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**Chapter 2 ****Problem solving and belief**

__Introduction__

The researcher, firstly, discussed mathematical problems, mathematical exercises and problem solving. Secondly, he discussed mathematics related belief systems. Lastly, he discussed the theoretical framework of the study.

__Mathematical problems and problem solving__

In this section, the researcher discussed some definitions of a mathematical problem, mathematical exercise and mathematical problem solving as suggested by various scholars. Differences between a mathematical problem and a mathematical exercise were also discussed. Some categories of mathematical exercises were discussed, identifying the kinds of mathematical exercises that would be used in this study. Some problem solving approaches, strategies (heuristics) and algorithms were discussed. Factors affecting problem solving, difficulties faced by learners in problem solving and characteristics of good problem solvers are some issues discussed in this section. At last, differences between expert and novice problem solvers were discussed.

__What is a mathematical problem?__

There has been different definitions of a ‘mathematical problem’ formulated over the years by different scholars in mathematics education (e.g., Andre, 1986; Xenofontos & Andrews, 2008; Branca, 1980; Blum & Niss, 1989; Carson, 2007; Focant, Gregoire & Desoete*,* 2006; Hoosain, 2001; Kee, 1999; Zanzali & Nam, 1997). A mathematical task can be referred to as a mathematical problem when a learner who faces it wants to solve it, has no an immediately available procedure to solve it and must, actually, attempt to solve it (Kee, 1999).

A mathematical problem can be defined as a situation in which a learner really wants to look for a solution to it, but does not know how to find it (Andre, 1986). Carson (2007) views a mathematical problem as a situation that is quantitatively expressed or otherwise, that is faced by a learner who wants to resolve it, but has no an immediate direct way to do it. Hoosain (2001) views a mathematical problem as a non-routine problem whose solution process requires more than ready-to-hand procedures or algorithms.

According to Hooisan (2001), a mathematical problem can be regarded as a task or experience which an individual encounters for the very first time of which he/she has no known procedures to handle it. In order to solve a mathematical problem, the individual should devise his/her own approach or method by utilizing the resources at his/her disposal such as the various skills, knowledge and strategies which were previously learned.

An analysis of mathematical problems done by Andre (1986) reveals that mathematical problems have four components: (1) The goal or goals (what you want to do in a situation); (2) The givens (what is available to you to start in a problem situation); (3) The obstacles (The elements or factors that get in the way of a solution); and (4) The methods or operations (The procedures that may be used to solve the problem). For a learner to effectively solve a mathematical problem, he/she should clearly identify these four components at the initial or approach stage of problem solving.

Though there is no a precise consensus on the definition of a mathematical problem from researchers as (Hoosain, 2001) noted, from the interpretations given by the authors such as Andre (1986); Carson (2007); Hoosain (2001) and Kee (1999), this researcher, however, observes that there seems to be a general agreement that a mathematical problem should be a situation that confronts an individual who desires a solution and for which an algorithm which leads to a solution is unavailable or not known by the individual. A mathematical problem, then exists if the situation is new and not recognisable by the potential problem solver, and the problem solver does not possess direct methods or algorithms that are enough to resolve the problem. This means that a mathematical question will not be considered a ‘mathematical problem’ if the individual has previously solved the problem or can easily solve the problem by applying algorithms that were previously learnt.

In this study, the researcher used the term „mathematical problem‟ to refer to a situation which carries with it either open questions or non-routine questions that challenge somebody intellectually who does not immediately possess direct methods that are enough to resolve the problem (Blum & Niss, 1989). Only mathematical problems were given to learners who participated in this study. In choosing the mathematical problems, the researcher attempted to ensure that the learners did not previously solve the mathematical problems or they did not previously learn the requisite direct methods or procedures to solve them

__What is a mathematical exercise?__

__What is a mathematical exercise?__

The Schoenfeld (1985) defines a mathematical exercise as a routine exercise in which a learner applies some learnt mathematical facts and procedures. The main purpose of a mathematical exercise is to enable the learner to master the relevant mathematical matter. The introduction of several similar worked examples in preparation for learners to attempt a mathematical exercise on their own eliminates or minimises the challenge of the task. In essence, the potential mathematical problems to learners are reduced to mere mathematical exercises.

