Nature And Structure Of Mathematics

Nature And Structure Of mathChapter 2Literature reviewIn this chapter, literature related to maths confidence, demonstration and care- figure start are reviewed. The chapter begins with an introduction to maths and the occurrence of f maps of lifeal changes and concerns in sulfur Africa. It examines the metacognitive activity reflection and its various facets a unyielding with affectional issues in math. and so, disparateiating amidst past and current interrogation, the concentrate on give be on how math confidence and contemplative survey process relates to the level of acquisition and per remainsance in math task-solving cropes. Concluding description give fol pocket-sized, illustrating the kindred amid reflection and maths confidence during problem-solving processes. 2.1 mathematics, its nature and structure math cigarette be seen as a combination of calculation expertness and reason out (Hannula, Maijala Pehkonen, 200417) and lowlife further be c lassified as an individual(a)s numeral appreciation. Mathematics is a process, fixed to a certain person, a topic, an environs or an idea (Hiebert Carpenter, 1992). Mathematics originated as a necessity for societal, technical and ethnical growth or leisure (Ebrahim, 20101). This desire led to the advance of concepts and theories in fiat to meet the require of various cultures throughout time. With its economic crisis in nature, architecture, medicine, telecommunications and info technology, the uptake of mathematics has everyplacecome centuries of problems and continues to fulfil the needs of problem- deciders to solve everyday problems. Although mathematics has changed throughout time, in its progress and catchs at that place are interwoven connections between the cognitive, con nonative and stimulated mental domains. The increase demand to process and apply cultivation in a in the south African society, a society characterised by increasing unemployment and im mense demands on schools, still awaits recovery and substance from these cognitive and metacognitive ch aloneenges (Maree Crafford, 2010 84). From a socio-constructivists perspective, arriveing, adapting and evolving much interwoven systems should be the aim and goal of mathematics training (Lesh Sriraman, 2005). gibe to Thijsse (200234) mathematics is an emotionally charged subject, evoking feelings of dislike, fear and failure. Mathematics involves cognitive and emotive component portions that machinate part of the epistemological assumptions, regarding mathematical discipline (Thijsse, 20027 that forget be discussed in the go aftering section. 2.1.2 Epistemological assumptions regarding mathematics learningEnglish (2007123-125) lays coldcock powerful ideas for developing mathematics towards the 21st century. well-nigh of these ideas complicate 2.1.2.1 A social constructivist view of problem-solving, planning, monitoring and communication2.1.2.2 Effective and crea tive reasoning skills2.1.2.3 Analysing and transforming composite selective information sets2.1.2.4 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 scholar, from a constructivist perspective, learning can be viewed as the expeditious process at bottom and influenced by the prentice (Yager, 199153). Mathematical learning is therefore an interactive sequel of the encountered information and how the learner processes it, establish on perceived nonions and existing sectionl fellowship (Yager, 199153). According to DoE (20033) competence in mathematics reproduction is aimed at integrating practical, foundational and reflective skills. While altering the pictures in learning, mathematics education was turned upside d make with the shift being towards instructing, administering and applying metacognitive-activity- found learning in schools as claim ed by Yager (199153) and Leaf (200512-18). This change and reform in education and education paradigms is illustrated in fingers breadth 2.1. Early 1900sEarly 1900s1960s mid-eighties1980s- 2000s1980s 2000sThe overarching get on with impact on education and therapy focussing on metacognitionIn Figure 2.1 Leaf (20054) states that the intelligence quotient (IQ) is one of the greatest paradigm dilemmas. This border on 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 unless left little or no room for early(a) slip substance of thinking in mental aptitude (Leaf, 20055). In contrast to the IQ-approach is Piagets approach, named later on its founder, Jean Piaget, who apposed the IQ-approach. Focussing on cognitive developing, he suggests timed stages or learning classs in a childs cognitive development as a prerequisite t o the learning process. Piaget exclaims that if a stage is overseen, learning will not take place. A third paradigm, the Information touch on age, divided problem-solving into tierce phases input, coded storing and output. Designed in an era where technological advances and computers entered schools and the school curriculum, 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 class-conscious order, and one phase essential be mastered before continuing to a more difficult task. Outcomes Based Education (OBE) was implemented after the 1994 national classless elections in mho Africa. Since 1997 school systems underwent drastic changes from the so called apartheid era. According to the revise National Curriculum Statement (2003) the curriculum is based on development of the learners full potential in a de mocratic southeast Africa. Creating womb-to-tomb learners are the focus of this paradigm. After unwinnerfully 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 (200512). 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, 200512-15). Above mentioned aspects are besides known to associate with exploit in mathematics problem-solving (Maree, Prinsloo Claasen, 1997a Leaf, 200512-15). 2.1.3 close to factors associated with execution of instrument in mathematicsLarge 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 mak e between mathematics performance factors in these developed and developing countries (Howie, 2005125). Howie (2005123) 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 farthermost less influential (Howie, 2005 124, Reynolds, 199879). According to Maree et al. (200585), South African learners perform inadequately due to a number of traditional approaches towards mathematics principle and learning. Maree (1997b95) likewise classifies problems in study orientation as cognitive factors, external factors, innate and intra-psychological factors, and facilitating subject content. One psychological factor in the Study orientation course 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 (2003138) documented a case where a psychological factor, dread, causes an impairment of mathematics action. A distillation of a study through with(p) by Wither (1998) reason out 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, 2003149). 