A small scale research based on a set of children was undertaken by me while I was in placement.
The research was based on children’s mathematical thinking and how teacher’s help develop children’s mathematical thinking. I will be using relevant quotes from appropriate theorists and literature to support my research.My small scale research is based on a group of 6 six children from keys stage one, year two, class of 28 children from a large urban primary school. The school is a newly built school based in East London, Leyton, opposite another primary school. It has a mixture of children from different races, cultures and religion.
The research consists of photocopies of children’s work, lessons plans, and evaluations of lessons, weekly lesson plans and transcripts. I chose to do my research on the higher ability children from the class as a lot of the children from the class speak English as a second language and find it difficult to explain themselves. Children from the lower ability find it difficult to understand the work and therefore are given much easier work compared to the higher ability.
The lower ability children also need a lot of help and support when they are doing their work. The six children that I chose to do my research on are Petar, Imran, Usayd, Zahran, Helin and Mimi. These children enjoy mathematics Petar and Usyad (see appendix 6)Mathematical thinking can be defined in various ways as it can be broken down into many sections. I believe that mathematical thinking is a logical/systematic way of thinking. It is developed through speaking and listening, group or collaborative work. Mathematical thinking is when children use and apply their skills.
I agree with the statement provided by the National Curriculum as it quotes alongside to what I believe mathematical thinking is. The National Curriculum states that mathematics trains children with a ‘uniquely powerful set of tools’ to understand and develop the world. These tools are: “Logical reasoning, problem solving and the ability to think in abstract ways.”DFES (1999) p.60-61Dr Collin Sparrow, lecturer in mathematics, states “Mathematics is not just a collection of skills; it is a way of thinking. it lies at a core of scientific understanding, and of rational and logical argument”DFES (1999) p.
60-61Both definitions, which are from the National Curriculum suggests how mathematics is a very intense subject which provides skills to children to be taken out of the class and used.Haylock, D (2001, p.1) refers to using and applying of mathematics as a part which should be “…At the heart of learning the subject”. Haylock’s quote is in conjunction with the National Curriculum as using and applying is a section found in the National Curriculum for both key stages.
Hopkins, C et al (1999) also states how mathematics thinking has three aspects involved which are: reasoning mathematically, mathematical communication, and solving problems.Speaking and listening plays a role in defining mathematical thinking, through speaking and listening children are given the opportunity to collaborate, expose and share ideas. Psychologist, Piaget (1970) felt that children’s speech did have an effect on their mathematical thinking and development. But he also finished off by saying that children’s egocentric speech only existed at the pre-operational stage. Whereas Vygotsky (1978) believed that speech was not something that disappeared, it was something that became more of an internalised thought which he labelled as the ‘inner speech’. Vygotsky believed that children used their ‘inner speech’ to help solve problems when they are stuck.
I agree with Vygotsky because through my small scale research I have noticed how children who are stuck either go off task or approach the teacher for help. Working with children, I have helped a lot of them because they approached me for help or I offered to help them.I have used a lot of speaking and listening techniques to help develop children’s mathematical thinking. From appendix 1 and 2 (samples of children’s work and lesson plans) you can see how the children have been speaking and listening to complete the activity.
Examples of activities are children making 3D shapes in pairs using straws and plastecine. In this activity children had to discuss their ideas first and decide which shape they were going to make first. By getting one child to hold the shape while the other one counted the sides, edges and faces, gave children the opportunity to collaborate. In the lesson plan for the 3D shape I got the children to talk about their shape in the plenary.
Another example of speaking and listening activity is when the children had to work transfer the time from the analogue clock into digital format. The children worked in pairs. The speaking and listening took place as one child made a time on the analogue clock and the other would try and tell the time, if they are correct they would write the time down in digital format.Children also worked with money. Here the speaking and listening strategy was used when the children acted in role play as a shop keeper and a shopper, the shopper would buy an item from the shop and the shop keeper would have to make sure either the right money is given to them or the right money is given back in change.
Children worked in their own ability group and were able to help each other out when one got stuck by showing which coin/s to use. By having the children act out in role play children were able to discuss and elaborate their mathematical understanding. Children were able to show their ways of making a certain amount using different coins. Some children already knew the value of certain coins and therefore were able to use and apply their existing knowledge as they were trying to buy or sell a product.In one activity (see appendix 1) I got children to find all the different ways of making a certain amount.
The higher ability group had to make the value of ï¿½10. The children came up with many different ways. Examples of what the children came up with by them self were: ï¿½10, ï¿½5+ï¿½5, ï¿½5+ï¿½1+ï¿½1+ï¿½1+ï¿½1+ï¿½1, ï¿½5+ï¿½2+ï¿½2+ï¿½1. But once I got the children to use pennies in making ï¿½10 a lot of the children were getting confused. Children worked as a group discussing and showing their ideas and strategies using coins.
In this activity children were able to demonstrate what they already knew about money and were challenging them to develop their understanding further. When I asked Usayd to put pennies into making ï¿½10 he easily took away one ï¿½1 coin and replaced it with 2 50 pence pieces.Hoyles (1985) also agrees how speaking also plays a role in developing children’s mathematical thinking. She believes there are two functions of talk.
The first function of talk is ‘communicative language’, which is when the child explains and justifies their strategy. The other is ‘cognitive talk’, this is when the child is given time to reflect back at their work.Piaget’s (1970) stage theory on children’s psychological development reflects on the cycle of learning. In the preoperational stage children are able to use symbols to represent their ideas. At this stage children are also egocentric which explains why children at that stage find it difficult to understand number, volume and length.Culture plays a role in helping develop children’s mathematical thinking. Bruner (1977) believed that cognitive development occurred from culture surrounding the child and the mental actions which occurred to process the information.
