A review of factors associated with young children's difficulties in acquiring age‐appropriate mathematical abilities

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A review of factors associatedwith young children's difficultiesin acquiring age‐appropriatemathematical abilitiesRoy Evans a & Kathy Goodman aa Roehampton Institute, LondonPublished online: 07 Jul 2006.

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A review of factors associated withyoung children's difficulties inacquiring age-appropriatemathematical abilities

ROY EVANS and KATHY GOODMAN

Roehampton Institute, London

(Received 26 May 1995)

INTRODUCTION

The term learning difficulty is frequently used synonymously with special edu-cational needs. In Britain as elsewhere, the experience of teachers is that suchdifficulties are often associated with poor attainment in one or other of thebasic skills or with problems of behaviour. Learning difficulties are neverthelesscomplex, often the outcome of several interacting factors. Some children mayshow such difficulties over relatively short periods of their school careers andnotably in relation to some subjects. For others their difficulties are persistent,long term and often inexplicable. The general direction of educational policy inEngland and Wales is to recognise that up to 1 in 5 children at some point in theirschool career may experience learning difficulties requiring special educationalhelp, and that about 2% of children will experience such severe difficulties thattheir access to the mainstream curriculum can only be assured by providingresources additional to those normally available in ordinary schools.

Over the last decade, learning difficulties associated with children's emotionaland behavioural problems have provoked increasing attention to the ways inwhich schools as well as individual teachers create active caring environments forchildren and organise whole school policies on behaviour in order to create aclimate in which children may derive the best possible benefit from the learningexperiences provided. Over the same period of time the national concern toraise standards in education has served to refocus attention on those childrenwho continue to show difficulty in learning and understanding key conceptsand behaviours relating either to initial literacy and/or initial numeracy. Despitesubstantial investment at national and local level in the development of pedagogywhich emphasises individualised, experiential and constructivist approaches tolearning, the difficulties which some children exhibit in acquiring basic ideas

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82 ROY EVANS and KATHY GOODMAN

concerning mathematics and in their response to texts, continue to challenge theprofessional knowledge of even experienced teachers. As teachers we recognisethat children do not all develop at the same pace and that for some, progressin all areas of the curriculum will be a slow and painstaking process. The mostperplexing problems often arise in relation to those children who in spite oftheir apparent intelligence, supportive home backgrounds and well resourcedteaching environments exhibit a marked difficulty with concepts which somemuch younger children appear to grasp quite easily. Such difficulties have fordecades been referred to as specific learning difficulties, in recognition of thefact that they do not characterise a child's general learning disposition but canadversely affect his response to specific sorts of tasks.

Inexplicable difficulties in responding to or in generating print have becomereferred to as dyslexic type behaviours. A similar term is occasionally used inrelation to children who have difficulties relating to aspects of mathematics,namely dyscalculia. Part of the purpose of this paper is to explore briefly theindicators of such difficulties for teachers, always recognising, however, that insome countries and amongst some professional groups the use of the term isfraught with difficulty. In Britain for example, dyslexia is regarded by someas an unnecessary term, by others as an unhelpful term, and by others asnecessary only insofar as it is a label which may be used to attract additionalteaching resources. Research by Miles (1983) indicated that not all dyslexicchildren show difficulties in mathematics, but as we shall see below children'spoor achievement in mathematics may nevertheless arise as a consequence ofchildren's poor language skills. A child's difficulty in one area may influence hisprogress in another area of the curriculum if the teaching method employedfails to recognise that particular approaches to teaching or the presentation oflearning tasks, can provide obstacles to a child. To put this another way, a child'sability to access the mathematics curriculum can be influenced adversely by poorlanguage skills if he is unable to read or understand the language of the taskthat is set.

If we were to take a systems approach to learning difficulties it would be possibleto suggest that the child's performance on a task at any particular point in timeis a complex function of many interacting factors. Figure 1 provides a simplifiedsystems' view of the ontology of a childs apparent mathematical difficulty, whereattainment on progress at any point in time is seen as the complex outcome oftransactions, amongst key factors in the close environment of the child. Issues ofpedagogy, task and context are relevant insofar as they provide either a nurturingenvironment to what the child brings to the situation, or denies opportunity forprogress as a consequence of the adverse effect of potentially modifiable factors.It is not the intention here to provide an inclusive account of the way in whicha child's response to the mathematics' curriculum is ultimately constructed, butit will be useful to focus on a number of major factors. Three such factors may,however, be taken as illustrative of the general complexity, each of these factorsbeing quite powerful in its determining effect. We would suggest that these threeare:-

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YOUNG CHILDREN'S DIFFICULTIES IN ACQUIRING MATHEMATICAL ABILITIES 83

Figure 1 A systems approach for inquiry into children's learning difficulties inmathematics.

