What Insights Can The Student Teacher Gain Education Essay

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According to Thompson (1997) the current structure of Primary mathematics has seen very little change since the appearance of the first mathematics curriculum development project placed in the 1960s. Throughout this essay I will be looking at research as well as my own observations in the classroom to ascertain whether recent changes have occurred within primary mathematics. For the purpose of this essay I am going to look at the area relating to a childs concept of number observing key points relating to early addition and place value with year 1 school children. Focusing in particular on number bonds and how it is taught in a primary school. I will then discuss traditional and more recent research from a variety of sources which will enable me to have an insight into effective teaching of number bonds. The traditional works of Vygotsky (1978) and his relationship between developing and learning and visual representation suggested by Carpenter and Moser (1983) will also be considered. The work from the Department of Education: Mathematical Vocabulary (2000) will be also be discussed.

Thompson (1997) suggested by the beginning of the twentieth century important development had taken place in the world of mathematics. New branches of the subject had been developed, and more refined and powerful techniques had gradually replaced established methods. These methods seen by Shuard and Rothery (1984) establish a more precise and unified language which helps children progress more confidently. Written communication is a major component of the methods of teaching mathematics used in most schools today. Textbooks, worksheets and workcards, either commercially produced or home made, form a integral part of the resources which teachers of mathematics use with their children.

Thompson (1997) explained there is a substantial amount of research evidence concerning the important part played by counting strategies in the development of children's number understanding. Young children hear the number words being used in a variety of different situations in their daily life, and these words vary in meaning according to the context in which they are used. However, their experience of actually saying the number words is more likely to have been gained in a counting context. Similarly, Dickson, Brown and Gibson (1984) agreed that many children develop notions of addition before they receive any formal instruction in school.

A structured approach to teaching and learning mathematical vocabulary to promote children's ability to use correct terminology as early as possible is an important component of any effective maths strategy. Mathematic Vocabulary: National Numeracy strategy (2000) is aimed at both teachers and support staff. It begins by outlining its purpose and intentions before analysing the different ways in which children demonstrate a poor understanding of mathematical language and explaining why mathematical language is important in the development of a child's thinking. This vocabulary provides strong words for the teacher to use in relation to counting and addition. However the vocabulary by itself will not have a significant impact on children's understanding of the concepts of addition. Carpenter and Moser (1983) concluded using vocabulary as a basis for teaching addition is more successful than teaching computational and visual skills first and then trying to apply them to solve problems. Children need to gain the knowledge and understanding of what number bonds are before seeing examples of number lines and games.

Ryan and William (2007) noted that some primary schools have begun to wonder if the role of activity with manipulatives and materials is still important as it once was. From observing number bonds being educated to a year 1 class during a mental oral starter, play and communication are seen as vital. Games and number beads are essential to help the children understand addition and the pattern of number bonds. Gelman and Gallistel (1986) states 'It is emphatically our view that the bases of mathematics are bodily and spatially situated, even if such activity does need to be mediated by language and communication before it becomes fully mature.' According to Hughes (1986) who worked with children on early addition, using denes blocks and cubes for children developing number bonds to ten can maintain their confidence. Children can construct number bonds to ten for themselves rather than teaching the bonds as can be seen in many maths workbooks. Hughes (1986) also suggested the next step for children is to make the link between the place value properties of the numbers and how this supports addition. After my close observations in the classroom it was noted that children who were encouraged to draw their own block of number bonds onto paper and being allowing to individually determine the whole number and relationship between each number bond were able to gain knowledge of place value.

Fuson et al. (1982) have shown that children learn the difference between counting words and non-counting words at a very early age. They asked a group of children between the ages of two and five to count collections of objects. All of the three-, four-, and five-year-olds and most of the two-year-olds used counting words on every occasion Hebbeler (1977) has researched the extent to which counting strategies are used by pre-school children to solve simple problems, and reports that the normal course of development for Kindergarten and first grade children is to progress spontaneously from counting to the use of number facts as a problem solving strategy. Similarly, Houlihan and Ginsburg (1981) have demonstrated that first years can apply non-taught counting algorithms in the solution of a number of problem solving types, and have suggested that young children select their counting strategies according to the size and familiarity of the numbers involved. Piaget (1952) argued that those children who had not reached the concrete operational stage of thinking, normally attained at about the age of seven, could not possibly understand the operations of addition. However, Hughes (1986) and Starkey et al. (1982), with their respective Box and Pennies tasks, have shown that children between the ages of three and five can perform addition tasks with some understanding, provided that the numbers involved are small, and the operations are carried out on real items in such a way that the objects and actions are visible to the children but the final total is not.

Research by Carpenter and Moser (1983) suggested children learn addition through different ways of representation from direct modelling with concrete objects through to using derived number facts. Visual representations are also used to develop children's understanding of addition and number bonds. Furthermore Barmby, Bilsborough, Harries and Higgins (2009) suggested that moving from representations similar to concrete objects that can be used, for example counters representations and more abstract representations such as the empty number line. Gravemeijer (1994) argues that the empty number line supports children in using their own informal strategies in addition. However, Barmby, Bilsborough,Harries and Higgins (2009) suggested that one of the most common problems with addition occurs through tallying with fingers or when using a number line where the first number or finger is counted, rather than counting on 'seven, eight, nine, ten; seven plus four is ten.' Therefore, children have to be aware of how to use representations.

