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This chapter looks at the literature around mathematical development. In order to address the research and key questions, the key themes from this were identified from various sources of literature. The analysis of various key theories of mathematical development will be critically reviewed as to gain a full appreciation of the chosen topic and knowledge of the work that has already been undertaken.
Mathematics plays a vital role in everyone's lives. Devlin (2000-cited in Pound, 2008, p.4), a mathematician, encapsulate this by stating that "Mathematics is not about number but about life". Acquisition of mathematical skill thus is vital as not only it enables citizens to participate effectively in everyday life but is also imperative to the life opportunities and achievements of individual citizens. Moreover, a report into mathematics education, "Making Mathematics Count-Inquiry into Post-14 mathematics" by Professor Adrian Smith (2004), states that problems with mathematics lead to the greatest disadvantages for the individual in the labour market and in terms of general social exclusion (Smith ,2004) .
Recent advances in neuroscience have enabled researchers to study brain development in ways that were not previously technologically possible. We are now aware that children's everyday experiences affect their brain development, by shaping the interconnections, or synapses, that develop between brain cells (Bruce, 2006). Bruce (2006) and Bruer (1999-cited in Fox, 2007, p.2) avers that the rate of human development and learning is most rapid in the first eight years of life. As experiences continue to enhance new brain growth, existing brain structure is refined (Bruce, 2006; Bruer, 1999-cited in Fox, 2007, p.2).
Conversely, recent studies of have also shown that in all societies, an array of economic, social and cultural background factors impact on a child's outcomes in later life (Jaeger and Holm, 2003; Cassen and Kingdon, 2007). Evangelou et al (2009) maintains that the development thus is convoluted by social-cultural contexts and by the architecture of the brain. She (ibid ) further affirms that early skill development is crucial for laying the foundation for lifelong education. This is coupled with research findings of the Organisation for Economic Cooperation and Development (OECD).
The OECD has long been involved in comparative surveys of early years care and educational provision, and its 2001 and 2006 report "Starting Strong" (OECD, 2001; 2006) states that the period from birth to 8 years is crucial in shaping children's lives, and that the high quality educational programs in the prior-to-school years facilitate the development of the child in all its dimensions and have long-lasting effects on the child's life (OECD, 2001). Moreover, evidence shows that early mathematical skills are important predictors of later achievement.
According to the National Council of Teachers of Mathematics (NCTM, 2007, p.1),
"Research on children's learning in the first six years of life validates the importance of early experiences in mathematics for lasting positive outcomes."
Indeed , a coordinated analysis of six large-scale longitudinal studies conducted by Duncan et al, (2007) found that best predictors of educational achievement at school entry is mathematics as not only it predicts later mathematics but also reading ability (Duncan et al., 2007). Furthermore, Pound (2004) avers that all human being are born mathematician. Well-replicated research (Baroody, 1987-cited in Smith, 2002, p.1; Butterworth, 2005) demonstrates that babies from as young as a few days old are able to differentiate between two and three dots on the cards, recognise adding and taking off a shape from the computer screen and distinguish between groups of one and two, two and three and three and four (Baroody, 1987-cited in Smith, 2002, p.1; Butterworth, 2005). Haylock et al (2007) elucidates that the importance of mathematics in individual's life provides ample reason for it to be part of the curriculum.
Mathematics is a key area within the curriculum, worldwide. It is one of the core subjects in the National Curriculum for England which is curriculum for primary aged children (DfEE, 1999) as well as is one of the six learning areas of the Early Years Foundation Stage Curriculum (EYFS) which is a statutory framework for children under the age of 5 (DCSF,2008). The Childcare Act 2006 gave legal status to the provision of six specific areas of learning and development for the knowledge, skills and understanding which all children (of all levels of ability and maturity) should be able to access (EYFS, 2008). Problem, Solving Reasoning and Numeracy is one of the learning areas of the EYFS which has been replaced from the term mathematical development of the Foundation Stage Curriculum Guidance (2000).It covers three aspects; Numbers as Labels and for Counting, Calculation and Shape Space and Measure (EYFS, 2008).