ONTOGENYSources of

Variations and

Influence

Child

History

Genetic inheritance.Congenitaldispositions.Prior learning.

Context

Characteristics

Temperament

Transactions

Complexmultilevelinteractionsand

transactions

Outcomes

Physical, cognitive,academic andaffective status.Progress throughthe curriculum.Confidence atmathematics

Parents/Caretakers

Home

Teacher

School

Values and beliefs.Level of education.Knowledge aboutchildren.

Physical resources.Economic Status.Structure of family.

Subject knowledge.Teaching competence.Teaching style.Knowledge ofchildren's learningand development.Classroommanagement.

EducationIdealogy.Organisationalculture.

Involvement.Care style.Support,Expectations.

Organisationalarrangements.Available space.Typical vs.non typicalcultural values.

Teaching practices.Pedagogy. Conceptof mathematics.Understandingof approachesto learning.Organisation forteaching. Attitudesto 'dyslexia'.

Class size. Wholeschool approaches tonumeracy, literacyand special needs.Available resourcesto support learning.

etc.

This should not be construed as exhaustive in terms of possible interactions (following Ramey, 1980).

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1. Characteristics relating to the child itself.

2. Characteristics relating to the teaching method i.e. pedagogy.

3. Characteristics of the subject of study e.g. mathematics. In drawing outthree major sets of characteristics above it is not suggested that all inevitablyoperate for any given child showing a particular learning difficulty. Suchcharacteristics can, however, be profitably considered by teachers when facedwith a child or children who are giving cause for concern as a result of theirfailure to achieve age-appropriate goals. With this in mind each of the abovemajor factors may be developed as follows:-

LEARNING DIFFICULTY IN MATHEMATICS WHICH MAY ARISE FROMCHARACTERISTICS OF THE CHILD

It is useful for teachers to consider whether one or more of the followingcharacteristics have some relationship to a child's perceived underachievementin mathematics. The list is not exhaustive and neither can we be sure about theparticular impact on any given child or the operation of any one or set of suchcharacteristics. It will also be apparent that some of the characteristics mentionedare derived, i.e. they are themselves the outcomes of prior learning. Nevertheless,the following may be helpful:

Poor self-image of himself or herself as a mathematician

Social psychological theory suggests to us that poor self image is as much anoutcome of learning as it is a significant factor in future learning. The child's viewof herself as poor mathematically, may become constructed through continuedfailure in mathematics. Children see themselves as failures in mathematics because'that has been their experience. They learn to fear mathematics because they seemunable to succeed at it and will consequently want to avoid the experience offailure. The challenge to the teacher in these circumstances is to help the childto build a more positive self-image by constructing opportunities for success. Thisis a function of task design.

Learning style

A considerable body of research in psychology and in education points to thefact that human beings vary in their cognitive style and in their approach toinformation processing. Much of this work represents cognitive style as a bi-polartrait or as a continuum. A great deal of interesting work has been done by Shipmanand Shipman (1985), Saracho (1983), Saracho and Spodek (1981), Witkin H.A.,Moore C.A., Goodenough D.R. and Cox P.W. (1977) and its relationship tomathematics has been explored by Vadiya S. and Chansky N. (1980).

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Saracho (1995) suggests that the attributes of field dependent and fieldindependent individuals are highly pervasive and consistent key features offield dependent individuals include: reliance on the surrounding perceptualfield, dependence on authority, strong interest in people, sensitivity to others,preference for collaborative tasks, experience of environment in a relatively globalfashion and by conformity to the effects of the prevailing field or context. Bycontrast field independent individuals are characterised as: perceiving objects asseparate from the field: able to abstract an item from the surrounding field andsolve problems that are presented and re-organised in different context, haveorientation to active striving, may be cold and distant, show social detachmentbut with clear analytical skills, and prefer tasks that permit them to work bythemselves.

