

Good the student answered that he/she is a good problem solver.

Uncertain the student answered that he/she does not know if he/she is a good problem solver.

Practice the student answered that he/she could get better at problem solving by practice.

Discriminated

practice the student answered that he/she would use discriminated methods of practice by detailing some specific procedures such as think about it, do it with a friend, read the question better.

The results are shown in Table 3.

Interview Questions	Gifted Group
Are you a good problem solver?	Good 47.5% Uncertain 52.5%
Do you think you can get better at problem solving?	practice 47% discriminated practice 53%
Are you good at solving some problems ?	Yes 95%
Do you have difficulty solving some sorts of problems ?	Yes 65%

Table 3. Gifted Students views of their problem solving ability.

Most students in this group indicate that they may be good at solving some sorts of problems and that they may have difficulty solving some sorts of problems. These responses indicate that the students recognise the need to consider the specific task in estimating their ability to solve a problem. This view of ability in relation to a specific task is taken up later.

2. What is the knowledge state of students in relation to the influence of dispositional factors in problem solving ?

The students answered questions to assess their beliefs about the way they approach a problem solving task or their orientation to a problem solving task.

Scores were given for answers students gave to interview questions:

5 When solving a problem, what do you do first?

6 What comes next ?

7 Why would you do that first ?

8 In solving a problem, what comes next ?

Each answer to questions, 5,6,7,8,was classified into the 4 phases of problem solving early, middle, late, control,( management). The number of responses for each category or event within each phase was totalled and a percentage of the total responses was calculated.

In the early phases a statement was classified as analysis if the child mentioned the need to understand what the problem was about.

In the middle phase a statement was classified as planning if the child mentioned a plan in problem solving.

A statement was classifies as selecting strategies if the child mentioned ways of trying to solve a problem and applying strategies if the child mentioned working out the problem.

In the late phase a statement was classified as evaluation if the child mentioned checking to see if an answer was correct.

It was classified as review if the child mentioned looking back over their work.

In the control phase a statement was classified as monitoring if the child mentioned checking how they were going with the problem.

Statements were classified as persistence if the child mentioned that they would keep going on the problem if they had difficulty.

A statement was classified as ask the teacher if the child mentioned asking the teacher for clarification or help.

The resultsof the student responses are shown in Table 4.

	Student responses	Total %
EARLY
Analysis	27
Representation	16.7	43.7
MIDDLE
Pianning	10
Selecting strategies	8.3
Applying strategies	19.4	37.7
SELF-MANAGEMENT LATE
Evaluate	6.9
Review	4.7	11.6
CONTROL/DISPOSITION
Monitor solution attempts	1.4
Persistence	2
Ask teacher	0
Unexplained	3.6	7

Table 4. Students in Gifted Group views of problem solving knowledge classified according to the Problem Solving Framework

From this analysis it can be concluded that the students in the gifted group view problem solving as working out what a problem is about and working out an answer.The responses suggest that most activity will occur in the early and middle phases of problem solving. There was a small emphasis on planning and selecting strategies to work out a problem. Little emphasis was given to metacognitive aspects of problem solving such as evaluation, review or monitoring.

Students were asked their views about the detail of the process involved in solving a problem in the questions:

In solving a problem, what steps do you take first?

When having difficulty what do you do?

The results are shown in Table 5.

In solving a problem what steps do you take first ?		When having difficulty what do you do?
Response	Gifted Group		Gifted group
Specific procedure or strategy	52%		53%
Vague or general strategy	48%		23%
Does not know if he/she has a strategy	0%		12%
Seek adult help	0%		12%

Table 5.

Percentage of students outlining the way a problem is approached and the way difficulty is handled.

Students in the gifted group outlined specific procedures for trying to solve a problem and for coping with difficulties in problem solving.

An example of specific procedures is shown in the following protocol:

Student 4 on beginning to solve a problem: The first time I take like little notes, the first I just read through it and the second time I read through it I take little notes and put down the important things and then I just work it out...

Student 4 on coping with difficulty: Well if I'm in a test and some problems are really hard to solve like in a maths. test a lot of them are really hard to solve and what I do then is I just skip them cos' other wise it'd be a waste of time and then when I'm finished I went back so I could spend more time on them.

These students indicate that they have a small degree of reliance on seeking adult help when striking difficulty in problem solving.

The students looked at the 9-Dots problem .

· · ·

· · ·

· · ·

Would you be able to solve this problem by drawing four straight lines to pass through all nine dots?

Each line has to be connected to the end of at least one other line.

