Literature on Special Educational Needs in Mathematics

Transcription

Literature on Special Educational
Needs in Mathematics
A bibliography with some comments
Olof Magne
Magne, O. (2003). Literature on Special Educational Needs in Mathematics: A bibliography with some comments. (4th Ed.) (Educational and Psychological 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 published 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 often prefer words beginning with the prefix “a” or ”dys”, such as “acalculia” or “dyscalculia”, 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 observed together with physical impairment of the individual, wrong learning
strategies, unwise teaching, social circumstances etc.? Mathematics learning has been subjected to many conflicting impulses, not always favourable. 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, mathematics, special education, special educational needs.
© School of Education, Malmö University
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
Affect and motivation ..................................................................... 34
3
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
Integration and inclusion............................................................... 224
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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
Thinking, active construction, and representation ........................ 408
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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 Bowen (2000), and Magne (1996b), see the references. Here and in the following 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 connection 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, particularly 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 ”problems in learning mathematics”, ”unsuccessful learning of mathematics” or
other similar expressions which can be treated as synonyms. I like to advise 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 achievement.
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 mathematical 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
definitions I want to refer to Magne (1998).
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Terms like Arithmasthenia (Ranschburg, 1905), Acalculia (Henschen,
1920) and Dyscalculia (Gerstmann, 1924) are restricted to low achievement in elementary 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 Dysmathematica, 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 practical 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 Administration 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 options 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, research 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 teachers experienced when they instructed their students. One consequence was
a belief that only some students can acquire mathematical knowledge and
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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 content deviation model (Magne, 1998). The remedy would be either to exclude 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 follow 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 therapists the obvious means to habilitate or rehabilitate a low achiever properly. In relation to special educational needs in mathematics, this approach
may be called the behaviour deviation model (Magne, 1998). The treatment consists of organic and 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 necessary 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 ethnomathematics 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 curriculum 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 natural settings of both student and teacher. The author looks upon mathematical knowledge as the dynamic interplay of mathematical 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 learning
(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, treatment 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 thoroughly study higher mental processes.
The author suggests that simple mathematics themes should be balanced 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 instruction 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’ characteristics 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 hierarchy but are placed on a par independent of each other. They may be coupled 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 bibliography 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
Ability and mathematics
Affect and motivation
Aggressivity, asocial behaviour, criminality
Bibliographies
Case studies
Communication
Concentration, attention, exertion
Content areas: General
Content areas: Problem solving (see also Problem solving)
Content areas: Number
Content areas: Calculation
Content areas: Geometry. Measurement. Money.
Content areas: Algebra. Functions.
Content areas: Statistics. Probability.
Content areas: Calculator, Computer
Curriculum, syllabus.
Diagnosis
Didactical considerations
Equality, inequality
Errors, misconceptions
Gender
Genetic conditions
Hyperactivity, impulsivity
Impairments, various multiply disabilities
Impairments, as to hearing impairments
Impairments, as to sight impairments
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Number
48
291
19
51
61
6
19
6
8
63
147
40
64
26
83
59
259
205
3
264
28
26
8
5
126
60
Impairments, as to cerebral palsy etc.
Individual variations, individualisation
Instructional aids
Integration and inclusion
Language and mathematics
Learning, instruction: theories
Low and high achievers
Mathematical competence in everyday life
Memory and mathematics
Mentally handicapped
Minorities
Movement/motility, motor skills and mathematics
Neurological or neuropsychological characteristics
Nonverbal disabilities
Perception and mathematics
Planning, strategy, metacognition of students
Problem solving
Social conditions in mathematics, ecology
Specific difficulties or underachievement in mathematics
Teaching and learning in preschool
Teaching and learning in primary education
Teaching and learning in secondary and post-secondary education
Teachers
Thinking, active construction, and representation
40
33
107
12
153
55
89
86
42
203
45
15
399
22
60
51
129
100
37
42
216
63
125
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Let us begin with quantitative notions based on the categories. First of all,
a few categories contain many times more documents than the least frequented 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)
Affect and motivation (291)
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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 instance:
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 reason 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 Impairments, 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 question. In the category Mentally handicapped 56 per cent of the documents
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mainly discuss skill in the four rules. The neurological or neuropsychological 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 qualitative 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 handicapped, phobic and emotionally disturbed students, respectively, may, as
