# Case Study Maths And Society Education Essay

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## Abstract

Mathematics consists of many words such as 'whole', 'differentiate', 'limit' and many more. It has been observed that mathematical terminology has a contextual meaning for students in everyday life. This causes issues with the interpretation of Mathematical terms in the context of the subject and consequently hinders the understanding of definitions and concepts.

This assignment analyses the issues with the language used in the teaching and learning of Mathematics and suggests approaches to alleviate these issues. It also explores how the issue of language competency can favour certain students compared to others based on their social background.

## Introduction

Language used in Mathematics causes implications in the teaching and learning of the subject. From reflecting on my experience, I have personally found the vocabulary used in both Mathematics and everyday life hard to comprehend in a Mathematical context and also observed issues that other peers were having with understanding the terminology. Additionally, I have observed in school that language is an issue but didn't realise the extent that it could hinder the learning of Mathematics, even for those that are able to access written and verbal instructions.

Whilst teaching, I have further observed how language used in Mathematics causes issues for even those that can speak English, as there are many words used in relation to the subject which are also everyday words, that causes confusion in understanding in a Mathematics context. This assignment explores the issues of language in the teaching and learning of Mathematics and how these can favour some social groups over others. It also suggests how these issues can be attempted to be resolved. In my opinion this issue is a major influence in the understanding of Mathematics which determines overall succession in the subject; hence I want to explore this area in more detail.

## Literature Review

This review explores and discusses the issues raised by the use of language in the teaching and learning of Mathematics, and focuses especially upon the problems encountered by learners, and the steps which practitioners may take to alleviate them. As Durkin points out, much of children's Mathematical education 'takes place in language' (Durkin, 1991, pg.4), and even mental or intuitive negotiation of mathematical problems by the individual is inevitably embedded in mathematical semiotics. It is argued here that the difficulties raised by language in Mathematics are multi-dimensional and can prevent learners from understanding what is said to them, or what is given to them in the form of written instructions by the teacher. These difficulties can hinder learners' efforts in working independently, by preventing them from accessing written instructional or text books. Since learners are mostly assessed through output orientated forms of assessment, those with language difficulties are at a disadvantage, especially if they cannot comprehend the questions. These difficulties can impede their performance and undermine their confidence in test situations. Consequently, this can have huge implications, both for the individual by harming their self-esteem and the institution, as it means that the school concerned will have poorer overall results, damaging their league-table position. Additionally, terminology used in the curriculum is constantly being altered, so practitioners have to adapt their practice and monitor learners' needs to ensure that students understand the new terms and methods.

## Literacy and Numeracy Standards

On various levels, underperformance in literacy can even have an enervating effect on quite able mathematicians at key points in their educational career. As Clarkson indicates, the inability to read texts at the speed required in test scenarios provides a key example of this (Clarkson, 1991, pg.240). Students that find it hard to interpret the question or take time to work out what is required, may know how to compute the answer to the problem but are restricted from answering all questions and finishing the paper due to time restraint. Alternatively, they may know a mathematical concept but cannot answer the question because it is phrased differently. For example, a student may be able to answer 'multiply 4 and 6' but not 'what is the product of 4 and 6' as they may not know that 'multiply' and 'product' mean the same thing. Clearly, the added pressure of 'exam stress' does not help, even though learners are usually given sufficient practice before the actual event under timed conditions. The important point here is that no amount of preparation on similar problems can remove the barriers inherent in a specific or unfamiliar problem. It is self-evident that written or spoken mathematical problems will usually present the most complex challenges for those whose literacy and numeracy skills are poorly aligned, or have developed unevenly. However, the difficulties experienced by such learners are not confined just to these areas.

In primary and secondary education, many problems which are written almost entirely in numerical form require some form of presentation in non-mathematical language, in order for the answer to be correctly construed. Even where no text is present within the question, the learner may still visualise either the problem or answer in prose form. It has to be conceded however, that it is in questions that are entirely written or verbalised that the learner may be unable to access the problem, therefore will be incapable of applying the required operations. However, in order to help learners meet these challenges, practitioners themselves must understand the learning processes which each individual undergoes. It is probable that the most important element within this is the careful monitoring and assessment of the learner's progress on a frequent, perhaps a daily or weekly basis. Practitioners should be attentive of those students who are not contributing to question and answer sessions, or are generally reluctant to offer answers to problems put on the board. These instances need to be addressed promptly, before the learner falls into a regular pattern of behaviour which is hard to eliminate.

