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Literature on Special Educational

Needs in Mathematics

A bibliography with some comments

Olof Magne

Magne, O. (2003). Literature on Special Educational Needs in Mathemat-ics: A bibliography with some comments. (4th Ed.) (Educational and

Psy-chological Interactions, 124). Malmö, Sweden: School of Education. Abstract: This documentation presents the findings of a survey of about 5,000 documents on low achievement in mathematics that have been pub-lished in one form or another during a long time, actually since 1886 and up to the new millennium. All these terms are defined as related to one and the same notion, namely low achievement in mathematics. For such low achievement some authors have used the term disability. Neurologists of-ten prefer words beginning with the prefix “a” or ”dys”, such as “acalcu-lia” or “dyscalcu“acalcu-lia”, thus referring to simple arithmetic only. For ordinary educational purposes the author prefers less defect orientated expressions, as special educational needs in mathematics. Is the low achievement ob-served together with physical impairment of the individual, wrong learning strategies, unwise teaching, social circumstances etc.? Mathematics learn-ing has been subjected to many conflictlearn-ing impulses, not always favour-able. The survey indicates a bias in the investigations. As to mathematical topics, simple arithmetic operations seem to have been most attractive to research. From a didactical view, the major part of the studies concerns extremely primitive learning conditions. Neurologists have often preferred to study simple neural deviations and sensori-motor processes connected with arithmetic. Future studies ought to go in for factor interplay between logical stuctures of mathematics, productive thinking of the student and social network of the environment, that is the factor-interplay model. Keywords: Bibliography, factor-interplay model, individualisation, mathe-matics, special education, special educational needs.

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This report can be read or downloaded from the following address: www.bit.mah.se/MUEP

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Contents

Introduction ... 7

Prelude ... 7

Definitions and terminology ... 8

Qualitative criteria ... 9

Choice of policy... 10

The thematic field of special educational needs in mathematics .... 10

A category system... 13

Suggested categories... 14

Characteristics of the students... 17

Qualitative interpretations... 18

Lubienski’s survey of mathematics education research 1982–1998 ... 24

Remarks on future studies... 24

References... 26

A survey of literature on special educational needs in mathematics: Suggested categories ... 29

Ability and mathematics ... 30

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Aggressiveness, behaviour disorders, criminality, and

mathematics ... 58

Bibliographies ... 60

Case studies... 65

Communication... 70

Concentration, attention, effort, exertion, and mathematics ... 71

Content areas: General... 73

Content areas: Problem solving (See also Problem solving) ... 73

Content areas: Number ... 74

Content areas: Calculation ... 79

Content areas: Geometry. Measurement. Money... 92

Content areas: Algebra. Functions ... 95

Content areas: Statistics. Probability ... 100

Content areas: Computers, Calculators... 102

Curriculum ... 110 Diagnosis... 116 Didactical considerations ... 143 Equality, inequality ... 163 Errors, misconceptions... 163 Gender ... 185 Genetic conditions ... 188

Hyperactivity, impulsivity etc... 190

Impairments, various multiply disabilities... 191

Impairments, as to hearing impairments ... 191

Impairments, as to visual impairments ... 203

Impairments, as to cerebral palsy etc. ... 208

Individual variations. Individualizing learning ... 211

Instructional aids... 215

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Language and mathematics... 225

Learning, instruction: theories ... 238

Low and high achievers in regular mathematics education compared... 243

Mathematical competence in everyday life... 253

Memory and mathematics... 265

Mentally handicapped, General ... 269

Mentally handicapped, “Idiots savants”... 288

Minorities... 293

Movement/motility, motor skills, and mathematics... 297

Neurological or neuropsychological characteristics of individuals... 298

Nonverbal disabilities ... 332

Perception and mathematics... 334

Planning, strategy, metacognition of students ... 339

Problem solving ... 345

Social conditions in mathematics. Ecology ... 357

Specific difficulties or underachievement in mathe-matics ... 366

Teaching and learning methods in preschool... 370

Teaching and learning methods in primary education ... 374

Teaching and learning methods in secondary, post-secondary education ... 391

Teachers ... 397

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Introduction

Prelude

Bibliographies of mathematical low achievement are found, for instance, in Fleischner and Garnett (1980), Matulis (1981), Fletcher and Loveland (1986), Grissemann and Weber (1989), König (1989)