In this study, learners attempted resolving mathematical questions that were new to them, which demanded originality and creative thinking. Learners devised their own strategic problem solving approaches they considered appropriate to solve the given mathematical problems. No similar worked examples were provided.

Difference between a mathematical problem and a mathematical exercise

The difference between a mathematical problem and a mathematical exercise is that, in a mathematical problem, an algorithm which will lead to a solution is unavailable to the problem solver, whereas, in a mathematical exercise, one determines the algorithm first and applies it in problem solving (Hooisan, 2001). When learners are given an opportunity to practice how to solve a mathematical problem, then, the mathematical problem becomes a mathematical exercise. Schoenfeld (2007) classifies this act as a „degradation‟ of mathematical problems into mathematical exercises. Mathematical problems are expected to pose a real challenge to the learner. Focant et al. (2006), Hooisan (2001), and Wilson et al. (1993) argue that a mathematical problem is relative to the individual(s) involved. A mathematical question or task that is a mathematical problem to one learner might be a mathematical exercise to another because of the absence of a real challenge (blockage) or acceptance of the goal to problem solving. This means that a mathematical situation can only be defined as a mathematical problem relative to specific learners.

In a way, based on Hooisan (2001) and Schoenfeld‟s (2007) view, one may argue that mathematical problems are non-routine, whereas mathematical exercises are routine (see section 2.2.4). A mathematical exercise is thus a mathematical question which a learner knows how to resolve immediately, whilst a mathematical problem is a mathematical question which requires a learner to apply creative thinking and resourcefulness in search of the appropriate problem solving approach. In another way, one may argue that a mathematical exercise can be either routine (procedural or algorithmic in nature) or non-routine (non-procedural or non-algorithmic in nature) as evident in school mathematics textbooks which present both routine and non-routine questions under the same heading „exercise‟ or „activity‟. Based on the definition of a mathematical exercise and the fact of degradation of mathematical problems into mathematical exercises stated by Schoenfeld (1985, 2007) (see sections 2.2.2 & 2.2.3), the classification of both routine and non-routine questions as an exercise done in school mathematics textbooks can be regarded as appropriate if learners were previously taught or learned by themselves how to solve similar questions. Otherwise, it is not appropriate if learners did not previously have an opportunity to learn how to solve questions of similar nature. In the light of this discussion, the researcher classified non-routine mathematical exercises as mathematical problems in this study after checking that the learners participating in the study were not previously taught how to resolve questions of similar nature.

This researcher, however, concedes that there are mathematical exercises that are non-routine simply because their resolutions are not obvious and learners have not practiced resolving them before. As such, a mathematical question qualifies to be a mathematical problem if it is novel and the learner cannot solve it immediately. According to Bunday (2013), a mathematical exercise serves to drill a learner in some technique or procedure and requires little, if any, original thought (because of the provision of worked examples of similar nature), while a mathematical problem requires thought on the part of the learner. When resolving a mathematical problem, the learner has to devise his/her own strategic attacks which might be subject to failure or success.

The distinction between routine/non-routine mathematical exercises and mathematical problems was of paramount importance to this study. If a learner could readily resolve a mathematical question posed by applying the previously practiced procedures, the objectives of the study would not have being met (see section 1.4). This would mean that the questions posed in the study were routine mathematical exercises and not non-routine mathematical exercises (or mathematical problems). Learners‟ approaches to real and suitable mathematical problems were expected to reflect, somehow, their mathematics-related belief systems as they search for solutions by applying their own original strategies. The solution to the mathematical problem should be an original product of the learner than a reflection of someone else‟s thought

__Categories of mathematical exercises__

__Categories of mathematical exercises__

__Routine mathematical exercise__

Routine mathematical exercises are mathematical questions which the learners solving them possess a previously established procedure for finding one. Brunning, Schraw and Ronning (1999) view a routine mathematical exercise as a well-defined problem whose single correct solution can be obtained by applying a well known guaranteed method or procedure. For example, solving a quadratic equation by using the quadratic formula produces one solution through pre-determined steps.