Saracho (1995) has pointed out that in consideration of the evidence concern-ing students' performance in academic situations related to field dependenceor independence the general evidence suggests that relatively field independentstudents perform significantly better in mathematics, sciences and engineeringthan field dependent students. In all of the studies reviewed by Saracho none ofthe field dependent children have been found to achieve better performance thanthe field independent students on a range of scholastic and attainment measures.Such studies it is argued suggest that cognitive style may influence the observeddifferences in performance and underline the importance that this construct hasfor education. There appears to be little doubt that existing studies suggest asuperiority of field independent individuals in mastering school subject matterrelated to mathematics learning.

A somewhat different approach has been suggested by Gordon Pask (1976) andhis colleagues who suggest two distinct cognitive styles and argues the existenceof two distinct learning strategies: serialist and holist. Serialists favour learningby small, well defined steps so that their knowledge and understanding buildsequentially — serialists build walls with bricks. On the other hand, holists tend towork in a more exploratory way, wanting to gain a feel for the overall frameworkbefore filling in the details — holists see walls; their bricks are defined bydrawing on higher order thinking skills and restructuring past knowledge andexperience.

A similar classification is used by Sharma (1989) who identifies two types ofmathematical learning personalities: the quantative and the qualitative learner.The quantative learner parallels the serialist and is characterised by the learnerfavouring teaching which emphasises sequence and procedure and which isdeductive and convergent in its approaches to mathematics. Problem solvingis likely to be approached by choosing and applying an appropriate 'recipe'based on stimulus equivalence. In contrast, the qualitative learner, adopts aholistic approach to learning mathematics, favours teaching which encouragesintuition over procedure and is inductive and divergent in the approach tomathematics.

Saracho's (op. cit.) work in particular points out the potential difficulty whereteachers' own cognitive style is at variance with that of pupils i.e. the field-

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independent teacher does not always recognise the learning needs of the field-dependent child. These differences in cognitive style, which become manifestedin learning style do not present insuperable difficulties for the teacher butthey do nevertheless need to be recognised in the design of learning tasks, thepresentation of materials and in the support that teachers offer to children withintheir class.

Poor language skills (including reading)

Under this heading it is possible to include a range of possible difficulties thatchildren may have, some of which relate to the richness of their receptive andor expressive vocabulary, their command of syntax and grammar, and theircomprehension skills both in relation to the spoken as well as the written word.

The teacher's use of natural language to aid in the construction of mathematicalconcepts with their associated imagery and vocabulary may be a source of difficultyfor some children. Further, since it has been argued that mathematics is in itselfa language, with its own vocabulary, grammar and highly condensed symbolsystem, children with poor language skills may experience difficulty with both themedium and the message. The richness of the child's language experience, andthe social-cultural context in which meanings have been acquired are both mattersof considerable importance for teachers to reflect upon. In contrast, the child withspecific reading difficulties may be able to complete the required mathematicaltask successfully, but it is the nature of the mathematical text which presents theproblem. The written text, graphic images and the layout of the page, are differentin mathematics texts from styles in other curriculum areas and demands specialreading skills (see e.g. Pimm, 1995; Durkin and Shire, 1991; Shuard and Rothery,1984).

Dyslexic type difficulties

Given the nature of many mathematical tasks it is perhaps not surprising thatperceptual problems and poor spatial discrimination are often associated withlearning difficulty in mathematics. Just as poor figure-ground discrimination willmake it difficult for the child to focus on a particular object or to see it as awhole, the child who has not developed an awareness of spatial relationships willbe at risk. For example, although the evidence is not conclusive, it does seemthat the identification and meaning of symbols in the first instance is likely to beproblematic. Further, since the basic spatial concepts of position and directionare needed in many mathematical tasks, insecurities over concepts of left orright, above or below, including figure-reversal are likely to result in confusion.Where poor spatial discrimination is accompanied by poor organisational skillsor immature motor control, underachievement in mathematics is the likelyconsequence, (see e.g. Miles, 1983 etc.). Also, since children with dyslexic typedifficulties often exhibit difficulty with making generalisations, it is not surprisingthat the study of mathematics, involving the search for and proof of generality,might be problematic for them. We have noted then that specific reading difficulty