According to Vygotsky (1978), teaching and learning is the very pathway through which cognitive, social, and affective development takes place. Learning is not development; however, properly organised learning results in mental development and sets in motion a variety of developmental processes that would be impossible apart from learning. These words by Lev Vygotsky address one of the most fundamental concerns for anyone dealing with children. However, according to Stevenson (1983) during the mid-1970s, there was a rise of the new cognitivist theory, learning itself ceased to attract attention. Discussion of learning was replaced by an interest in the discovery of deep universal laws of mental machinery that were presumed to be hardly affected by any external influences. It proposed all that matters are the characteristics of a predetermined and virtually unchangeable general storage capacity that are thought to be best revealed through research on the memorization of meaningless information for example strings of letters or numbers outside schooling experiences or any other learning practices in which the child is engaged, children in experimental classes gradually develop a genuine learning motivation.

Although Vygotsky (1978) provided theoretical arguments about adults and children working in the Zone of Proximal Development, Murphey, Selinger, Bourne and Briggs (1995) states there is little empirical evidence of what this might look like in practice. Bruner (1985) argues that Vyogotskys theory implies that through scaffolding the learning task, the tutor or peers make it possible for the child to internalise knowledge and invert into a tool for conscious control. Bruner (1985) points out an important point being the importance of the teacher modelling a task to engage the children's interest. This may be easily accomplished when the task is a practical one, observing teaching of number bonds songs and rhymes of the patterns of numbers are used regularly to engage the children. These practical tasks can be demonstrated by the teacher to the pupils so they can gain a sense of what they need to accomplish.

From observations in my year one classroom when teaching number bonds, mental oral starters are always used to help recap this addition skill. Pair games, cuisenaire rods and sum thing beads are used to reinforce the learning of the children. Rhymes and songs can also help the children recite the relationship between each number bond. My observations lead me to believe a picture is worth more than many words, having observing the children draw number bonds on paper using circles or bar diagrams (see figure 1). For the children to imagine each circle to be a pile of blocks, and think of the bar as the blocks lined up in a row. Even a young child who does not understand math notation can clearly see the connection between these numbers: the whole (10) has been pulled apart into two piles (6) and (4), and the piles can be pushed back together to make the whole. After observing a mental oral starter the message from the teacher was made clear that the goal was not for her to memorize specific math facts, but that she understand and be able to use the concept of taking a number apart and then putting it back together.

Number lines were also used during the mental oral starters. This enabling the children to using Number Facts but also to model these on number line related to the appropriate number sentence.

(Class IWB activity)

This example illustrates number facts set to show number bonds to 10 with 7+3 =10 highlighted. This number bond is also modelled on the number line starting at 3 and showing the jump of 7 to 10. As the counters are changed on number facts the setting on the number line can be changed to match.

Asking children to write all the number bonds onto paper enables the children to see the relationship between each bond. After conversations with the class teacher this method allows the children to always relate back to their work, allowing them to work independently, some children work more efficiently as independent other than as a class during a mental oral starter.

+ = 10

+ = 10

+ = 10

+ = 10 (Class worksheet activity)

+ = 10

+ = 10

+ = 10

+ = 10

+ = 10

+ = 10

Looking at research gathered by Carpenter and Moser (1983) and his study of visual representations I have observed a variety of different teaching aids of visual representations that enable the children to use concrete objects to allow effective learning visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education.

From research and my own observations in the classroom I can explain visual representations and how they are used in the classroom. This can include manipulatives, pictures, number lines, and graphs of functions and relationships. Representation approaches to solving mathematical problems include pictorial for example diagramming, concrete, verbal and mapping instruction. When teaching number bonds the teacher tends to use concrete materials such as Cuisenaire blocks, beads, base-ten blocks or unifix cubes to model the mathematical concept to be learned, then demonstrates the concept in representational terms. For example drawing pictures, and finally in abstract or symbolic terms such as numbers, notation, or mathematical symbols.

After introducing the concept of number bonds to the children with concrete manipulatives, the teacher would model the concept in representational terms, either by drawing pictures or by giving students a worksheet of unfilled addition number bonds and asking the children in the class to fill in the blanks.

In the Researching Effective Pedagogy in the Early Years Project (2003), the researchers found that pre-school children do best when they are engaged in activities that make them think deeply, particularly 'environments that encourage sustained shared thinking between adults and children make more cognitive, linguistic and social-behavioural progress' (Blatchford., Sylva, 2002)

Furthermore the studies referenced above highlight clear links between activities that lead to both high levels of involvement and high quality outcomes for learning. Sustained, shared thinking and high levels of cognitive and linguistic progress are also clearly linked.

