Source Integration Task Essay Example | Topics and Well Written Essays - 1000 words

Source Integration Task - Essay Example This has also been maintained that it compels high schools to reduce the range of subjects they teach which is also shown as a significant limitation of the current entrance exam system. Further, university entrance exams of the day cause unreasonable stress on young people. To sum all these arguments up, the university entrance exams should be abolished because they reduce education to the simple memorizing of information, force high schools to reduce the range of subjects they teach, and cause unreasonable stress on young people. University entrance exams also result in limiting the content that high schools teach to their students. Significantly, one of the most essential defects of the current entrance exam system has been that the high schools are forced to reduce the range of subjects they teach as they need more to focus on the university entrance exam syllabus. It is natural for every high school to expect its students to enter the best schools for their further studies. In the attempt to help the students pass the entrance exams and secure admission in the best schools, the high schools are forced to concentrate on the topics of these exams and ultimately sacrifice broad range of subjects. That is to say, these high schools are forced to dedicate the depth of learning to focus on exam content only in order to secure pass in the entrance exams. However, it cannot be ignored that such an educational practice contributes too little for the overall development of the students. On the other hand, such narr ow-based education limits the benefits of education and it does more harm than good. It is indubitable that a system of broad-based education is needed by young learner’s intellectual development and it is important to provide such a curriculum to the overall development of the students. Therefore, it is important to realize the current

Law of Obligations(Tort) LLB Essay Example | Topics and Well Written Essays - 1000 words

Law of Obligations(Tort) LLB - Essay Example Frances (and his parents), depending upon the effect of the incident on the child, may also have a claim for psychological damage against the nursery brought about by the latter's negligence. In the recent case of Jones v BBC, 2007 WL 2187023 (QBD), where Jones, a freelance sound recordist for defendant BBC claimed that he suffered personal injury when a windmill rotor fell onto his back causing severe spinal injury rendering him paraplegic. In ruling for the claimant, the court stated that since BBC's safety crew had identified a risk of the falling mast, a discussion before filming should have been made to warn the crew not to go beneath it. But the safety crew did not give the warning. Such failure of BBC, through the safety crew, is considered negligent which caused Jones' accident. Thus, the BBC was liable for Jones' injuries. Also, the cameraman and Jones worked as a team because their equipment was linked. Jones with his equipment was following the cameraman who had decided to pass beneath the mast thereby leading Jones into the hazardous area. The cameraman was then in breached of his duty of care and the BBC was vicariously liable for that negligence. In Wilsons & Clyde Coal Company, Limited v English, [1938] A.C. 57, the House of Lords stated as follows: " primarily the master has a duty to take due care to provide and maintain a reasonably safe system of working in the mine, and a master, who has delegated the duty of taking due care in the provision of a reasonably safe system of working to a competent servant, is responsible for a defect in the system of which he had no knowledge" By the Jones and Wilsons cases, it is clear that the employer is under a duty of care to provide the employee with competent fellow employees including a qualified medical personnel, properly maintained site and facilities, and to provide a safe place and system of work. The question of whether the employer breached that duty of care depends on the standard of care owed by the employer to its employee and whether it has taken reasonable steps considering the circumstances. (Latimer v A.E.C. Ltd.[1953]) In Jones, the breach of the employer's duty consists in B BC's failure (through its safety crew) to discuss with the cameraman and Jones the risk of the falling mast and to warn the cameraman and Jones in unequivocal terms that they must not go beneath it. In Wilsons, the breach by the employer consists of its failure to provide competent fellow employees, properly maintained mine and equipment, and to provide a safe place and system of work. In the case of Ina here, the failure of the employer to provide competent fellow employees and to properly provide and maintain a safe place and system of work which caused the employee's disability to work for three (3) months constitute a breach of the standard of care required of the employer. Jack's negligence in leaving the drill on the floor in a place where thirty (30) toddlers freely roam about constitutes a negligent act for which the employer must be held vicariously responsible. The nursery cannot invoke the defence that Jack is merely a hired self-employed carpenter because as the court st ated in the Jones case the BBC had clearly assumed a responsibility for the health and safety of freelancers when they were working on BBC productions that was equivalent to that of an employer to a direct employee. Jack should have been more careful with his tools because it can reasonably be expected

Path of Light Essay Example | Topics and Well Written Essays - 500 words

Path of Light - Essay Example Depending on the surface and the position of the reflective surfaces images can be inverted and in other cases depending on the arrangement of the surfaces form many images. The structure and path that the light follows is therefore dependent on the surface and the origin of the light at the source. The reflective surfaces differ from curved surfaces to mirrors that reflect the light and form an inverted image (Keller et al 29). In addition, if the image is reflected across two mirrors there is formation of many similar images that are formed through reflection. Curved surfaces on the other hand form images depending on whether the surface is concave or convex with each surface forming its own unique form of image (Kokhanovsky 107). This is as a result of the path that the light takes once it hits the curved surface and the eventual image that is illustrated on a screen. Similarly there are images that are formed from shiny surfaces that are brighter and more reflective while the dar ker images form unique and distinct images depending on whether the surface can let any light pass through. The surface is therefore an important part in understanding the path of light since it forms different images depending on the surface. There are different methods that can be used to make an experiment and establish the path of light and how the light travels from one source to another. One of the experiments uses cardboards that are placed in a straight line with holes on the same point. The cards are used to show that when one illuminates light at one end the light is seen at the other end when the cards are arranged in a straight line. However, if one of the cards is located on a different position or the hole on the cards are not in congruence then there is no image formed at the other end. The light does not move to the other cards since it does not travel in

