Promotie Ed de Moor
Van Vormleer naar Realistische Meetkunde
Een historisch-didactisch onderzoek van het meetkundeonderwijs aan kinderen van
vier tot veertien jaar in Nederland gedurende de negentiende en twintigste
(From `Vormleer' to Realistic Geometry: A historical-didactic research
into geometry teaching to children of four to fourteen years of age during the
nineteenth and twentieth centuries in the Netherlands)
Utrecht, April 26, 1999. Time 12:45.
De handelseditie van het proefschrift Van vormleer naar realistische
is nummer 33 in de reeks CD-b Wetenschappelijke Bibliotheek.
708 pagina's, illustraties, foto's, tabellen, drie tijdbalken, bijlagen,
notenapparaat, personenregister, uitgebreide literatuurlijst.
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From `Vormleer' to Realistic Geometry describes the history of the development
of geometry teaching to children of four to fourteen years of age in the
nineteenth and twentieth centuries in the
Netherlands.1 Historical aspects are
dealt with in Parts A, B and C. Part D contains the report and the analysis of
an empirical study on the current state of geometry teaching in 1995 in the
eighth grade of primary school (twelve-year-olds). Viewpoints of those directly
involved in geometry teaching at primary school level are also discussed in Part
D, based on three questionnaires. These are primary school teachers active in
the eighth (highest) grade, teacher educators from pabo's (Pedagogische Academie
Basis Onderwijs, Colleges of Education) and school advisors.
The research that constitutes the basis of Parts A, B and C is of a historical
didactic nature. That means that the curriculum, teaching methods, motivation
and goals of geometry as a school subject form the focal point of the study in
relation to relevant historical developments in science, in society and in
Part A contains the history of vormleer in primary and infant schools in the
nineteenth century. In addition to the concepts of `word' and `number',
Pestalozzi also took the concept of `shape' as a starting point for education.
Based on this concept, he created the school subject Formenlehre at the
beginning of the nineteenth century. The main aim of this `geometry' was
learning how to `anschauen', but also to promote thinking, speaking, working
systematically and aesthetic development.2
Learning vormleer as it was developed
in Die Elemente der Form und Größe (Elements of Form and
Size), by Joseph Schmid, began with dots and line segments. An important aspect
of this pure form of vormleer was the synthetic construction and ordering of
configurations of dots and line segments under specific conditions, which is
now called `geometrical combinatorics'.
In their educational work, J.F. Herbart, F.A.W. Diesterweg and F.W.A. Fröbel
also used the concept of Anschauung as a keystone and each developed their own
methods for informal geometry teaching. In the Netherlands, the
teachers-cum-educators P.J. Prinsen, R.G. Rijkens and B. Brugsma were active in
propagating Pestalozzi's vormleer. The teacher Dirk van Dapperen, who had been
taught by Pestalozzi, wrote a textbook founded on Schmid's ideas, which was to
determine the character of vormleer in the Netherlands until the middle of the
However, vormleer was not popular with Dutch teachers because of its ambiguous
goals and formal character. In the first half of the nineteenth century, the
subject was taught in no more than 10 percent of all primary schools. It was
nevertheless made compulsory under the Education Act of 1857, as a matter of
fact as a main subject. The influential official administrator D.J. Steyn Parvé,
Minister J.R. Thorbecke's right-hand man, contributed to this by especially
recommending Herman Bouman's textbook. Bouman's views originated in Diesterweg's
ideas. The latter saw vormleer simply as visual and practical geometry. After
1857, Schmid's pure vormleer had, to all intents and purposes, been dispensed
with, and textbooks with a more visual character were introduced, concentrating
more on applications. About 1875, an important contribution was made by the
renowned mathematics educator Jan Versluys. He regarded the subject as visual
geometry and drew up a lucid and practical syllabus, primarily based on the idea
that vormleer should be a visual introduction to formal geometry in secondary
education. Despite clarification of the goals and despite improved textbooks,
for many teachers the subject remained a disaster area.
During the Dutch industrial revolution the demand for better educated craftsmen
increased. Under pressure from business and industrial circles and because of
the increased consciousness of the cultural importance of drawing, a growing
body of opinion wanted this subject, which was already taught in 25% of all
schools, to be amalgamated with vormleer. In the summer of 1889, these practical
deliberations unexpectedly led to the abolition of vormleer by the Dutch
Parliament. It was axed as a compulsory subject and drawing took its place. All
attempts over the best part of a century to make geometry a feature of Dutch
primary education had thus failed.
