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ORIGINAL ARTICLE
Concrete models and dynamic instruments as early technology
tools in classrooms at the dawn of ICMI: from Felix Klein
to present applications in mathematics classrooms
in different parts of the world
Maria G. Bartolini Bussi •Daina Taimina •
Masami Isoda
Accepted: 18 October 2009 / Published online: 7 November 2009
FIZ Karlsruhe 2009
Abstract Most national curricula for both primary and
secondary grades encourage the active involvement of
learners through the manipulation of materials (either
concrete models or dynamic instruments). This trend is
rooted in the emphasis given, at the dawn of ICMI, to what
might be called an experimental approach: the links
between mathematics, natural sciences and technology
were in the foreground in the early documents of ICMI and
also in the papers of its first president, Felix Klein. How-
ever, the presence of this perspective in teaching practice is
uneven. In this paper, we shall reconstruct first an outline
of what happened in three different parts of the world
(Europe, USA and Japan) under the direct influence of
Klein. Then, we shall report classroom activities realized in
the same regions in three different research centres: the
Laboratory of Mathematical Machines at the University of
Modena and Reggio Emilia, Italy (http://www.mmlab.
unimore.it); the pedagogical space of Kinematical Model
for Design Digital Library at Cornell, USA (http://kmoddl.
library.cornell.edu/); and the Centre for Research on
International Cooperation in Educational Development at
Tsukuba University, Japan (http://math-info.criced.tsukuba.
ac.jp/). They have maintained the reference to concrete
materials (either models or instruments), with original
interpretations that take advantage of the different cultural
conditions. Although in all cases the reference to history is
deep and systematic, the synergy with mathematical mod-
elling and with information and communication technolo-
gies has been exploited, not to substitute but to complement
the advantages of the direct manipulations.
Keywords Models Instruments ICT ICMI
Felix Klein
1 Introduction
The relationships in teaching between mathematics and
the concrete materials of the real world have a long
history that does not suit the space of a single paper. We
shall limit ourselves to some classical quotations (Cas-
telnuovo, 2008). Jan Comenius (1592–1670) defended in
Didactica Magna (1657) the relevance of the manipula-
tion of concrete things in every individual experience of
knowledge construction:
‘‘Everything must be presented to the senses as much
as possible; to wit, the visible to the eye, the audible
to the ear, odors to the sense of smell, the tastable to
the taste, and the touchable to the sense of touch; and,
whenever something can be grasped by more than
one sense at one time, let it be presented to them at
one time. One may, however, if the things themselves
M. G. Bartolini Bussi (&)
Dipartimento di Matematica, Universita
`di Modena e Reggio
Emilia, Via Campi 213/B, Modena, Italy
e-mail: bartolini@unimore.it;
mariagiuseppina.bartolini@unimore.it
URL: http://www.mmlab.unimore.it
D. Taimina
Department of Mathematics, Cornell University,
Ithaca, NY 14853-4201, USA
e-mail: daina.taimina@cornell.edu; dtaimina@lanet.lv
URL: http://www.math.cornell.edu/*dtaimina/
M. Isoda
Graduate School of Human Comprehensive Science,
University of Tsukuba, 305-8572 Tsukuba, Japan
e-mail: msisoda@human.tsukuba.ac.jp;
msisoda@muc.biglobe.ne.jp; isoda@criced.tsukuba.ac.jp
URL: http://math-info.criced.tsukuba.ac.jp/
123
ZDM Mathematics Education (2010) 42:19–31
DOI 10.1007/s11858-009-0220-6
cannot be presented, use representations of them,
such as models and pictures. […] It is a mistake to let
rules in an abstract form go before, and afterwards
explain them in examples. For the light must go
before him for whom it is intended to shine. […]
Whatever is to be done, must be learned by doing it.
Mechanics do not detain their apprentices for a long
time with meditations: they put them to work at once,
that they may learn to forge by forging, to carve by
carving, to paint by painting, etc. So the pupils should
also learn at school to write by writing, to speak by
speaking, to count by counting, etc. Then the schools
are workshops filled with the sound of work’’
(Comenius, 1657).
These general methodological rules were, at the time of
Comenius, largely ahead of the teaching practice, as (at
least in Europe) there were no public educational institu-
tions for everybody. One century later in France, another
voice was raised, with explicit reference to the teaching of
geometry linked with ‘reality’ (whatever that means).
Alexis Clairaut (1713–1765) discussed the issue as follows:
‘‘Some authors put after each basic statement (of
geometry) its practical use: yet in this way they
establish the usefulness of geometry, without making
geometry learning easier. Because if any statement is
before its use, the mind can come at sensible ideas
only after having struggled with abstract ideas’’
(Clairaut, 1741).
He then suggested his method to evade the above
drawbacks:
‘‘I have planned to find all that could have given rise
to geometry; and I have managed to explain its princi-
ples as naturally as possible, like early inventors’’
(Clairaut, 1741). Hence, he claimed that in geometry
teaching it is necessary to start from measuring land (the
Greek etymology of ‘‘geometry’’: measurement of earth
or land).
These educational principles were available, although
neither widely shared nor widely applied, when, for the
first time in Europe, in the first article of the France
National Constitution (1791), the right to a system of
public instruction was stated: ‘‘Public instruction for all
citizens, free of charge in those branches of education
which are indispensable to all men.’’