Do you think you could solve this problem?

Would you be good at solving this type of problem?
Gifted Group

81% could do it if they tried

81%could be good at this

problem (experience).

The responses indicate a positive disposition toward the problem shown. This indicates that the students believe that they will be able to do the problem if they try.

The problem is a mathematical problem and the students may have more experience and knowledge of solving problems of this type. In this case explicit knowledge may act as an orienting factor.

The students were asked to consider the following social problem solving situation :

A friend of yours is being bullied in the schoolyard by a year 7 and asks you to help them . Would you be able to solve the problem?

Would you be good at solving this type of problem?

Gifted group

81% would be able to solve the

(problem (experience).

81% may be good at this

problem (experience).

The responses indicate a strong disposition for social problem solving of this sort. Experience is cited as the main reason for the positive disposition.

In both specific situations presented the students have a strong view of their possible success.

R.Q.3 To what extent do students indicate that they can exert control over their problem solving performance ?

The students were asked the following question in the self-report questionnaire following the problem solving task:.

Why did you succeed on the problem?

The responses were categorised into attributions for success on the problem solving task.

Percentages were calculated for the answers to this question. The results are shown in Table 6.

Stable	Internal	External
	Ability	Task
	28.5%	28.5%
Unstable	Effort	Luck2
	28.5%	14.5%

Table 6. Attributions of students in the gifted group for success on the problem solving task.

The responses show that children make attributions across the four categories.

The responses of the students show a small reliance on external factors of task and. luck. 57% of attributions were to internal factors of ability and effort indicating that the students place greater emphasis on their own control of their problem solving performance.

Attributions to factors such as effort or ability are under the internal control of the student.

4. Do students describe problem solving in a domain general way or a domain specific way ?

A key point of interest is whether problem solving is viewed in a more general or more specific way. Information was sought about this by asking students to rate their capabilitiesas a good problem solver in each of 6 curriculum areas.

The student ratings were scored so that a higher score reflected a higher rating of capability.

The ratings scale was a 7 point lickert scale:

6=very true 3=sometimes true 0=not true

The results of this analysis are shown in Table 7.

	French	Language	Mathematics	Music	Resource
Language	-.09
Mathematics	.06	.26
Music	-.03	.37	.00
Resource	.43*	.31	.31	.44*
Social	.23	.25	-.18	.14	.48*

*p<.05

Table 7. Spearman correlation co-efficients between student ratings of their ability to be a good problem solver in each of 6 curriculum areas.

The results show that there were no very strong correlations. The highest correlation is only moderate in strength.

The correlations suggest that the students have a domain-specific view of their capabilities in problem solving. The students did not transfer their self-efficacy beliefs for ability as a problem solver from one curriculum area to another. If students rated their ability to be good problem solvers in one curriculum area at a high level , this rating did not predict that the student would rate their ability at a similar level in another curriculum area.

The self-ratings of the students show that the students generally rated their ability at a low level to be a good problem solver in a general sense in social situations. The ratings differ from student answers to the questions about their belief in their ability to solve a specific Mathematics -type problem and a specific social-type problem about bullying at school.

It may be that students did not understand the general term social problem solving but did understand the problem situation presented as it was within their experience.

The students do not transfer their ratings of self-efficacy as a problem solver across different curriculum areas.This supports the previous statement that the students have a domain specific view of their problem solving.

5. What is the relation between knowledge of students and their actions as exhibited on a problem solving task ?

The Robotic Shopping Trolley Problem Solving Task.

Materials

The childen were supplied with seven pieces and were required to use all the pieces in the construction. The pieces were:

a bucket

a garbage can

a roof guttering elbow

a suction plunger

a metal lampshade

a mechanical hand

a skateboard

The constraints:

Create a robotic shopping trolley that is able to:

Collect all the things your family needs in a week.

Pay for the goods.

Deliver them to your car.

Do all these things by itself.

Spend five (5) minutes on this problem. Think aloud as you do it.

Each student attempted to solve the invention problem. This task was chosen to chart the course of each student's problem solving activity. The students engaged in think aloud procedures as they worked on the problem.

It was anticipated that the students would make dispositional comments as they approached the task, that they may try to represent the problem, consider its constraints and plan a course of activity to use in their solution attempt.

Through analysing the transcribed protocols it would be possible to determine if each student was aware of the processes described in the problem solving framework.

If the students were aware of these processes it was anticipated that they would pay attention to them in the actual problem solving situation.