Caddell (2003) highlights the significant influence of Piaget theory's on early childhood mathematics. According to Piaget's constructivist theory (1952-cited in Ojose, 2008), children's ability to think and understand develops through a series of four stages;- The Sensorimotor stage ; the Pre-operational Stage ; the Concrete operational stage and the Formal operational stage (Ojose, 2008). It maintains that logic underpins mathematical development and thus adults should provide learning materials and activities that involve the appropriate level of motor or mental operations for a child of given age. For example it stipulates that for children in sensorimotor stage which is a period of birth to 2 years old, hands on activities e.g. movement and toys should be provided as children in this stage develop through the use of their senses. For children in Pre-operational Stage, 2 to 7 years old, it asserts to use visual aids, actions and hands on activities as children in this stage cannot mentally manipulate information .Piaget also proposed that a child acts on his own environment for learning (Ojose, 2008). However, Piaget's theory has been profoundly critiqued by various theorists e.g. Jerome Bruner and Lev Vygotsky as his theory is based on observations of his own three children. Because of his small and unrepresentative research sample it is difficult to generalize his findings to a larger population. Moreover, it disregards the social effect on children's development (Ojose, 2008).
Although Bruner (1966-cited in Wood, 2004, p.56), likewise Piaget, believes that mathematical development follows the enactive - iconic, symbolic sequence. However, unlike Piaget, Bruner (1966-cited in Wood, 2004,p.??) argued that social factors play active role on child development and learning. He (ibid ) emphasize that role of adult is crucial in children's development and learning, describing as scaffolding a child's learning, putting a scaffold around the child's learning to support the child until the child can operate independently at that level (Wood, 2004). This is supported by Vygotsky's (1978-cited in Wood, 2004,p.?) social constructivist theory which also regards social interaction with peers and adults crucial in children's development and learning. He (ibid) stresses on the active role of the adult in maximizing children's intellectual development and avers that learning occurs in the Zone of proximal development which represents the difference between what the child actually knows and what the child can learn with the assistance of a more knowledgeable other (Wood, 2004).
Liebeck (1990-cited in Haylock, 20007,p.110) purposes to use 'ELPS Model' suggesting that there are four stages in which children learn mathematics. These are through, 'Experience', 'Language', 'Pictorial representation' and 'Symbolic representation'. Children develop abstract thought by exploration of objects such as toys. Later, they will start to recognise words and pictures representing that experience and will then associate written symbols with the object. She (ibid) avers that practitioners should apply this concept when assisting children in their mathematical development (Haylock, 2007).
Early Years Foundation Stage Curriculum (EYFS, 2008) affirms that play underpins all development and learning for young children. EYFS (2008, p.) stipulates that the
"developmental of mathematical understanding should include the use of stories, songs, games and imaginative play"
Although play enhances all round development of children, Holton (2001) and Tucker (2003) avers that play and math's are "partners" as learning mathematics through play enables the child to perceive that math's is useful, enjoyable, social and co-operative. Moreover, it helps to develop positive dispositions for learning mathematics (Holton, 2001; Tucker, 2003). However, Blatchford (2009) argues that to support development, play needs quality adult involvement as it enhances and extends children's learning. Furthermore, according to research conducted by Young-Loveridge et al (1995-cited in Gifford, 2005, p.2) on 4 years old children, who were videotaped for 70 hours during their independent play in the Nurseries, it was found that children used only 1.6 percent of their time on mathematical skills.
EYFS (2008, p.?) endorses that
"Well planned, purposeful activity and appropriate intervention by practitioners will engage children in the learning process"
Moreover, EYFS (2008) is a child-centred curriculum which is based on individual child's needs and interests. It stipulates that all the learning experiences, including mathematics, should reflect the interest and needs of the children. Gifford (2005) avows that when learning experiences are based on children's interest, children are intrinsically motivated to develop and learn. Furthermore, all the six areas of learning and development outlined in the EYFS Practice Guidance are intrinsically connected and EYFS (2008) elucidates that no area should be considered in isolation to the detriment of others. Thus, it avers that
"Children must be supported in developing their understanding of Problem Solving, Reasoning and Numeracy in a broad range of contexts in which they can explore, enjoy, learn, practise and talk about their developing understanding" (EYFS,2008,p.14)
Pound (2008) avers that mathematical contexts should reflect, explore and link with children's everyday experiences. This she (ibid) claims provides a positive ethos and atmosphere in which children are motivated to learn in. The Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools led by Sir William (2008) endorse this notion and affirms that practitioners should develop children's mathematical skills and concepts by presenting them with real life problems which are embedded in meaningful contexts such as during snack time in the Nursery, practitioners can ask children "how many cups do you think we need" etc as by doing this practitioners can intrinsically motivate the children to develop mathematical skills.