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or dyslexic behaviour can operate from at least two perspectives to influencea child's progress in mathematics. Firstly, such difficulties may be generallyindicative of poor spatial discrimination, orientation, sequencing and problemsof direction. Where this is the situation some aspects of the maths curriculum willbe problematic for the child but text generally may represent significant difficulty.Secondly, where the text is inaccessible to the child it may in itself be responsiblefor the child's apparent difficulty in acquiring appropriate mathematical skills.Modifying the way in which learning tasks are designed and presented can forsome children ensure that their reading difficulty does not present difficultieswith mathematics as well. Where the child exhibits dyslexic type behaviour, it isunlikely that the child's response to text will significantly improve in the absenceof carefully structured special help which recognises the need to emphasisethe multi-sensory nature of 'reading'. Normal classroom approaches to literacydevelopment which are successful with most children may not enable the dyslexicchild to progress. In the absence of structured interventions, development ofliteracy skills and mathematical skills are at risk.

Resistance to the use of the term 'dyslexia' by many teachers and teachereducators has two major sources. On the one hand the importance of the'condition' is minimised through the argument that not all poor readers are'dyslexic', and that in general poor literacy skills may be accounted for in termsof language experience, prior learning or inappropriate pedagogy. On the otherhand little use is made of experimental studies of 'dyslexics' by clinicians suchas Pavlidis (1990), Connors (1986), Miles (1983), Wilshire (1987) which leave littleroom to doubt that inexplicable reading difficulty for some children has its originsin neurological and/or structural anomalies in the left hemisphere. The possiblereasons for such anomalies, whether hereditary, congenital or acquired is lesssignificant than the fact that they can be identified earlier than the child wouldnormally begin to show reading difficulties. In other words the basis for futuredifficulties can be identified, through the work of Pavlidis for example, at a muchearlier date than that at which the problem calls attention to itself. The importanceof screening for dyslexia is important so that appropriate pedagogical measuresmay be taken in order to ameliorate its effects and prevent it impacting onother areas of learning such as Mathematics. Most teachers do not have access tothe sophisticated neuro-science screening techniques required for the analysis ofeye-movement (Pavlidis, 1980, 1985, 1990), and ironically in the U.K., additionalresources for children with learning difficulties usually are triggered only whenthey have demonstrated failure in school. However, indicators to possible dyslexictendencies do exist and can be employed by teachers from the point of schoolentry, or before, as alerting mechanisms to future literacy difficulty in the absenceof suitably structured intervention.

Table 1 gives a brief list of Indices of Dyslexia most of which are amenableof observation prior to the stage at which most children learn to read. It is notnecessary for any single child to show all of these indicators for the teacherto be alerted to possible future difficulties and to embark on suitably enrichedpreventative work.

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Table 1 Indices of Dyslexia.

1. Directional Confusion (Left-Right)

2. Spontaneous Writing and Spelling Impairment

3. Finger-Differentiation Problems

4. Visual-Perception Deficiencies

5. Handedness and Cerebral Dominance

6. Weakness in Memory Storage

7. Maternal and Natal Factors

8. Motor Dysfunction

9. Delayed Maturation

10. Delayed Speech Development

11. Familial or Inherited Disability (Genetic Factors)

12. Sex Difference

13. Language Delays

Table 2 Symptoms of Dyslexia.

1. Discrepancy between intellectual level and performance at spelling

2. Bizarre spelling

3. Confusion of'b' and 'd' in either reading, writing or both

4. Difficulty over distinguishing left and right

5. Difficulty in repeating polysyllabic words, such as "preliminary" or statistical

6. Difficulty in repeating digits in reverse order

7. Difficulty in repeating months of the year, especially in reverse order

8. Inability to do subtractions except with "concrete" aids

9. Difficulty in memorising arithmetical tables

10. "Losing the place" when reciting tables

11. History of clumsiness

12. Late walking or late talking

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Table 2 provides a list of characteristics often associated with the classroombehaviour of 'dyslexic' children. With children who are showing failure inreading, writing, spelling or in mathematics, existing pedagogy is evidently notworking. Such failure is a plea to the teacher to re-examine teaching style as wellas the content of tasks set for children and their manner of presentation.