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 (199951) 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 learnerAffective issues revolve around an individu als environment within different systems and how that individual matures and interact within the systems (Lombard, 199951 Beilock, 2008339). In these systems it appears that learners have a validatory or oppose attitude towards mathematics (Maree, Prinsloo Claasen, 1997a). Beliefs near ones own capabilities and that success cannot be linked to effort and hard work is seen as affective factors in problem-solving (Dossel, 19936 Thijsse, 200218). Distrust in ones own hunch, not shrewd how to correct mistakes and the lack of personal effort is regarded as factors that facilitate mathematics anxiety, manifested by the learner (Thijsse, 200236 Russel, 199915).2.1.3.2 Some associated environmental factorsTimed test environments such as oral exam/testing situations, where answers must be prone quickly and verbally are seen as environmental factors that facilitates underachievement in mathematics. Public contexts where the learner has to express mathematical thought in bowel movemen t of an audience or peers may likewise be seen as an environmental factor limiting performance. 2.1.3.3 Some associated factors during the process of instructionKnowledge or so study methods, implementing different strategies and domain specific noesis is seen as factors that influence performance in mathematics. It seems as though performance is measured jibe to the learners ability to apply algorithms dictated by a higher strength figure such as parents or teachers (Russell, 199515 Thijsse, 200235). Thijsse (200219) agrees with Dossel (19936) and Maree (1997) that the teachers maintenance to the right or wrong dichotomy, stresses the fact that mathematics education can in addition be associate with performance. A brief discussion on mathematics problem-solving will now follow.2.2 Mathematics problem-solvingA mathematics problem can be outlined as a mathematical based task indicating realistic contexts in which the learner creates a copy for solving the problem in various circumstances (Chalmers, 20093). Making decisions within these contexts is only(prenominal) 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 (20032) managing uncertainty in ones understanding is essential in mathematical problem-solving. Lester and Kehle (2003510) fear that mathematical problem-solving is currently get more complex whence in old years. Therefore problem-solving continues to win 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 (2007764) statesThe pendulum of curriculu m change again swings back towards an emphasis on problem-solving.Problem-solving is exclamatory as a method involving inquiry and decision making (Fortunato, Hecht, tag end Alvarez, 199138). 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, 200310). 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, 2007764). 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 re depend through on mathematics problem-solving.2.2.1 Some research done on mathematics problem-solving in the pastStudies 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 , 2007764). Good problem solvers were generally compared to unretentive problem-solvers (Lester Kehle, 2003507) while Schoenfeld (1992) suggested that the former not only knows more mathematics, only when also knows mathematics differently (Lesh and Zawojewski, 2007767). The nature and development of mathematics problems are also widely researched (Lesh Zawojewski, 2007768), 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, 2007768). Confirming the use of th ese strategies Zimmerman (19998-10) describe dimensions for academic self-regulation by involving conceptual based questioning utilise 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 storehouse, information processing and mathematics problem-solving of the individual learnerIn the 1970s problems were seen an approach from an initial state towards a goal state (Newell Simon, 1972 in Goldstein, 2008404) involving search and adapt strategies. 2.2.2.1 Working entrepot as an aspect of problem-solvingThe working memory is essential for storing information regarding mathematics problems and problem-solving processes (Sheffield Hunt, 20062). Cognitive effects, such as anxiety, lop off processing in the working memory system and underachievement will follow (Ashcraft Hopko Gute, 1998343 Ashcraft, 20021). These intrusive thoughts , like worrying, overburden the system. The working memory system consists of cardinal components the psychological articulatory loop, visual-spatial sketch pad and a central executive director (Ashcraft Hopko Gute, 1998344 Richardson et al, 1996). 2.2.2.2 Problem-solving persona of the mathematics learnerThe 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 abide by in mathematics problem-situations (Lesh Zawojewski, 2007767 Zimmerman, 1994). Clearly learners problem-solving personality affects their achievement. According to Thijsse (200233) learners who trust their intuition and perceive that intuition as insight into a rational mind, kinda than emotional and irrational feelings, are more confident. The intermixture 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 factorsAlthough cognitive and metacognitive processes are compared in literature, Lesh and Zawojewksi (2007778) argues that mathematics concepts and higher order thinking should be studied correspondingly and interactively. describeing individual trends and behaviour patterns or feelings, could relate to mathematics problem-solving success (Lesh Zawojewksi, 2007778). 2.4.1 acquaintance processes during mathematics problem-solvingNewstead (199925) describes the cognitive levels of an individual as being either convergent (knowledge of information) or divergent (explaining, justification and reasoning). 