,

Lubienski and Bo-wen (2000), and Magne (1996b), see the references. Here and in the fol-lowing context, the reader will mainly be referred to Magne’s (1996b) list of literature for full references to the cited works.

These surveys of the literature are, by no means, complete catalogues of all existing documents. First of all, there are many records of research or practical work that are not printed or in other way offered to the public. Secondly, many documents are available in languages that are difficult to understand or even to register. Thirdly, some documents refer to infrequent conditions, complaints or impairments and, thus, are seldom available to persons outside a very exclusive circle of specialists.

This presentation shows the findings of Magne’s survey of about 4,300 documents on low achievement in mathematics that have been published in one or another form during a long time, actually since 1886 and up to the 1990’s.

The author has chosen documents from the following areas in connec-tion with mathematical low achievement:

• philosophy • neuropsychology • sociology • education • technology.

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Specifically chosen themes are for instance: Educational and social needs, physical impairment, affect and motivation, competence and ability, mathematical content, teaching and learning, diagnosis and achievement. The bibliography is justified by the fact that it is still a very tedious practice to find relevant literature in this particular field by looking for documents in libraries’ data bases or free search with IT techniques, par-ticularly before the present decade.

Definitions and terminology

In this presentation I mainly use special educational needs in mathematics as a comprehensive technical term, or shorter, special mathematical needs. For colloquial use I suggest the least defect oriented expressions as ”prob-lems in learning mathematics”, ”unsuccessful learning of mathematics” or other similar expressions which can be treated as synonyms. I like to ad-vise people not to use defect oriented words like mathematical disability or acalculia.

Definition: All these expressions are defined as related to one and the same notion, namely low achievement in mathematics.

It is totally irrelevant which are the possible causes to the low achieve-ment.

The definitions of the condition in view are often based upon:

• either a statistical measurement, such as 1 to 1.5 standard deviation units below the mean of a normal population as to a defined mathe-matical type of task,

• or a certain fixed criterium, irrespective of population, for example that the student shall achieve 90 per cent of a certain measurable aspect of mathematics.

However, the literature displays a great variety of terms and definitions. The total number of terms is great. For further discussion of terms and

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Terms like Arithmasthenia (Ranschburg, 1905), Acalculia (Henschen, 1920) and Dyscalculia (Gerstmann, 1924) are restricted to low achieve-ment in eleachieve-mentary arithmetic. Etymologically, the word dyscalculia is a linguistic monster, being a compound of of the Greek element ”dys-” and the Latin element ”calculus”. Magne (1988) introduced the term Dys-mathematica, or Dysmathematics because it covers the widest possible range of topics in school matematics.

According to the author, low achievement is a social construct. It is not a fact but a human interpretation of relations between the individual and his environment. Special educational needs in mathematics must be looked upon from a relativist view. The assessment refers to a set of achievement elements and depends on the learning or instructional criteria due to the prevailing educational conditions, traditions or legislation in a given school system. Obviously, standards may vary from one school system to another. The condition may refer to a given occasion or to a defined period of the individual’s life. Low achievement may be assessed by different criteria in and outside the school systems.

The expression Special educational needs in mathematics is often used for the condition when a student fails in his/her efforts to master one or several main areas of mathematics according to set standards. Applied to education, the low achievement is often general, related to the whole set of mathematical topics or even to all other school subjects as well. The low achievement sometimes refers to one mathematical topic but not to others. In exceptional cases the special need is observed in mathematics only, a case which is often called underachievement or specific educational needs in mathematics.

