# Nature And Structure Of Mathematics

4288 words (17 pages) Essay in Mathematics

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### 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 individual’s 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 1900’s |
Early 1900’s |
1960’s – 1980’s |
1980’s- 2000’s |
1980’s – 2000’s |

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 Piaget’s 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 child’s 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 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 individual’s 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 one’s 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 one’s 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 learner’s 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 teacher’s 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 one’s 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 1930’s (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 1970’s 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 problem’s 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 al’s (2001) framework for reflective practice. The first model is that of Gibbs (1988).

### i Gibbs’s (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 reflexive practice.

### iiiRolfe et al’s 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, “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 |
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 correlation 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 individual’s 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 and 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

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