Following from discussions on vormleer, education circles were becoming
increasingly aware of the difficulties involved in setting up formal
entry-level teaching in geometry, the subject which, since Euclid's
Elements, seemed ideally suited for learning to reason. Educators
wondered more and more often whether twelve was not too early an age to
start geometry, according to Euclid's logic-deductive method. The
principle of Anschaulichkeit was (re)discovered as an essential general
teaching aid and was not only propagated in geometry
teaching.3 At the
same time, Diesterweg propagated heuristic or discovery learning. In
particular, the genetic principle was (re)discovered for geometry,
initially in a historical-genetic sense. As early as the eighteenth
century, A-C. Clairaut had suggested taking the historical development
of geometry --the practical discipline of surveying and astronomy-- as a
starting point for developing a learning strand. But towards the end of
the nineteenth century, the psychological-genetic principle, i.e. taking
the cognitive development phases of the child as a basis, also grew in
importance. In the Netherlands, it was chiefly Jan Versluys who
initiated the systematic study of the teaching of mathematics.
Part B explores the period between 1890 and 1970, in which geometry was no
longer compulsory in primary schools (six to twelve-year-olds). There was
however evidence of geometry activities in nursery schools.
That is the reason why the work of Friedrich Fröbel is subjected to a
mathematical-didactical analysis in Part B. With his so called `gifts'
(playthings for learning), Fröbel laid the foundation for a practical geometry
programme for nursery schools. Thanks to the efforts of an educator from Leiden,
Wybrandus Haanstra, this had an enormous influence on Dutch nursery education.
Mathematical aspects remained implicit, however, and the developments had no
influence on normal primary school curricula, from which geometry had been
expunged since 1889. This study has shown that practical Fröbel materials and
activities were a new impetus for curriculum development during the
As a sequel to developments in teaching methods at the end of the
nineteenth century, Part B also investigates activities and
contributions made by a number of practitioners and thinkers in the
field of early geometry teaching at secondary level, notably those of W.
Reindersma, T. Ehrenfest-Afanassjewa, E.J. Dijksterhuis, P.M. van Hiele,
D. van Hiele-Geldof and H. Freudenthal.
The principle of activity-based learning was given new meaning at the start of
the international Reform Movement at the end of the nineteenth century,
particularly in the form of discovery learning by experimenting in real-life
situations. For the first steps in geometry, this implied reconsidering the
logic-deductive method. Felix Klein and other learned mathematicians and
educators disagreed with this approach and stood up for a visual, informal
introduction to geometry. International initiatives were taken by scientific
mathematicians to improve geometry teaching. In the Netherlands, the mathematics
teachers J. Kleefstra and G. Wolda attempted this with their geometry textbooks,
but these individual initiatives were of marginal importance only. The first
educator to publish a popular geometry textbook which moved away from
traditional views was Reindersma. With his theoretical observations, this
pioneer became an authority in his field.
In 1924, Tatiana Ehrenfest-Afanassjewa published the brochure Wat kan en moet
het meetkundeonderwijs aan een niet-wiskundige geven? (What Can and Should
Geometry Teaching Offer a Non-Mathematician?), in which she argued the case for
an intuitive introduction to geometry. This led to heated discussions with
Eduard Jan Dijksterhuis which were published in the first volume of a Dutch
journal for the teaching of mathematics, Euclides. Dijksterhuis also wanted to
change mathematics education, but from an epistemic point of view, thus
retaining and even reinforcing its formal character. Ms Ehrenfest was not given
much support. Geometry education remained very much as it was, but the strictly
logical approach was more or less disregarded in the classroom.
Ms Ehrenfest carried on with her work, however, and in 1931 she published her
Übungensammlung (Collection of Problems), in which she substantiated her ideas
from a practical point of view. This book contains a unique collection of
problems of an entirely different nature from the traditional ones. It presents
everyday geometrical phenomena that could be examined by children of ten years
old or even younger. Accordingly, these problems served to stimulate children's
intuitive notions of geometric concepts and properties, thus forming a basis for
later formal and systematic work. This publication, too, which Ms Ehrenfest
called an introductory course, had little direct effect on geometry education at
first, but was to play an important part at a later stage.