Posamentier and Schulz (1996), and Kirkley (2003) view routine mathematical exercises as well-structured problems that always use the same step by step solution. Some distinguishable characteristics of these mathematics questions are that they have a solution strategy that can be predicted, have one correct answer, and contain all information needed to solve the problem. Such types of mathematical questions were not suitable for use in this study, as they could not distinguish learners‟ different problem solving approaches or behaviours. Different problem solving strategies learners used to resolve non-routine mathematical problems were interpreted as a manifestation of different belief systems in this study.

Landa (1983) views routine mathematical exercises as algorithmic problems. An algorithmic mathematical question or task enables the problem solver to resolve it by following a predefined sequence of operations. Landa (1983) posits that, for any given algorithmic mathematical question or task, a learner can clearly state all the mental processes he/she has to undergo in order to successfully solve it. It is clear from this discussion that algorithmic tasks limit learners‟ independent mathematical reasoning or creativity; and might promote development of, for example, a belief that a mathematical problem has one exact correct answer that can be obtained by applying predetermined procedures.

Non-routine mathematical exercise

Non-routine mathematical exercises are mathematical questions which the learners attempting to solve them possess neither a known answer nor a previously established (routine) procedure for finding one (Branca, 1980). Non-routine mathematical exercises are referred to as non-routine mathematical problems or simply mathematical problems in this study. It is important to note that mathematical questions which are non-routine to somebody may be routine to another. There were attempts in the current study to ensure that the mathematical problems were non-routine as far as possible to the participating learners by checking that they have not attempted to solve the problems previously and that they have not been taught standard methods of solving the types of problems involved.

Brunning et al. (1999) classify a non-routine mathematical exercise as an ill-defined mathematical problem that has several acceptable solutions that can be obtained by several unique strategies. There is no a single strategy that is universally agreed upon to resolve a non-routine problem. Each new unique non-routine problem requires different new approach strategies to resolve it. Kirkley (2003) views non-routine mathematical exercises as ill-structured problems that have vague and unclear goals. The problems are characterised by having multiple perspectives, goals and solutions; solution that is not well defined, predictable or agreed upon, and some needed information, often, must be determined for effective problem solving. In a similar way, Polya (1985) classifies non-routine mathematical problems into two categories: (a) Problems to find, the principal parts of which are the unknown, the data, and the condition; and (b) Problems to prove which comprise a hypothesis and a conclusion.

Chapter 1

1.1. Introduction

1.2. Background to the study

1.3. Statement of the problem

1.4. Research question

1.5. Justification of the study

1.6. Definition of key terms

1.7. Structure of thesis

Chapter 2 Problem solving and beliefs

2.1. Introduction

2.2. Mathematical problems and problem solving57

2.4. Theoretical framework

2.5. Summary and conclusion of the chapter

Chapter 3 Methodology

3.1. Introduction

3.2. Research design

3.3. Population and case selection

3.4. Data collection instruments

3.5. Data collection procedure

3.6. Ethical considerations

3.7. Summary and conclusion of the chapter

Chapter4 Data analysis

4.1. Introduction

4.2. Data presentation and analysis

4.3. Analysis of mathematics-related beliefs questionnaire

4.4. Analysis of mathematics non-routine problem solving test

4.5. Analysis of interview schedule, open-ended questionnaire And retrospective questionnaire

4.6. Analysis of the relationship between belief systems and Approaches to problem solving

4.7. Summary and conclusion of the chapter

Chapter 5 Research findings

5.1. Introduction

5.2. Learners‟ approach strategies to mathematical problem solving

5.3. Learners‟ mathematics-related belief systems

5.4. Relationship between learners‟ mathematics belief systems and their approach to mathematics problem solving

5.5. Summary and conclusion of the chapter

Chapter 6 Discussion of the results

6.1. Introduction

6.2. Utilitarian, exploratory and systematic learners‟ approach to mathematical problem solving

6.2.1. Utilitarian believer‟s approach to mathematical problem solving

6.3. Approach to problem solving versus belief systems

6.4. Summary and conclusion of the chapter

Chapter 7 Summary, conclusions and recommendations

7.1. Introduction

7.2. Summary of the study

7.3. Conclusion

7.4. General recommendations

7.5. Recommendation for future studies

7.6. Limitations of the study

7.7. Reflections on my intellectual journey

References

Appendices

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