What is human nature Essay Example | Topics and Well Written Essays - 750 words

What is human nature - Essay Example The issue of human fulfillment is an issue that has never really been understood to its fullest since man has learnt to think. Writers and philosophers of every time have attempted to answer this question and no final word is available to mankind.The fact of the matter lies in the fact, that man cannot achieve fulfillment if he does not think; this, even as a proposition is considered to be impossible. By goodness of undertaking things, dealing with them, and ultimately being deceived by them, it becomes all the more evident to man that there is a reality that exists.This reality can be identified with, and the highest goodness can subsequently be attained by means of thinking. Contemplation gives insight, and ultimately self-actualization. There can be no better strength than realizing one's true potential of existence. All our endeavors or desires primarily follow from the necessity of our intrinsic nature. This concept given by him refers to the identification of the power that th e human being yields and associates from within himself. Whatever one might seek or ask, it is basically a product of how one identifies with the concept of self. The most significant goodness here becomes knowing about oneself. A person, who realizes himself, is truly the most successful of all.Desires, passions and restraints of the material world, can only be understood with reference to the person. Therefore, if and when a person appreciates his own personal standing and inception, it subsequently becomes the greatest source of endowment and fulfillment for the person. The fundamental factor here is about insisting on oneself and never imitating. This again is a propagation of the belief that an individual should at all times try to be what he/she is, and not impose a figure, picture of personality which is not a true depiction of the same. If one has the tendency to doubt one's self-image, then true actualization of self can never be there. The concept of uniformity of behavior and thought has always triggered the minds of thinkers. However, it is interesting that both of these present a similar stance on the said subject. It is not naturally desirable for anybody to maintain an existing thread of ideas and/or actions. To be able to appreciate and attain the highest order of fulfillment, the greatest goodness here would be to not pay heed to social and extraneous challenges, and keep representing one's own personality as it is. If in the face of conflict, one tends to shy away from the true sense of being, then it can never really come out. Under normal circumstances, it is always simple to portray oneself; however, the true test of character is when there is opposition, and yet still a person can attain goodness by means of beings what he truly is. Though this conflict may seem ironically contrary to the actual concept of life itself, yet this line answers probably the greatest mystery of life - being bad. The ultimate attainment in life therefore would be in the fact that one can appreciate its true essence, and not be afraid of the process of good or bad itself. It is only after this, that one would be able to overcome all the subsidiary issues of life, and would proceed to a stance wherein he would seek to discover the power within himself. Once this is accomplishment, nothing else would seem difficult. Despite being in different eras of time and possessing differing fortes of frame of references, yet we see a noteworthy comparison between their views on the said subject. Again, the interesting thing is that it is not a man of a particular culture, creed, time or race that is referred to. Their concepts are so intense, deep and universal, that even at this hour they can be identified with, and a sense of fulfillment may be sought if one wishes so. On the significant role played by nature in the behavior of man, both these opinions interject. Though nature has been a sort of subjective reality for the sake of many philosophers,

Interest groups and Political parties Essay Example | Topics and Well Written Essays - 500 words

Interest groups and Political parties - Essay Example Common good can be referred to as the goodness of an event or a phenomenon that is shared by all communities of a given society. Thus, keeping in view the roles played by interest groups and political parties, it can be stated that both the groups are conducive in working towards common good of a particular society and those elements which make up that society. However, this statement holds certain pre conditions, i.e. the interest groups are not self centered and political parties do not strive for gaining power only over all others. In other words, it is the intent behind the formation of these groups which comes into play and is considered extremely important. Motives, actions and desires yield goodness as long as the driving force behind them remains noble. Same is the case with interest groups and political parties. The reason of stating that the two set of groups are conducive in attaining or working towards the common good is that these groups provide a platform to the general public around through which people can raise their voices, share their opinions, take part in decision making process, highlight their concerns and problems and reach such decisions which are acceptable and beneficial for all. In this regard, work of Rousseau holds a special place. According to him, people should be "sovereign" and the government should only be there to implement laws. He believed that laws shall be enacted by direct democracy, which followed that each person should directly be involved in law making through a platform.

