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‘Language is an integral part of most learning, and oral language in particular has a key role in classroom teaching and learning. The ethos of the ‘speaking and listening' mathematical classroom is of vital importance. We constantly need to challenge the traditional views of how mathematics is most effectively taught...'
(Speaking, Listening and Learning Mathematics Supplement - Medway Council)
Traditionally mathematics has been a teacher-led subject. According to Solomon and Black (2008, p. 76) mathematics ‘Perhaps more than other subjects…has suffered from the ‘chalk and talk' approach'. They imply that pupils believed there was only one way of obtaining a correct answer in mathematics and this would be the teacher's way. However, this meant that although, children would know how to effect a mathematical operation, they would not have an understanding of the concept and an ability to apply it outside the maths classroom (Pimm, 1987). Recent reform in the field of mathematics education has tried to counter this and promote the role of classroom discussion in supporting children's learning.
The introduction of the National Numeracy Strategy (NNS) in 1999 signalled a significant change in the teaching of mathematics, both in terms of its focus and its pedagogical practice (Pratt, 2006). It meant the arrival of the daily numeracy lesson, the introduction of the three part lesson (oral/mental starter, main, plenary) and, more importantly for the purpose of this essay, one of the key principles of the NNS is ‘direct teaching and interactive oral work with the whole class and groups' (DfES 1999).
This was just the start of a government initiative to make pupils ‘active partners in their learning' (DfES, 2003, Excellence and Enjoyment page 10) and to raise pupil talk as an important aspect of learning with the publication of ‘Speaking, listening, learning: working with children in Key Stages 1 and 2' in 2003.
This seems to be built on a constructivist approach to learning. Being an active learner, as required by Excellence and Enjoyment (DfES 2003), means not just moving around the classroom or manipulating objects (although they might be involved) but being able to interpret and understand new experiences (Barnes, 2008 p. 2); we learn by relating new ideas to our existing view of the world. In order to be able to do this, language plays an important role and Vygotsky's theories show that language helps develop thought At first the language is interpersonal, between the child and others and then it becomes intrapersonal, i.e. it is internalised as thought (Jones & Brader-Araje, 2002). Therefore it is by talking that the child can work out his train of thought and consolidate his learning. This involvement of others is known as social constructivism.
Naturally, the danger can be that by verbalising their mathematical thinking, children may be reinforcing their own misconceptions by the force of their own explanation (Pirie, 1991). However, the opportunity to speak out loud makes the teacher aware of this, and gives the opportunity to address misconceptions on the spot (Clarke, 2009).
How teachers respond to pupil interaction is crucial in determining the confidence pupils have in being able to say anything (Clarke, 2009). The interactive teaching approach where class discussion takes place requires a classroom atmosphere where everyone feels comfortable in exchanging their views and expressing their ideas. Bain (2001) states that ‘Within mathematics this involves children explaining, exploring, hypothesising, predicting, testing and talking about their mathematics amongst themselves and with their teachers. Mistakes are treated as new discoveries and can be made with neither fear of reprimand nor of ridicule.'
In order to be able to express their ideas in the maths lesson and increase their understanding of maths, mathematical vocabulary needs to be introduced during the lesson (Wickham, 2008). However, studies by Pimm (1987) and cited by Lee (2006) emphasise that the language of maths does not correspond to how children would normally speak. That the language of maths is a barrier to children's understanding of mathematical concepts and means that children may feel that they cannot ‘do' mathematics (Lee, p. 12). An example this writer can think of is that while children in KS1 are able to understand about ‘sharing', when this is expressed as ‘division' it needs to be explained. Pupils need to learn how to use mathematical language in order to have control over the concepts of maths. The NNS booklet ‘Mathematical Vocabulary' (2000)
emphasises the need for children to learn mathematical vocabulary in order to be able to take part in classroom activities and to be able to talk about mathematical concepts in order to be able to progress in their understanding (p. 1). Turner and McCullouch (2004) assert that teachers need to ensure that language has a shared meaning and that without shared meaning the children may develop misconceptions by misunderstanding the use of vocabulary or not being able to access the explanations. This is particularly important in the case of children for whom English is an additional language (EAL). The writer paid particular attention to vocabulary about halving and doubling (sharing/diving by 2 and multiplying by 2) (see appendix 3) and ensured that the pupils understood the language by asking children to the front to model the explanations and also by questioning.