Qualitative criteria

The author has subjected the bibliography to a qualitative categorization although the reader may be better assisted by himself or herself if he/she elaborates a register from this biographical material fitting his/her own interests or study purposes. There are many conceivable criteria for such

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categorizing attempts, due to various value judgments, ideologies or prac-tical purposes.

It may be useful to introduce some examples of criteria.

Choice of policy

One important comprehensive value perspective has to do with the official choice of policy for education and socialization of individuals (Magne, 1996a). For the State’s Administrative Unit or Authority for Education, the point of departure may be looked upon as a mandate for the Administra-tion to work out systems for transmitting or impressing a certain uniform, regimented civil preparedness upon the minds of all citizens.

A second mandate to consider could be founded on a principle of op-tions for the individual to satisfy his/her personal interests and talents. This mandate may very well clash with the official uniformity mandate and, therefore, be opposed to administrative legislation.

A possible third mandate may refer to groups of citizens to give them opportunities, in an informal or formal sense, to attain certain individual and group competences in their respective activities, according to their interests.

The thematic field of special educational needs in

mathematics

A second comprehensive value perspective concerns the thematic field of special educational needs in mathematics.

One starting-point is the mathematical subject matter. In that case, re-search stresses the students’ difficulties to attain certain objectives in the various main areas of mathematics. Apparently, in the traditional school systems this was a usual foundation for investigating the trouble the teach-ers experienced when they instructed their students. One consequence was

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that other students had better do something else. In relation to low achievement in mathematics this approach has been described as the con-tent deviation model (Magne, 1998). The remedy would be either to ex-clude the student from mathematics, or to assign to the students tasks on an appropriate optional level of mathematical complexity.

A contrasting starting-point is to base the corresponding research on biological conditions, for instance to study neuronal impairment, and fol-low up on the consequences for mathematical achievement. Some early anatomical and histological investigators worked from this starting-point. Today we find that neuropsychologists like to use this mode of thought. They seem to believe that, by diagnosing brain functions, tuition will get a foundation for sound strategies and consequently give educators and thera-pists the obvious means to habilitate or rehabilitate a low achiever prop-erly. In relation to special educational needs in mathematics, this approach may be called the behaviour deviation model (Magne, 1998). The treat-ment consists of organic and treat-mental therapy.

Obviously, research has so far been unable to prove that organic lesions play more than a marginal part in the learning of mathematics even for low achievers. It is usual to find that the supposed proportion of brain injuries among students with special mathematical needs amounts to no more than about 20 per cent of those with special mathematical needs. This fact makes it unrealistic to fully accept both the content deviation model and the behaviour deviation model to one hundred per cent. It also seems nec-essary to concede that mathematical knowledge is not an objective reality but constructed by the human mind. The content deviation model seems to be valid only when due consideration is taken to the learner’s mind, beliefs and attitudes. On the other hand, the behaviour deviation model seems to have no validity on its own, namely if the subject matter of mathematics is disregarded. Content and behaviour balance each others.

A third starting-point is to include the social prerequisites that trigger the reactions of the thinking/learning students when they are brought face to face with mathematics topics. After the recent introduction of ethno-mathematics a great deal of arguments have been suggested in favour of alternative mathematics (see for instance D’Ambrosio’s contributions). It has been urged that, to some extent, the traditional formal didactics shall

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be supplemented with a more dynamic approach characterised by a social or sociological bearing. According to this starting-point we get a balance between mathematics, behaviour and social environment.

This third model appears to be more realistic than the two previous attempts to describe the thematic field of special educational needs in mathematics. It has been called the factor-interplay model (Magne, 1998). Summing up these thematic perspectives, research as well as curricu-lum innovation and teaching practice are approached from the notion of a complex vector space where, among other factors, three main vectors are considered, namely the mathematical contents, the pupil’s individuality and the social environment (network).