Hitler Was a Weak Dictator Essay Example for Free

Hitler Was a Weak Dictator Essay How far do you agree with the view that in the years 1933-39, Hitler was a weak dictator? Hitler has been categorised as a weak and a strong dictator by structuralist and intentionalist historians respectively, intentionalist for the importance of Hitler, and structuralist for the importance of those around him. Timothy Mason, a British Marxist historian, is an example of a structuralist historian, believing that Hitler was heavily reliant on those around and supporting him. There is also Ian Kershaw, who believed in the ‘Working towards the Fuhrer’ ideology, where Hitler exploited the attitudes of the German people in an attempt to have them work towards him. Timothy Mason, a British Marxist historian was certainly believed that Hitler was a weak dictator, in agreement to the question. Hitler was meant to be an all-powerful Fuhrer, being the only person who really knows what Germany and Germans want and need, yet he as Mason describes, â€Å"†¦had a preference for creating new organs of state to carry out specific projects.† If Hitler had been an all-powerful Fuhrer, he would not have required new organs to help him carry out tasks, he would have just been able to dictate and organise everything himself. This was shown in the massacre of the Night of the Long Knives This shows that Hitler in fact was a weak dictator, who did not have control over everything that happened in Nazi Germany, and he needed the support of the Nazi backroom to help him run the country. Mason also described, in about how Hitler would give jobs to other people to carry out. He realised that he wasn’t cut out to be leading certain jobs, and therefore he chose, ‘the right man for the job’. This is no more apparent than in the appointment of Goebbels as Propaganda Minister. Hitler wanted someone better than himself to arrange the propaganda appearances and meetings for himself, and Goebbels was clearly the best man for the job, as shown by his arrangement of the Night of the Long Knives in 1933, a highly successful propaganda campaign. Hitler was not capable of doing such things himself, proving that he is a weak dictator, by leaving jobs to others. Hitler was a dictator entirely dependent on his reputation, and therefore was indecisive when it came to making decisions, as Feuchtwanger describes, ‘Hitler often refused to take decisions, especially when a decision might damage his popularity†¦Ã¢â‚¬â„¢ It was well known that Hitler’s popularity paved the way for his success, as was shown in results in General elections. However, a strong dictator would not have to worry about the decisions they made, because whilst popularity would have been a cause for concern, their position would have been so secure that worrying would not be an issue, however, in this case where Hitler is clearly agitated and worried over the support he is receiving, it shows that his position as dictator is not as strong as it could be. Kershaw writes about the central role of Hitler to the running of Germany and the Nazi system. He was the, ‘†¦focal point of the Nazi system of rule.’ Every dictator needs to be the focal point of whatever Government they are the head of, and therefore Hitler’s proving of being the front focus of Nazi rule, shows how strong a dictator he was. He was the focal point and face of every Nazi policy, such as the Decree for the protection of People and State. Hitler used communists as an excuse for bringing in new laws and he was central to the policy making and was the face of the new policy. Werner Wilkins was a State Secretary in Germany during the Nazi regime, and he felt that Hitler was neither a weak or strong dictator, but that Germany was in fact Working towards the Fuhrer. This would mean that Hitler was not necessarily important in the running of everyday politics, such as a normal dictator, but he would however let people interpret his policies how they wish. This would not lead to a weak or strong dictator theory, but a society where everyone was working towards the Fuhrer’s will. Werner says how, â€Å"†¦everyone worth a post in the new Germany has worked best when he has, so to speak, working towards the Fuhrer.† This means that everyone in the country no matter what Hitler did was working towards his will, as it is believed he is the only one who knew what Germans truly wanted. In conclusion, Hitler was seen to be a weak dictator, as he was very afraid from the real threat created by losing his popularity, which would have turned the people against him, and whilst he had still not consolidated power fully, particularly at the start of the period, he was always under the threat of being thrown out of power again. Because of this, he had to curb some of his will and policies, and any threat to his popularity was handed over to someone else, whose loss in popularity would not have affected Hitler’s personal rule.

