Nieuwe Wiskrant 31-1/september 2011 33

IntroductionMathematics teachers and researchers of didactics ofmathematics are constantly looking for ways toimprove the teaching of mathematics. Recent world-wide reseach supports the idea that student under-standing is more effective when using manipulativematerial (Heddens; Canny, 1984; Fennema, 1973; Suy-dam, 1984). The important role that visualizationplays in the case of geometry makes the use of con-crete manipulatives particularly appropriate for itsteaching. There is wide support that this use facilitatesthe construction of representations of geometric con-cepts in young children (Taylor et al., 2007; Grenier &Grenier, 1986; Gerhardt, 1973; Prigge, 1978). It hasalso been shown that manipulatives are an essentialaid in learning geometry for older students as well,especially those at lower level of geometric knowledge(Clements & Battista, 1992; Fuys, Geddes & Tischler,1988).

Other researchers, however, warn that concretemanipulatives are not sufficient to guarantee meaning-ful learning (Baroody, 1989) and point out the com-plexity of the matter. For example, studentssometimes learn to use manipulatives only in anautomatised manner or without recognizing mathe-matics in it, and although performing the correctsteps, learn little from it. Clements and McMillen(1996) point out that teachers and students shouldavoid using manipulatives without careful thought,since they alone are not sufficient, but they must beused to actively engage children’s thinking withteacher guidance.

Recent research also supports the use of virtualmanipulatives, which have often replaced concreteones in the teaching geometry due to the increasinguse of computers. For instance, in Reimer & Moyer(2005) it is shown that working with virtual manipula-tives helps children develop conceptual and proce-

dural knowledge in geometry, and that thesemanipulatives are more motivating than paper andpencil tasks. Other studies (Sarama & Clements, 2004;Sarama et al., 2003) show that computer manipula-tives help children of different ages learn variousgeometry concepts.

In this text we focus on a collection of about 150physical models present at the University of Gronin-gen. By physical models we mean here concrete models,often made of plaster or string, which represent cer-tain geometric objects. The author’s thesis (Polo-Blanco, 2007) reports the study of some of these mod-els from both a historical and a mathematical perspec-tive. Since this work was completed, the models havebeen the topic of several master theses of mathemat-ics students in Groningen. One of these investigationsresulted on finding the origins and mathematicalmeaning of a set of thread models on the Groningencollection, and has been reported in Top and Weiten-berg (2011).

Most models at this university were sold by the Ger-man companies L. Brill and M. Schilling in the secondhalf of the nineteenth century, and will be the focus ofthis text. We discuss the origin of the models, themotivation behind their construction, as well as otheraspects like the mathematics behind them and theirpedagogical use.

Mathematical models in Germany andKlein’s roleIn Europe, the use of concrete mathematical modelsand dynamical instruments for higher education wasalready common during the seventeenth and eigh-teenth century, but received a new impulse in thenineteenth century (Maclaurin, 1720). We find anexample of this at the polytechnic schools in Germanyduring the second half of the nineteenth century(Dyck, 1892; Fischer, 1986) where collections of

Physical models for the learning of geometry

Tegenwoordig worden in de wiskundeles materialen, plaatjes of modellen ingezet om hetbegrip van leerlingen van een concreet niveau naar een formeler en abstracter niveau tebrengen. Nieuw is dit niet: in de achttiende en negentiende eeuw werden op diverse Ne-derlandse universiteiten al fysieke modellen gebruikt om het meetkundeonderwijs voorstudenten aanschouwelijk te maken. De Spaanse didacticus Irene Polo-Blanco, in Ne-derland gepromoveerd op dit onderwerp, doet in het Engels verslag.

34 Physical models for the learning of geometry

mathematical models were constructed, for instanceto simplify the visualization of some geometricobjects (Polo-Blanco, 2007).