In the first half of the nineteenth century, a German
educator, Friedrich Froebel (1782–1852), launched a
practice of active methods for children. He designed
open-ended instructional materials called the gifts, with
complementary occupations (Le Blanc, n.d.). These were
for use both in kindergarten and in school, and gave
children hands-on involvement in practical learning
experiences through play. Foundational to the develop-
ment of the gifts was the recognition of the value of
playing with blocks. Through proper use of the gifts, the
child progresses from the material to the abstract: from
the volumetric lessons offered by blocks, through the
two-dimensional planar ones elucidated by play with
parquetry tiles (flat, geometrically patterned wooden
shapes), to deductions of a linear nature drawn from
stick laying and to the use of the point in pin-prick
drawings. Points, in turn, describe a line, and the child
completes the logic by returning from 2D to the 3D
realm of volume through joining small malleable peas
with toothpicks and onto solid three-dimensional work in
clay.
Although Froebel’s work was mainly oriented to young
children, his approach was representative of an atmosphere
shared with contemporaneous mathematicians in Central
Europe. For instance, Gaspard Monge (1746–1818), a
mathematician deeply involved in French revolution, sup-
ported the creation of beautiful models of surfaces.
‘‘Monge is known as the father of differential
geometry and his efforts in the early 1800s to classify
surfaces by the motions of lines, along with his
descriptive geometry for representing three-dimen-
sional surfaces in two-dimensions, led naturally to the
construction of elaborate models made of tightly
stretched strings. One of his students, Theodore
Olivier (1793–1853), built some of the most beautiful
concrete models of mathematical concepts ever
made. He also made some money in the process: the
models were expensive. Olivier sold them to the
emerging technical schools in the United States,
which were attempting to emulate the example of
Monge and the Ecole Polytechnic.’’ (Mueller, 2001).
It is in this atmosphere that one must situate the work of
Felix Klein (1849–1925) and the other European founders
of ICMI.
This paper is divided into two parts organized around
two main issues:
•First part: the influence of Felix Klein at the dawn of
ICMI in different parts of the world (Europe, USA,
Japan).
•Second part: present applications in today’s classrooms
of the above seminal ideas in the same parts of the
world.
The former concerns the historical roots; the latter
concerns the present utilization of concrete models and
dynamic instruments, which are complementing yet neither
contrasting nor excluding the recourse to information and
communication technologies.
20 M. G. Bartolini Bussi et al.
123
2 First part: concrete models and dynamic instruments
at the dawn of ICMI
2.1 The European roots and the contribution of Felix
Klein
The use of concrete models and dynamic instruments was
common in Europe in the seventeenth and eighteenth
centuries (see for instance, Maclaurin 1720), but it had a
new impulse in the nineteenth century. In particular, the
second half of the nineteenth century was a time when
many new mathematical ideas were born, combining
together previously separated parts of mathematics.
Mathematicians began to build intricate models out of
wood, string and plaster. In Germany, the main advocate
for the use of concrete models and dynamic instruments
was Klein. In many ways, Monge in France and Klein in
Germany set the standards for how mathematics was taught
in Europe, Northern America and the Far East in the
nineteenth and early twentieth centuries (Klein & Riecke,
1904; see also Schubring, 1989). One of the most thorough
approaches in mathematics, to use concrete models and
dynamic instruments in education and research, is the
famous collection of models in Go
¨ttingen. This model
collection already had a long history when Felix Klein and
Hermann Amandus Schwarz (1843–1921) took over the
direction of the collection. Especially under the direction of
Klein, the collection was systematically modernized and
organized for the education of students in geometry and
geodesy. The first clear indication of an interest in model
building had appeared in the ‘‘Monthly reports of the Royal
Prussian Academy of Science in Berlin’’ in 1873, where it
was described that Ernst Eduard Kummer (1810–1893)
presented a plaster model of the Steiner surface, which he
had constructed himself. A wide production of models
began in the 1870s, when Felix Klein and Alexander Brill
(1842–1935) founded a laboratory for the construction of
models and instruments at the Munich Technische Ho-
chschule. The production and detailed study of models was
one of the purposes of the problem sessions directed by
Klein and Brill at the Royal Technical University in
Munich. Some of the models, which were really con-
structed only as exercises or examples, showed that these
types of visual aids were not at all superfluous; on the
contrary, they were of great value (Schilling, 1911).
Klein required his students to make models in connec-
tion with their dissertations on algebraic surfaces. Klein,
Brill and their students built a number of the models that
ultimately became a part of the collection marketed by
Brill’s brother Ludwig, the owner of a publishing firm in
Darmstadt. When Ludwig Brill took over the sales of the
models, they were put together in series, each being
accompanied by a mathematical explanation. Other
producers also offered models, devices, and instruments for
mathematics, physics and mechanics. In 1892, under con-
tract of the newly established German Mathematical
Society, Walter von Dyck (1856–1934) assembled a com-
plete catalogue of such products (Fischer, 1986). He was
one of the creators of the Deutsches Museum of Natural
Science and Technology in Munich, and he was also
appointed as the second director of the museum in 1906.
The Deutsches Museum was the first of its kind and its
ideas were soon copied by other science museums around
the world.