Nature And Structure Of Mathematics

Nature And Structure Of Mathematics Chapter 2 Literature review In this chapter, literature related to mathematics confidence, reflection and problem- solving are reviewed. The chapter begins with an introduction to mathematics and the occurrence of educational changes and concerns in South Africa. It examines the metacognitive activity reflection and its various facets along with affective issues in mathematics. Then, differentiating between past and current research, the focus will be on how mathematics confidence and reflective thinking relates to the level of achievement and performance in mathematics problem-solving processes. Concluding description will follow, illustrating the relationship between reflection and mathematics confidence during problem-solving processes. 2.1 Mathematics, its nature and structure Mathematics can be seen as a combination of calculation skill and reasoning (Hannula, Maijala Pehkonen, 2004:17) and can further be classified as an individuals mathematical understanding. Mathematics is a process, fixed to a certain person, a topic, an environment or an idea (Hiebert Carpenter, 1992). Mathematics originated as a necessity for societal, technological and cultural growth or leisure (Ebrahim, 2010:1). This desire led to the advancement of concepts and theories in order to meet the needs of various cultures throughout time. With its imprint in nature, architecture, medicine, telecommunications and information technology, the use of mathematics has overcome centuries of problems and continues to fulfil the needs of problem-solvers to solve everyday problems. Although mathematics has changed throughout time, in its progress and influences there are interwoven connections between the cognitive, connotative and affective psychological domains. The increasing demand to process and apply information in a South African society, a society characterised by increasing unemployment and immense demands on schools, still awaits recovery and substance from these cognitive and metacognitive challenges (Maree Crafford, 2010: 84). From a socio-constructivists perspective, developing, adapting and evolving more complex systems should be the aim and goal of mathematics education (Lesh Sriraman, 2005). According to Thijsse (2002:34) mathematics is an emotionally charged subject, evoking feelings of dislike, fear and failure. Mathematics involves cognitive and affective factors that form part of the epistemological assumptions, regarding mathematical learning (Thijsse, 2002:7 that will be discussed in the following section. 2.1.2 Epistemological assumptions regarding mathematics learning English (2007:123-125) lays down powerful ideas for developing mathematics towards the 21st century. Some of these ideas include: 2.1.2.1 A social constructivist view of problem-solving, planning, monitoring and communication; 2.1.2.2 Effective and creative reasoning skills; 2.1.2.3 Analysing and transforming complex data sets; 2.1.2.4 Applying and understanding school Mathematics; and 2.1.2.5 Explaining, manipulating and forecasting complex systems through critical thinking and decision making. With emphasis on the learner, from a constructivist perspective, learning can be viewed as the active process within and influenced by the learner (Yager, 1991:53). Mathematical learning is therefore an interactive consequence of the encountered information and how the learner processes it, based on perceivednotions and existing personal knowledge (Yager, 1991:53). According to DoE (2003:3) competence in mathematics education is aimed at integrating practical, foundational and reflective skills. While altering the paradigms in learning, mathematics education was turned upside down with the shift being towards instructing, administering and applying metacognitive-activity-based learning in schools as claimed by Yager (1991:53) and Leaf (2005:12-18). This change and reform in education and education paradigms is illustrated in Figure 2.1. Early 1900s Early 1900s 1960s 1980s 1980s- 2000s 1980s 2000s The overarching approach with impact on education and therapy focussing on metacognition In Figure 2.1 Leaf (2005:4) states that the intelligence quotient (IQ) is one of the greatest paradigm dilemmas. This approach is designed in the early twentieth century by F. Galton and labelled too many learners as either slow or clever. The IQ-tests did assess logical, mathematical and language preference and dominance in learners but left little or no room for other ways of thinking in mental aptitude (Leaf, 2005:5). In contrast to the IQ-approach is Piagets approach, named after its founder, Jean Piaget, who apposed the IQ-approach. Focussing on cognitive development, he suggests timed stages or learning phases in a childs cognitive development as a prerequisite to the learning process. Piaget exclaims that if a stage is overseen, learning will not take place. A third paradigm, the Information processing age, divided problem-solving into three phases: input, coded storing and output. Designed in an era where technological advances and computers entered schools and the school cur riculum, information processing was seen as comparing the learner with a microchip. Thus, retrieving and storing data and information was seen as a method to practise and learn as being the focus of learning. This learning took place in a hierarchical order, and one phase must be mastered before continuing to a more difficult task. Outcomes Based Education (OBE) was implemented after the 1994 national democratic elections in South Africa. Since 1997 school systems underwent drastic changes from the so called apartheid era. According to the Revised National Curriculum Statement (2003) the curriculum is based on development of the learners full potential in a democratic South Africa. Creating lifelong learners are the focus of this paradigm. After unsuccessfully transforming education in South Africa, a need still exists to challenge some of the shortcomings of the above mentioned paradigms. An Overarching approach is an aided paradigm proposed by Leaf (2005:12). The Overarching approach focuses on learning dynamics or in other words, what makes learning possible. This paradigm utilizes emotions, experiences, backgrounds and cultural aspects in order to facilitate learning and problem-solving (Leaf, 2005:12-15). Above mentioned aspects are also known to associate with performance in mathematics problem-solving (Maree, Prinsloo Claasen, 1997a; Leaf, 2005:12-15). 