As De Corte and Verschaffel have argued, there are five stages to be successively implemented when solving written problems. Firstly, a complex 'text processing' activity occurs, involving the analysis of the problem. Secondly, the subject considers the appropriate operations in order to find the 'unknown element' in the representation, which is performed in the third stage. The formulated answer is then located in the original representation, whilst in the fifth and last stage, the reflective learner 'verifies' their solution by reviewing its feasibility (De Corte, E., and Verschaffel, 1991, pg.118). The overall success of this process is dependent upon two interdependent factors, namely that,

• 'Word problems that are solvable using the same arithmetic operation, can be described in terms of different networks of concepts and relationships…'

• Constructing an appropriate internal representation of such a conceptual network is a crucial aspect of expertise in word problem solving. (De Corte and Verschaffel, 1991, pg.119)

The individuals' implementation of these stages also depends on whether the question was constructed around a 'change', 'comparison', or 'combination' problem. Change problems involve changing the value of a quantity due to an event or situation, combination problems relate to quantities that are considered either separately or together and comparison problems are the comparisons or differences between amounts (De Corte and Verschaffel, 1991, pg.119). The important point here is that the learner negotiates the problem intellectually, and the more complex it is, or the more stages it involves, the more difficult it is for students to do so successfully. In other words, no matter what written or calculator operations are required, the learner will first attempt to put the various elements of the problem together into some kind of logical sequence in order to visualise the eventual output, i.e. the answer. As an example of this, calculator based questioning allows the use of digital calculators in problem solving and in examination contexts relieves the learner of undertaking the required operations. However, initially they must obviously determine what those operations should be. There are plenty of instances where the learner's consideration of the problem has proved inaccurate and has been misunderstood, leading to incorrect answers, even obtained on a calculator as the wrong operations were carried out. The overall point is that learners think about problems by visualising terms like 'add', 'divide' etc, in order to help them decide on the correct explanation. In semiotic terms, the instruction is the sign, which in-turn symbolises the 'signifier' or meaning. If the learner's linguistic capabilities are not sufficiently developed, even the absence of text cannot really help them and they will find it difficult to even interpret symbols.

## Spoken and Heard Mathematics

Similar kinds of problems can attend the understanding of spoken Mathematics questions or instructions, and, as Orton and Frobisher indicate, some classroom practices may exacerbate this. They specifically suggest that learners who have difficulty in interpreting verbalised concepts are frequently offered more practice at written versions of them, effectively steering them off into an epistemological tangent, which causes them to take the wrong direction in terms of the methods required. This is unbeneficial to learners as more written examples cannot necessarily help to solve the problems inherent in aural or verbal Mathematics comprehension. There are different kinds of problems involved, which need to be addressed in specific ways. As Orton and Frobisher explain, the act of articulating our thoughts not only offers a greater chance of communicating our understanding to others, but 'allows us to better understand what we are saying.' (Orton and Frobisher, 2002, pg.59). The corollary to this is that learner's require ample opportunity to speak about Mathematics in a structured environment, something which an emphasis on pencil and paper methods, and output orientated assessment can deny them and can affect the learning of the subject.

There are many benefits for speaking about Mathematics in the classroom, specifically so that students can communicate their thoughts and ideas which would give practitioners an insight into the thought processes of students, consequently helping them to understand their students. According to the research of Zack and Graves, positive outcomes have been demonstrated where the practice is encouraged (Zack, V. and Graves, B., 2001, pg.229). In other words, the more learners are allowed to speak about Mathematics, the more opportunity they have to correct their own errors and reflect on their thinking. The other dimension which needs to be considered here is that of the social context. Learners have to develop the confidence to engage in classroom dialogues with their peers and the teacher. Arguably, those students who experience the greatest difficulties in spoken and heard Mathematics will be the most reserved about doing this. Consequently, it will be evident for practitioners themselves to rapidly become aware of those learners who are least likely to volunteer answers and become involved in problem solving activities and discussions. It is then their responsibility to support the individual in visualising participation as a target, and devise the appropriate strategy. However, this problem is obviously exacerbated when the underlying issues are embedded in literacy rather numeracy comprehension. As primary practitioners will be particularly aware, the literacy and numeracy curriculum run parallel to each other, rather than converging in a structural way; they have their own developmental stages, and these do not take account of cross-curricular needs. In other words, a learner who is having difficulties with mathematical text will not necessarily find any directly relevant support in their literacy work. This implies that the practitioner must keep up-to-date in the context of numeracy teaching, whilst ensuring that the learner is also on track with their staged mathematical development.