Blatchford (2009) assert that practitioners can smooth the progress of young people, by giving them confidence in their own abilities and encouraging positive ambitions for educational achievement by intervening in children's learning. She (ibid) further postulates that practitioners should 'sustained shared thinking' with the children, which she (ibid) defines as
"an episode in which two or more individuals (children together, or adults and children) 'work together' in an intellectual way to solve a problem, clarify a concept, evaluate activities or extend a narrative etc. Both parties must contribute to the thinking, and it must develop and extend." (Blatchford, 2009, p.1).
Blatchford (2009) avers that strategies of sustained shared thinking include showing genuine interest, recapping, suggesting, clarifying, encouraging, speculating, reciprocating, modeling and asking open ended questions. Moreover, Pound (2008) maintains that open ended questions such as; How are they same / different? How did you find that out? Should be asked by the practitioners as they extend children mathematical thinking skills. However Wood (1998-Tomlinson, 2000, p.7) argues that developmentally appropriate questions should be asked as asking challenging questions to those children who are not developmentally ready may impact on the adult/ child interaction and thus learning. Tomlinson (2000, p.8) avows that when interacting with children, especially during group activities, practitioners should differentiate as "students who are same age differ in their readiness to learn, their interestsâ€¦."
Tucker (2005) affirms that the learning environment within the setting plays a crucial role in a child's learning. An effectively designed environment has potential for positively influencing all areas of children's development (EYFS, 2008). However, Worthington and Carruthers (2006) argues that if learning environment is mathematical, then mathematics happens. They (ibid) affirm that wide variety of mathematical resources both indoor and outdoor should be provided to the children.
Nevertheless, Clements and Sarama (2002) argue that for deep level learning to take place resources should be stimulating yet challenging. They (ibid) postulate on the use of ICT. ICT resources such as computer software's and calculators etc provide instantaneous feedback to the children .Moreover computers can act as catalysts for social interaction (Clements and Sarama, 2002).
Worthington and Carruthers (2006) concurrently claim that psychological environment plays an equal part on children's development and learning. Fisher (2002) supports this notion and affirms that opportunities for children to talk to both with their peers and their practitioner are the most important element of the learning environment. He (ibid) affirms that positive encouraging comments from a practitioner will help the children perceive themselves as competent mathematicians.
Pound (2008) avows that mathematics is a subject that people can respond to in a very negative way. Consequently, they then pass on their negative feelings to those they are teaching. Moreover, Perry and Docket (2008) sustain that low levels of content knowledge and the resulting lack of confidence about mathematics hinders teacher's ability to create opportunities for engaging children in the mathematical learning embedded within existing activities. However Bobis et al and Gifford (2004), affirms that this attitude of teachers could be them being part of a "cyclical process" (Gifford, 2004, p.2). Teachers who themselves were poorly taught mathematics or view mathematics negatively, as a result, may lack confidence in teaching it in an innovative ways. Thus they may rely on more formal and structured methods such as worksheets which Fisher (2002) avers do not give accurate views of what children can actually do.
In order to maintain children's confidence in mathematics and prevent them from leaving the early years setting with feeling of inadequacy and anxiety towards the subject, Gifford (2004, p.2) asserts that practitioners should "monitor their attitude towards maths and to guard against negative attitude". She (ibid) posits that practitioners should make teaching numeracy fun and lively to keep children motivated.
EYFS (2008) affirms that parents should be seen as partners in children's learning as they are "children's first and most enduring educators "(EYFS, 2008, p.??).
Longitudinal studies, Desforges et, al (2003) and EPPE (2003) , shows that children whose parents are involved show greater social adjustment, greater mental health and greater social competence .Moreover parental activities conducted at home such as singing nursery rhymes, cooking, shopping makes an important difference to children's attainment (Desforges et al, 2003; EPPE, 2003).