The Munich collection was considered so important that
later Klein exhibited the models on the occasion of the
World’s Columbian Exposition 1893 in Chicago. Models
from the collection of L. Brill were later purchased by
many mathematics departments throughout Europe and the
USA (Parshall & Rowe, 1991).
Later, Klein, as the first president of the International
Commission on Mathematical Instruction, also supported
in secondary schools in a very strong way the recourse to
models, instruments and practical work. He reaffirmed the
importance of work with models and instruments also in
his books: Elementary mathematics from an advanced
standpoint (Klein, 1924, 1925), the famous series for sec-
ondary mathematics teachers, to ‘‘put before the teacher, as
well as the maturing student, from the view-point of
modern science, but in a manner as simple, stimulating and
convincing as possible, both the content and the founda-
tions of the topics of instruction, with due regard for the
current methods of teaching (Klein, 1908, preface to the
first edition, published in Klein, 1924, p. III).’’
The influence of the title of this series was large: in
Germany, courses were opened for prospective teachers
with this exact name; the same name is still used in Italian
universities for the epistemological courses in the educa-
tion of prospective mathematics teacher.
Just to give an example of Klein’s approach, we may
quote what he wrote about the practice in calculating with
integers (Klein 1924), after having described in detail the
famous mechanical calculator Brunsviga (see Fig. 1).
‘‘Let us consider for a moment the general signifi-
cance of the fact that there really are such calculating
machines, […]. In the existence of such a machine we
see an outright confirmation that the rules of opera-
tion alone, and not the meaning of the numbers
themselves, are of importance in calculating, for it is
only these that the machine can follow; it is con-
structed to do just that, it could not possibly have an
intuitive appreciation of the meaning of the numbers.
[…] Although it is not historically authenticated, still
I like to assume that when Leibniz invented the cal-
culating machine, he not only followed a useful
Concrete models and dynamic instruments as early technology tools 21
123
purpose, but that he also wished to exhibit, clearly,
the purely formal character of mathematical
calculation.’’
For this very reason, Klein wished that every teacher of
mathematics should become familiar with calculating
machines, and that it ought to be possible to have it dem-
onstrated in secondary instruction. In this case, Klein
defended the recourse to mechanical instruments as a
means to become familiar with the very abstract distinction
between the syntactic and the semantic aspects in
mathematics.
Several dynamic instruments were discussed also in the
second volume on Geometry (Klein, 1925): for instance, a
polar planimeter (p. 14), a mechanism to perform affine
transformations (p. 75), a linkage to realize a circular
inversion and, hence, to guide a point so that it will
describe a straight line (p. 100). This last instrument,
called, after Charles-Nicolas Peaucellier (1832–1913), the
Peaucellier straight line mechanism, is represented in
Fig. 2.
In the same years, Franz Reuleaux (1829–1905), a
professor of mechanical engineering in Berlin, built a large
collection of 800 mechanical models in Berlin and mar-
keted 350 of them to universities around the world.
Unfortunately, much of these collections were destroyed
during World War II, but some originals and reproductions
of these models can be found in the Deutsche Museum in
Munich, the University of Hannover, Kyoto University,
Moscow’s Bauman Technical School, Karlov University in
Prague and possibly in some other places we do not know
yet. The largest collection of these models is in Cornell
University where there are 220 (from the originally
acquired 266) Reuleaux models (see Fig. 3). We shall
illustrate this point later.
Reuleaux believed that there were scientific principles
behind the invention and the creation of new machines,
what we call ‘‘synthesis’’ today. This belief in the primacy
of scientific principles in the theory and design of machines
became the hallmark of his worldwide reputation, espe-
cially in the subject of machine kinematics (Moon, 2002).
Hence, Reuleaux was a champion of the application of
mathematics. Reuleaux also devoted serious attention to
education and the role of mathematics:
‘‘The forces of nature which advance taught us to
look to for service are mechanical, physical and
chemical; but the prerequisite to their utilization was
a full equipment of mathematics and natural sciences.
This entire apparatus we now apply, so to say, as a
Fig. 1 Brunsviga calculating machine (courtesy of New Beginning
Antique at http://www.newbegin.com)
Fig. 2 Peaucellier straight line mechanism (http://KMODDL.library.
cornell.edu)
Fig. 3 Reauleaux kinematic model display in Sibley Hall, Cornell
University in 1887. Scientific American cover
22 M. G. Bartolini Bussi et al.
123
privilege. […] The instruction in the polytechnic
school has of necessity to adopt as fundamental
principles the three natural sciences—mechanics,
physics, and chemistry, and the all-measuring master
art of mathematics’’ (Reuleaux, 1876).
Franz Reuleaux incorporated mathematics into design
and invention of machines in his work Kinematics of
machinery. For mathematicians, he is best known for the
Reuleaux triangle, which is one of the curves of constant
width (see Fig. 4.) This curved triangle can be seen in some
gothic windows; it also appears in some drawings of
Leonardo da Vinci (1452–1519) and Leonhard Euler
(1707–1783), but Reuleaux in his Kinematics gave the first
applications and complete analysis of such triangles, and he
also noticed that similar constant-width curves could be
generated from any regular polygon with an odd number of
sides. (Taimina & Henderson, 2005).
