A Wealth Of Mathematical Skills And Knowledge Education Essay


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Mathematics is described as a universal language Vorderman, et al., 2011, and yet my observations suggest that it can be the language of multiplication that obstructs progression towards the application of the written algorithm to broader concepts. An example of this was evident during a Year 6 maths lesson where a small group of pupils were struggling to carry out multiplication of larger numbers using known facts. The pupils, for whom English was their second language, were unable to apply their mastery of the 'times tables'. A discussion with the pupils yielded the information that they had not processed the facts they had learned as being multiplication calculations (Appendix A) as they had not connected the different terms being used. Quinnell & Carter (2011) describe this as an example of a phrase being 'mistakenly attached to a specific mathematics operation, preventing flexibility of application'.

Vygotsky (1962) places much emphasism on the adult's role of language provider and this theory is supported by my subsequent discussion with teachers who voiced a possible source of the pupils' misconception. All staff confirmed that early multiplication facts are routinely referred to as 'times tables' with an assumption that pupils would connect the term with multiplication. This, combined with the evidence from the numeracy lesson, inspired my increased interest and enquiry into the possibility that pupils held other language based misconceptions.

One significant difficulty was demonstrated during a lesson where Year 2 pupils were working towards achieving the National Curriculum objective of understanding 'multiplication as repeated addition' (DfEE, 1999). Despite a long history of repeated addition being a method used to introduce younger pupils to multiplication (McArdle, Clemson, W. & Clemson, D. 2002), the lesson evaluation (Appendix B:i) demonstrates that the lower ability pupils (BAR) had significant difficulties with linking the two operations as a result of them considering each operation as being unconnected (Appendix C). This supports the theory of Wallace and Gurganus' (2005), who suggest that each operation should be taught separately. Conversely, all other pupils were able to understand the connection (Appendix B:ii), and later progressed towards relating the taught strategies to division as repeated subtraction, therefore supporting the guidance offered by the DfES that states that pupils should be taught to connect ideas and concepts, particularly in relation to multiplication and division. Sousa declares that 'pattern inferencing' makes connecting concepts a natural progression (2008), and so the BAR pupils' difficulties could emanate from gaps in their conceptual understanding of multiplication, resulting in confusion regarding the terminology and further supporting the previously mentioned claims of Quinnell & Carter (2011).

Walls (2009) suggests that mathematical language difficulties could be a result of the specialised nature of mathematical language. Skemp clarifies this further by describing differences between words used in everyday life and as mathematical terms, as a 'faux amis' (Skemp, 1977:1), as in the example of the use of the word 'table'. With pupils gaining mathematical understanding through connecting ideas with real life (Haylock & Cockburn, 2003), it is therefore possible that language related difficulties can be overcome through explicit use and explanation of mathematical vocabulary, and through developing pupils understanding and application of multiplication algorithms (Harris, 2001). For example, by expanding their understanding of the relationship between multiplication and division, the 'faux ami' regarding the use of terms such as 'lots of' could be prevented.

My research supports the belief that all pupils need to be exposed to a wide range of mathematical vocabulary (Clarke, 2010). Moreover, Kennedy, Tipps & Johnson state that combining the accurate terminology with pupils own language provides more opportunities to connect ideas with new mathematical concepts (2007). In view of this, and the knowledge that it is the responsibility of teachers to identify and address children's difficulties (Forlin, 2012), I informed my colleagues of my findings in order to allow them to evaluate and adapt their own use of the language associated with multiplication (Appendix D).

In addition to inconsistencies with mathematical language, the strategies used to teach multiplication can be conflicting, with some teachers and pupils preferring a transmission, or procedural, style of teaching, and others a conceptual approach. Difficulties arise when only one method is offered, particularly as an overall understanding of multiplication involves both, as in the example of the relationship between repeated addition and multiplication as being procedural, and the relationship between multiplication and division as conceptual (Nunes, Bryant & Watson, 2009).

Conceptual teaching is supportive of the constructivist method as it allows pupils to form associations between different aspects of arithmetic and real life (Nunes, 2001). It teaches pupils to form relationships between concepts, described by Skemp as relational understanding (2002:16). A previously mentioned instance of where relational understanding would have supported further progression is the example of the Year 6 pupils who were struggling to multiply larger numbers. Had these pupils learned the distributive properties of multiplication, and understood the relationships between their known facts, they might have been able to adapt their existing schema in order to progress (Barmby, et al., 2009). Instead, the pupils had only instrumental understanding of the multiplication tables and were therefore unable to transfer their knowledge.