2.1.3 Some factors associated with performance in mathematics Large scale international studies, focussing on school mathematics, compare countries in terms of learners cognitive performance over time (TIMSS, 2003 PISA, 2003). A clear distinction must be made between mathematics performance factors in these developed and developing countries (Howie, 2005:125). Howie (2005:123) explored data from the TIMSS-R South African study which revealed a relationship between contextual factors and performance in mathematics. School level factors seem to be far less influential (Howie, 2005: 124, Reynolds, 1998:79). According to Maree et al. (2005:85), South African learners perform inadequately due to a number of traditional approaches towards mathematics teaching and learning. Maree (1997b:95) also classifies problems in study orientation as cognitive factors, external factors, internal and intra-psychological factors, and facilitating subject content. One psychological factor in the Study Orientation in Mathematics questionnaire (SOM) by Maree, Prinsloo and Claasen (1997b) is measured as the level of mathematics confidence of grade 7 to 12 learners in a South African context. Sherman and Wither (2003:138) documented a case where a psychological factor, anxiety, causes an impairment of mathematics achievement. A distillation of a study done by Wither (1998) concluded that low mathematics confidence causes underachievement in mathematics. Due to insufficient evidence it could not prove that underachievement results in low mathematics confidence. The study did indicate that a possible third factor (metacognition) could cause both low mathematics confidence and underachievement in mathematics (Sherman Wither, 2003:149). Thereupon, factors manifested by the learner are discussed below. Academic underachievement and performance in mathematics is determined by a number of variables as identified by Lombard (1999:51); Maree, Prinsloo and Claasen (1997); and Lesh and Zawojewski (2007). These variables include factors manifested by the learner, environmental factors and factors during the process of instruction. 2.1.3.1 Some associated factors manifested by the learner Affective issues revolve around an individuals environment within different systems and how that individual matures and interact within the systems (Lombard, 1999:51 Beilock, 2008:339). In these systems it appears that learners have a positive or negative attitude towards mathematics (Maree, Prinsloo Claasen, 1997a). Beliefs about ones own capabilities and that success cannot be linked to effort and hard work is seen as affective factors in problem-solving (Dossel, 1993:6; Thijsse, 2002:18). Distrust in ones own intuition, not knowing how to correct mistakes and the lack of personal effort is regarded as factors that facilitate mathematics anxiety, manifested by the learner (Thijsse, 2002:36; Russel, 1999:15). 2.1.3.2 Some associated environmental factors Timed testing environments such as oral exam/testing situations, where answers must be given quickly and verbally are seen as environmental factors that facilitates underachievement in mathematics. Public contexts where the learner has to express mathematical thought in front of an audience or peers may also be seen as an environmental factor limiting performance. 2.1.3.3 Some associated factors during the process of instruction Knowledge about study methods, implementing different strategies and domain specific knowledge is seen as factors that influence performance in mathematics. It seems as though performance is measured according to the learners ability to apply algorithms dictated by a higher authority figure such as parents or teachers (Russell, 1995:15; Thijsse, 2002:35). Thijsse (2002:19) agrees with Dossel (1993:6) and Maree (1997) that the teachers attention to the right or wrong dichotomy, stresses the fact that mathematics education can also be associate with performance. A brief discussion on mathematics problem-solving will now follow. 2.2 Mathematics problem-solving A mathematics problem can be defined as a mathematical based task indicating realistic contexts in which the learner creates a model for solving the problem in various circumstances (Chalmers, 2009:3). Making decisions within these contexts is only one of the elementary concepts of human behaviour. In a technology based information age, computation; conceptualisation and communication are basic challenges South Africans have to face (Maree, Prinsloo Claasen, 1997; Lesh Zawojewski, 2007). Problem-solving abilities are needed and should be developed for academic success, even beyond school level. According to Kleitman and Stankov (2003:2) managing uncertainty in ones understanding is essential in mathematical problem-solving. Lester and Kehle (2003:510) fear that mathematical problem-solving is currently getting more complex then in previous years. Therefore problem-solving continues to gain consideration in the policy documents of various organisations, internationally (TIMSS, 2003; SACMEQ, 2009; PIRLS, 2009; Moloi Strauss, 2005 NCTM, 1989) and nationally (DoE, 2010; DoE, 2010: 3). As Lesh and Zawojewski (2007:764) states The