However, research studies (Lareau ,1987-cited in West, 2007; Starkey et al, 2000) additionally shows that social class differences may be implicated in the extend to which children are involved in mathematical experiences at home. In their study Starkey at el (2000) found that middle-class parents reported providing more mathematical activities to their children than did working-class parents. Nevertheless, these research studies also eloquently state that all parents, regardless of their socio-economic status, want their children to do well in school (Lareau ,1987-cited in West, 2007; Starkey et al, 2000 ).
DCSF (2009) affirms that pupils from deprived backgrounds are more likely to have parents and carers who had bad experiences of school themselves and who have low levels of education. Thus they may lack confidence and knowledge in helping with their children's learning. The finding published in the Every Parent Matters (2007 p. 10) documents clearly states that issues with numeracy, literacy can act as barriers to parents supporting their children's learning. It report that almost one half of adults (17 million) in UK have difficulty with numbers. This insinuates the importance of school to support parent's basic skills. Research conducted by Starkey et al (2000) with Head Start Families in America found that low-income parents were willing and able to support this area of their children's development once they were provided with the training to do so. The support that parents provided to their children through the intervention was effective in enhancing the development of children's informal mathematical knowledge. Indeed, intervention children developed more extensive mathematical knowledge than a comparison group of low-income children.
Pond (2008) avows that early year's practitioners play crucial role in supporting parents become confident about supporting their children's mathematical learning. She (ibid) proclaim that practitioners can help parents by creating positive home learning environment by putting in place strategies such as organising mathematics workshops , lending maths books and games . However, Gifford (2004) and Pound (2008) concurrently argues that for practitioners to be able to talk comfortably and knowledgeably to parents about their children's mathematical learning, they need to gain sufficient confidence themselves. Moreover, McMillan (2005) avows that lack of teacher training could be a barrier in effective parental involvement. Gifford (2004) avows that for practitioners to become confident, it is imperative for them to take part in continued professional development. Muijis et, al (2008 p. 195) asserts that
"where teachers are able to reflect, access new ideas, experiment and share experiences within school cultures and where leaders encourage appropriate level of challenge and support there is greater potential for school and classroom improvement."
Assessment is an integral part of the practitioner's role. EYFS (2008) explains how practitioners need to create enabling environments to observe children in order to establish their needs and what they can do, known as formative assessments. Pound (2008) avows that as observations provide invaluable insights into children's development, it should be done over time as single observation can not determine their ability or understanding. Moreover, she (ibid) maintains that to determine children's mathematical development, practitioners should observe them during wide range of activities, not just mathematical. As though through mathematical activities such as number puzzles and threading beads practitioners can find out about the children's knowledge, they may not inform practitioners about their understanding.
EYFS (2008) affirms that planning should start by observing children and should integrate their existing knowledge and understanding .Moreover, it stipulates that observations should derive from differing viewpoints -all who are involved and concerned with the child, including the parents as robust research on early years settings, Researching Effective Pedagogy in Early Years (REPEY) shows the significance of formative assessment to best outcomes for children. It stipulates that
the more knowledge the adult has of the child, the better matched their support and more effective the subsequent learning (REPEY,200?).
However, a study by Aubrey (1997-cited in Caddell, 2003, p.6) looked at the extent to which early years practitioners took account of children's existing mathematical knowledge and understanding. It found that practitioners were unaware of existing mathematical knowledge and understanding. Moreover, it was not used to inform teaching experiences at school entry decisions. This she (ibid) claim then impacts on summative assessment of a child's development / learning.
Progress Towards Profile, is form of a summative assessment for pre-reception aged children (BCC, 2009). It consists of 13 scales which cover all six of the areas of the EYFS and are derived from development matters specified in the EYFS Practice Guidance (BCC, 2009; EYFS, 2008).
To conclude, there were many key issues outlined throughout the literature review such as, the teaching and learning strategies, the mathematical learning environment, assessment and planning and parental involvement. Key themes of the literature review showed how the practitioners and parents should work together to provide overall effective practice.
The next chapter discusses the research methodology used within the case study supported by relevant literature.