Visual clues were used to enhance the learning experience and these are useful for all children, not just EAL children, as they reinforce the language being used.
Pupils and teachers gain much from listening to one another; in fact listening is where Assessment for Learning comes into action (Lee, 2006). If pupils can hear one another then they can listen to what is being said and build up an understanding of mathematical concepts. Pupils hear someone else's ideas and can judge how far the ideas coincide with their own. They can then self-assess and ask themselves questions about whether they agree or disagree with what is being said. If they then ask follow-up questions they are adding to the knowledge being built up in the classroom and are in a position to receive feedback from their peers or the teacher. If the conversation is developing misconceptions, the teacher can intervene. ‘When teachers actively listen to their pupils they can know what their pupils are thinking and understanding and can intervene with appropriate learning activities' (Lee, 2006 p. 23).
During BSE1, the writer was in a Year 2 class of 30 pupils divided into 3 ability groups.
The first lesson the writer will refer to was a lesson on word problems (see appendix 1).
The lesson plan shows several examples of Speaking and Learning (S&L) activities -
mental and oral starter; teacher's questioning; pair work; teacher working with a group; 2 pupils showing the class how they solved their word problem in the plenary. It would appear that the lesson was productive from an S&L perspective.
In the main activity the children were asked to be ‘maths detectives'. It is evident from the plan that the teacher posed questions but these questions appear to be looking for one definite answer. There is also a ‘problem card' ready so that the children themselves do not have to discuss how to go about solving a word problem but have the steps ready for them. When posing questions, the teacher needs to ask questions that can be explored; encouraging the pupils to think and explore them is part of AfL (Lee, 2006). According to the NNS, the children will have come across word problems in Year 1; this perhaps would have been an opportunity to assess prior learning and assess the stage the pupils were currently at - explore their current understanding and check for any gaps or misconceptions.
According to Black et al (2004), many teachers do not plan and conduct classroom dialogue in a way that allows pupils to learn. Their research has shown that many teachers do not allow pupils enough time to think of an answer after posing a question. Often, if they do not receive an immediate reply, teachers ask another question or answer the question themselves. Consequently, the dialogue is short lived. If teachers increase the waiting time, this would help pupils become more involved. Black and William (1998) also found that generally only a few pupils in a class answer the teacher's questions. The rest of the class, feeling that they are unable to answer the questions as rapidly as these few, do not attempt to answer. The teacher is then out of touch with the understanding of most of the children in the class. Several researchers, such as Black and William (1998) mentioned above, Myhill (2006) and Clarke (2009) suggest a ‘no hands up' policy so that all children feel that they may be called upon and therefore reflect upon the question.
EAL children become confident when listening to question and answering sessions as they hear not only their teacher but their peers speak and it helps build up their vocabulary (St. Lewis, 2009
In pair work, the writer was constrained by how the class teacher pairs up the children in the maths class. The writer found that pair work can be problematic in that one partner may be the more dominant and therefore not allow the other to formulate ideas or neither partner attempts to answer the question (see evaluation Appendix 1). This is contrary to what is supposed to happen as the children are meant to be listening and responding to each other. (http://nationalstrategies.standards.dcsf.gov.uk/node/305713?uc%20=%20force_ujare talk partners). Studies have shown that changing talk partners on a regular basis, e.g. weekly, works well (Clarke, 2009). If the pairings are changed regularly, children get the opportunity to talk to children that they might not normally have interaction with (Lancashire County Council, 2006) and new friendships may evolve.
Medway Council in their Maths Supplement (available at www.public.merlin.swgfl.org.uk/.../Speaking%20and%20Listening/speaking__and_listening_Medway.doc) suggest changing partners on a half-term basis so that there can be a build up of confidence and trust.
