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Becoming numerate requires that children acquire a wealth of mathematical skills and knowledge, and are able to combine the two in ways that allow them to apply concepts in everyday life (Askew, et al. 1997).One of the fundamental skills of the learning journey is for children to become proficient at calculating using each of the four operations, both individually and collectively (Ofsted, 2011). For some, this mastery of arithmetic can be hampered by many issues. In order to compare theory and my own personal experiences, I have elected to focus on multiplication and within this, issues regarding the use of mathematical language and contrasting teaching strategies.
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'. Identifying language based problems, and the reasons behind them, is important in order to ensure that pupils' difficulties can be addressed and that progression is not delayed or even prevented (Molina, 2012).
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 mentioned misconception. All staff confirmed that early multiplication facts are routinely referred to as 'times tables' with an assumption that pupils would associate the term with multiplication, thus suggesting that teachers might have previously compounded the pupils inability to make the connection and reinforcing research stating that teachers must model the use of appropriate vocabulary (International Academy of Education, 2009). 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. However, 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 (2006) that states that pupils should be taught to connect ideas and concepts, particularly in relation to multiplication and division. Sousa (2008) declares that 'pattern inferencing' makes connecting concepts a natural progression, 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 (1977:1) clarifies this further by describing differences between words used in everyday life and as mathematical terms, as a 'faux amis', 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 'groups 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 (2007) state that combining the accurate terminology with pupils own language provides more opportunities to connect ideas with new mathematical concepts. 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 had learned the multiplication tables as 'rules without reason' (Mueller, Yankelewitz & Maher, 2010), 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 that 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). While observing a Kumon training session I was alarmed at the absence of social interaction during the session (Appendix E). Although this type of tuition is progressive and personalised to each pupil's current level of attainment, the omission of collaborative learning between the pupils and tutors is in direct opposition to theories that place great emphasis on the benefits of social interaction in mathematical language acquisition (Bruner, 1983: Gross, 1992).
Nevertheless, as a result of this training, the three pupils were all able to quickly recall multiplication table facts and were 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 associated multiplication facts, nor use them as foundations on which to build further understanding (Appendices F & G). 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 her to reason using the facts, 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. Despite benefits to pupils who prefer a transmission method of teaching (Alexander, 2009), teaching purely instrumental understanding can be detrimental to relational or creative learners (Coelho, 1998). Critics suggest that this theory has been overlooked in research for the 2012 National Curriculum review (Pollard, cited in Paton, 2012: National Union of Teachers, 2012) and that the review (Gove, 2011) is designed with an inappropriate 'implied pedagogy' that is firmly based on procedural teaching (ACME, 2012).
In relation to multiplication, despite no explicit mention of rote learning, the National Curriculum Review implies that memorising multiplication and other facts will ensure that all pupils master acquisition and instant recall by the age of 9 (DfE, 2013). With only a brief mention of conceptual understanding (DfE, 2013:53), the review appears undoubtedly in favour of instrumental teaching, despite past research establishing relational understanding as being more beneficial to longer term progress (Skemp, 1987). 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 H).
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, supporting Skemp's theory that some aspects of Mathematics might be better learned instructionally (Skemp, 2002:9). 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 alone was no longer sufficient. Klingberg (2012) suggests that this could be the result of the multiplication facts being learned and stored in the long term memory as 'language-related' facts, thus explaining the pupils' inability to use the facts to develop mathematical reasoning.
The differences in learning styles and attainment levels of the pupils studied, symbolises the inconsistencies in theory and research surrounding the teaching of multiplication. With our current Government's thinking leaning towards an instrumental approach, and Ofsted (2012) advocating a more holistic approach to teaching, it is imperative that teachers are aware of the advantages and disadvantages of both, as illustrated in Appendix I. Moreover, Bloom's Taxonomy defines that knowledge is just the first element in mastering a concept (Appendix J), and so if 'Concepts are the substance of mathematical knowledge' (NCTM, 1989), then facts without understanding surely must mean mathematics with no substance. Being numerate needs to involve mastery of procedures and concepts that, when understood and carried out efficiently, become just one element of a working memory that can 'use and engage in mathematics in ways that meet the needs of that individual's life as a constructive, concerned, and reflective citizen' (OECD, 2003).