Nature And Structure Of Mathematics

Nature And Structure Of mathsChapter 2Literature reviewIn this chapter, literature related to maths agency, reflection and task- solving atomic number 18 reviewed. The chapter begins with an penetration to maths and the occurrence of educational changes and concerns in S discoverh Africa. It examines the metacognitive activity reflection and its various facets along with affectional issues in math. Then, differentiating betwixt past and current research, the tenseness go out be on how maths arrogance and reflective thought process relates to the level of achievement and performance in mathematics problem-solving processes. Concluding description get out follow, illustrating the relationship between reflection and mathematics confidence during problem-solving processes. 2.1 Mathematics, its nature and structureMathematics butt end be seen as a combination of calculation attainment and reasoning (Hannula, Maijala Pehkonen, 200417) and can further be classified as an mortals numeral savvy. 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, expert and cultural growth or leisure (Ebrahim, 20101). This desire led to the advancement of concepts and theories in localize to meet the needs of various cultures throughout prison term. With its imprint in nature, architecture, medicine, telecommunications and knowledge technology, the theatrical role of mathematics has overcome centuries of problems and continues to fulfil the needs of problem- convergent thinkers to solve everyday problems. Although mathematics has changed throughout time, in its progress and influences there argon interwoven connections between the cognitive, connotative and affectional 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 difficult systems should be the aim and goal of mathematics education (Lesh Sriraman, 2005). According to Thijsse (200234) mathematics is an emotionally charged subject field, evoking feelings of dislike, fear and failure. Mathematics involves cognitive and affective factors that form part of the epistemological assumptions, regarding numeral recording (Thijsse, 20027 that will be discussed in the following section. 2.1.2 Epistemological assumptions regarding mathematics learningEnglish (2007123-125) lays mastered 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, inventning, monitoring and communication2.1.2.2 Effective and creative reasoning skills2.1.2.3 Analysing and t ransforming building complex data 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 fashioning.With emphasis on the learner, from a constructivist perspective, learning can be viewed as the active process within and influenced by the learner (Yager, 199153). Mathematical learning is therefore an interactive consequence of the encountered information and how the learner processes it, establish on perceivednotions and existing personal knowledge (Yager, 199153). According to DoE (20033) competence in mathematics education is aimed at integrating practical, siteational and reflective skills. While fixture the tropes 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 (199153) and Leaf (200512-18). This change and tame in edu cation and education paradigms is illustrated in Figure 2.1. Early 1900sEarly 1900s1960s 1980s1980s- 2000s1980s 2000sThe overarching approach with impact on education and therapy focal point on metacognitionIn Figure 2.1 Leaf (20054) 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 in like manner many learners as either slow or clever. The IQ-tests did assess logical, mathematical and language preference and dominance in learners but leftfield little or no room for other ways of thinking in mental aptitude (Leaf, 20055). In contrast to the IQ-approach is Piagets approach, named after its founder, Jean Piaget, who apposed the IQ-approach. counseling 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 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 hierarchical order, and one phase mustiness 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 revise 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 unsucces sfully 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, get under ones skins, backgrounds and cultural aspects in order to expedite learning and problem-solving (Leaf, 200512-15). Above mentioned aspects are also known to associate with performance in mathematics problem-solving (Maree, Prinsloo Claasen, 1997a Leaf, 200512-15). 2.1.3 Some factors associated with performance in mathematicsLarge musical 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, 2005125). Ho wie (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 far 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 teaching and learning. Maree (1997b95) 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 deliberate as the level of mathematics confidence of coterie 7 to 12 learners in a South African context. Sherman and Wither (2003138) documented a case where a psychological factor, perplexity, causes an impairment of mathematics achievement. A distillation of a study through w ith(p) 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, 2003149). Thereupon, factors manifested by the learner are discussed below. Academic underachievement and performance in mathematics is determined by a number of variables as place 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 argument.2.1.3.1 Some associated factors manifested by the learnerAffective issues revolve around an psyches environment within different systems and how that individual matures and interact within the systems (Lombard, 199951 Be ilock, 2008339). In these systems it appears that learners demand a positive or cast out attitude towards mathematics (Maree, Prinsloo Claasen, 1997a). Beliefs about ones own capabilities and that success cannot be linked to effort and hard extend is seen as affective factors in problem-solving (Dossel, 19936 Thijsse, 200218). Distrust in ones own suspicion, not knowing how to slouch mistakes and the overlook of personal effort is regarded as factors that facilitate mathematics solicitude, manifested by the learner (Thijsse, 200236 Russel, 199915).2.1.3.2 Some associated environmental factorsTimed testing environments such as unwritten 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 asso ciated factors during the process of instructionKnowledge 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 high authority figure such as parents or teachers (Russell, 199515 Thijsse, 200235). Thijsse (200219) agrees with Dossel (19936) and Maree (1997) that the teachers upkeep 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-solvingA mathematics problem can be defined as a mathematical based task indicating virtual(prenominal) contexts in which the learner creates a model for solving the problem in various circumstances (Chalmers, 20093). Making decisions within these contexts is only one of the elementary concepts of homophile 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 infallible 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 getting more complex then in previous years. Therefore problem-solving continues to gain consideration in the policy documents of various organisations, inter nationally (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 curriculum change again swings back towards an emphasis on problem-solving.Problem-solving is emphasised as a method invo lving inquiry and decision making (Fortunato, Hecht, molecule 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 undermentioned 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 pastStudies on learners performance in mathematics and how their beh aviours 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 customaryly compared to poor problem-solvers (Lester Kehle, 2003507) while Schoenfeld (1992) suggested that the former not only knows more mathematics, but 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 these strategies Zimmerman (19998-10) describe dimensions for academic self-regulation by involving conceptual based ques tioning 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 store, 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 memory board 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, disrupt 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 three components the psychological articulator y loop, visual-spatial sketch pad and a central executive (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 succeed 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, 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 Cogniti ve 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. Identifying individual trends and behaviour patterns or feelings, could relate to mathematics problem-solving success (Lesh Zawojewksi, 2007778). 2.4.1 Cognition 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, th at may or may not have worked in the past, is linked with experiences ( proscribe or positive) which provide a framework for reflective thinking. Reflection 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 interaction. 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) 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 reflective sendAccording 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), tails-model (2000) for structural reflection and Rolfe et als (2001) framework for reflective practice. The scratch model is that of Gibbs (1988).i Gibbss (1988) model for reflectionGibbs 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 particul ar 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 maneuver 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 three and four.Reflect on and identify factors that influence your actionsFigure 2.3Johns model for reflective practice generatorAdapted from John (2000)The model in Figure 2.3 is divided into two phases. Phase 1 refers to the recal l 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 reflexive practice.iiiRolfe et als model for reflexive practice.According to Rolfe e t 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 2Re flection 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 barrier 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 anyObserve 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 worked 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, M aijala Pehkonen, 200417). A study done by Ma and Kishor (1997) confirmed belief, as an affect on mathematics achievement, being weakly check 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 formed when the learner engages in mathematical problem-solving (Hannula, Maijala Pehkonen, 20043). It plays an important role in attitudes and emotions due to its cogni tive 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 period 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 must be done in isolation) (Hannula, Maijala Pehkonen, 200417). Faulty beliefs about problem-solving allow fewer and fewer learners to take mathematics courses or to revert 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 worryAnxiety, an aspect of neuroticism, is often linked with personality traits such as conscientiousness 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 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 le arners 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 tensity 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 dodging-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, 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 instead 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, avoida nce 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, habitual excuse from the classroom and difficulty of verbally expressing oneself (Maree, Prinsloo Claasen, 1997a). Mar