In the Thirties the Wiskunde Werkgroep (Mathematics Study Group) was founded as
a study group of the Dutch branch of the international New Education
Under the inspiring leadership of Ms Ehrenfest this group of mathematics
teachers was to remain active up to the Seventies, and should be seen as the
platform for education pioneers of that period. Educational studies inspired by
this group were carried out on pre-geometry teaching. In the new mathematics
curriculum of 1958 this group was also instrumental in creating the possibility
of an intuitive start to geometry in secondary education.
In the Fifties, Pierre van Hiele published his theory of cognitive levels of
learning in mathematics, for which he achieved national and international
renown. His wife, Dina van Hiele-Geldof tested this theory in a practical
teaching experiment in geometry in the first secondary school year. Although the
programme was based on the traditional geometry curriculum, a number of Ms
Ehrenfest's ideas were also used.
New Math entered the international forum in the 1960s. This approach to
mathematics education used formal mathematical structures as a starting point,
and subsequently adapted them to secondary and even nursery and primary school
levels. Psychologists such as J. Piaget, J. Bruner and the
mathematician-cum-psychologist Z. Dienes supported this reform, and their
experiments and theories confirmed that psycho-mathematical structures in
children could be developed from an early age. They worked in accordance with
the psychological-genetic principle, but tended to neglect the
historical-genetic component. New Math also influenced secondary school teaching
in the Netherlands, so that in 1968 geometry was abolished as a separate
subject. It was replaced by a type of transformation geometry which itself
prepared the way for arithmetized geometry using vectors. Hence, the aspect of
Anschaulich-keit disappeared from geometry teaching.
From 1945 onwards, Hans Freudenthal became increasingly involved in mathematics
education in the Netherlands, particularly as chairman of the Wiskunde
Werkgroep. Internationally, too, he was making a name for himself. Although at
the beginning he agreed, to some degree, with New Math, he soon began to oppose
it and became a formidable adversary of this revolution. In his view, New
Math was reprehensible, especially for primary education.
Traditionally speaking, assigning formal value to mathematics, and in
particular to geometry, was an important means of justifying the
teaching of this subject. This argument has been shown to have been an
almost constant factor in the last two centuries, even though there were
periods in which the practical value of mathematics took a more
prominent position. Under the influence of the metamorphosis of science,
society and economy after the Second World War, the school of thought
stressing the formal value of mathematics began to take a back seat. In
the 1950s, Freudenthal, who at the time attached little credence to the
formal value, instigated a public discussion with Ms Ehrenfest who, on
the contrary, trusted in the teaching of mathematics to develop
proficient cognitive patterns. In the last chapter of Part B, the
historical question of the justification of mathematics, above all its
formal value, is examined separately.
The main conclusion of Part B is that between 1890 and 1970 a foundation was
laid for reforms which since 1970 have taken place in Dutch mathematics
education. People of note, such as Ms Ehrenfest, kept this process going and by
establishing the Wiskunde Werkgroep in the Thirties, for the first time created
an educational platform from which developments could find their way into
everyday teaching. Although at the time developments in thinking about geometry
were directed in the main towards the level of the first secondary school year,
they do prove to have been of considerable consequence for primary schools as
The history of the genesis of realistic geometry for primary schools and the
transition year of secondary schools is addressed in Part C. This period covers
about twenty-five years and began around 1970.
At the end of the Sixties, New Math seemed to be making its way into Dutch
primary education as well. Under the inspired guidance of Edu Wijdeveld and Fred
Goffree, a number of teacher educators at Colleges of Education founded the
`Wiskobas' movement (Wiskunde op de basisschool, Mathematics in Primary Schools)
in 1968. Wiskobas' aim was to adapt reforms in mathematics education to the
Dutch situation and traditions. In 1971, they were given a seat in the newly
founded IOWO (Instituut voor Ontwikkeling van het WiskundeOnderwijs, Institute
for the Development of Mathematics Education). Freudenthal was the first
professor to become director of this institute, which later changed its name to
In the years between 1971 and 1981, a new model curriculum for primary schools
was developed at the IOWO, in addition to a new teacher training programme and
materials for in-service training. The practical materials were backed up by
exploratory developmental studies in a design school. This work was
scientifically validated by Adrian Treffers in a publication about the goals in
mathematics education. Since 1979, the term `realistic mathematics education'
has been used to describe the effects of this school of thought.
Correspondingly, the term `realistic geometry' is used.