Table 8. Comparing the views of Problem Solving in the contempory theoretical literature; the Key Competency of Solving Problems and student views and actions when problem solving


PhaseContext	Context General Specific	Context General	Context General/ Specific
EARLY	Analysis	Clarification	27%	6%
	Represent	Framing	16.7%	6%

MIDDLE	Planning	Anticipation	10%	7%
	Selection	xxxxxxxxx	8.3%	5%
	Application	xxxxxxxxx	19.4%	21%

LATE	Evaluation	Evaluation	6.9%	12%
	Review	Evaluation	4.7%	20%

CONTROL/ DISPOSITION	Monitor	xxxxxxxxx	1.4%	6%
	Persist.	xxxxxxxxx	2%	7%
	Ask T.	xxxxxxxxx	3.6%	10%

Students in both groups expressed the view that most of their problem solving activity would occur in the early and middle phases of problem solving.

There was little explicit attention to planning problem solving attempts or to selecting strategies to work on a problem. Little mention was made of the self-management aspects of the problem solving process.

Most activity occurred in this problem solving task in the middle phase of problem solving. There was considerable evidence of control over the final part of the problem solving process. This finding contradicts the views of the students. It appears that the students do not recognise the management skills which they possess and use implicitly

To be able to review the course of the problem solving activity children need criteria with which to judge their problem solving strategies.

The responses of the students in the interview indicate that explicitly very few of the students recognised the need to use such criteria to review their problem solving on this task.

The protocols were analysed for evidence of review.

Strong evidence of review was judged as present if the child checked all 4 constraints ;

weak evidence of review was judged as checking less than 4 constraints;

no evidence was judged when the child did not appear to check the constraints.

The results of this analysis are shown in Table 9.

No. Strong evidence Weak evidence No evidence

children of review of review of review

21 60% 40% 0%

Table 9. Evidence of review on the robotic shopping trolley problem

All of the students in the gifted group exhibited evidence of review on the problem solving task.

Students answered the question in the self-report following the robotic shopping trolley problem: Do you think you solved the problem ?

To assess the coherence of the solution of the robotic shopping trolley problem the researcher checked each completed trolley to see that all of the parts were connected and that the problem constraints were satisfied.

Table 10.Coherence of the problem solution of the Gifted Group

Student Assessment that the problem is solved 66.5%

Researcher Assessment that the problem is solved 62%

The views of the students in the gifted group were mostly accurate when compared with their performance. Of the 6 students who did not connect all of the parts and explain how the parts related to each other, 4 believed that they did not or may not have solved the problem. Two of the 6 students who did not solve the problem considered that they did not solve it.

Conclusions

Problem solving knowledge is composed of both explicit knowledge which can be stated by the student and implicit knowledge which is evident in the actions of the student when problem solving.

1. Students recognise that generic skills are part of the problem solving process and specific skills relevant to particular content areas are also part of the problem solving process.

2. Dispositions are an integral part of the process of problem solving. The students in the gifted group show more reliance on internal factors than external factors in their problem solving views. These students were mostly successful at the problem solving task.

3. Many students in the gifted group did not report strong incidence of management in their problem solving, though these students showed considerable evidence of self- management on the problem task.

4. Students in the gifted group used the cues in the problem to review the completion of the task.

5. Strategies and self-management are an important part of problem solving and should be included in any description of problem solving. Students need to be taught about these things as they do these things implicitly but do not value them.

The findings from the study which is described in this paper support the view that problem solving is the interaction of general and specific problem solving knowledge in the context of a specific problem.

Recent curriculum documents such as The Key Competencies discuss problem solving as a collection of general skills that can be taught in a general way across the curriculum.This view is incomplete in that specific skills and knowledge are necessary in solving specific problems. Experience influences the dispositional attitudes of students when engaging in a problem. Both are important aspects of problem solving and need to be considered in teaching problem solving in classrooms.

The results of this study support the view that problem solving knowledge involves knowing and doing.This notion is stated in the Key Competencies Report.

The Committee has strongly emphasised the notion that competence is the capacity to 'do' something rather than just 'know' something (Mayer,1992 :8).

However, the Key Competencies outline of Solving Problems is incomplete as these doing aspects are omitted :

selecting strategies to work on a problem

applying strategies to work out a problem

monitoring of problem solving activity

persisting when striking difficulty

asking the teacher for clarification.

The outline of Solving Problems should be revised before it is introduced into classrooms.

Implicit and explicit problem solving knowledge interact and should be taught in conjunction. In this study the students identified as gifted managed both aspects of knowledge and were successful problem solvers because of it.

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