The corresponding remediation would be to give social tuition with freedom for the educator and the student to select mathematical options according to the student’s interests and personal circumstances in the natu-ral settings of both student and teacher. The author looks upon mathemati-cal knowledge as the dynamic interplay of mathematimathemati-cal subject matter, individual activities and social opportunities.

The author accentuates the complexity of mathematics learning. An additional value perspective refers to research on specified varieties of special mathematical needs per se. The three aspects of mathematics learn-ing

(a) a “mathematical goal” (seen from a mathematical point of view), (b) a “personal predisposition” (seen from a behavioral point of view)

and

(c) a “social need” (seen from a socio-educational aspect)

calls for a good diagnosis and a well defined programme of tuition, treat-ment and/or habilitation.

From all three aspects, a multi-factored research programme seems to be recommended. As to mathematical topics, the author recommends that we should devote more studies to complex than to simple subject matter. From a socio-educational view, the studies should rather concern complex productive learning than simple learning skills. Behavioural research

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should not limit itself to simple sensori-motor processes but more thor-oughly study higher mental processes.

The author suggests that simple mathematics themes should be bal-anced against complex themes, that simple sensory-motor processes and complex central neural processes should be compared, and that we should weigh pros and cons between simple learning and complex learning. The student’s retention tends to be optimal if the learning and instruc-tion is based upon thinking strategies and constructive activities. Thus, it is the student’s own efforts to learn that shall be ascribed the central position in mathematics education. Also the student with special educational needs in mathematics learns through his/her own efforts with the aid of social tuition.

A category system

What can be found in a bibliography of special educational needs in mathematics? In order to answer this question the author has carried out some analyses of the content of the bibliography. The method was to study the documents and assess the main character of each one.

It must be stressed that each document has been chosen because it deals with some aspect of special educational needs in mathematics or, seen from another aspect, low achievement in mathematics. These aspects may refer to variables of mathematical content, variables of students’ character-istics or variables concerning environmental networks.

After assessing the distinctive features of the documents a descriptive category system was worked out. The categories do not constitute a hierar-chy but are placed on a par independent of each other. They may be cou-pled together in optional combinations by the reader.

The method to categorize the documents involves certain validity and reliability error factors that cannot be easily controlled. However, the bib-liography is available to every researcher. Everybody has an opportunity to analyse the material and, thus, to construct a category system according to his/her own purposes.

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With this in mind it is possible to quantitatively interpret the research situation in the field of special educational needs in mathematics.

The number of documents in the respective categories is shown in the following list, displaying the frequences after the October 2000 revision.

Suggested categories

Categories

Number

Ability and mathematics 48

Affect and motivation 291

Aggressivity, asocial behaviour, criminality 19

Bibliographies 51

Case studies 61

Communication 6

Concentration, attention, exertion 19

Content areas: General 6

Content areas: Problem solving (see also Problem solving) 8

Content areas: Number 63

Content areas: Calculation 147

Content areas: Geometry. Measurement. Money. 40

Content areas: Algebra. Functions. 64

Content areas: Statistics. Probability. 26

Content areas: Calculator, Computer 83

Curriculum, syllabus. 59 Diagnosis 259 Didactical considerations 205 Equality, inequality 3 Errors, misconceptions 264 Gender 28 Genetic conditions 26 Hyperactivity, impulsivity 8

Impairments, various multiply disabilities 5

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Impairments, as to cerebral palsy etc. 40 Individual variations, individualisation 33

Instructional aids 107

Integration and inclusion 12

Language and mathematics 153

Learning, instruction: theories 55

Low and high achievers 89

Mathematical competence in everyday life 86

Memory and mathematics 42

Mentally handicapped 203

Minorities 45 Movement/motility, motor skills and mathematics 15

Neurological or neuropsychological characteristics 399

Nonverbal disabilities 22

Perception and mathematics 60

Planning, strategy, metacognition of students 51

Problem solving 129

Social conditions in mathematics, ecology 100 Specific difficulties or underachievement in mathematics 37

Teaching and learning in preschool 42

Teaching and learning in primary education 216 Teaching and learning in secondary and post-secondary education 63