Geometry took up a prominent position in the model curriculum developed by
Wiskobas. The leading authors were F.J. van den Brink, J.H. ter Heege and L.
Streefland. Although some authors were at first still influenced by New Math
structuralist and empirical trends, a specific Wiskobas geometry did evolve from
1973 onwards, the cornerstone of which was formed by everyday geometric
phenomena. Emphasis was placed on `observing, doing, thinking and seeing', as
Goffree described the Wiskobas geometry concept in 1975. This was a concise
recapitulation of the didactic principles of Anschaulichkeit, concrete
operating, understanding and insight, which did indeed take shape in the actual
teaching materials. Sometimes existing teaching material was revived, such as
Fröbel's blocks, but activities designed with this material had a new and
realistic quality. The curriculum, remarkably, had much in common with Ms
Ehrenfest's view on the intuitive introduction to geometry and it also had a
striking resemblance to several activities described in her
Übungensammlung from 1931.
Influenced by this work of Wiskobas, between 1976 and 1981 at the iowo A.
Goddijn, M. Kindt and G. Schoemaker developed a great many geometry teaching
units for secondary school transition classes at the iowo. The aspect of
sighting was central here, so that this approach is often referred to as `vision
In the 1980s, the effects of the Wiskobas work became apparent in the new,
realistic textbooks for primary schools. Geometry was also given a place in
them. In the textbook series Rekenen en Wiskunde (Arithmetic and Mathematics) by
K. Gravemeijer et al., geometry was combined with measuring and ratio to become
a fully-matured learning strand. However, in teaching practice the topic was
often seen as a side-issue or an incidental subject. Halfway through the
Eighties, at the request of the nvorwo (Nederlandse Vereniging voor Ontwikkeling
van het Reken- WiskundeOnderwijs, Dutch Association for the Development of
Mathematics Education), a study was carried out into the possibility of setting
up a national curriculum for realistic mathematics education in primary schools.
This resulted in the publication of a programme draft Proeve I by A. Treffers,
E. de Moor and E. Feijs in 1989 in which geometry was given a place as well.
Meanwhile, the slo (Stichting voor Leerplan Ontwikkeling, National Institute for
Curriculum Development), as well as other experts and representatives in the
field had been asked by the Minister of Education to define achievement goals
(to be attained by all students) for primary education. Proeve I was taken as a
starting point for mathematics, and an abridged version of the goals for
geometry was also adopted. Whilst the report in question was being drawn up, the
geometry goals were all but removed in 1989. Since 1993, however, these
achievement goals for primary education, including those for geometry, have been
adopted officially. In the national standard final test for primary schools
drawn up by the Cito (Centraal Instituut voor ToetsOntwikkeling, National
Institute for Educational Measurement), items on geometry are now included.
At the start of the 1990s, the Ministry required that a new mathematics
curriculum be developed for the first three to four years of secondary
education. Here was the opportunity to regenerate geometry education, which had
been languishing since 1968. The ideas of vision geometry from the Seventies
were further developed, so that the realistic approach took shape for this
level, too. Be that as it may, there is no continuity between primary and
secondary education in relation to vision geometry.
It can be concluded that realistic mathematics education originated in the
period from 1970 to the present day. Preparatory geometry in particular has been
given a complete face-lift as a subject, the main aim of which is to actually
grasp space and its phenomena. It is extraordinary that, having been absent from
primary schools for over a century, this subject is now back again.
Six research themes are the principle constituents of this study. In Part E,
the first five are answered and discussed. The sixth question arises from Part D
and concludes this summary. Here follows a summary of the answers to the first
The historical analysis shows that the development of geometry education in the
last two centuries is distinguished by the search for suitable subject matter,
teaching methods and goals. The first three research questions are based on
these three aspects. What evidence is there of constant and what of variable
As far as subject matter is concerned (question 1), the Euclidean tradition was
continually pressed into service. This constant factor was highly effective in
blocking possibilities for imbuing geometry education in primary schools with
new ideas. Scientific developments in geometry in the nineteenth century were
also ineffective in improving the matter at the primary school level. Although
Pestalozzi and his assistants wanted to move away from the Euclidean framework
at the beginning of the nineteenth century, they too reverted to the traditional
way of introducing geometry by using dots, lines and shapes. The spatial play
school material for infants developed by Fröbel in the mid-nineteenth century
was a different matter: this material was suitable for further development for
the benefit of the older pupil. Unfortunately, this did not come about. It was
only around 1930 that Ms Ehrenfest's ideas were instrumental in ousting
Euclidean tradition. Nevertheless, it took almost half a century before any
effect was noticeable.