Teachers' may need help in recognising children's mathematical explorations and their creative thinking, when they make observations evidenced through their play, talk and representations After having a conversation with my class teacher it has become clear that when observing and teaching the class during mathematics and mental oral starters the teacher would like more help from support or assistant staff . This support is needed so they can plan to extend children's learning, based on what they have seen, and to build on what children already know, understand and can do. If the teacher has the extra support from the assistant staff, the teacher will be able to concentrate more on the teaching method that Vygotsky (1978) suggested of the child and teacher working in the Zone Proximal Development.

As mentioned I have observed children learning about the abstract written symbolism of mathematics by making connections between their own early mathematical marks and the symbols to which they are gradually introduced in school. My observations can be backed up by research from Worthington and Carruthers, (2003) who has developed the theory of multi-competent stage.

Visual representations were recognised by Vygotsky(1978) as powerful cultural tools that support learning within socio-cultural contexts. Number lines used on the active whiteboard and games used for number bonds allow the children to work on the carpet and talk to their neighbour and discuss the answers and ideas. Therefore producing cognitive learning in an effective way.

Research shows how children's addition mathematics can be inventive and joyous, reminding us of the creativity of young children so long as the classroom teacher has the confidence and support to achieve such representations. Viewing children's representations from a positive perspective provides opportunities for children to explore and make decisions about their own chosen forms and has the power to support deep levels of cognitive challenge, of rich language discussions and high levels of creativity. These processes underpin the recommendations of the curriculum for addition, yet as the research has shown children who are not yet aware of the representation stage may encounter problems with addition which could include tallying with fingers.

From observations I have gained a greater understanding of mathematics and how mathematics should be taught. There is a large potential for developing high levels of cognitive challenge and creativity in mathematics through encouraging children's mathematical addition games and visual pictures. Also it is crucial that teachers should listen to children's voices and to recognise value and support children's thinking and visual representation in addition. Failure to do so will mean that opportunities for creative thinking in mathematics will continue to be limited. This has significant implications for young children's understanding of addition, number bonds. From observing different ways of teaching number bonds and from conversations with the class teacher I have developed an understanding of how number bonds are effectively taught in the classroom. Using the correct vocabulary with children stops confusion and children begin to understand the correct meaning of each mathematical word. The use of visual sources and active games increase the learning and interest for the children, allowing written pictures and diagrams to be presented.

References

Barmby. P, Bilsborough,L. Harries,T. and Higgins,S. (2009) Primary Mathematics: Teaching For Understanding. New York: Open University Press. Pp 30 - 40

Bruner, J. (1985). Vygotsky: a historical and conceptual perspective. In J. V. Wertsch (Ed), Culture, Communication and cognition: Vygotskian perspectives. Cambridge: Cambridge University Press.

Carpenter,T.P. and Moser,J.M. (1983)The acquisition of addition and subtraction concepts, in R. Lesh M.Landau (eds), Acquisition of mathematical concepts and processes. New York: Academic Press, pp 7-44

DfES (2000) Mathematical vocabulary: National Numeracy Strategy. Department of Education. London: Crown Copyright.

Dickson,L, Brown,M and Gibson,O (1984) Children learning Mathematics: A Teachers Guide to Recent Research. London:Cassell Education Ltd.

Fuson, K. Richards, J and Briars, D (1982) The acquisition and elaboration of the number word sequence, in C. Brainerd (ed), progress in Cognitive Development research Vol. 1: Childrens logical and Mathematical Cognition. New York: Springer Verlag, pp.33-92

Gelman, R., & Gallistel, C. R. (1986). The child's understanding of number. Cambridge, MA: Harvard University Press.

General teaching council for England (2003) Researching effective pedagogy in the early years. Research for teachers: TLA resources.

Gravemeijer,K. (1994) Educational Development and developmental research in mathematics education, Journal of Research in Mathematics Education, 25 (5): 443-82

Hebbeler,K: (1977),Young Childrens addition. Journal of Childrens Mathematical Behavior (4) 108-121

Houlihan, D.M and Ginsburg, H.P (1981) The addition methods of the first and second grade children, Journal for research in Mathematics Education 12(2) 95-106

Hughes, M (1986) Children and Number. Oxford: Blackwell Publishers.

Murphy, P. Selinger,M. Bourne,J and Briggs,M (1995) Subject Learning in the Primary Curriculum. New York: Biddles Ltd.pp 211-219

Piaget, J.-P. (1952). The origins of intelligence in children. International Universities Press, New York.

Ryan, J and Williams, J (2007) Childrens Mathematics 4-15: learning from errors and misconceptions. Maidenhead: Open University Press.

Shuard,H and Rothery,A (1984) Children Reading Mathematics. London: athenaeum Press. Pp 1-3

Siraj-Blatchford, I., Sylva, K; Muttock, S., Gilden, R., and Ball, D., (2002) Researching Effective

Pedagogy in the Early Years', DfES Research Brief, No: 356.

Starkey, P and Gelman, R. (1982) The development of addition and subtraction abilities prior to

formal schooling in arithmetic, in T.P. Carpenter, J.M. Moser and T.A.Romberg (eds) Addition

and Subtraction: A cognitive Perspective, (Hillsdale, New jersey, Erlbaum).

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Worthington, M. and Carruthers, E. (2003a) Children's Mathematics: Making Marks, Making Meaning. London: Paul Chapman Publishing.

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