Reuleaux classified his mechanisms using an alphabet,
that is, assigning letters to different groups of his mecha-
nisms. In that way, he stressed that each individual
mechanism was like a letter in an alphabet and, on com-
bining them together, we would get words and sentences,
which denote machines. His style of classification resem-
bles later classification ideas used in topology and theo-
retical computer science. The largest number (39) of
Reuleaux mechanisms is in the so-called S-series: straight
line mechanisms (see Fig. 3)(http://kmoddl.library.cornell.
edu/model.php?cat=S). Changing circular motion into
straight line motion had been a challenge to technology
since ancient times (Kempe, 1877). This problem was
crucial to James Watt (1736–1819) when he was working
on improving the steam engine (Taimina, 2005b; Hender-
son & Taimina, 2005b).
Klein defended the importance of models and instru-
ments to illustrate a theory, against the disposition of pure
mathematicians. In the same years, these ideas were shared
by the officers of ICMI. At the fifth International Congress
of Mathematicians in 1912, the section on didactics
received ICMI reports examining the state and trends of
mathematics teaching. In particular, the report of the Sub-
Commission A (mathematics in secondary education)
focused on ‘‘Intuition and experiment in mathematical
teaching in secondary schools’’ (Smith, 1913). Eugene
David Smith (1860–1944) discussed ‘‘contemporary
developments aimed at providing an ‘intuitive’, ‘percep-
tual’, ‘experiential’ and ‘experimental’ base for the subject
(p. 611), through ‘applying mathematics seriously to the
problems of life, and…visualizing the work’ (p. 615).
This, then, represented a foundational theme for the ICMI.
[…]. The main areas which the Sub-Commission singled
out […] were geometrical drawing, graphical methods,
practical measuring and numerical computation’’ (Ruthven,
2008). Further details may be found in Giacardi (2008).
2.2 Klein’s influence on mathematics education
in the USA: the Cornell collection
In 1857, the US Military Academy ordered 26 Olivier
models, of which the Department of Mathematical Sci-
ences still has 24 (Shell-Gellasch & Acheson, 2007). In this
way, the first common schools attempted to emulate the
‘‘objective’’ practices of their German counterparts. Sets of
geometric solids were sold to the new schools with claims
that they would help teachers present a common curricu-
lum. School boards and government commissions formal-
ized the arrangement, making models a required
component of mathematics instruction in many states.
Business was so good during the nineteenth century that
model makers were able to diversify into much more
lucrative catalogues of Mathematical Apparatus, which
included everything from the latest in ‘‘noiseless’’ drawing
slates and elegant ‘‘pointing rods’’ to hand-crafted
‘‘numerical frames,’’ made of the finest woods. There was
eventually a backlash, however, as teachers began to
complain that the expensive, and increasingly complicated,
apparatus was driving the curriculum. It is also of interest
to look at the response to the educational ‘‘technology’’ of
the time, i.e. concrete models in American colleges. The
use of concrete models in the classroom caused the same
kind of divisive debates that surround modern visualization
Fig. 4 Reuleaux triangle rotating in the square (photo Prof. F. Moon)
Concrete models and dynamic instruments as early technology tools 23
123
technology. Then, as in the present, usage varied from
place to place and from instructor to instructor. The most
enthusiastic endorsements in the report come from teachers
who used the models to help students visualize problems in
three-dimensional calculus and the ‘‘higher surfaces’’ of
descriptive geometry (Mueller, 2001).
Although Klein declined several opportunities to teach
in the USA, he nevertheless had a long-lasting influence on
American mathematics. The terrain was fertile, as Cornell
University in 1882 had acquired a collection of mechanical
models designed by Reuleux (see Fig. 2). The first Chair of
the Cornell Mathematics Department, James Edward Oli-
ver (1829–1895), went to study mathematical physics in
Cambridge; but, after hearing enthusiastic accounts of
Klein’s lectures, Oliver wrote to Klein from Cambridge to
ask whether it would be possible for him to spend some
time in Go
¨ttingen. One of the things in which he was
particularly interested was ‘‘to get pretty fully Klein’s ideas
as to methods of teaching, topics and courses of study, and
promising directions for original research by my young
men’’. Hence Cornell emerged as a prime sphere of Klein’s
influence in the USA (Parshall & Rowe 1991). Oliver sent
his student, Virgil Snyder (1869–1950), to study with Klein
in Go
¨ttingen. After the World’s Fair in Chicago (1893) and
the famous Evanston lectures, Klein travelled to the USA,
and one of the points of his interest was Cornell University,
where he was hosted by Oliver (Parshall & Rowe 1991).
As mentioned earlier, Cornell hosted the famous Rea-
uleaux collection, and hence the attitude towards experi-
mental approaches was very well established. We consider
later the pedagogical revival of this collection in the twenty
first century.
