The Essence of Mathematics - Essay Example

?‘The essence of mathematics is not to make simple things complicated, but to make complicated things simple (S. Gudder, .’  Mathematical thinking is an abstruse term which can be defined in a variety of ways, and broken down into several sections. It is believed that mathematical thinking is the process of logical thought, as highlighted by DFES (1999) is a process of ‘Logical reasoning, problem solving and the ability to think in abstract ways (DFES, 1999, p.60-61).’ Development of this method of thought can be aided through the art of speaking and listening, alongside group and collaborative work. Children using and applying their skills equates into mathematical thinking, as agreed by the National Curriculum (1999) which states that ‘mathematics trains children with a ‘uniquely powerful set of tools’ to understand and develop the world (Directgov, 2010).’ The National Numeracy Strategy, introduced by the Department of Education and employment, lays down a framework for the curriculum to be followed in all schools while teaching mathematics for children attending Reception to Year 6 (National Numeracy Strategy, 1999). Compiled in 1999, The National Numeracy Strategy set out four main approaches to the teaching of mathematics, viz.: Dedicated mathematics lessons every day Direct teaching and interactive oral work with the whole class and groups An emphasis on mental calculation Controlled differentiation, with all pupils engaged in mathematics relating to a common theme (National Numeracy Strategy, 1999). The National Numeracy Strategy has outlined guidelines for numeracy to be taught nationwide which has to be utilised by teachers within the classroom (National Numeracy Strategy, 1999). It suggests strategies for the effective use of time, incorporating planning time for teachers and states the timeframes of taught mathematical lessons. It is imperative that children enjoy mathematics through structured activities, enabling them to build on knowledge previously taught. It is suggested by Haeckel (1906) that ‘To really learn and master mathematics, one’s mind must go through the same stages that mathematics has gone through during its evolution (Haeckel, 1906).’ An important theoretical framework underlying mathematical thinking is the focus on promoting mental calculations or cognitive calculations. In order to calculate either cognitively or use written algorithms, it is necessary to have basic numeracy skills. These include: remembering and recalling number facts, relationships of numbers, and problem solving using mental visualization and/or previously learned strategies. Mental mathematics is the foundation to all mathematical methodologies. The ability to count or at least know place value is all cognitive during which memory is used to recall numerical facts or obtain new ones, therefore this skill should be nurtured and emphasised. There are several educationalists who are proponents of encouraging children to engage in more mental calculations rather than just solving problems on paper. In this regard, Askew (1998) states that, ‘Ultimately, mathematics is a mental activity. While practical mathematics can help children develop mental images, written work on its own is not sufficient (Askew, 1998)’. This statement agrees with the principle laid down in the National Numeracy Strategy, which states that “an ability to calculate mentally lies at the heart of numeracy (National Numeracy Strategy, 1999)’ It is of truth, and it is of little use if a child can complete a page of sums, but does not know how to tackle a problem that has not been written down. The National Numeracy Strategy emphasizes that pupils need to be given opportunities to develop flexible methods of working with numbers mentally that enables them to use known information to derive facts that they cannot recall (National Numeracy Strategy, 1999).  Differentiation within education largely relates to the differences between comparable things, for example, building a picture for the children that a cuboid is different to a cube or that a twenty pence piece is different to a fifty pence piece, although their shapes are similar. Mental mathematics comes into play here also, building cognitive pictures suitable for kinaesthetic and dynamic learners who respond well with visualisation. Controlled differentiation is also an opportunity to verbally explain in a direct teaching environment, where clarifying questions are asked and the children can greatly learn from this.  These four recommendations are interlinked, they work well together. Their inclusion in this strategy is justified, because the four are necessary for effective teaching and learning. They cover all types of learners, stimulate cognitive, social, physical and emotional education and most importantly, give structure to a daily lesson and make it interesting. One of the strategies to facilitate mathematical learning within the classroom is to use the aid of support staff such as Classroom Assistants. The Classroom Assistant should aim to support all the children in the class, some of whom might need targeted help and others might need access to the assistant throughout the day. Support staff should try and remain one step behind, allowing the child to take calculated risks and therefore allow cognitive challenge. As Fox (1993) comments, it is a difficult task to maintain the balance between giving support and promoting independence (Fox, 1993).  This involves being clear about ones expectations and firm in ones directions without pressurising the child, however, sensitivity should direct the teacher if and when to intervene (Lorenz, 1999, p.19).  