Teachers 125 Thinking, active construction, and representation 27

Let us begin with quantitative notions based on the categories. First of all, a few categories contain many times more documents than the least fre-quented categories. This indicates that some themes are popular among researchers, possibly because they are looked upon as being more relevant for studies than other categories. The following categories or groups of categories are represented by a large number of documents. Here are some examples:

Content areas, indicating specification of non-achievement (437) Neurological or neuropsychological characteristics (399)

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Errors, misconceptions (264) Diagnosis (259)

Impairments (231) – multiply 5, hearing 126, sight 60, palsy 40 Mentally handicapped (203)

In fact, there are surprisingly few documents on each kind of impairment. Some categories are represented by a small number of documents, for in-stance:

Equality, inequality (3) Communication (6) Hyperactivity (8)

Integration and inclusion (12) Aggressivity (19)

Concentration, attention, exertion (19) Genetic conditions (26).

Gender (28)

Individual variations, individualization (33)

Specific difficulties or underachievement in mathematics (37)

It is difficult to understand these frequences from any understandable rea-son than a bias of the authors.

If we consider the various school levels it is remarkable that studies of primary education are much more frequent than studies of preschool and secondary or tertiary education.

Another example of possible bias is the preference of simplest possible arithmetic over more sophisticated processes. An interesting finding is that the main part of the studies is devoted to the four rules, usually simple exercises with small natural numbers. Thus, in the categories of Content areas a third of all documents refer to this skill. In the categories of Im-pairments, as to sight, hearing and cerebral palsy, skill in the four rules is treated by 63, 55, and 70 per cent, respectively, of all documents in

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ques-mainly discuss skill in the four rules. The neurological or neuropsy-chological studies nearly always refer to extremely simple calculations. Such proportion between simple skills and all other mathematical topics is absurd.

Apart from this illustrative quantification of the bibliography qualita-tive interpretations appear to be of considerable interest.

Characteristics of the students

Students with similar characteristics may have similar needs. But relevant research is scarce. The blind, deaf, physically disabled, mentally handi-capped, phobic and emotionally disturbed students, respectively, may, as kindred groups show similar, typical learning reactions due to their spe-cific physical or mental dispositions.

• Blind students are at a disadvantage because they have a handicap to visualise.

• Deaf students may be linguistically impaired.

• Emotionally disturbed students may display a disordered state of mind.

• Physical impairment often inhibits the growth of form perception and geometric notions.

According to other studies, mathematical low-achievement and under-achievement appear as complex and multi-factored disabilities. The litera-ture indicates that many children ought to be treated not only with respect to the learning needs in mathematics but, in addition, to several other needs as well. We may find it well-advised to look at each individual stu-dent as unique. One inference of this analysis is that planning for mathe-matical tuition must emphasise both learner and subject matter, both teacher and student, both individual and environment. The relationship between student and school organisation is also very complex. Some mathematics special needs students are unable to adapt themselves to de-mands in a national curriculum. A special programme should be worked

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out. The tuition can often be conducted in the classroom. Severe learning needs may or may not be able to be met in the regular classroom.

However, few studies concern these complexities of the learning condi-tions for the individual student.

Qualitative interpretations

From the present categorization of the documents in the Magne bibliogra-phy (1996b) it seems possible to make a few inferences concerning the many authors’ selection of their topics.

A. Notions on the nature of mathematics learning

1. This bibliography indicates that authors seem to look at special needs mathematics learning as identically the same as mathematics content according to prescribed curricula for the regular school systems.

2. Neuropsychological studies nearly always comprise mathematical areas of the most elementary types of number notions or calculations.

3. In neuropsychological studies it is usual to find confusion between rea-ding, writing and simple arithmetic.

4. The most frequently investigated part of mathematics is arithmetic, most common is addition with one- or two-digit numbers (numerals). It is usual that authors study and describe unrealistic, unsophisticated tasks. Social or ethnic backgrund is rarely considered. Students’ future social needs are seldom discussed.