Lack of mathematical experiences in the home

Mathematical concepts develop through a variety of experiences in and out ofschool. Indeed, since mathematical education really begins in the first year of achild's life, early and subsequent mathematical experiences in the home may affectattitude to and achievement in this curriculum area. The child-centred homewhich nurtures the child's curiosity, need for activity and developing languagewill not only be influencing the child's attitude to learning but invariably, andperhaps inadvertently, provide opportunities for such mathematical experiencesas classifying, comparing, matching, ordering and quantifying. A lack of suchconcrete experiences may affect the child's formation of mathematical concepts,and elementary counting activities. Furthermore, the use in the home by parentsand older siblings of calculators and mathematical aids also has influence on thechild's general disposition to mathematical experiences.

Further opportunities for developing enriched conceptual structures are lost inhomes where family members scarcely use, are seen to use or indeed avoid usingmathematics.

It is always the teacher's role to find out where the child's need lies andto provide relevant experiences to help the child to progress. Where themathematical learning difficulty can in some part be attributed to the homeenvironment it is essential that the teacher not only organises the curriculum sothat mathematical learning needs are met but maintains high expectation of thechild's mathematical potential. This will mean providing the opportunity for play,experimentation, developing the language necessary for mathematical thoughtand using and applying mathematical skills, knowledge and understanding inpurposeful contexts.

Different cultural backgrounds

General research suggests that ethnic minority and bilingual pupils may experi-ence learning difficulty in mathematics as a result of several factors. These includenot only the widely acknowledged influences of the 'hidden' curriculum, teachers'attitudes and expectations, classroom organisation and interactions, but also theovert curriculum.

Although mathematics might be thought to be a universal activity, it is notculture-free. Increasingly, and particularly since the 1988 International Congressof Mathematical Education in Budapest, there has been much debate on how therelationship between mathematics, social contexts and cultural influences haveshaped the mathematics curriculum. It is worth noting that the International

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Study Group on the Psychology of Mathematics Education has two workinggroups being respectively concerned with social aspects of mathematics andcultural aspects of mathematics. The Study Groups have in fact undertakenresearch and publications since 1985. If content areas of mathematics, the contextsin which the mathematics is presented and the resources marginalise pupils fromdifferent cultural backgrounds, these pupils are denied the opportunity to makea relationship between their lives and classroom mathematics and are likely toexperience learning difficulty.

Since mathematics not only has its own peculiar language patterns andvocabulary but is notorious for attaching specialised meaning to everyday words,it may present the bi-lingual learner with particular difficulty. That difficultymay not, however, be with the mathematics itself, and again it is importantthat the pupil's potential as a learner of mathematics be acknowledged andaddressed. Teachers can do this by recognising that the use of the child's firstlanguage is important for learning and providing opportunities for collaborativework where the cognitive strain of working in a different language is removed.The teacher can also be instrumental in developing the child's confidence inmathematics by allowing the child time to formulate and communicate ideas, andaccepting the meaning of the communication without criticism of the form of thecommunication.

The general thrust towards equality of opportunity and entitlement haslead to attempts to celebrate cultural differences. This has resulted in some'multiculturalisation' of the mathematics curriculum and the sensitisation ofteachers and publishers. Although the teacher may have little control overwider social issues the teacher is charged with the responsibility of providing acurriculum which can be accessed by pupils from minority cultures, and whichboth utilise and celebrate different cultural perspectives.

Gender differences

Although statistical analysis of test and examination results suggest that success atmathematics is evenly distributed between boys and girls until the age of fifteen,girls are under represented in the take-up of the study of mathematics at sixteenplus and eighteen plus. Whereas this may not in itself suggest learning difficultyas a consequence of gender differences it does indicate under-participation on thepart of girls. There is no evidence to suggest that boys are more able than girlsbut there is evidence to suggest that boys and girls approach mathematics withdifferent experiences and attitudes. The APU (Assessment of Performance Unit(1988)1 findings showed that, at ages eleven and fifteen, girls were less confidentin mathematics than boys, over-rated the difficulty of mathematics, and morefrequently attributed their success to luck. As the wealth of literature concerningthis phenomenon indicates, such differences are, in no small part, due to thecomplex inter-relation between sociological and psychological factors. What is

' in JofFe, L. and Foxman, D. (1988). See References. See ??? (1991).