Brunner’s theory opposes to Piaget’s theory. Bruner believed that children could be taught effectively at any stage of development which supports our spiral curriculum. “A Curriculum as it develops should revisit these basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.”Hansel (2005) p.5Culture did contribute in the lessons to help children develop their mathematical thinking.
An example would be when I asked the children to think about how long they thought a minute was (see appendix 2). By thinking of all the things that take a minute to do make the children reflect back on things they do regularly, things they do at home, at school, outside etc. This activity did reflect on children’s culture as children came up with different things that could be done in a minute. The activity used children’s own experience and developed further from there. Children were able to use and apply their own existing knowledge.Having the right setting helps develop children’s mathematical thinking this is because from experience I have noticed that children can perform better with the right setting.
An example of this is when the children did role play with a shop keeper and customers (see appendix 3). I worked with the lower ability children and noticed how a child who finds difficulty in adding and subtracting was doing really well. He recognised a lot of the coins and was capable of adding up small sums of coins to make the right amount without any help.
The child was able to help his friends that were giving him the wrong amount to purchase an item. However if the same sum was to be done on pen paper, he would have needed help to complete. He also managed to help his friends that were working with him on making the right amount and using the right coins. By helping his friends when they got the amount wrong he was able to demonstrate logic and reasoning.
It also shows how culture plays a part of children’s mathematical thinking. This child obviously must be experienced in handling money outside school and therefore was familiar with the coins.From research Lave (1988) identified how setting played a role in developing children’s mathematical thinking. For example Lave noticed how shoppers in a supermarket apply their mathematical knowledge in calculating supermarket context; they were almost 100 per cent accurate.
Nevertheless the same shoppers given the same calculations on pen and paper, their accuracy declined to two thirds.It is very important to have role play, models, images and resources for children when they are doing their activity. An example of role play is given in the above paragraph and it shows how the child was engaged and developed mathematical thinking through making different amounts using coins, whereas on pen and paper he would have found this difficult. Through props children are given the opportunity to learn from experience.Children made their own analogue clocks (see appendix 2) which became their own resource when they were working on time. They also had the opportunity to play with their clocks (see appendix 1) and transfer the time on the digital format. Without this resource it would have been difficult for children to imagine the time and write the time down.
They would also not be able to make links between the digital format and the analogue format. By having their individual analogue clock, children were able to work in smaller groups, which avoided conflict because they did not have to share as much and decide who’s turn it is and who has had the clock for the longest amount of time.Also having the children to make their own 3D shapes by looking at other 3D shapes (see appendix 1) gave them hands on experience on 3D shapes. As it was a new subject for them, children were finding it very difficult to draw what they had made in a 3D version.
If they did not make the 3D shapes then they would not have known at all how to draw, at least they made an attempt in drawing their 3D shape. It would have been more difficult for the children to imagine and draw.Teachers I believe play a huge role in developing children’s mathematical thinking. As the main expert in the class room they have more control over the subject taught. Teacher’s decide when and how to teach the children.
Yackel and Cobb (1996) research on sociomathematical norms which is “…Acceptable mathematical explanation and justification” (Yackel and Cobb, 1996, p.
461) explains that in order for children to develop sociomathematical norm, they would need to be able to “…
Interpret explanation in terms of action on mathematical objects that were experientially real to them” (p.461). As the teacher’s role, it would be to act as a contributor who would challenge and allow certain aspects of children’s mathematical activity. It is through whole class discussions as to when the teacher makes sense of their responses. Yackel and Cobb (1996) believe that sociomathematical norms are essential within the discussions taken in the classroom as they create mathematical ideas.As the teacher I devised lessons for children that I knew would help develop children’s ideas and knowledge. Through experience I had noticed how children work better and are more focused when the maths session is in the morning. I noticed in the afternoon, children can be very hyper and unsettled.
This makes it difficult to control and manage the class and also takes longer for the children to complete their work. If you compare some of the evaluations (see appendix 3) you will notice how the sessions which took place in the afternoon, the children were a little hyper and I did lose control over them couple of times. However the sessions which took place in the mornings worked well.Going back to the essay title I have demonstrated the needs of developing children’s mathematical thinking. As there are many sections of mathematics it becomes necessary as a teacher to develop children’s mathematical thinking through all sections in all the various ways available. Having taught children through all the various ways I feel I have managed to teach to the utmost extent and children have learnt and developed to their maximum of understanding. It is important that teachers take into account of differentiation and children’s abilities when planning and teaching so children do learn and can develop their mathematical thinking.BIBLIOGRAPHY:Bottle, G (2005) Teaching Mathematics in the Primary School.
London: Continuum.Bruner, J (1977). The Process of Education, Second Edition. Cambridge, MA: Harvard University Press.DFES (1999). The National Curriculum. London: DFES.
Hansen, A (2005). Children’s Errors in Mathematics. Exeter: Learning Matters LtdHaylock, D (2001).
Mathematics Explained for Primary Teachers.2nd Edition, London: Paul Chapman Publishing.Hopkins, C, Pope, S and Pepperell, S (2004). Understanding Primary Mathematics.
London: David Foulton Publishers.Hoyles, C (1985). What is the point of group discussions in mathematics? Educational Studies in Mathematics.
16: 205-14Lave, J (1988). Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge, Cambridge University Press.Piaget, J (1970). The Science of Education and the Psychology of the Child. New York: Crossman.
Vygotsky, L.S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA: Harvard University Press.
Yackel, E and Cobb, P (1996). Journal for Research in Mathematics Education. 27 (4): 458-77