5. Investigations are rare in the field of preschool, secondary and univer-sity mathematics learning although a promising literature was emerging during the latest decade.

6. In connection with ethnic research, a few authors have lately discussed options how to learn alternative and informal mathematic topics, what may be called “social mathematics”.

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B. On mandates to mathematics special education

1. The overwhelming majority of authors in this bibliography seem to start from the presumption that mathematics for the special needs stu-dent should contain fixed standard topics and exercises, presented in one and the same order, and have the same structure and function as for all other students. The usual standpoint of the authors seems to be that official curriculum standards ought to be followed in order to remediate the mistakes of the special needs student adequately.

2. Individual programmes and group work are surprisingly seldom de-scribed, discussed or analysed in connection with students with special mathematical needs.

3. There is an apparent lack of studies on how teacher education should take place in order to prepare the trainees to take care of students with special mathematical needs.

4. Very few studies deal with the so-called didactogenic problems, for instance conditions in school administration, curricular construction, classroom organisation, instruction methods etc. that may elucidate the origin of special mathematics needs.

C. The thematic interests of the authors

1. The most common approach to the various topics seems to be to pro-mote mastery of the mathematical content (may indicate tuition accord-ing to the content deviation model).

a. Diagnostic studies are frequent. Diagnostic research oftens aims at finding achievement levels of special mathematical needs students or relating these students’ results to normalisaed data or to fixed standards.

This is particularly obvious in studies concerning achievement of students with visual and hearing impairment. Investigating errors and their diagnosis is also popular in documents on mentally handi-capped students. Apparently, studies of mathematical learning among mentally retarded students are mainly directed towards nu-meration, calculation or visual perception, not mathematical think-ing/reasoning by the students.

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b. Error studies are very common. Surprisingly, they may be assessed to more than two fifths of all the cited studies. A usual procedure is to describe errors as related to various defects in a student who is in-capacitated to automatise skills in arithmetic prescribed by authori-ties.

c. Also neuropsychological studies seem to refer to student defects re-lated to basic mathematical formal standards rather than socio-educational reaction patterns (indicating belief in both the content deviation model and the behaviour deviation model).

2. In some studies mathematical errors are interpreted as results of errone-ous automatisation, slips of the tongue, wrong concentration, verbal misinterpretation etc. This may indicate belief in the behavior deviation model.

3. Many authors presuppose circumscribed organic impairment as the main cause of low attainment, exclusively, in spite of obscure founda-tion, for instance supposed lesions to the nervous system (behaviour deviations).

4. Anxiety is a very common theme, usually in one or both of two varie-ties: mathematics anxiety and test anxiety, where innate behaviour, neu-ronal disposition, learning conditions and/or instructional conditions are looked upon as instrumental to the anxiety. The building up of mathe-matics anxiety is sometimes looked upon from the behaviour deviation model or the content deviation model but seldom with reference to the factor-interplay model.

5. There is a lack of interest among authors to relate special mathematics needs to socio-educational system factors.

6. Motivation, attribution and belief theories as well as the self-efficacy conception have been considered as causal factors despite doubtful evi-dence. An increasing number of authors tends to take up these themes.

D. Interest in specific traits or special needs

1. Apart from neurological studies, the majority of the documents deals with general conditions among ordinary students in regular classes,

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usually very mild disabilities or educational needs with small discrep-ancies from statistical standards.

2. Mental handicap is the most frequently studied form of special need. These studies most often deal with mild learning disability. Severe mental handicaps are rather neglected.

3. A remarkably small number of authors have tried to investigate rela-tions between language disabilities (e.g. dyslexia) and cases of under-achievement in mathematics.

4. There has been a limited interest in underachievement in mathematics. In some countries results from this research seem to have started ani-mated discussions.