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clear, however, is that girls are at risk of disaffection from and disablement inschool mathematics.

Dyscalculia

In 1974 Kosk suggested that some mathematics learning difficulty stemmedfrom "a genetic or congenital disorder of those parts of the brain that are thedirect anatomical-physiological substrate of the maturation of the mathematicalabilities adequate to age, without a simultaneous disorder of mental functions"(p. 165). The term used to describe this phenomenon was 'developmentaldyscalculia'. The area remains controversial not least because mathematics (asopposed to calculation) requires the use of many different intellectual skillswhich are not exclusive to mathematics. We have touched on this in relationto dyslexia.

LEARNING DIFFICULTY IN MATHEMATICS RELATING TO CHARACTERISTICSOF THE TEACHER/TEACHING METHOD

The range of teaching methods in mathematics can also be presented as a bi-polarcontinuum ranging from expository methods of a deductive kind to spontaneousdiscovery method. Although the exclusive use of one of these extremes is highlyunlikely a predominance of one over the other could lead to children being taughtby a method dissonant with their learning style. For example, expository methodscan lead to glib verbalisation of concepts (the square on the hypotenuse is equalto the sum of the square on the other two sides) and the idea that mathematics ismerely memory work. Since it is the teacher who has control over direction andpace of the mathematics, children can be discouraged from initiating exploringand discussing mathematics. In contrast, discovery learning can lead to unclearconcepts and inaccurate generalisations being formulated.

The work of Skemp (1976) is of particular relevance here. He suggests atheoretical framework which identifies two distinct kinds of learning whichlead to knowledge about mathematics: instrumental and relational learning.Instrumental learning is dependant on external input and is acquired whenmathematics is presented as a series of successive stages, with no opportunity forthe learner to make relationships between these stages and with no appreciationof the wider goal. In contrast, relational learning consists of building conceptualstructures (schema) where each part of the schema is a growth point for furtherrelationships to be developed.

Teaching for instrumental learning will draw on a pedagogy which is trans-missive and didactic and, since pupils can be taught to get the right answersto routine questions if they are presented in a familiar way, is likely to yieldimmediate, assessable and apparently successful results. Teaching for relationalunderstanding, will draw on a pedagogy where the locus of control rests with the

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learner, where the teacher uses his or her skills to choose appropriate activitiesand to generate a spirit of enquiry so that prior knowledge and understandingcan be used to generate new knowledge and skills.

Without forming the relationships which are the essence of relational under-standing a pupil cannot deduce or check half remembered facts by referenceto secure knowledge. Hence the learning difficulty may be the direct result ofan over-reliance on a teaching style which has led exclusively to instrumentallearning.

Similarly, a child's learning difficulty may result from the lack of experiencewith 'concrete' materials from which to build mental imagery and abstractcommon mathematical structure.

As has been discussed earlier, learning difficulty in mathematics may be theresult of pupils being unable to make connections between mathematics andtheir own experiences. Even where there may be some success at 'disembedded'mathematical activity it is likely that the learner will not be able to apply hisor her mathematical knowledge to solve problems. Since it can be argued thatpupils are more likely to understand mathematics if it is developed in relationto a problem, which is itself meaningful, it would seem imperative that pupilsbe given the opportunity to be involved in problem solving. The irony here,of course, is that traditionally it has been thought that children with learningdifficulty in mathematics need much practice in basic skills so that they can thenlearn to apply these to problem solving. Inadvertently the teacher may have beendenying the very medium the children needed to develop their mathematicallearning.

CHARACTERISATION OF THE SUBJECT

The views of mathematics gained by our pupils during their school career dependon the content, guidelines and ideologies presented in the official curriculum,but more so on the individual teacher's personal beliefs about mathematics,teaching mathematics, the curriculum, the nature of children's learning and theappropriate methods of assessment.

It is possible to sketch three "caricatures" of the ideological positions of typicalteachers who will drive pupils actions and opinions in different directions, andwho will consequently influence their attitudes and achievements.

Utilitarian

This regards mathematics as an unquestioned body of useful knowledge, factsand skills. Teaching mathematics consists of becoming a skilled instructor, theknowledge itself is the authority whereby the teacher motivates the pupilsby reference to their future needs. The learning of useful basic skills at theappropriate level are the necessary preliminary to certification, which shows that

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the pupil has acquired the skills through practice and can demonstrate how tosolve problems by "correct" methods.