The benefits of partner talk are that some children may feel more confident in expressing their answer as part of a partnership (Lancashire County Council, 2006). Partner work also encourages pupils for whom English is an Additional Language (EAL) to attempt to express themselves. Having a talking partner helps the EAL child develop language skills and organise their ideas (Primary National Strategy 2003).
Group work is another important aspect of Speaking and Listening (Primary National Strategy, 2003). Hill's studies (2008) found that noisier members of the group became calmer whilst the quieter ones participated more. The collaborative thinking helped his pupils complete their tasks and also enhanced their learning. He also found that children can show their group work skills in a variety of lessons as pupils gain more confidence in working in groups. Bilsbrough, Hawes and Dixon (2008) suggest that teachers consider the size of the group; if the group is too large then it might not give the opportunity for all children to speak. They also recommend that a group has a mix of different personalities and that children need to be in a group where they feel confident to talk.
These are important aspects to consider when delivering a numeracy lesson. However, in the writer's experience on BSE1, grouping in numeracy is generally by ability and perhaps one does not obtain the level of interaction and construction of knowledge that the Primary National Strategy and other studies suggest. Also, a teacher is sometimes constrained by a school's ethos. The writer's placement school gave great importance to evidence of work in workbooks. Every week, the Headteacher required a sample of workbooks to be sent to his office in order to review work done and pupils' progress. Much of the work done in the numeracy lesson was individual work.
The writer did have some opportunities to work with groups on a more interactive basis. The opportunity to work with the higher ability group on halving and doubling (Appendix 2), offered the opportunity to check the children's understanding of these concepts. The writer evaluated that the pupils understood halves and doubles but were unable to comprehend the meaning of three quarters. The next step would be to use this as AfL and plan how to extend the children's knowledge.
One issue with group work - or even partner talk - is that sometimes the class can become a little noisier than usual. The writer had a pupil in her class who was diagnosed with Attention Deficit Hyperactivity Disorder (ADHD) and the increase in noise levels in the class upset the pupil (appendix 3). Turner and McCullouch (2004) say that the increase in noise levels is perhaps unavoidable but this is something the teacher should be aware of and try to contain, especially in the presence of more sensitive pupils.
In schools language is the main tool for teaching and learning across the curriculum and maths is certainly no exception. It is through language that children can manipulate and express their ideas. We no longer have a culture where children are taught to learn to learn mathematical concepts just because ‘that is the way' as children need to be able to connect the mathematics with other curriculum subjects and the wider world (Turner & McCullouch, 2004). Talking in mathematics helps the children reflect on their thinking and helps them explore and form new understanding.
However, there appears still the need for work to be done in the area of Speaking and Listening. Myhill (2006) points out that teachers tend to dominate whole class teaching episodes because they feel under pressure to cover curriculum objectives and achieve pre-specified goals. This view is also held by Pratt (2006) who points out the apparent inconsistency between the NNS asking the teachers to make use of children's contributions whilst simultaneously ensuring that lessons have certain outcomes.
Clarke (2008) appears to be optimistic that supporting teachers' development in Speaking and Listening will reinforce its importance as a valuable tool to be used in AfL and raise children's attainment.
There is also the anticipation of The New Primary Curriculum (2009) due to be implemented in schools in 2011, which promotes communication, working together, and talking as means of giving pupils the skills so that they can become successful learners. The section on Mathematical Understanding specifies the requirement for individual and group work and one of the aims is also that children be able to ‘articulate their thinking in discussion'. Perhaps teachers will be given more opportunity to implement Speaking and Listening in their classroom and will move away from the ‘chalk and talk' approach that Solomon and Black (2008, p 76) mention towards an environment of
Talk for Maths http://www.teachers.tv/video/28610
Does the National Numeracy Strategy address gender issues within the learning of mathematics? Bain, R
PHILOSOPHY OF MATHEMATICS EDUCATION JOURNAL 14 (2001) PoME journal 14, May 2001
DfES : Excellence and Enjoyment: learning and teaching in the Primary Years
Solomon, Y. & Black, L. (2008) 'Talking to learn and learning to talk in the mathematics classroom' in N. Mercer and S. Hodgkinson (eds) Exploring Talk in School: Inspired by the Work of Douglas Barnes London: Sage