5. Aggressiveness has seldom been studied in connection with special educational needs in mathematics

6. Mathematical attainment and needs of socially and emotionally dis-turbed students have attracted only slight attention. This also holds true for criminality and psychotic conditions.

7. There is also a lack of studies on the problem how students with emo-tional or social disturbances learn mathematics.

8. Few studies are devoted to the role of concentration, staying power etc. 9. Many authors have tried to describe mathematical skills of visually and

hearing impaired persons but few have investigated remediation of their mathematics achievement.

10.Only few investigators have taken up the issue how mathematics is learnt by disabled students in ethnical or cultural minorities.

E. Causes for mathematical disability

1. Although the study of symptomatology of special educational needs in mathematics seems to be very popular theme there are few effective studies which have succeeded to demonstrate clear causes for difficulty to learn mathematics. According to the accessible literature mathemat-ics special needs students display very complex behaviour profiles. 2. Some authors claim that they have found a causal relationship between

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occipital and parietal lobes of the left hemisphere. Some parts of the right hemisphere are described as sites of geometrical skills and possi-bly problem solving skills.

3. As to more complex mathematical topics there is no proved relationship between neuronal functions and mathematical achievement.

4. There are only vague unverified hypotheses concerning causal genetic factors and mathematical ability.

5. Interrelations also seem to be uncertain between social conditions and mathematical ability and mathematical achievement.

6. Some authors claim that they have found circumstantial evidence that some types of instruction can cause low achievement in mathematics. It has been maintained that one such condition might be mechanical train-ing, based upon associationistic learning principles.

F. Educational principles and remediation methods

Few authors in the bibliography have written about methods how to treat low achievers in mathematics. Most of these methods may described as remediation. These methods can be roughly categorised as medical, educa-tional, psychotherapeutic and social (sociological). It seems to be a re-markable fact that quite a few authors write about successful treatment or instruction methods but, nevertheless, seldom describe their methods. Another remarkable fact is that these same authors specify their diagno-sis of mathematical “difficulties” with a wealth of details, but seem rather reluctant to specify learning objectives or instructional procedures.

The authors usually write about the mild complaints. Mostly, they de-scribe mild learning disabilities treated with various forms of educational and social tuition, for instance the educationally subnormal children. Usu-ally little is written how they are tutored. There is very little research re-ported on the education of severe cases, and apparently this refers to all sorts of impairments, disabilities or handicaps. Severe and moderate diffi-culties of all kinds are sometimes given a mixed treatment. But there are few reports on the efficacy of the various methods.

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grave controversies concerning adequacy of treatment principles, for in-stance

• associationism/behaviorism – psychodynamism, • instruction – learning,

• individualistic approach – social interaction, • drill – constructive thinking,

• frontal instruction – individualised discourse, • whole-class mediation – small group tuition, • external motivation – internal motivation.

Many authors try to justify their irreconcilable contrasting opinions by reference to educational philosophies, theories of cognition and political ideas. In some cases, the testing outcome is exhaustingly documented but without sufficient information of the previous learning and instruction procedures of the conducted research projects which, as a consequence, result in inconsistent research documentation. Most authors present reports on a small scale and/or refer to extremely specific situations. There are few reports on large-scale, well-designed projects on relearning of special edu-cational needs in mathematics which could be looked upon as pioneer work and serve as model projects or prototypes for future studies. There are a few exceptions. The following examples may be mentioned: The Finnish-Swedish mathematics clinics experiment (Ikehäimo and Magne et al.) and the Hungarian realistic, experimental teaching in special schools (Tarnai and Varga et al.).

A critical comment might be that, unfortunately, researchers on relearn-ing seem to have selected experimental problems and procedures more or less by trial and error and examined the outcome haphazardly. The re-search development may have let circumstances decide in an undirected way, and passing fancies may have been more influential than well-structured presumptions.

Psychologists, neurologists and mathematics educators seem to share related perspectives, but this is often done in awkward manner. What Shaughnessy says about controversies on psychological research in

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prob-ability and statistics may be applicable to mathematics special needs stu-dents (1992), that is to say that some researchers seem to the same high degree be busy “muddying the primeval waters of cognition with murky educational experiments that may distort their observation paradigm” (op. cit. p. 469).

