“Making sense of mathematics” by Swan. M (2003) Demonstrates three key themes throughout the piece of writing: children need to know the meaning of mathematical processes and understand them to become fluent, connections need to be made to prior knowledge and that there should be some degree of cognitive conflict that encourages children to discuss their findings and methods to further build on their understanding. The themes connect to support children in making sense of tasks presented to them. The key points will be discussed in terms of importance and their requirements in providing children with the ability to use other texts to support themselves in understanding mathematics; how each one is needed for the other to be successful. This argument will be reinforced with personal school experience placement, theoretical and professional understanding.
Connections between their prior knowledge and learning need to be made. Haylock. D & Cockburn. A (2003) state that children need to build connections of new experiences to their previous knowledge. The growth of children’s understanding is the construction of cognitive connections; therefore, their understanding is more stable and clear as stated by Swan. Swan. M (2003) further articulates that children should be supported and encouraged to create their own meanings and links. Children that build on prior knowledge may develop more independence in their learning and allows them to discuss with their peers which methods would be best. This contradicts the idea that mathematics can’t be enjoyable or interesting as children will gain confidence through reflection and discussion. Anthony. G & Walshaw. M (2009) suggest that to make sense of a new process or skill, pupils need to make connections to their existing mathematical understanding and that these connections help them recognise the relationship between mathematics and real life. They further state that when given opportunities to apply their mathematical skills to everyday scenarios pupils learn the value of mathematics and how it contributes to other areas of knowledge and they will start to view mathematics as relevant and interesting. Piaget (1936) described these connections between ‘schemas’ as a ‘set of linked mental representations…which we use to both understand and respond to situations’. Children may need time to adjust to a new idea and to relate is to concepts they may already possess, according to Swan. M (2003), as this allows them to make the connections required to absorb the additional information and for it to make sense. Haylock. D & Thangata. F (2007) explain that teachers need to give time for children to understand procedures and make connections to solidify their understanding, this is done by using suitable language; therefore, making connections contributes to their understanding. Swam. M (2003) explains that when these methods, equations and processes are not supported by meaning then children are left with unconnected rules and ideas that are quickly forgotten. This theory is backed up by personal experience and theoretical and professional understanding; during experience as a teaching assistant, it was evident that some children made the connections easily, whereas other children needed more support in creating the connection. When the children were supported in making the connection then their classroom experience was deemed more positive.
Swan. M (2003) believes that understanding mathematics evolves from understanding that is based on physical actions through more abstract meaning therefore children can use their physical actions to develop a more intellectual understanding. Swan. M (2003) applies this theory to multiplication, explaining that ‘multiplying’ derives from ‘repeatedly adding’; therefore, allowing children to build on their prior knowledge. This suggests that children need to build on their understanding and meaning of their previous knowledge, without this, children will simply be learning to memorise without a clear or useful meaning as to why they are gaining this information. Cockburn. A (2007) supports this, stating that compared to simply memorising, the theory of understanding is vital to succeed, this can be achieved, this can be achieved through providing children with practical experiences and opportunities to create their own understanding behind the meaning. If children are not given reason as to why these processes need to be understood to apply them to everyday life, then the interest and desire to engage in the class will be lost. The Conversation journal states that ‘Cultivating this sense of relevance and interest in mathematics has been shown to be a key contributor to a child’s future success in this area. Children benefit from developing an authentic mathematical understanding by constructing meaning from their own efforts and reflecting on their work, when they observe other ideas and strategies as suggested by Cockburn. A (2007). When children are given opportunities to discuss mathematics it contributes to their understanding (Haylock & Cockburn 2003) as it allows them to gather other ideas and methods from their peers to build on what they believe and understand. Swan. M (2003) suggested that fluency should be justified by meaning, as it supports children is remembering processes as well as developing an understanding to the necessity of mathematics. The idea that ‘meaning’ is permanent and that it grows and evolves throughout education and life is used to further support Swan. M (2003) with the notion that mathematics needs to have a meaning for children to have the eagerness to learn. During school experience, the meaning of mathematics was never brought to the attention of pupils but simply an ideology that without mathematics one would be unable to go far in their future. This can lead to risks of pressure and anxiety in the pupils causing the subject to become a chore rather than a future tool. Hoffman. B (2010) suggested that the biggest impact of mathematics anxiety relates to performance, as a strong negative correlation exists between mathematics anxiety and mathematics achievement. Swan. M (2003) states that curiosity should be the aim and that no judgment must be placed upon the children to avoid the idea that answering questions create a feeling of fear and embarrassment. Learners should be provided with a range of representations and examples to develop a deeper understanding as suggested by Cockburn. A (2007) a without these, children will have no link to the use of the processes they are learning, and their engagement and sustained attention may diminish.
