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partial underachievement at each level of intelligence, there has been so lit-
tle proof in the past that this definition is feasible that it has now been rejec-
ted for dyslexia as well (Grissemann & Weber, 1982). Moreover, it seems to
make little sense for pedagogical reasons to dismiss all students who do not
fall under this definition of discrepancy, but are nevertheless in need of in-
dividual help in the field of arithmetics.
1.2 Research and Explanatory Approaches
Psychodiagnostics. Psychodiagnostics, which is oriented toward test metho-
dology, emerged from the problem of selection, that is, the need to identify
appropriate versus less appropriate candidates for a specific demand. It
created the construct of "intelligence," which seemed to justify selection and
assignation to certain school types. Although different cognitive abilities are
considered to determine intelligence within the various intelligence models,
R
. Biehler, R. W. Scholz, R. Sträßer, B. Winkelmann (Eds.),
Didactics of Mathematics as a Scientific Discipline, 291-301.
© 1994 Dordrecht: Kluwer Academic Publishers. Printed in the Netherlands.

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almost all of them have in common that components like spatial intuition,
short-term and long-term memory, language factors, and calculatory compe-
tence must be included. However, in testing methodology, this creates the
problem that the requirement of having factors of intelligence correlate only
in a small way eliminates the very components that are responsible for cal-
culatory competence, for the latter must prove to be independent of the
other factors of intelligence. This impeded a clarification of mathematical
ability by methodology alone. Thus, Ginsburg's assessment of this approach
is devastating:
By contrast, standard tests are of little value . . . . They yield unhelpful labels like
"low mathematical aptitude," and worst of all, they fail to reveal children's
strengths. The tests say nothing specific about what the child can do and about
how instruction should proceed. All this is positively harmful to the child who has
trouble learning. (Ginsburg, 1977, p. 149)
Due to the simplistic idea about the causes of dyscalculia, it was impos-
sible to establish elaborate didactical-methodological approaches to the
problem. Curricular aids were derived from test items in the vein of associa-
tionism, and appropriate exercises to improve arithmetical competence were
developed. This kind of task analysis thus mostly led to a simple drill-and-
practice unit that subdivided the subject matter to be learned into small
steps. Resnick characterizes the behaviorist methods developed from the
psychodiagnostic approach as follows:
[Skinner
]
and his associates showed that "errorless learning" was possible
through shaping of behavior by small successive approximations. This led natu-
rally to an interes t in a technology of teaching by organizin g practice into care-
fully arranged sequences throug h which the individua l graduall y acquires the el-
ements of a new and complex performanc e withou t makin g wron g responses en
route. This was translated for school use into "programmed instruction" – a form
of instruction characterized by very small steps, heavy prompting, and careful se-
quencing so that children could be led step by step toward the ability to perform
the specific behavioral objectives. (Resnick, 1983, pp. 7-8)
Special education. Special education tackles some factors of ability relevant
for the learning of young children and elementary school students in an iso-
lated way in specific assistance programs. The genesis of problems with
calculating is mainly seen in factors like (a) disturbance of the body schema;
(b) visuomotor integration disturbances; and (c) spatiovisual weakness of
grasp and representation (Johnson & Myklebust, 1971). This approach, ho-
wever, remains attached to a defectological view and is insufficiently speci-
fic with regard to subject matter and content to be able to derive detailed
statements about how arithmetical contents are learned.
Success is expected here of orthopedagogical exercise treatments concen-
trating on symptom clusters, which are made responsible as a basis for dis-
turbing learning or processing and integrating information. Besides these
measures, which, while focusing on the particular ability deficits, are not
specific with regard to content, special education tries to respond organiza-
GIFTED AND RETARDED STUDENTS
292

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tionally to the problem of impeded achievement. Students are integrated
into special classes or schools, absolving a slowed-down version of the ele-
mentary school curriculum. Special methodological approaches have not
been developed for arithmetic, the only basis being a "weeding-out" hypoth-
esis, that is, the subject matter curriculum is lightened and disburdened of
"superfluous" contents. Underhill, one of the major representatives of
American special education, considers only memory a factor relevant for
calculus:
Children with Low Mathematics Ability
Some children master mathematics concepts more slowly than do the majority.
They have difficulties conceptualizing mathematical ideas, understanding math-
ematical generalizations, and remembering computational procedures and basic
facts when such ideas are presented in conventional ways. There are many rea-
sons why children experience difficulties ... suffice it here to say that learning
styles, socioeconomic status, heredity, and teacher skill are but a few of the fac-
tors which are positively correlated with pupil sucess in mathematics.
Characteristics: Children with low mathematics ability are usually characterized
by low-normal intelligence, poor reading achievement, and poor memory. They
are frequently from homes of lower-class culture and may have parents whose
backgrounds reflect poor mathematics achievement and attitudes.
A. Since intellectual development is correlated with mental age, expect slow chil-
dren to perform at a less abstract level than bright and average children.
B. Since low-ability students progress more slowly, allow them more time at the
concrete and semiconcrete levels of concept development.
C. Use a multi-sensory approach to accomodate various learning styles.
D. Prevent practising of errors through careful developmental work and short sets
of practice exercises.
E. Introduce only one new development and more time for practice. (Underhill,
1977, pp. 39-40)