Teaching methods (question 2) developed at a continuous but tardy pace. The
principles of genetic development, the heuristic approach, the activity
principle and Anschaulichkeit appear to be constant factors in the history of
geometry teaching. In almost all attempts at reform of geometry education, the
starting point was to be found in dissatisfaction with the formal approach. Time
and again, a link was sought with the child's way of thinking in its own
environment. The didactic principles themselves underwent further growth and
sophistication, yet it remained difficult to find the right methodological
means. For all that, under the influence of a few scientists and practitioners,
these didactic principles have gradually found an educational audience.
The study of constancy and change in the goals (question 3) of geometry
education has shown that this problem was primarily dominated by justifications
of the value of the subject itself. Generally speaking, in the first half of the
nineteenth century the subject was especially valued for its formality. For the
following fifty years, practical values were preponderant, and this same cycle
was repeated in the twentieth century. The reasons for this are to be found to a
large extent in developments in society and science. In the first half of the
nineteenth century, the emerging preparatory school with its aspirations to
educate children to become reasonable-thinking citizens played a part. After
that, the industrial revolution had a more practical impact on education.
Following on from that, the first half of the twentieth century was a period in
which education expanded, again with the aim of giving working-class children
better schooling. Technological progress after the Second World War also
affected education, so that mathematics education was increasingly justified on
the grounds of its applicability. Arguments such as preparatory value, but also
personal value and aesthetic development were still put forward to justify math
The fourth research question centres around those processes which led to the
official introduction and eventually abolition of vormleer/geometry. By
examining textbooks, journals, archive documents, inspection and parliamentary
reports, an attempt has been made to understand the often peculiar decisions
taken in this matter. A remarkable and constant factor is that up until now
innovations were almost always introduced by individuals who then played a
tenacious part in the process. Final decisions as to introduction or abolition
of geometry were almost always taken at the highest policy levels, in the last
century even without prior expert advice having been called in. Even when advice
was requested, for example in 1968 when the New Math approach was introduced at
secondary school level and the old geometry disappeared for good, the government
still took the final decision in that matter. The extent to which personal
interventions did indeed play a role became apparent from the course of events -
as they were unravelled by this study - leading to the near abolition of
geometry from the achievement goals for primary schools in 1989. It seems that
accidental circumstances and personal influence often swayed the final decisions
to a larger extent than would have been expected by looking at the actual
The fifth research question concentrates on the discussion as to whether the
historical survey undertaken can also be of value for the present day. It is
true that there is a certain friction between scientific geometry and an
intuitive introduction to the subject. What is also true is that no proof could
be given for justifying geometry on the grounds of its formal value and it took
considerable effort to demonstrate its practical value using concrete teaching
material. The study also shows that pre- and in-service training are
indispensable if a new subject is to be introduced successfully. Scientific
research into learning processes in young children learning geometry has so far
been incidental. The few experiments that have been done clearly show that
significant didactic and practically applicable ideas can be gleaned from such
studies. If anything can be learned at all from this study, then it is that to
successfully introduce a new geometry approach in teaching, a balance must be
struck between the scientific, social and educational significance of this
Part D offers an answer to the sixth research question. Here the results of an
empirical study on the status of realistic geometry education in 1995 are
discussed. In that year, 923 twelve-year-olds were given a geometry test at the
end of their primary schooling. The test was developed by the author of this
study, in association with assistants at the Freudenthal Institute (J. Menne)
and the Cito (J. Janssen and J-M. Kraemer). The evaluation and the compiling of
a report were also done cooperatively.
The average correct score for the geometry test, which was developed for this
study, was 62%. The score of the national standard final arithmetic test was 71%
for this group. The scores were as follows for the various aspects of geometry:
orientation and localization (63%), sighting and projecting (53%), spatial
reasoning (64%), transforming (68%), constructing, measuring and calculating
(56%) and area and scale (85%). In this test boys performed significantly better
than girls. The partial correlation between geometry and arithmetic is weak
A qualitative analysis was made of the work done by the pupils on a number of
problems. Performance on localizing, simple symmetry problems, restructuring
figures and spatial perception was satisfactory to good, but performance was
weak bordering on very weak as soon as constructing or spatial reasoning was
required. The pupils had little grasp of the concept of angles in particular. As
no data on the amount of geometry taught or the textbooks used were available,
the quantitative and the qualitative results have to be interpreted with due
Surveys were also conducted among three groups involved in geometry teaching:
teachers in the highest classes of the primary school, mathematics teacher
educators at Colleges of Education, and school advisors. The response from
primary school teachers was 81% (N = 192), teacher educators 50% (N = 138) and
school advisors 37% (N = 156).