This viewpoint regards mathematics as a body of knowledge tried and testedthrough time and experience, which is to be transmitted efficiently to pupils.There is no room for questioning the body of knowledge, and the authority andall the choices and the methodological and pedagogical decisions lie in the handsof the teacher.

Structural

Here, mathematics is regarded essentially as a more abstract exercise with thenecessary (and obvious) applications. The emphasis is on an appreciation of theabstract mathematical structure which lies behind and guarantees the success ofthe particular skills and their applications. The teacher has a duty to pass onthis body of knowledge whereby pupils will understand how to solve problemsby appreciation of the structural links between the mathematical models whichtypify different problem areas. Motivation of pupils and the basic pedagogicalapproach is through demonstrating examples of the typical problems whichdisplay these structures. Assessments for the pupils are intended to demonstratetheir knowledge of the structures in the context of solving problems, and anyerrors are carefully examined to discover structural misunderstanding.

Here again, the content of mathematics itself is unchallenged, and the "bodyof knowledge" must be passed on. However, the aim is much more intellectualappreciation of the structural beauty of the subject, and practical problems aresolved by recognising the typical models which are appropriate. As above, theauthority lies with the teacher, as expert mathematician and custodian of theknowledge.

Social

Mathematics is a personalised activity which has implications for society. The"knowledge" is owned and passed on by the culture the individual inhabits, anddifferent parts of this "knowledge" are appropriate in different situations andat different times. Teaching mathematics consists of facilitating the individualpupils personal exploration through discussion of the content and questioning theassumptions involved. It is necessary for the teacher to try to become aware of thepupils needs and difficulties. The curriculum needs to encourage pupil centredcreativity, and a critical social awareness, and is necessarily varied to becomeaccessible to different cultural groups. Learning involves active exploration, andan exercising of choice in methods of approach to problems, and even to the typesof problems themselves. The ultimate aim of the teacher is to enable the pupil tobecome an independent and confident mathematician aware of their own abilitiesand limitations within a particular context.

This is in strong contrast to the views above. However, this view, or variationsof it, are having a strong influence on the debates about the curriculum and

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on teaching methodology today. In the context of this discussion on learningdifficulties in mathematics, it may be more appropriate to consider the merits ofat least part of this standpoint.

The three views described above, while they may not truly represent anyparticular individual teachers opinion, ideology or action, serve to focus ourdiscussion on the way in which the subject of mathematics is characterised, andby implication, the ways in which the curriculum is presented in school. Clearly, amore sensitive approach to the subject itself, opening up a debate among teachersof children with learning difficulties as to it's nature, and the appropriate contentand methodology to be employed is a necessary part of our approach to thecomplex problems involved in teaching mathematics to children who displaylearning difficulties.

CONCLUSION

The purpose of this paper has been to draw attention to the complex issuessurrounding the apparent failure of an individual child to make progress insome facet of the mathematics curriculum. Attention has been drawn to thecomplexity of interactions which combine to enable or handicap the acquisition ofdesirable skills and abilities. Attention in particular has been drawn to three sets offactors which may usefully be considered by the teacher who is concerned aboutprogress in mathematics. It has been proposed that the child may have dyslexictype difficulties which affect not only his ability to access written material butwhich also may manifest in apparent difficulties with seriation, visual perceptualdiscrimination and spacial orientation. Whilst such difficulties may impede thechild's progress, they will prevent it only insofar as teachers do not adjustthe design of learning tasks and their own teaching methods to take intoaccount such special educational needs. Appropriate pedagogy can assist the childtowards significant learning in mathematics. The importance of teaching styleand teaching method has been emphasised throughout the paper and we havenoted that even where the character of the subject is presented as a possiblereason for underachievement, hereto such characterisation frequently resides inthe mind of the teacher and influences the way in which the subject is presentedto pupils. A sensitive re-appraisal of such issues by all concerned in the teachingof mathematics will do much to reduce the anxiety and stress that many youngchildren face when presented with mathematical tasks.

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Documents

A review of factors associated with young children's difficulties in acquiring age‐appropriate mathematical abilities