The great period of model building started in the1870ss, when Ludwig Brill, brother of Alexander vonBrill, began to reproduce and sell copies of somemathematical models. A firm under the name L. Brillwas founded in 1880 for the production of modelsand was taken over in 1899 by Martin Schilling whorenamed it. By 1904, Martin Schilling had already pro-duced twentythree series of models. Schilling’s 1911catalogue (1911) describes forty series consisting ofalmost four hundred models and devices. But by 1932Martin Schilling informed the mathematical instituteat Göttingen that “in the last years, no new modelsappeared” (Schilling, 1911).

These models often represent objects from differen-tial geometry, algebraic surfaces or instruments forphysics. In Schilling’s catalogue the models are givena name and a short mathematical explanation. In somecases this is accompanied by a drawing, as shown infigure 1.

fig. 1 Left: drawing of model nr. 1, Series XII: from Schilling(1911). Right: model nr. 1, Series XIII from the Groningen collec-tion.

The case of the Clebsch diagonal modelDuring the second half of the nineteenth century,many models were constructed under the direction ofFelix Klein. He developed his love for geometricmodels under the influence of Alfred Clebsch. Theidea of model building reached many universities inEurope. Klein also brought the topic to the UnitedStates when he presented many of the models, includ-ing a model of the Clebsch diagonal surface, at theWorld Exhibition in Chicago in 1893. The Clebschdiagonal surface is defined by the equations

and was discovered by Alfred Clebsch in 1871. It hadbeen proved previously in 1849 by Arthur Cayley andGeorge Salmon that every smooth cubic surface con-tains precisely twenty-seven lines over the complexnumbers. On a Clebsch diagonal surface all the

twenty-seven lines are also real. This means in partic-ular that the twenty-seven lines can be represented ina model. Figure 2 shows such a models of this surfaceconstructed in 2005 by the Spanish sculptor CayetanoRamírez López.

fig. 2 Clebsch diagonal surface by C. Ramírez López. Universi-ty of Groningen.

The model might be appreciated by many for itsbeauty, but why should one use it to study and under-stand this surface? Wouldn’t having a nice drawing ora computer dynamic representation be sufficient? Webelieve that having the physical model might help inunderstanding some of the properties of this surface,such as the complexity of the configuration of itslines. The twenty-seven lines intersect in a specialmanner, which was described by the Swiss mathema-tician Ludwig Schläfli in 1858 by defining the conceptof double six. A double six is a set of twelve of thetwenty-seven lines on a cubic surface:

where each line represented above only intersects thelines which are not on the same column, nor on thesame row. That is, does not meet for , does not meet and meets for . This con-figuration is difficult to understand and to visualize

x13

x23

x33

x43

x53

+ + + + 0=

x1 x2 x3 x4 x5+ + + + 0=

a1 a2 a3 a4 a5 a6

b1 b2 b3 b4 b5 b6

ai aj i j aibi ai bj i j

Nieuwe Wiskrant 31-1/september 2011 35

just by looking at the definition, or even by using acomputer representation. However, having the modelin your hands might really help to get a feeling of thisstructure. In the model of the Clebsch diagonal sur-face, the six lines above have been drawn in green(light grey) and the six lines in red (dark grey). Bymoving the model around one can observe forinstance that a green line never meets another greenline and that it only intersects five of the red ones.Another property of the lines is that there are pointswhere three lines of the cubic surface intersect (calledEckard points) which can also be found in the model byturning it around.

Figure 3 shows a fragment of Klein’s lecture at theChicago Exhibition where he discussed the construc-tion of the model of this surface. His words alsoreflect the important pedagogical purpose he attrib-uted to the models:

fig. 3 Lecture by Klein at the Chicago Exhibition in 1893.

A Dutch perspectiveMost Dutch universities keep some collection ofBrill’s or Schilling’s models. We shall briefly describesome of them, namely the ones located at the Univer-sities of Amsterdam, Leiden and Utrecht, and the con-dition in which they were kept at the time thisinvestigation took place (between 2005 and 2007).1. University of Amsterdam. The University of Amster-

dam keeps the largest collection of mathematicalmodels in the Netherlands. It consists of morethan 180 models, most of them from the collecti-ons of Brill and Schilling. The University Museumof Amsterdam has made a detailed inventory oftheir collection. Most of the plaster models andmany of the string ones have been restored. Moredetailed information on this collection can befound in the online catalogue1 under Bijzondere Col-lecties.