One of the main questions in all three surveys was whether one agreed that
geometry should be an achievement goal in primary schools. The answer to this
question was very positive: 93% of primary school teachers, 99% of teacher
educators and 95% of school advisors answered in the affirmative.
According to primary school teachers, there is sufficient geometry in current
textbooks. According to them, about 70% of what is in the textbooks is actually
put to effect. Slightly more than 60% of this group state that they had no or
insufficient training in geometry at college. However, there is little call for
further in-service training (28%).
The total time spent doing mathematics in the four-year course at Colleges of
Education averages 360 hours. One third is face-to-face instruction (lessons),
the rest is private study. On aggregate, geometry takes up about 10%.
Nevertheless, it is considered to be an important subject by the teacher
Advisors think that only 50% of the geometry available in textbooks is actually
used in primary schools. Three quarters of this interviewee group believe that
geometry for primary school teachers is as difficult or more difficult than
arithmetic. They also estimate that these teachers make light of the importance
of geometry as opposed to arithmetic. In general, they think that primary school
teachers have little proficiency in geometry.
Both the survey conducted among teacher educators and that among school advisors
show that geometry in primary schools is still not considered a fully-developed
The author has drawn up a general proposal for a realistic geometry curriculum
for primary school children aged four to twelve. This proposal has been
presented to twenty experts. Their reactions trigger further reflection on
goals, subject matter and form of a primary school geometry curriculum.
In conclusion, the following ten recommendations have been made.
1. More detailed historical sociological research, especially on the
Wiskobas group of the Seventies, should be done into the life and work of
prominent individuals in mathematics education mentioned in this study.
2. Renewed historical didactic research is needed into mathematics
education in the Netherlands during the nineteenth and twentieth century.
3. The importance of Fröbel and Haanstra for Dutch nursery education should be
4. Prior to education development projects, a historical-didactic analysis
should be undertaken.
5. A conference should be convened with a number of experts with the aim of
reaching a consensus on basic goals, content and form of a realistic primary
school geometry curriculum.
6. A comprehensible description of a programme should be included in the school
7. For primary schools, geometry tests other than multiple choice tests should
8. Short in-service courses in realistic geometry should be developed for
mathematics coordinators in primary schools.
9. Geometric reasoning in children of four to fourteen years old should be
studied. The connection between logic and spatial intelligence, the effects of
specific geometry teaching, the differences between boys and girls and the use
of geometry computer programmes should also be examined.
10. Geometry curricula in primary and secondary schools should be coordinated.
Pestalozzi pioneered the idea of introducing geometry as a foundation subject
for young children, side by side with language and arithmetic. Fröbel managed to
construct geometric material and activities suitable for the youngest children.
It is only in the last few decades that geometry topics have been available that
can be used in an informal geometry curriculum for children aged four to
fourteen. The author of this study is of the firm opinion that putting such a
curriculum into practice in primary schools would contribute to the development
of spatial reasoning in children and that this would be good preparation for
more formal geometry. Implementation will only succeed, however, if geometry is
given a substantive position in the primary school curriculum and if sufficient
emphasis is placed on pre- and in-service training of teachers.
1. The literal meaning of vormleer is `the study of shapes'.
2. Pestalozzi's conception of Anschauung was wide-ranging. `Intuitive
perception' could be a translation of the German word, but this does not cover
all meanings. In this study the term is restricted to (1) the physical
observation of concrete (mathematical) objects and shapes, (2) their (drawn)
representations (figures), (3) the perception of a mental image of such an
object, and (4) the mental operations involving these mental images.
3. We speak of Anschaulichkeit of a problem or more generally of a text when it
is posed in such a form that it gives the opportunity to create mental images
and visual understanding. Usually this is done by means of drawings, schemes,
diagrams or other visual aids.
4. `Vision geometry' is based on looking at (observing), perceiving,
representing and explaining spatial objects and spatial phenomena, in which the
idea of the straight line as a vision line (sighting) and a light ray plays a