pendulum of curriculum change again swings back towards an emphasis on problem-solving. Problem-solving is emphasised as a method involving inquiry and decision making (Fortunato, Hecht, Tittle Alvarez, 1991:38). Generally two types of mathematical problems exist: routine problems and non-routine problems. The use and application of non-routine problems, unseen mathematical processes and principles are part of the scope of mathematics education in South Africa (DoE, 2003:10). Keeping track of and on the process of information seeking and decision making, mathematics problem-solving is linked to the content and context of the problem situation (Lesh Zawojewski, 2007:764). It seems as though concept development and development of problem-solving abilities should be part of mathematics education and beliefs, feelings or other affective factors should be taken into account. In the next section a discussion will follow regarding past research done on mathematics problem-solving. 2.2.1 Some research done on mathematics problem-solving in the past Studies on learners performance in mathematics and how their behaviours vary in approaches to perform, was the conduct of research on mathematics problem-solving since the 1930s (Dewey, 1933; Piaget, 1970; Flavell; 1976; Schoenfeld, 1992; Lester Kehle, 2003; Lesh Zawojewski , 2007:764). Good problem solvers were generally compared to poor problem-solvers (Lester Kehle, 2003:507) while Schoenfeld (1992) suggested that the former not only knows more mathematics, but also knows mathematics differently (Lesh and Zawojewski, 2007:767). The nature and development of mathematics problems are also widely researched (Lesh Zawojewski, 2007:768), especially with the focus on how learners seeand approach mathematics and mathematical problems. Polya-style problems involve strategies such as picture drawing, working backwards, looking for a similar problem or identifying necessary information (Lesh Zawojewski, 2007:768). Confirming the use of these strategies Zimmerman (1999:8-10) describe dimensions for academic self-regulation by involving conceptual based questioning using a technique called prompting. Examples of these prompts are questions starting with why; how; what; when and where, in order to provide scaffolding for information processing and decision making. 2.2.2 Working memory, information processing and mathematics problem-solving of the individual learner In the 1970s problems were seen an approach from an initial state towards a goal state (Newell Simon, 1972 in Goldstein, 2008:404) involving search and adapt strategies. 2.2.2.1 Working memory as an aspect of problem-solving The working memory is essential for storing information regarding mathematics problems and problem-solving processes (Sheffield Hunt, 2006:2). Cognitive effects, such as anxiety, disrupt processing in the working memory system and underachievement will follow (Ashcraft; Hopko Gute, 1998:343; Ashcraft, 2002:1). These intrusive thoughts, like worrying, overburden the system. The working memory system consists of three components: the psychological articulatory loop, visual-spatial sketch pad and a central executive (Ashcraft; Hopko Gute, 1998:344; Richardson et al, 1996). 2.2.2.2 Problem-solving persona of the mathematics learner The learner, either an expert or novice-problem-solver is researched on his/her ideas, strategies, representations or habits in mathematical contexts (Ertmer Newby, 1996). Expert learners are found to be organised individuals who have integrated networks of knowledge in order to succeed in mathematics problem-situations (Lesh Zawojewski, 2007:767; Zimmerman, 1994). Clearly learners problem-solving personality affects their achievement. According to Thijsse (2002:33) learners who trust their intuition and perceive that intuition as insight into a rational mind, rather than emotional and irrational feelings, are more confident. The variety of attributes, such as anxiety and confidence, is included in reflective processes either cogitatively or metacognitatively which will be discussed in the next section. 2.3 Cognitive and metacognitive factors Although cognitive and metacognitive processes are compared in literature, Lesh and Zawojewksi (2007:778) argues that mathematics concepts and higher order thinking should be studied correspondingly and interactively. Identifying individual trends and behaviour patterns or feelings, could relate to mathematics problem-solving success (Lesh Zawojewksi, 2007:778). 2.4.1 Cognition processes during mathematics problem-solving Newstead (1999:25) describes the cognitive levels of an individual as being either convergent (knowledge of information) or divergent (explaining, justification and reasoning). 2.3.2 Metacognition 2.3.2.1 Components of metacognition 2.3.2.2 Past research done on metacognition The Polya-style heuristics on problem-solving strategies, mentioned previously, is noted by Lesh and Zawojewski (2007:368) as an after-the-fact of past activities process. This review process between interpreting the problem, and the selection of appropriate strategies, that may or may not have worked in the past, is linked with experiences (negative or positive) which provide a framework for reflective thinking. Reflection is therefore a facet of metacognition. 2.3.3 Reflection as a facet of metacognition Reflection, as defined by Glahn, Specht and Koper (2009:95), is an active reasoning process that confirms experiences in problem-solving and related social interaction. Reflecting can be seen as a transformational process from our experiences and is effected by our way of thinking (Garcia, Sanchez Escudero, 2009:1). 