2.3 Klein’s influence on mathematics education
in Japan: mathematization in curricula
Japan was under the influence of Chinese mathematics until
the sixteenth century; the influence of European mathemat-
ics started in the middle of the nineteenth century from the
UK, France, Germany and USA (Isoda, 2004). To develop
new academy and school mathematics, some mathemati-
cians and mathematics educators studied mathematics in
Germany at the end of the nineteenth century and at the
beginning of the twentieth century. When they came back,
they became engaged in developing an integrated curriculum
as part of the Klein movement using the ideas such as in Perry
(1913) and Sanden (1914). This movement resulted in the
establishment in 1919 of the Japan Society of Mathematical
Education for the improvement of secondary school math-
ematics. Several textbooks based on this movement had been
published, such as those by Kuroda (1920), and several
contents had been experimentally taught at the secondary
school attached to the Tokyo higher normal school (origin of
the University of Tsukuba) with respect to calculus (see
Sanden 1914; Kuroda 1927) and projective geometry. Yet,
the integration into curricula was postponed because of the
earthquake, which had burned the capital Tokyo in 1923.
After the earthquake, the integration of the mathematics
curriculum was completed during WWII. It was done in the
curriculum reform of 1942 and published as textbooks in
1943 with the key word of mathematization (sugakuka in
Japanese). In the textbooks, a number of mechanical
instruments were treated as the subject matter for mathe-
matization. The textbooks’ style had a workbook format with
open-ended problems enabling students to learn by them-
selves with the support of their teachers during the air alerts
of the war. At first, the textbooks focused on the construction
of mental objects, which should be mathematized; later, it
became more sophisticated mathematically, repeatedly on a
spiral sequence: similar mathematical situations were
explored again and again for developing mental object in
mathematics.
Figure 5represents an example of mathematization
related to the design of the cap of an electric lamp. In grade
6 (11-year-old students), students explore how to draw the
development of the section of cylinder experimentally by
using a set of different viewpoints such as front, side and
top views and by rotating the viewpoints on the radius of
the bottom part of the circle for their drawing. Here, stu-
dents study the methods for analysing solids by projection
onto planes. Later, the same situation is re-explored for
studying conic sections in grades 9 and 10. The same
mental object, which the situation explored, will be re-
explored in grade 11, with the recourse also to trigono-
metric functions.
Another example is taken from the 1943 textbooks
(approved by Monbusyo, 1943) for 14-year olds with the
aim of realizing mathematization with open-ended
approach from the situation to elementary geometry and
from elementary geometry to analytic geometry. There
Fig. 5 The first step of
designing the cap of an electric
lamp (Monbusyo, 1943)
24 M. G. Bartolini Bussi et al.
123
were designed shifts from everyday mechanisms (Fig. 6)to
the geometrically simple mathematical instruments
(Fig. 7), then to geometrically deduced mathematical
instruments (Fig. 8) and finally to algebraic representations
(all the figures are taken from Monbusyo, 1943). This
sequence beginning from Fig. 7is the same as in the
textbook by Franz van Schooten (1615–1660), De organica
Conicarum Sectionum Constructione (1646), which first
treated conics plane geometrically and algebraically,
instead of as the section of a cone.
Mechanical instruments were well integrated in the
textbooks around World War II, but were gradually lost
after the war, and only the pantograph has remained after
the modernization of the 1960s–1970s curriculum because
of the introduction of more algebraic approaches such as
linear algebra.
2.4 Klein’s influence on mathematics education
in Italy: the collections in mathematical institutes
Italian mathematicians shared Klein’s interest in activities
with concrete models and dynamic instruments. Between
the nineteenth and twentieth centuries, many models and
instruments were available in nearly all the universities, at
the mathematical institutes, where the education of both
professional mathematicians and secondary mathematics
teachers took place an example is shown in the Fig. 9. Still,
in the early decades of the twentieth century, the impor-
tance of the reference to intuition (developed from the
consideration of models) was a feature of the so-called
Italian school of algebraic geometry. Guido Castelnuovo
(1865–1952) presented their work in this way:
‘‘We had created (in an abstract sense, of course) a
large number of models of surfaces in our space or in
higher spaces; and we had split these models, so to
speak, between two display windows. One contained
regular surfaces for which everything proceeded as it
would in the best of all possible worlds; analogy
allowed the most salient properties of plane curves to
Fig. 6 How does the point F move? (Monbusyo, 1943, vol. 3, p. 2)
Fig. 7 The locus of the point F (Monbusyo, 1943, vol. 3, p. 2)
Fig. 8 How does the point C move? (Monbusyo, 1943, vol. 3, p. 2)
Fig. 9 A plaster model of Steiner surface (courtesy of G. Ferrarese at
the Dipartimento di Matematica, Universita
`di Torino)
Concrete models and dynamic instruments as early technology tools 25
123
be transferred to these. When, however, we tried to
check these properties on the surfaces in the other dis-
play, that is on the irregular ones, our troubles began,
and exceptions of all kinds would crop up…With the
aforementioned procedure, which can be likened to the
type used in experimental sciences, we managed to
establish some distinctive characters between the two
surface families’’ (Castelnuovo, 1928)(http://www.
icmihistory.unito.it/portrait/castelnuovo.php).
Some decades later, Luigi Campedelli (1903–1978)
sharply commented that a reader could have read the above
vivid description of concrete models, ignoring the paren-
thesis ‘‘in an abstract sense, of course’’ (Campedelli, 1958).