## Staged Development in Literacy and Numeracy

Meanings and values are not only acquired through the curriculum or in the classroom, and each individual will have a pre-formed collection of perceptions, however, not all may be accurate. The amount of exposure and comprehension of Mathematical language varies extremely between learners, depending upon their cultural, social and family background, which causes differences in learning behaviour. Despite these variations, as Clarkson indicates, learners need to be secure in the alternative usages which often surround identical operations (Clarkson, 1991, pg.241). This problem may have cultural origins for some groups of learners, or as Orton and Frobisher point out, may stem from the fact that much Mathematical terminology has alternate meanings in everyday language, examples include; 'chord', 'relation' and 'segment' (Orton and Frobisher, 2002, pg.55). It is important that the teacher understands whether the learner has problems with literacy or numeracy, or both. However, it can be difficult for the practitioner to tell whether mathematical or literacy problems are preventing learners from progressing. As Clarkson points out, 'reading and comprehension are two distinct abilities which must be mastered.' (Clarkson, 1991, pg.241). There is certainly no simple correlation between ability in literacy or standard written/spoken English and achievement in Mathematics.

## Language Competency

Language competency is an issue for students who speak English as a foreign language, causing them to underachieve in Mathematics. In order to read text books and understand verbal instructions, students must work within the language of instruction. Educational progress is enhanced depending on whether a student's first language is that of their instruction or not and this clearly affects those from lower social backgrounds.

Mathematics has many words particular to the subject, for example, 'integral, differentiate, matrix, volume and mass'. This can be confusing for non-native English students, as they have to learn new meanings in the context of Mathematics (Zevenbergen, 2001, pg.15-16). The same word can be interpreted in different ways by non-native students, causing misunderstandings which affects learning. For example, the word 'times' is generally related to the time on a clock, not to multiplication and the words 'hole' and 'whole' sound the same but have different meanings, meaning a whole number in Mathematics (Gates, 2002, pg. 44).

Practitioners may find this lack of language background can make a Mathematics class difficult to teach. Conversely, accomplished young mathematicians with poor English skills can access the universal languages of number and operations with comparative ease so the question to be asked is; what kind of Mathematics problems are at issue? According to Pimm, logograms, pictograms, punctuation symbols and alphabetic symbols can facilitate extensive, but not total mathematical communication (Pimm, 1987, pg.180). As Orton and Frobisher indicate, it is up to the practitioner to determine the extent to which mathematical problems need to be graduated for individual learners and it cannot be assumed that their experiences and needs will be identical (Orton and Frobisher, 2002, pg.54). For example, understanding that the difference between two numbers is something produced when one is subtracted from another may be difficult to understand for learners who have not encountered that style of problem before.

## Setting by ability

In Mathematics, setting is used to group students according to their ability and students take exams depending on what set they are in, which determines the maximum grade they can achieve. This seems unfair for lower setted students, whose full potential may not have been realised and who surely deserve the chance to achieve a higher grade.

Students with language issues may work more slowly or misunderstand questions and hence, be setted in a lower-level group, which is clearly unfair. Therefore, those children with the language competency and additional external help are in favour of learning Mathematics more successfully. However, even these students struggle with certain terminology.