Children make mistakes, it is a vital part of their learning and understanding, without mistakes better methods cannot be developed nor the desire to achieve higher standards. Mistakes occur for several reasons such as lapses in concentration, memory or when put under pressure. Children that make mistakes whilst calm and well-motivated may have a deeper routed problem as explained by Swan. M (2003). Over the years the idea of making a mistake was considered a ‘bad thing’ and that it should be avoided. Present day teaching suggests that without making mistakes children will not develop their understanding and will simply disconnect from the class due to worry of being ‘wrong’. The NCTEM states that teaching for mastery will give children a deeper understanding and contradicts the idea that some people ‘just can’t do maths’ and through this method of teaching children who do not grasp the concept will be given deeper meanings. Swan. M (2003) states that children should be given time to reflect and discuss their ideas and confront any inconsistencies they may have come across, this allows the cognitive conflict they have encountered to be challenged by themselves then possibly with their peers through Assessment is for Learning. Haylock. D & Thangata. F (2007) describe cognitive conflict as a discrepancy or conflicting idea that is different from what was predicted or that which conflicts with existing understanding and that resolving this conflict can lead to increased knowledge and understanding. Overcoming this conflict can be done by children discussing how they came to that conclusion and reflecting on what other methods they could use to improve their process. Encouraging this method of learning is important for children as it will build, from an early age, the idea that it is considered a ‘good thing’ to make mistakes and learn from them. Haylock. D & Thangata. F (2007) further stated that pupils are more likely to make errors in mathematics than in any other area of the curriculum as the CNE explained that The Debilitating Anxiety Model suggests a link between maths anxiety and maths performance is driven by anxiety’s devastating consequences on learning and recalling maths skills. To back this up, introducing the idea that the concept of cognitive conflict being useful to children will encourage them to have more input and discussion as they do not face the ‘fear’ of being wrong and possibly being ridiculed by their peers. From personal experience, when a wrong answer was given the teacher was quick to either correct this themselves or move onto another child for the correct answer. This impacted negatively on the child creating a feeling of embarrassment and withdrawal from further interaction or input. Cockburn. A (2007) suggests that misconceptions made in primary school are difficult to resolve in secondary, the use of cognitive conflict should therefore be used correctly for the children to resolve their mistakes. For a child to resolve a conflict they should be provided with the opportunity to discuss with their peers and if necessary the teacher. Swan. M (2003) stated that teachers should “create the itch before we offer to scratch” which implies that children should be allowed to attempt their tasks and make mistakes before discussing and debating their methods to their peers or challenge each other’s ideologies and methods. This creates a desire for children to devise the correct conclusion amongst themselves before the teacher gets involved; developing their independent learning. Cockburn. A (2007) suggests that the use of cognitive conflict to promote cognitive development emphasises the importance of existing knowledge as children can use what they already know to help them better understand what steps that have possibly missed.
As discussed throughout the assignment, the three key themes and theories support one and other; if children are not provided the opportunity to make mistakes through cognitive conflict, then they will not be able to explore their own methods or discuss this with their peers therefore they will not gain a deeper understanding or meaning behind the process. Without this understanding it will be difficult to make the necessary connections, which build upon their prior knowledge. Swan. M (2003) states that teachers must explain this concept to the children and why mistakes are acceptable, so they can further their understanding and make the connections required as when working as a teaching assistant, children that were not informed as to why mistakes were acceptable still had the concept that it was unacceptable to make mistakes which was seen to cause slight anxiety in some of the children. These themes are therefore crucial in developing a solid foundation of understanding and a positive attitude towards mathematics as a subject. Cockburn. A (2007) explains that children must construct and refine their process until it is effective and efficient. Without this, children may have inadequate foundations. This statement brings together the points raised that without constructing; making connections, and refining; making mistakes, children will not be able to create a stable foundation; a solid meaning and understanding within mathematics.
From the piece of writing “Making sense of mathematics” by Swan. M (2003) and the support of the other sources, ideas discusses should be implemented within primary schools; it has demonstrated a positive and stable approach for a child to build a solid foundation of knowledge to then work from. The evolution of teaching has come a long way as a few years ago children tended to be thought of as an ’empty vessel’ as stated by Locke. J (1689) that children were viewed as a ‘tabula rasa’ translating into blank slate, for teachers to fill with information, whereas now teachers build from the children’s prior knowledge they will have gained from their home environment. From these new methods children will have a more fulfilling and enjoyable school experience where mathematics is equally interesting and engaging as their other subjects.