As Fletcher-Campbell (1992) notes, it is a waste of valuable resources if the Classroom Assistant remains with a pupil needlessly, and it also serves to highlight the difference between the pupil and his or her peers which might cause the pupil requiring extra help to feel inferior to his contemporaries (Campbell, 1992). Gravelle (1996) suggests that focused support for children with English as a second language should also be provided in order to give children the opportunity to use their heritage language within the classroom (Gravelle, 1996). Another similar strategy is to employ directed teaching. Directed teaching has various sub categories, such as directing, instructing, demonstrating, discussing, consolidating, and summarising. These are all key features to ensure effective mentoring. Without direct teaching, lessons would be lecture-like and uninteresting. Directed teaching allows the teacher to interact orally with the class, small groups or individuals. This gives the opportunity for pupils to share their methods with their peers. It is suggested by A. Bandura (1977) that “Learning would be exceedingly laborious, not to mention hazardous, if people had to rely solely on the effects of their own actions to inform them what to do”. This statement supports my personal opinion that children do learn new ways of calculating mentally and physically, and can get a better grasp on numeracy as a whole when exposed to a social environment, and in this example, directed teaching. In the Foundation stage, directed teaching provides the ground for mental mathematics. Counting with the class or looking at shapes together helps cognitive processes early on by visualizing and imitating each other. ICT has been shown to develop collaborative learning and therefore develop mathematical language and learning.  The Cockcroft report, 1982 (Appendix D) stresses the importance of discussion and of using the correct mathematical language.  Children often get confused due to its ambiguous nature and so it is essential that children are taught the correct terminology at an early age.  Computers encourage pair and group work and much software is aimed at collaborative work.  Children in most schools have to work together due to the limited amount of computers, but this can be used as an advantage.  Wegerif et al (1998) agrees stating that direct software can support discussion and reasoning.  He found that the intervention could move classes from 50th out of 100 to the top 30. He believed that software needs to challenge, have a clear purpose, on screen prompts, no features that encourage turn taking and multiple choice questions (Higgins, 2004, p.170). There is a lot of evidence that shows the effectiveness of discussion and peer support.  Vgotsky states that with the help of their peers children enter into a ‘zone of proximal development’ which allows them to achieve their full potential. ICT can be used very effectively in mathematics during experimental learning, where the role is reversed and children ‘teach the computer’. Much research suggests that when children explore and experiment for themselves they truly learn. Teachers we need to be aware of children’s capabilities and allow them to explore. Williams and Easingwood (2004), state that ‘a good teacher is often a teacher who is prepared to take risks’ (p36).  Ownership can be liberating for children and gives teachers new insights into the way children learn. Additionally teachers can learn a great amount from the children.  They often know a lot of the software better than the adults due to being brought up on computers.  This can be a good opportunity for children to teach the class, or tutor their peers, and this has been shown to be a very effective way of supporting children’s maths skills, as it allows them to explain their ideas of skills and concepts, which consolidates their learning. The teaching of mathematics is the teaching of concepts, which is why many children struggle.  These concepts must be taught in a way children understand in order to avoid misconceptions.  Dean (1992) identifies ‘therefore it is important to set up appropriate situations which lead to the development of specific concepts.’  According to Bruner teachers must be inventive in order to translate information to levels that are appropriate to those who must understand it.  He says knowledge should be ‘coded’ so it is useable by children.  When it is the concepts that children understand and not just rules to apply to calculations misconceptions are less likely to occur. Thus, in conclusion, importance of numeracy in the modern world can be summed up in the words of the famous cognitive scientist, Iddo Gal, who states that: ‘Numeracy is not the same as mathematics. It is an aggregation of skills, knowledge, beliefs, dispositions, habits of mind, communication capabilities, and problem solving skills that people need in order to engage effectively and autonomously in quantitative situations arising in life and work (Iddo, 2001).’   Bibliography Askew, M. (1998). Teaching Primary Mathematics: Hodder and Stoughton Bandura, A. (1977). Social Learning Theory. New York: General Learning Press. Cockcroft, (1982) http://www.educationengland.org.uk/documents/cockcroft/ Dean, J. (1992) Organising Learning in the Primary School Classroom. London Routledge DFES (1999). The National Curriculum. London: DFES Directgov. (2010). Understanding the National Curriculum. Retrieved January 10, 2011, from Direct.gov: http://www.direct.gov.uk/en/Parents/Schoolslearninganddevelopment/ExamsTestsAndTheCurriculum/DG_4016665 G. Fox (2003) Handbook for Learning Support Assistants London, David Fulton        Gifford, S. (2003) Mathematics Teaching, issue 184, p 33-38 Iddo, G. (2001) The Essentials of Numeracy, London Routledge Haeckel, E. (1906) The evolution of man—a popular scientific study. London: Watts Harries, (2000). Mental Mathematics for the Numeracy Hour. London, David Fulton Haylock, D. (1995) Mathematics Explained for Primary Teachers, Paul Chapman Lorenz, S. (2002) Effective In-Class Support. London, David Fulton National Numeracy Strategy. (1999, March). Framework for teaching mathematics from reception to Year 6. Retrieved January 11, 2011, from http://www.lancsngfl.ac.uk/curriculum/math/getfile.php?src=1139/intro.pdf&s=!B121cf29d70ec8a3d54a33343010cc2 S. Gudder. (2010). S. Gudder quotes. Retrieved January 2, 2011, from Thinkexist.com: http://thinkexist.com/quotation/the_essence_of_mathematics_is_not_to_make_simple/219215.html Williams, J & Easingwood (2004) ICT and Primary Mathematics.  A Teachers Guide.  Oxon.  RoutledgeFalmer   Read More