2.3.3.1 Development of reflective thinking Thinking about mathematics problems and reflecting on them is essential for interpreting the given problems provided details about what is needed in order to solve the problem (Lesh Zawojewski, 2007:368). Schoenfeld (1992) mentions an examining of special cases for selecting appropriate strategies from a hierarchical description, but Lesh and Zawojewski (2007:369) argue that this will involve a too long (prescriptive process) or too short conventional list of prescribed strategies. Lesh and Zawojewski (2007:770) rather suggest a descriptive process to reflect on and develop sample experiences. The focus should be on various facets of individual persona and differences, such as prior knowledge and experiences, which differs between individuals. 2.3.3.2 Expansion models for reflectivepractice According to Pletzer et al (1997) applying reflective practice is a powerful and effective way of learning. Three models for reflective practice exist: the reflective cycle of Gibbs (1988), Ertmer and Newby (1996), Johns-model (2000) for structural reflection and Rolfe et als (2001) framework for reflective practice. The first model is that of Gibbs (1988). i Gibbss (1988) model for reflection Gibbs model is mostly applied during reflective writing (Pugalee, 2001). This model for reflection is exercised during problem-solving situations by assessing first and second cognitive levels. A particular situation, such as in Figure 2.2, when the learner has to solve a mathematical problem is described by accompanying feelings and emotions that will be remembered and reflected upon. A conscience cognitive decision will then be made determining whether the experience was a positive (good) otherwise negative (bad) emotion, or feeling. By analysing the sense of the experience a conclusion can be made where other options are considered to reflect upon. (Gibbs, 1988; Ertmer Newby, 1996) iiJohns (2000) model for structural and guided reflection This model provides a framework for analysing and critically reflecting on a general problem or experience. The Johns-model (2000) provides scaffolding or guidance for more complex problems found on cognitive levels three and four. Reflect on and identify factors that influence your actions Figure 2.3Johns model for reflective practice Source:Adapted from John (2000) The model in Figure 2.3 is divided into two phases. Phase 1 refers to the recall of past memories and skills from previous experiences, where the learner identifies goals and achievements by reflecting into their past. This could be easily done using a video recording of a situation where the learner solves a problem. It is in this phase where they take note of their emotions and what strategies were used or not. On the other hand, phase 2 describes the feelings, emotions and surrounding thoughts accompanying their memories. A deeper clarification is given when the learner has to motivate why certain steps were left out or why some strategies were used and others not. They have to explain how they felt and the reason for the identified emotions. At the end the learner should reflect between the in and out components to identify any factor(s) that could have effected their emotions or thoughts in any way. A third model is proposed by Rolfe et al (2001), known as a framework for reflex ive practice. iiiRolfe et als model for reflexive practice. According to Rolfe et al (2001) the questions ‘what? and ‘so what? or ‘now what?, can stimulate reflective thinking. The use of this model is simply descriptive of the cognitive levels and can be seen as a combination of Gibbs (1988) and Johns (2000) model. The learner reflects on a mathematics problem in order to describe it. Then in the second phase, the learner constructs a personal theory and knowledge about the problem in order to learn from it. Finally, the learner reflects on the problem and considers different approaches or strategies in order to understand or make sense of the problem situation. Table 2.1 demonstrates this model of Rolfe et al (2001) in accordance with the models of Gibbs (1988) and Johns (2000) as adapted by the researcher. It shows the movement of thought actions and emotions between different stages of reflection (before, during and after) in problem-solving. Table 2.1Integration of reflective stages and the models for reflective practice Stage 1 Reflection before action Stage 2 Reflection during action Stage 3 Reflection after action Descriptive level of reflection (planning and describing phase) Theory and knowledge building of reflection (decision making phase) Action orientated level (reflecting on implemented strategy-action) Identify the level of difficulty of the problem and possible reasons for feeling, or not feeling, â€Å"stuck†, â€Å"bad† or unable to go to the next step. Pay attention to thought and emotions and identify them. Describe negative attitude towards mathematics problems, if any Observe and notice expectations of self and others: like parents, teachers or peers Evaluate the positive and negative experiences Analyse and understand the problem and plan the next step, approach or strategy Perform the planned action Awareness of ethics, beliefs, personal traits or motivations Recall strategies that worked in the past. Reflect on the solution, reactions and attitudes Source:Adapted from Johns (2000), Gibbs (1988) and Rolfe et al (2001) 2.3.3.3 The reflection process While some research claims, seeing and doing mathematics as useful in the interpretation and decision making of problem-solving processes (Lesh Zawojewski, 2007), a more affective approach would involve feelings or the feelings about mathematics(Sheffield Hunt, 2006), in other words, affective factors. 