2.3.2 Metacognition2.3.2.1 Components of metacognition2.3.2.2 Past research done on metacognitionThe Polya-style heuristics on problem-solving strategies, mentioned previously, is noted by Lesh and Zawojewski (2007368) 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 ( minus or positive) which provide a framework for reflective thinking. look is therefore a facet of metacognition.2.3.3 Reflection as a facet of metacognitionReflection, as defined by Glahn, Specht and Koper (200995), is an active reasoning process that confirms experiences in problem-solving and related social inter satisfy. Reflecting can be seen as a transformational process from our experiences and is effected by our way of thinking (Garcia, Sanchez Escudero, 20091).2.3.3.1 Development of reflective thinkingThinking 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, 2007368). Schoenfeld (1992) mentions an examining of special cases for selecting appropriate strategies from a hierarchical description, but Lesh and Zawojewski (2007369) argue that this will involve a too long (prescriptive process) or too short conventional list of prescribed strategies. Lesh and Zawojewski (2007770) kinda 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 gets for reflective practice sessionAccording 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 morphologic reflection and Rolfe et als (2001) framework for reflective practice. The first model is that of Gibbs (1988).i Gibbss (1988) model for reflectionGibbs model is mostly applied during reflective th eme (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) differently negative ( corky) emotion, or feeling. By analysing the sense of the experience a end point can be made where other options are considered to reflect upon. (Gibbs, 1988 Ertmer Newby, 1996)iiJohns (2000) model for structural and guided reflectionThis 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 lead and intravenous feeding.Reflect on and identify factors that influence y our actionsFigure 2.3Johns model for reflective practiceSourceAdapted 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 a ny way. A third model is proposed by Rolfe et al (2001), known as a framework for reflexive 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 practiceStage 1Reflection before actionStage 2Reflection during actionStage 3Reflection after actionDescriptive 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, stuck, 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 come and notice expectations of self and others like parents, teachers or peersEvaluate the positive and negative experiencesAnalyse and understand the problem and plan the next step, approach or strategyPerform the planned actionAwareness of ethics, beliefs, personal traits or motivations Recall strategies that worke d in the past. Reflect on the solution, reactions and attitudesSourceAdapted from Johns (2000), Gibbs (1988) and Rolfe et al (2001)2.3.3.3 The reflection processWhile 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 mathematicsRapidly ever-changing states of feelings, moderately stable tendencies, internal representations and deeply valued preferences are all categories of affect in mathematics (Schlogmann, 20031).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, 200417). 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 correlation between their belief and achievement in mathematics. Beliefs and are related to non-cognitive factors and involve feelings. According to Lesh and Zawojewski (2007775) the self-regulatory process is critically affected by beliefs, attitudes, confidence and other affective factors. 2.4.1Beliefs as an affective factor in mathematicsBelief, in a mathematics learner, form part of constructivism and can be defined as an individuals understanding of his/her own feelings and personal concepts f ormed when the learner engages in mathematical problem-solving (Hannula, Maijala Pehkonen, 20043). It plays an important employment in attitudes and emotions due to its cognitive nature and, according to Goldin (20015), 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, 20008) developed over a long uttermost of time (Mcleod,1992578-579). Beliefs about mathematics can be seen as a mathematics world view (Schlogmann, 20032) and can be divided into four major categories (Hannula, Maijala Pehkonen, 200417) 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 mu st be done in isolation) (Hannula, Maijala Pehkonen, 200417). Faulty beliefs about problem-solving allow few 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, 20008). 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 careAnxiety, an aspect of neuroticism, is often linked with personality traits such as painstakingness and agreeableness (Morony, 20102). This negative emotion manifests in faulty beliefs that causes anxious thoughts and feelings about mathematics problem-solving (Ashcraft Hopko Gute, 1998344 Thijsse, 200217). Distinction can be made between the different types of anxieties as experient by learners across all age groups. Some of these anxieties incl ude 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, 19992). 2.4.2.1 Mathematics anxietyThe 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 purpose 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 stretch 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, cog nitive assessments and transaction with these reactions. A number of researchers expand the concept of mathematics anxiety to include facilitative and debilitative anxiety (Newstead, 19982). It appears that Ashcraft Hopko Gute (1998343) 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 and preferably focus on the task at hand.2.4.2.2 Symptoms for identifying mathematics anxietyMathematics anxiety is symptomatically described as low (feelings of loss, failure and nervousness) or high (positive and motivated attitude) confidence in Mathematics (Maree, Prinsloo Claasen, 1997a7). Dossel (19936) and Thijsse (200218) 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, 198733 Thijsse, 200217). 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, 1977205 Maree, Prinsloo Claasen, 1997 Newstead, 1998 Thijsse, 200216). Panicking, anxious behaviour and worry manifests in the form of nail biting, crossing out correct answers, frequent excuse from the classroom and difficulty of verbally expressing oneself (Maree, Prinsloo Claasen, 1997a). Mar