Lubienski’s survey of mathematics education

research 1982–1998

Our results mainly seem to agree with Lubienski’s (2000) inventory of mathematics education research. Lubienski utilized the ERIC database on Silver Platter CD ROM containing 510,241 abstracts. She identified in-formation from 23,000 items published in 48 journals between January 1982 and early 1998 and counted and categorised 3,011 articles which provide a broad look at mathematics education research. This study differs from our study insofar as it is about the whole field of mathematics, but she interested herself to an intensive analysis of the following three subthemes, namely gender, ethniticy, class and disability. 623 of the 3,011 mathematics education articles related to at least one of the four ”equity” categories. Most attention was given to gender with 323 articles, disability was second with 193 articles and then followed ethniticy with 112 and class with 52 articles. The overall result of her survey was ” a broad, rough picture showing a body of research that gives considerabe focus to cogni-tion and achievement, primarily in Grades K–12, with significant attencogni-tion to integers and problem solving. In relation to equity, the results appear mixed” (p. 631).

Remarks on future studies

A recommendation would be that prospective investigations should be better organised, focus on cardinal issues and build upon carefully pre-pared cooperation between universities and nations.

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Special educational needs in mathematics are disregarded both in po-litical intercommunication and international research and development programmes. Lately, problems of special educational needs in mathematics have appeared at a few international conferences, for instance ICME and EASE.

There is a great need of increased research. The author suggests that research starts with Magne’s definition of the special educational needs. In addition, the author recommends that future studies should consider Magne’s qualitative criteria concerning choice of policy (p. 10-11) and thematic fields of mathematics (p. 11–14) or similar preconceived models. The author therefore suggests that the factor-interplay model (p. 13) seems to constitute a sound foundation for didactical philosophy, research policy, administrative decisions, curricular planning and actual classroom work. Regarding the matematical subject matter it is important to expand our knowledge on low achievement in mathematics to other school systems than primary education. We should investigate retention of subject matter curricula in preschool, secondary and tertiary education, universities and all types of continued education for disabled students.

Regarding the students and their activities we must map out the relation between mathematical subject matter and varieties of personality traits. The literature provides meagre information on concentration, cognitive perception and memory, productive thinking, planning strategies, aggres-sivity, hyperactivity, associated with mathematical subject matter. On the other hand, literature abounds in descriptions of sensori-motor processes, affectivity, intelligence and elementary language habits. But even these themes need deeper analysis according to aforementioned principles and other up to date philosophical and didactical principles.

Regarding the social environment in which the special needs students create their mathematical knowledge we notice proportionally little inter-est. The list on page 10–11 shows few documents on Criminality (0), Communication (6), Equality, inequality (3), Hyperactivity (8), Integration and inclusion (12), Concentration, attention, exertion (19) and Aggressiv-ity (19). Some essential themes are represented by a moderate number of references, as Individual variation, individualization (33), Teaching and

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learning in preschool (42), Minorities (45), and Social conditions in mathematics and ecology (78).

Although the literature displays many documents on physically im-paired or otherwise disadvantaged students, there is a lack of penetrating studies concerning the connection between their mathematical needs and learning conditions. There are extensive studies on the quality of life and social competence of socially disadvantaged persons. But it appears from these studies that solid analyses have been neglected concerning the rela-tion between mathematical subject matter curricula and learning oprela-tions for the special needs student.

A consequence of these reflections is to study a change from traditional subject matter curricula to an alternative “social mathematics” curriculum that concentrates on mathematical topics of present and future social rele-vance for the special needs students, maybe a prevocational experience. The author suggests that the ecological network idea is useful to help the disabled student to create an appropriate study programme together with teachers, classmates and, outside the class, parents and family. This approach may also vitalise the social climate in a more general sense.

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