2.4 Affective factors in mathematics Rapidly changing states of feelings, moderately stable tendencies, internal representations and deeply valued preferences are all categories of affect in mathematics (Schlogmann, 2003:1).Reactions to mathematics are influenced by emotional components of affect. Some of these components include negative reactions to mathematics, such as: stress, nervousness, negative attitude, unconstructive study-orientation, worry, and a lack of confidence (Wigfield Meece, 1988; Maree, Prinsloo Claasen, 1997). Learners self-concept is strongly connected to their self-belief and their success in solving mathematics problems is conceptualised as important (Hannula, Maijala Pehkonen, 2004:17). A study done by Ma and Kishor (1997) confirmed belief, as an affect on mathematics achievement, being weakly correlated with achievement among children from grade 2 to 8. However, Hannula, Maijala and Pehkonen (2004) conducted a study on learners in grade 7 to 12 and concluded that there is a strong correlatio n between their belief and achievement in mathematics. Beliefs and are related to non-cognitive factors and involve feelings. According to Lesh and Zawojewski (2007:775) the self-regulatory process is critically affected by beliefs, attitudes, confidence and other affective factors. 2.4.1Beliefs as an affective factor in mathematics Belief, in a mathematics learner, form part of constructivism and can be defined as an individuals understanding of his/her own feelings and personal concepts formed when the learner engages in mathematical problem-solving (Hannula, Maijala Pehkonen, 2004:3). It plays an important role in attitudes and emotions due to its cognitive nature and, according to Goldin (2001:5), learners attribute a kind of truth to their beliefs as it is formed by a series of background experiences involving perception, thinking and actions (Furinghetti Pehkonen, 2000:8) developed over a long period of time (Mcleod,1992:578-579). Beliefs about mathematics can be seen as a mathematics world view (Schlogmann, 2003:2) and can be divided into four major categories (Hannula, Maijala Pehkonen, 2004:17): beliefs on mathematics (e.g. there can only be one correct answer), beliefs about oneself as a mathematics learner or problem solver (e.g. mathematics is not for everyone), beliefs on teaching mathematics (e. g. mathematics taught in schools has little or nothing to do with the real world) and beliefs on learning mathematics (e.g. mathematics is solitary and must be done in isolation) (Hannula, Maijala Pehkonen, 2004:17). Faulty beliefs about problem-solving allow fewer and fewer learners to take mathematics courses or to pass grade 12 with the necessary requirements for university entrance. Beliefs are known to work against change or act as a consequence of change and also have a predicting nature (Furinghetti Pehkonen, 2000:8). Affective issues, such as beliefs, generally form part of the cognitive domain, anxiety (Wigfield Meece, 1988), which will be dealt with in the next section. 2.4.2 Anxiety Anxiety, an aspect of neuroticism, is often linked with personality traits such as conscientiousness and agreeableness (Morony, 2010:2). This negative emotion manifests in faulty beliefs that causes anxious thoughts and feelings about mathematics problem-solving (Ashcraft; Hopko Gute, 1998:344; Thijsse, 2002:17). Distinction can be made between the different types of anxieties as experienced by learners across all age groups. Some of these anxieties include general anxiety, test or evaluation anxiety, problem-solving anxiety and mathematics anxiety. The widespread phenomenon, mathematics anxiety, threatens performance of learners in mathematics and interferes with conceptual thinking, memory processing and reasoning (Newstead, 1999:2). 2.4.2.1 Mathematics anxiety The pioneers of mathematics anxiety research, Richardson and Suinn (1972), defined mathematics anxiety in terms of the affect on performance in mathematics problem-solving as: Feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations This anxious and avoidance-behaviour towards mathematics has far reaching consequences as stressed by a number of researchers (Maree, Prinsloo Claasen, 1997; Newstead, 1999; Sheffield Hunt, 2006 Morony, 2009). Described as a chain reaction, mathematics anxiety consists of stressors, perceptions of threat, emotional responses, cognitive assessments and dealing with these reactions. A number of researchers expand the concept of mathematics anxiety to include facilitative and debilitative anxiety (Newstead, 1998:2). It appears that Ashcraft; Hopko; Gute (1998:343) and Richardson et al (1996) see mathematics anxiety in the same locale as the working memory system. Both areas consist of psychological, cognitive and behavioural components. Although they agree on the same components, Eysenck and Calvo (1999) states that it is not the experience of worry that diverts attention or interrupts the working memory process, but rather ineffective efforts to divert attention away from worrying a nd instead focus on the task at hand. 2.4.2.2 Symptoms for identifying mathematics anxiety Mathematics anxiety is symptomatically described as low (feelings of loss, failure and nervousness) or high (positive and motivated attitude) confidence in Mathematics (Maree, Prinsloo Claasen, 1997a:7). Dossel (1993:6) and Thijsse (2002:18) states that these negative feelings are associated with a lack of control when uncertainty and helplessness is experienced when facing danger. Unable to think rationally, avoidance and the inability to perform adequately causes anxiety and negative self-beliefs Mitchell, 1987:33; Thijsse, 2002:17). Anxious children show signs of nervousness when a teacher comes near. They will stop; cover their work with their arm, hand or book, in an approach to hide their work (May, 1977:205; Maree, Prinsloo Claasen, 1997; Newstead, 1998 Thijsse, 2002:16). Panicking, anxious behaviour and worry manifests in the form of nail biting, crossing out correct answers, habitual excuse from the classroom and difficulty of verbally expressing oneself (Maree, Prinsloo Claasen, 1997a). Mar