According to Watson, it is a matter of 'social justice' to teach Mathematics to all children as their achievement in the subject is judged throughout their life and participates in determining future prospects. Grades achieved in Mathematics affect future studies and career paths; for example, to enter university, usually a minimum of GCSE grade C is required, and this requirement varies depending on the course (Watson, 2006). Therefore, as a result of setting, 'those in lower sets are less likely to be entered for higher tiers' (Day, Sammons and Stobart, 2007, pg. 165), consequently harming their future study and job opportunities. Also, some children have an advanced grasp of Mathematics due to an advantaged background, parents' help or private tuition so setting is unfair as it is biased towards early developing children or those who have been given extra help outside of the classroom.

In schools, the setting system is supposed to be purely based on ability level. However, in reality, streaming could be decided upon for other reasons. For example, two areas of prejudice encountered can be social class and ethnic dimensions (Capel and Leask, 2005, pg. 155). Bartlett, Burton and Peim point out that often 'lower class students were deemed to have a lower intellectual ability than middle class peers purely due to unrelated social issues such as accent or parents' jobs.' (Bartlett, Burton and Peim, 2002, pg. 182) Sukhnandan and Lee (1998) comment on the fact that lower-ability sets consist of high number from low social-class backgrounds, ethnic minorities, boys and children born in the summer, who are at a younger age for their school year. Sukhnandan and Lee believe that setting in this way causes 'social divisions'. (http://www.tes.co.uk/article.aspx?storycode=81217).

Therefore, it appears that language competency is being used as a major factor in determining which set students are placed in and consequently impacts achievement in Mathematics.

## Conclusion

In conclusion, it may be argued that there is an ongoing need to re-assess how learners internalise the mathematical concepts conveyed in language. Practitioners have acknowledged that semiotics, or the relationship between language, symbolism and thought, impacts the way in which learners interpret information. For example, as Pimm indicates, regarding the concept of negative numbers, 'involves a metaphoric broadening of the notion of number itself…among mathematicians, the novelty becomes lost with time, and with it the metaphoric content of the original insight of useful extension. It becomes a commonplace remark - the literal meaning.' (Pimm, 1987, pg.107). Although Mathematics tends to pursue positivist or absolute outcomes, it involves much that is abstract; quantities, frequencies, probabilities etc, are all events or values that occur independently of the need to visualise them, or calculate and record them. The need to do so is usually derived from the need to understand or control events which have happened in the past, are happening now, or predict what will happen in the future. As discussed, individuals must match their own internal understanding of a particular problem with its communicable value, either in language, text, or numbers, however, firstly they must make the appropriate link. As Lee indicates, there are distinct social and communicative advantages when learners are allowed to articulate their understanding of these concepts (Lee, 2006, pg.4). Moreover, as Morgan observes, the disempowerment of individuals who lack the necessary control over language continues to cause concern and registers the need for further research (Morgan, 1998, pg.5). One of the principal issues arguably lays in drawing the distinction between linguistic and conceptual difficulties, and deducing the relationship between the two. As De Corte and Verschaffel have argued, learner's errors in word problems are often 'remarkably systematic', resulting from 'misconceptions of the problem…due to an insufficient mastery of the semantic schemes underlying the problems.' (De Corte and Verschaffel, 1991, pg.129). Therefore, further research into the origins of such problems and the means of addressing them is required.

As many practitioners will know from experience, the worst scenario is 'global' failure of understanding, where the learner cannot even articulate why they do not understand. In other words, they cannot begin to solve the problem because they have not understood the question. In these cases, the teacher needs to spend time with the individual concerned, which is not always easy or feasible in a classroom scenario. It is important to note that; the earlier problems are diagnosed, and the appropriate support put in place, the better it is. Unfortunately, there is no universal solution which can be applied here; it is simply good assessment practice, effective planning and the sensitive framing of problems which can gradually break down the problems involved.

Having explored this area in-depth, language competency does pose implications in understanding Mathematics, consequently favouring certain social groups. In my opinion, practitioners should regularly monitor learners to determine whether the individual is progressing or requires additional needs. Language competency is not a substantial enough reason for restricting how high a student can achieve and by using this as a factor in setting is clearly unfair. Sets should be formed and amended regularly, based upon student progress and mathematical ability to ensure there is no bias on social background. More individual support should be made available through an expansion of the appropriate budgets, so that the necessary action is not compressed into normal lesson timetabling and students can receive the maximum support possible of their needs, to enhance their succession in Mathematics.