2. University of Leiden. The Mathematical Institute ofLeiden University keeps a collection of about 100models distributed between the library and roomnumber 210 of the Mathematical Institute. Mostmodels are preserved in good condition, althoughliterature, references or a local catalogue of the col-lection have not been found.

3. University of Utrecht. The Department of Mathema-tics in Utrecht keeps about twenty mathematicalmodels. They are in a reasonably good condition.A few of them are on display in the library of theDepartment, while the rest are kept in a closed partof the library. Some of the broken models are inthe University Museum.

One might wonder how universities in the Nether-lands acquired a collection of models. How did thesemodels become known in the first place? For exam-ple, were there advertisements from the Brill or Schill-ing companies in Dutch scientific journals of thatperiod? A quick search through the volumes of theyears 1875 to 1910 of the Dutch mathematical journalNieuw archief voor wiskunde shows no advertisementsconcerning the models. However, several advertise-ments of Brill and Schilling’s collections of modelsappeared in the renowned American Journal of Mathe-matics during the great period of model building (seefigure 4). Collections of models were bought by manyUniversities in Europe, including some Dutch univer-sities.

fig. 4 Advertisements in the American Journal of Mathema-tics: July 1890.

As it can be appreciated in the figure, models wereadvertised to be used for Higher Mathematical Instruction.Although the most important reason to buy thesemodels seems to have been of a didactic nature, oneof the few concrete pieces of evidence for this that we

aibi

36 Physical models for the learning of geometry

have been able to find, consists of the photographshown below (figure 5). In this photograph, datingfrom 1938, Professor Van Dantzig is giving one of hislectures at Delft Polytechnic School. On his tablethere are two string models. One of them can be iden-tified as model nr. 5 from Series IV in the Schillingcatalogue and shows a hyperbolic paraboloid. On theblackboard one sees some of the equations definingthis surface.

fig. 5 Van Dantzig during one of his lectures at Delft Polytech-nic in 1938.2

The work of the author’s thesis (Polo-Blanco, 2007)focused on the collection of models at the Universityof Groningen which we briefly describe next.

The collection at the University of Groningen The collection of physical models at the University ofGroningen consists of about 150 pieces and severalquestions concerning them were investigated. Someof these questions were: When where these modelsbought? Who decided to buy the models? What wastheir use? A search of the archives of the city of Gro-ningen revealed the answer to the first two questions.In the financial records of the faculty, it was discov-ered that the Department of Mathematics received ayearly subvention of ƒ100,-- for buying mathematicalmodels. Schoute spent this budget of 100 guildersbetween 1882 and 1912. A bill for one of these pur-chases is shown in figure 6. Schoute’s successor Bar-rau continued, after Schoute’s death in 1913, to buymodels for another three years.

As for the use of the models in Groningen during thattime, we wondered for instance whether the modelswere shown during lectures. Also, were the modelsappreciated for their artistic value? Where were themodels kept? Were they on display? In trying to findinformation about this, the author interviewed sevenalumni of the University of Groningen who studiedmathematics between 1930 and 1960. From the inter-views, one can draw some conclusions. Although it

seems that J.C.H. Gerretsen (professor of geometry inGroningen between 1946 and 1977) showed modelsduring some of his lectures, their didactical use doesnot seem to have been impressive enough as toremain in the mind of these alumni. However, a doc-ument found in the archives of the Faculty of Mathe-matics and Sciences shows that Gerretsen was at leastinterested in the maintenance of the mathematicalmodels. This document3 concerns the move of theoffice of Gerretsen and the Mathematical Library tothe Mathematical Institute on the Reitdiepskade in1957. It shows Gerretsen’s plan for this new building.One of the seven items where Gerretsen describes thenew structure of the building states that (translatedfrom Dutch) the room behind the lecture room is destined tobecome the room for the models.