In the following decades, under the influence of the
Bourbaki’s trend, the collections of models fell into
neglect, were warehoused and even destroyed. Recently,
the remains of some of the collections have been restored
and arranged in display cases as relics of the past. A
complete catalogue of the still existing collections of these
kinds of models and instruments has been prepared by
Palladino (n.d.), who visited all the departments of math-
ematics in Italy to check for their existence and present
condition. We shall consider later the use of these kinds of
instruments and models in today’s classrooms.
2.5 Different approaches
These three examples show different approaches to the
complex relationships between mathematics as a mental
activity and the exploration of the concrete materials of the
real world. To sum up, as a first approximation, one may
compare the real world on the one hand and the mathematical
world on the other hand. Klein, as evidenced by the above
quotation of Leibniz’ intentions, tends to consider concrete
models and dynamic instruments as representations of
mathematical concepts and processes. They can also be used
to solve problems in the real world, but this does not seem to
be the main issue. Rather concrete and dynamic exploration
of models and instruments (realized in the real world) may be
useful for teachers, at all school levels, to foster both math-
ematical understanding and the production of conjectures.
This is evident also in the above quotation from Castelnuovo
(1928) regarding collections in the mathematical institutes,
the purpose of which is to train both professional mathe-
maticians and mathematics teachers.
On the contrary, Reuleaux (1876) championed the impor-
tance of applied mathematics, with emphasis on the interac-
tion between mathematics, mechanics, physics and chemistry.
Japanese mathematicians and mathematics educators in
the early decades of the twentieth century have chosen an
integration between the two approaches, which are both
represented: the cultural approach, championed by Klein,
with reference to the history of mathematical ideas, and the
application approach (mathematization) with reference to
the mathematical modelling of concrete instruments. In
principle, they are not incompatible with each other; rather
they show the complexity of relationships between math-
ematics and the world of concrete experience. Gabriel
Koenigs (1858–1931) expressed well the complex links
between pure and applied mathematics, referring to a
pantograph for homotheties.
‘‘The theory of linkages is supposed to start in
1864. Surely linkages were used also earlier: a
dedicated and precise scholar might track down
them in the most ancient times. One might discover
in this way that each age has in hand, so to say,
yet without awareness, the discoveries of future
ages: the history of things often anticipates the
history of ideas. When in 1631 Scheiner published
for the first time the description of his pantograph,
he certainly did not know the general concepts
contained as germs in his small instrument; we
claim that he could not know them, as they are
linked to the theory of geometric transformations,
that is a theory typical of our century and gives a
unitary stamp to all the made advances’’ (Koenigs,
1897).
3 Second part
3.1 Use today in mathematics classrooms
In the first part of this study, we have reported some his-
torical information about one of the richest periods in the
recent history of mathematics education, as far as the
relationships between pure and applied mathematics are
concerned, in three different continents (Europe, America
and Asia). This reconstruction meets the needs of a his-
torical issue about resources and technologies in the
International Commission on Mathematical Instruction.
Yet, it would be misleading to avoid reporting about the
present use of concrete models and dynamic instruments
described above in the same countries, where specific
research centres have been continued or created anew.
Actually, the historical traditions dating back to Klein and
other scholars have been resumed in three different places,
where the three authors of this paper developed important
programs in mathematics education involving teachers and
classrooms.
26 M. G. Bartolini Bussi et al.
123
3.2 Use today in mathematics classrooms: KMODDL
pedagogical space at Cornell University (USA)
Taking advantage of the Reuleaux collection, in 2002, the
Cornell University started to develop KMODLL (kinematic
models for design digital library). Now these models can be
explored on the Web site http://kmoddl.library.cornell.edu/
or http://kmoddl.org. On the Web sites, besides having still
images of models, there are historical information and
interactive movies that allow a viewer to explore how these
models work. The Web site also has scanned rare books
that are important in the history of technology. A signifi-
cant part of this project works in connecting concrete
models and dynamic instruments, on the one hand, and
mathematical ideas behind these, on the other hand, for the
purpose of using them in the classroom. Teaching materials
have been developed and can be found in the section of the
Web site called Tutorials. There are also stereo litho-
graphic files that allow 3D printing of some of the models.
This material is available for teachers who wish to intro-
duce these activities in their classroom.
Cornell Faculty in Mechanical Engineering, Mathemat-
ics and Architecture are using the KMODDL Web site in
the classroom to teach mathematical principles of mecha-
nisms as well as machine design and drawing. Mathemat-
ical ideas from this collection have found their place in the
geometry textbook (Henderson & Taimina, 2005a).
The initial evaluation of the KMODDL in an under-
graduate mathematics class has confirmed the usefulness of
various physical and digital models in facilitating learning,
and revealed interesting relationships among usability,
learning and subjective experiences of the students (Pan
et al. 2004).
Reuleaux models have been also the object of a 9-week
interdisciplinary project ‘Exploring Machine Motion
Design’ in area schools for grades 7–9, carried out several
times in the last few years. The choice of exploring kine-
matic models was determined by the possibility to take
actual Reuleaux models into the schools. During this pro-
ject, students learned about the history of engineering and
the role of mathematicians in it. One of human’s oldest
mechanical devices is gears. The earliest written records on
gearing are dated from about 330 BC in the writings of
Aristotle. He explained gear wheel drives in windlasses,
pointing out that the direction of rotation is reversed when
one gear wheel drives another. The most probable uses
were in clocks, temple devices and water lifting equipment.