Further discussions with the pupils yielded the information that they had learned the multiplication facts through rote learning, and had been taught them by both their parents and by tutors outside of the school environment. Three of the pupils had attended 'Kumon' training. Kumon training is a traditional method of learning where pupils memorise mathematical facts and procedures through a process of repetition (Mukisa, 2011). As a result of this training, the three pupils were all able to quickly recall multiplication table facts were all far more competent than any of their peers at this type of instant recall. Furthermore, the pupils were all eager to share their knowledge and it became apparent that their ability to answer questions at speed was having a positive impact on their confidence levels. However, observations during daily Maths lessons showed that the pupils were neither able to derive or recall associated division facts, nor use their existing knowledge in wider contexts (Appendix E). This example provided me with a basis for further exploration of the advantages and disadvantages of rote learning of multiplication facts and concepts.

Rote learning of multiplication facts, or rote memorising as it is described by Skemp (2002:35), can enable pupils to reach an answer more quickly than through relational thinking (Skemp, 1977:8). However, this method of learning also attributes only one notational representation of each calculation, and therefore neither facilitates development of understanding of the concepts of multiplication, nor the subsequent ability to apply it (Harries & Barmby, 2007). Furthermore, although instant recall of the multiplication facts proved to be beneficial to the confidence levels of the pupils previously mentioned, failure to learn them can also ignite some pupil's perceptions of themselves as being ineffective mathematicians (Caron, 2007).

One such example of this is an eleven year old girl who, despite intensive support from her parents, still was unable to commit the multiplication facts to memory. She therefore struggled to transfer the facts to broader concepts, with the result being reduced levels of self-confidence in all subjects. The girl, Child A, was a self-professed visual-spatial learner (VSL) and it soon became apparent that she had previously been taught as an auditory-sequential learner (ASL). Auditory-sequential learners learn through memorisation of words, therefore suggesting that they are able to memorise facts through rote learning, however visual-spatial learners need to visualise images and, although retrieval might be slower, can be considered as conceptual learners (Golon, 2008). By implementing a series of lessons using visual strategies, Child A was able to visualise enough multiplication facts to enable the transfer to broader concepts, with the results being raised levels of confidence and attainment across the entire subject of mathematics.

Child A's difficulties reinforce the notion that teachers need to ensure that their teaching of multiplication incorporates a wide range of styles and strategies. Teaching purely instrumental understanding can be detrimental to relational or creative learners (Coelho, 1998), and this theory has become a focus for criticism of the 2012 National Curriculum Review (Gove, 2011a). The review is based on claims that numeracy levels in Britain fall below those of Asian countries (Gove, 2011b). However, researchers do not consider the new draft curriculum as being the answer, suggesting that it is designed with an inappropriate 'implied pedagogy' that is firmly based on procedural teaching (ACME, 2012).

In relation to multiplication, the National Curriculum Review implies, despite no explicit mention, that using rote learning strategies will ensure that all pupils master acquisition and instant recall of the facts by the age of 9 (DfE, 2012:11). With only a brief mention of conceptual understanding (DfE, 2012:1), the review appears undoubtedly in favour of instrumental teaching, despite past research establishing relational understanding as being more beneficial to longer term progress (Skemp, 2002:9). In order to examine this theory in context, I compared the progress data of six Year 6 pupils, three of whom had received instrumental tuition outside of school, and three who appeared to be relational learners (Appendix F).

The data shows that the instrumental learners excelled during their first few years of schooling, and all three achieved higher attainment levels than the relational learners. However, while the relational learners continued to make sustained progress, the progress of the instrumental learners slowed as the curriculum became more demanding, suggesting that their rote memorisation of facts was no longer sufficient. Observations of the pupils during subsequent lessons showed that their ability to commit multiplication facts to memory did not allow them to use them as foundations on which to build further understanding.

The differences in learning styles and attainment levels of the pupils studied, symbolises the inconsistencies in theory and research surrounding the teaching of multiplication. The pupils attainment at different points suggests that a combined approach of instrumental and relational teaching of multiplication should enable all learners to learn multiplication facts, while understanding them and being able to make links with broader concepts.

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