fig. 6 Bill for the acquisition of mathematical models by G.Schoute in 1908.4

However, it seems from these interviews that G.Schaake, Gerretsen’s predecessor, was not interestedin the didactic aspects of the models, and was proba-bly not interested in the models at all. Some questionsremain unanswered. What was the location of the col-lection of models before 1957? Lectures were givenon the first floor of the Academy Building. Accordingto de Jager, this was the case until 1957, when an Insti-

Nieuwe Wiskrant 31-1/september 2011 37

tute of Mathematics was formed at the Reitdiepskade.As the mathematics lectures were given in the Acad-emy Building until 1957, one could conjecture thatthere was already a collection of models kept in thisbuilding when it caught on fire in 1906. Since the Gro-ningen collection consists mostly of models withSchilling’s stickers and not Brill’s (like the collectionsof other Dutch universities), one may guess that amajor part of the mathematical models was destroyedin the fire of the Academy Building, especially themodels of the Brill collection, and that the big pur-chase of 1908 came from the wish to replace the pre-vious collection.

Another part of the work in the author’s thesis con-sisted on making a complete inventory of the Gronin-gen collection and a restoration of the plaster models.As a result of the former, a web page was designed5

where a photograph of each model appears, accompa-nied by a short mathematical explanation. As for therestoration of the plaster models, the sculptorCayetano Ramírez López took care of that in 2005.The project, funded by the University Museum andthe Department of Mathematics, consisted of a resto-ration and a cleaning of almost eighty plaster models.One of these models is shown in figure 7, both beforeand after the restoration. For more information thereader may consult the complete report on the resto-ration (Ramírez López, 2005).

fig. 7 Plaster model before and after the restoration.

ConclusionThe German collections of mathematical models werevery popular among the Universities in Europe at theend of the nineteenth century, in particular at Dutchuniversities. Although the models were originallyadvertised as Models for the higher mathematical instruction,the account presented in this text shows that the ped-agogical purpose of the models from the Groningencollection was almost forgotten already by the 1930s.

However, we find some evidence of how this old tra-dition can be resumed today. Research concerning the

models in Groningen carried out as part of studentsmaster thesis is an example of this (Top & Weiten-berg, 2011). Another one can be found in Bussi et al.(2010), where an account is given of the reproductionof many classical instruments as digital models usingdynamic geometry software like Cabri, as well as theiruse as teaching material. But, as is pointed out:

There is no claim that concrete models and dynamic instru-ments may be replaced by their digital copies with no loss. [...]a deep analysis of the changes (if any) in both didactical andcognitive processes when a concrete object is replaced by a di-gital copy is yet to be performed (Bussi et al., 2010).

We believe that both physical and computer manipu-latives, as well as other meaningful representations,might help students to construct geometric conceptsand discover some of their properties, and hope toencourage teachers to benefit from them by recaptur-ing this classical tradition.

Irene Polo-Blanco,University of Cantabria, Spain

& Freudenthal Instituut, Universiteit Utrecht

About Irene PoloIrene Polo Blanco graduated in Mathematics in 2002from the University of Groningen and from the Uni-versity of the Basque Country (Spain). In 2007 shereceived her Ph.D. from the University of Groningenunder the supervision of Jan van Maanen, Marius vander Put and Jaap Top. She currently has a teachingposition at the University of Cantabria (Spain) in thegroup of Didactics of Mathematics. From May tillAugust 2011 she obtained a Spanish research grantwhich she enjoyed working at the Freudenthal Insti-tute in Utrecht.

Noten[1] http://opc.uva.nl.[2] Photograph of Van Dantzig, personal archives G.

Alberts (1938).[3] Plan for the structure of the new Mathematical In-

stitute building in Reitdiepskade, May 1957, Gro-ningen Provincial Archive, Inv. Nr. 94.

[4] Register of effected payments in 1908. Groningen Pro-vincial Archive, Inv. Nr. 487.

[5] http://www.math.rug.nl/models.

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