The Romans and Greeks made wide use of gearing in
clocks and astronomical devices. Gears were also used to
measure distance or speed. One of the most interesting
relics of antiquity is the Antikythera machine, which is an
astronomical computer. Mathematical studies of gears were
begun by Nicholas of Cusa (1401–1464) who, around
1450, studied cycloids. The famous painter Albrecht Du
¨rer
(1471–1528) also was interested in cycloids and he dis-
covered epicycloids. The students in the project recalled
that some of them played in their childhood with a toy
called ‘Spirograph’; such toys are still available in some art
museum stores. Students were surprised to learn that one of
the most important problems in the development of early
technology was seemingly based on a simple question: how
to draw a straight line (see Taimina, 2005b). This led the
class discussion to some geometry of inversions and con-
struction of linkages. Discussion on linkages was continued
during a field trip to the Cornell Robotics Laboratory,
where a group of researchers were working on designing
evolutionary robots and testing these devices by asking the
robots to re-create linkages in the kinematic model col-
lection (see Taimina, 2005a). Exploring the history and
mathematics of the universal joint helped students to see
connections with spherical geometry that is a neglected
topic in school curricula. At the end of this project, students
were asked to create their own machine motion design,
using as parts of their design models in the digital kine-
matic mechanism collection.
The 9-week interdisciplinary project could not cover all
the riches of the kinematic model collection, so students
submitted questions they had on the history of mechanisms
and particularly on its connections with geometry. The goal
of the project was to give students an opportunity to learn
about basic elements of mechanisms, so that they could
apply their knowledge to design their own machines. When
this project was taught for the second time, students were
able to use interactive computer design that involved ele-
ments from the historic kinematic model collection as
building blocks for their own geometrical motion design.
3.3 Use today in mathematics classroom: CRICED
at Tsukuba University (Japan)
As stated above, in Japan, the capacity of using mechanical
instruments in the standard classroom has strongly
diminished in the last decades. The use of mechanical
instruments was rediscovered in 1990s to implement
explorations within dynamic geometry software (DGS)
inspired by the history of mathematics (for an example, see
the following section). A project on mechanical materials
such as LEGO (Isoda et al. 2001; Isoda & Matsuzaki,
2003) was established in 1996. Within this project, online
teaching materials were developed (Isoda, 2008) in con-
nection with Bartolini Bussi (1998), drawing on a number
of historical textbooks. This project has influenced the
revision of Japanese textbooks.
Isoda et al. (2006) developed the free software ‘dbook’
in xml format to produce e-textbooks including mathe-
matical tools in classroom used together with an interactive
Concrete models and dynamic instruments as early technology tools 27
123
board (Isoda, 2008). It enables us to use DGS and digital
traditional instruments, such as a compass, on an e-text-
book through Web sites while still keeping the traditional
chalkboard classroom teaching approach (Fig. 10).
Figure 10 shows an e-textbook, which was developed
from the textbook by van Schooten (1646). As shown by
Isoda (2008), dbook was used for graduate students of the
Federal University of Rio de Janeiro to find the intuition
emerging in the activity with the textbooks. For example,
the instrument in Fig. 10 (Fig. 7) is mechanically and
mathematically the same as in Fig. 8, if we have geomet-
rical intuitions. The participants understood well the exis-
tence of this intuition, which van Schooten had. But,
without specific activity, students lost it when algebra got
the upper hand over geometric intuition.
3.4 Use today in mathematics classroom:
the Laboratory of Mathematical Machines
in Modena (Italy)
In Italy, the importance of teaching the use of concrete
models and dynamic instruments was defended by charis-
matic teachers such as Emma Castelnuovo (2008). Within
this tradition, in the University of Modena and Reggio
Emilia, a rich collection of more than 200 concrete models
and dynamic instruments was constructed in a carpentry
workshop, taking a leaf out of the phenomenology of
geometry from the classical age to the nineteenth cen-
tury. The collection is now stored in the Laboratory of
Mathematical Machines (http://www.mmlab.unimore.it). It
is always increasing, as new models and instruments are
built every year. Briefly, a mathematical machine (con-
cerning geometry) is a tool that forces a point to follow a
trajectory or to be transformed according to a given law.
Familiar examples of mathematical machines are the
standard compass (that forces a point to go on a circular
trajectory), the instruments described by Klein (1924,
1925) and reported above, the dynamic instruments of the
Cornell collection and van Schooten’s drawing devices
used in the CRICED project.
The implementation of hands-on activities on mathe-
matical machines, aiming at producing conjectures and
constructing proofs about their functioning and mathe-
matical properties, has been reported elsewhere (e.g.
Bartolini Bussi & Pergola, 1996; Bartolini Bussi, 1998;
Maschietto, 2005). There is also space for modelling
activity, when dynamic geometry software is introduced
(Bartolini Bussi, 2001).