kindred groups show similar, typical learning reactions due to their specific 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 underachievement appear as complex and multi-factored disabilities. The literature 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 student as unique. One inference of this analysis is that planning for mathematical 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 demands 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 conditions for the individual student.
Qualitative interpretations
From the present categorization of the documents in the Magne bibliography (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 reading, 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 university 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 student 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 described, 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 promote mastery of the mathematical content (may indicate tuition according 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 handicapped students. Apparently, studies of mathematical learning
among mentally retarded students are mainly directed towards numeration, calculation or visual perception, not mathematical thinking/reasoning by the students.
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2.
3.
4.
5.
6.
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 incapacitated to automatise skills in arithmetic prescribed by authorities.
c. Also neuropsychological studies seem to refer to student defects related to basic mathematical formal standards rather than socioeducational reaction patterns (indicating belief in both the content
deviation model and the behaviour deviation model).
In some studies mathematical errors are interpreted as results of erroneous automatisation, slips of the tongue, wrong concentration, verbal
misinterpretation etc. This may indicate belief in the behavior deviation
model.
Many authors presuppose circumscribed organic impairment as the
main cause of low attainment, exclusively, in spite of obscure foundation, for instance supposed lesions to the nervous system (behaviour
deviations).
Anxiety is a very common theme, usually in one or both of two varieties: mathematics anxiety and test anxiety, where innate behaviour, neuronal disposition, learning conditions and/or instructional conditions are
looked upon as instrumental to the anxiety. The building up of mathematics anxiety is sometimes looked upon from the behaviour deviation
model or the content deviation model but seldom with reference to the
factor-interplay model.
There is a lack of interest among authors to relate special mathematics
needs to socio-educational system factors.
Motivation, attribution and belief theories as well as the self-efficacy
conception have been considered as causal factors despite doubtful evidence. 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,
20
usually very mild disabilities or educational needs with small discrepancies 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 relations between language disabilities (e.g. dyslexia) and cases of underachievement in mathematics.
4. There has been a limited interest in underachievement in mathematics.
In some countries results from this research seem to have started animated discussions.
5. Aggressiveness has seldom been studied in connection with special
educational needs in mathematics
6. Mathematical attainment and needs of socially and emotionally disturbed 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 emotional 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 mathematics special needs students display very complex behaviour profiles.
2. Some authors claim that they have found a causal relationship between
some very simple arithmetic skills and neurological lesions to frontal,
21
3.
4.
5.
6.
occipital and parietal lobes of the left hemisphere. Some parts of the
right hemisphere are described as sites of geometrical skills and possibly problem solving skills.
As to more complex mathematical topics there is no proved relationship
between neuronal functions and mathematical achievement.
There are only vague unverified hypotheses concerning causal genetic
factors and mathematical ability.
Interrelations also seem to be uncertain between social conditions and
mathematical ability and mathematical achievement.
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 training, 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, educational, psychotherapeutic and social (sociological). It seems to be a remarkable 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 diagnosis 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 describe mild learning disabilities treated with various forms of educational
and social tuition, for instance the educationally subnormal children. Usually little is written how they are tutored. There is very little research reported on the education of severe cases, and apparently this refers to all
sorts of impairments, disabilities or handicaps. Severe and moderate difficulties of all kinds are sometimes given a mixed treatment. But there are
few reports on the efficacy of the various methods.
To a great extent the bibliography reveals that there are various kinds of
22
grave controversies concerning adequacy of treatment principles, for instance
• 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 educational 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 relearning seem to have selected experimental problems and procedures more or
less by trial and error and examined the outcome haphazardly. The research development may have let circumstances decide in an undirected
way, and passing fancies may have been more influential than wellstructured 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 prob-
23
ability and statistics may be applicable to mathematics special needs students (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 information 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 cognition and achievement, primarily in Grades K–12, with significant attention
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 prepared cooperation between universities and nations.
24
Special educational needs in mathematics are disregarded both in political 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, aggressivity, 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 interest. 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 Aggressivity (19). Some essential themes are represented by a moderate number of
references, as Individual variation, individualization (33), Teaching and
25
learning in preschool (42), Minorities (45), and Social conditions in
mathematics and ecology (78).
Although the literature displays many documents on physically impaired 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 relation between mathematical subject matter curricula and learning options
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 relevance 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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Literature on Special Educational Needs in Mathematics

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