Consider, for instance, the device for drawing a parabola
by Bonaventura Cavalieri (1598–1647). In Chapter XLVI
of Lo specchio ustorio (the burning mirror, Cavalieri,
1632), Cavalieri described a method for tracing a parabola
by means of hard instruments, made up of rulers and set
squares. In the following figures, the original drawing by
Cavalieri (Fig. 11) is paired with a wooden copy of it
(Fig. 12), shown in the same orientation to foster the
identification of the same components. The tracing point is
I (where a pencil lead R is put). The point A is fixed. The
Fig. 10 The e-textbook
developed from van Schooten
(1646) by using dbook
28 M. G. Bartolini Bussi et al.
123
bar KL has a fixed length (evident in the wooden copy) and
slides in the rail, dragging the set squares KLN and KIA,
aAnd the ruler AI slides on the point A. During the motion,
R (in I) traces an arch of parabola. In Cavalieri’s drawing
also the symmetric arch is drawn, in spite of the practical
impossibility of drawing it, without modifying the set
squares.
The wooden copy has been built in the Laboratory of
Mathematical Machine drawing on Cavalieri’s design. This
copy is usually given to students (undergraduate prospec-
tive teachers), asking them to produce a digital model by
means of Cabri software. Actually, by means of dynamic
geometry software such as Cabri, it is possible to produce a
construction that may be moved like the original to produce
a parabolic arch. This is a true modelling task, where a
concrete object is analysed to build its mathematical
model, not described by equations but by analogical
reproduction.
The process may be reported as follows:
First part: in the world of the dynamic instrument
(a) to look at the instrument; to catch the possibility of
motion;
(b) to identify the fixed elements (the rail; the point A;
the right angles KIA and KLN; the length KL), using,
if necessary, measuring instruments;
(c) to understand that some elements are not essential in
this kind of modelling (e.g. the rulers’ thickness; the
cracks in the bars; the screws).
Second part: in the geometrical world (Cabri Geometry)
(a) to draw (in the Cabri screen) the rail and the fixed
points; to choose a fixed line segment to model the bar
KL; to identify a point to be used to direct the motion
(it may be either a free point, if any, or a point bound
to the rail);
(b) to draw the digital copy of the instrument around the
chosen elements; this has to be done following the
software logic on the one hand, and the geometrical
properties of elements, on the other hand. So, for
instance, when one knows point A and point K, to find
point I, one needs to construct a right angle, which
can be done using the features of semicircles…
Third part: back in the world of the dynamic instrument.
(a) to interpret the digital copy of the instrument; to drag
it;
(b) to verify whether the fixed rulers maintain fixed
length; to check whether the arch is drawn (see
Fig. 13);
(c) to check whether the digital copy of the instrument
works in the same way as the concrete one; to analyse
limits (e.g. is the same arch traced?) and potentialities
(may the fixed line segment KL, a ‘parameter’ of the
construction, be changed?) What happens if this
parameter is changed?
Changing the length of some line segments in the digital
model has different consequences:
Fig. 11 The original drawing
Fig. 12 A wooden copy
A
L
K
I
A
L
K
I
Fig. 13 Two different frames of the same digital instrument
Concrete models and dynamic instruments as early technology tools 29
123
•changing the line segments containing AI and IK
changes the length of the arc on the same parabola;
•changing the length of the line segment KL changes the
width of the parabola.
A practical application of the modelling process is
realized when it is necessary to build a new concrete
model: the measure of the wooden board and of the bars
may be designed carefully before cutting them, to obtain a
well proportioned arc.
4 Concluding remarks
The three examples show clearly how an old tradition,
rooted on Klein’s views, may be resumed today. In all
cases, the reference to the history of mathematics is explicit
and brought to the students’ knowledge, concrete models
and dynamic instruments are available for students’ real
manipulation and complementarities between cultural
aspects and modelling are pursued.
Last, but not least, information and communication
technologies are introduced, although in different forms
and with different aims:
•In the Cornell project, professional movies show the
actual functioning of the precious historical instruments
and stereo lithographic files allow 3-D printing of some
of the models; moreover, other interactive simulations
allow Web exploration of dynamic instruments;
•In the CRICED project, historical books are weaved
together with interactive dynamic simulations;
•In the mathematical machines project, dynamic simu-
lations are not only available on the Web, but also are
objects of specific teaching and learning activities with
prospective mathematics teachers offering a non-trivial
context for mathematical modelling.
There is no claim that concrete models and dynamic
instruments may be replaced by their digital copies with no
loss. Trivially, the digitalization of instruments allows
them to become widely available: where there is an access
to the Internet one can play with these models interactively.
Yet a deep analysis of the changes (if any) in both didac-
tical and cognitive processes when a concrete object is
replaced by a digital copy is yet to be performed.
As mentioned earlier, mathematical modelling is not the
only element, but an important element of all three research
programmes. In this study, we have shown that modelling
and application can be paired within an approach that does
not neglect, but rather emphasize, the cultural aspects
of mathematics, going back to the prominent founders
of modern mathematics and taking advantage of the
increasingly wider diffusion of information and commu-
nication technologies.
Acknowledgments We are very grateful to David W. Henderson
for his help with the final version of this manuscript. M.G. Bartolini
Bussi was funded by the project PRIN 2007B2M4EK on ‘‘Instruments
and representations in the teaching and learning of mathematics:
theory and practice’’. D. Taimina was partially funded by National
Science Foundation (research grant DUE-0226238) and Institute of
Museum and Library Services (research grant LG-30-04-0204-04). M.
Isoda was funded by the project JSPS 17300243 on ‘‘Developing
Mathematics Exhibition for Science Museum’’.
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