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Researching into the History ofTeaching and Learning Mathematics:the State of the ArtGert SchubringPublished online: 23 Jan 2007.

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Paedagogica HistoricaVol. 42, Nos. 4&5, August 2006, pp. 665–677

ISSN 0030-9230 (print)/ISSN 1477-674X (online)/06/040665–13© 2006 Stichting Paedagogica HistoricaDOI: 10.1080/00309230600806955

Researching into the History of Teaching and Learning Mathematics: the State of the Art1

Gert SchubringTaylor and Francis LtdCPDH_A_180638.sgm10.1080/00309230600806955Paedagogica Historica0030-9230 (print)/1477-674X (online)Original Article2006Stichting Paedagogica Historica424/5000000August 2006GertSchubringgert.schubring@uni-bielefeld.de

Relevance of the Field

First of all, we should address the question: Why study the history of mathematicsinstruction? Evidently, such research is not undertaken for pure curiosity. Since thepresent situation is the product of a historical process, the evolution informs themathematics educator regarding political, social and cultural constraints to improvingmathematics instruction. Practically all the research questions in mathematics educa-tion have a historical dimension that too often, however, remains implicit, or istreated too superficially. Research can be improved by explicit consciousness for thehistory of teaching and learning mathematics. And, what is probably even moreimportant, the history of mathematics instruction should constitute one of the dimen-sions of the professional knowledge of mathematics teachers. In order to be able tohandle the problems they encounter in their professional life, mathematics teachersshould know how their profession emerged historically, how it developed and whichtypes of problems were encountered during this development, and what obstacles hadto be overcome for the effective establishment of mathematics teaching. The historyof their own profession should, hence, constitute part of what has been called the‘meta-knowledge’ of mathematics teachers and should therefore constitute anelement in the education of mathematics teachers.2 Within the desirable and in someplaces realized historical component in this teacher training, there should also be

1 This report was prepared in discussion with the other members of the team for Topic StudyGroup 29: Yasuhiro Sekiguchi (Japan), Hélène Gispert (France), Hans Christian Hansen(Denmark), Herbert Khuzwayo (South Africa).

2 Cf. Arbeitsgruppe Mathematiklehrerbildung, ed. Perspektiven für die Ausbildung derMathematiklehrer. Köln: Aulis-Verlag Deubner, 1981; Schubring, Gert. Die Entstehung desMathematiklehrerberufs im 19. Jahrhundert: Studien und Materialien zum Prozeß der Professionalisierungin Preußen (1810 – 1870). Weinheim: Beltz, 1983.

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courses on that specific history. We are of course aware that it is difficult to introduceadditional elements in teacher training programs.

Present State

Despite this systematic importance, one has to admit that this field of investigationhas not yet been developed far as a scientific endeavor. Although a number of relevantstudies were already published during the nineteenth century, the studies relating tothe various countries are rather scattered: they were in general undertaken for certainoccasions, and not pursued systematically. Even in international handbooks onmathematics education there are no sections on history. Interested scholars areworking in isolation. In fact, due to the lack of communication there are no sharedstandards of research, and methodology is rather weak, many studies relying on‘hand-drafted’ ad-hoc approaches. However, we can name here four exceptions to thegeneral situation of absent communication:

1. The Netherlands, where for ten years there has been the Historische KringReken– en Wiskunde Onderwijs (HKRWO), which organizes yearly conferenceson the history of mathematics teaching in the Netherlands, and where there areconsequently a considerable number of ever more systematic investigations;

2. Japan, where there has for four years even been a society for the history ofmathematics instruction that publishes a journal of its own, so that there is alarge number of relevant publications;

3. In April 2004, there was a symposium in Italy on the history of mathematicsteaching in Italy;

4. In May 2004, the first symposium in Portugal was held on the history of mathe-matics teaching in that country.

Issues of Methodology

The lack of communication is particularly acute on the international level. Almost allof the studies are dedicated to historical issues within a certain culture or within acertain state or nation, without ever going beyond the frame and the restraints of thatparticular context. And there are very few studies that address comparative issuesregarding the history in different countries or cultures. It is this lack of internationalcommunication that has to be overcome in order to achieve an exchange regardinggeneral patterns underlying the particular histories, and to establish shared standardsof research. It is thus the primary goal of the team organizing this Topic Study Group(TSG) to promote and to achieve communication.

The low emphasis on methodology may be caused by what proves to be an illu-sion: the idea that research into the history of mathematics instruction presents aneasy task, that this history is just a collection of facts which are observable withoutdifficulties, and that one only needs to ‘collect’ these facts. This is in particular theview of the history of mathematics instruction as a series of administrative decisions

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that supposedly were transformed into practice. According to this perspective, thehistory basically is a history of the curriculum, of the syllabus, managed by centralistauthorities.

Even if the broad spectrum of historical issues is reduced to the syllabuses, the realproblem is whether, and how, centralized decisions were implemented in schoolpractice, and this opens up again the immense range of dimensions relevant to thehistorical development.

In fact, the history of the teaching and learning of mathematics constitutes aninterdisciplinary field of study; the principal disciplines concerned are the history ofmathematics and the history of education, but the science of history contributes aswell. Moreover, sociology is quite essential, in particular sociology of religion.

Realizing the complexity of our field of study, we might even say that it requires aneven more complex methodology than the history of mathematics. Clearly, mathe-matics history is a part of cultural, political and social history, too, but the contentsof mathematics and the evolution of its concepts occupy a far more extended domainwithin mathematics history than within the history of mathematics instruction.Compared with this rather dominant role of mathematical ideas and concepts, thehistory of teaching and learning mathematics constitutes a social reality that needsincomparably more social categories to reveal its dimensions.

Even the entity corresponding to the structured set of mathematical concepts,namely ‘school mathematics’, is far from being just a derivation or a projection of the‘savoir savant’ as Yves Chevallard pretended3 – well to the contrary, school mathe-matics develops as a product of numerous interactions, and even pressures, from andbetween various sectors of society.

But what complicates the research in our field even more is the fact that mathemat-ics never appears in educational systems in an independent way but always functionswithin structures which are characterized by a compound of several school disci-plines. This means that mathematics teaching and learning is always dependent onother factors that it is barely capable of influencing.

Yet, despite this fundamental and structural dependence on a concert of disciplinesthat in general exhibit no peaceful coexistence, the perhaps most considerabledeficiency of the large majority of studies in our field is that they treat mathematicsas an isolated teaching subject, without regarding relationships, dependencies andhierarchies in the system defining school learning.

It should be evident by now that marked progress in research necessitatesmethodological reflection and refinement. A decisive resort in doing so is presentedby comparative issues – not only comparative studies of the history of various schooldisciplines within a given educational system, but even more importantly comparativestudies on the history of mathematics instruction in different states and differentcultures.

3 Chevallard, Yves. La transposition didactique: du savoir savant au savoir enseigné. Grenoble:Éditions La Pensée Sauvage, 1985.

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It is quite natural that most research pursued or ongoing is concentrated on thehistory within a given nation or a given culture as the history of mathematics teachingand learning first and foremost constitutes part of the educational history of thatcountry or culture. But in order not to end up with a collection of separate, isolatedhistories without interconnections, one has to establish relations between thedifferent histories and to reveal what is ‘general’ in them and what constitutes, say,cultural, or social, or political peculiarities. Practically all questions in our historicalfield deserve comparative studies. In a later section, examples from recent researchwill be mentioned.

The approaches and results from comparative education hence provide an essentialtool for an international history of teaching and learning mathematics, in order tograsp national specificities as well as overall and global trends. Of particularmethodological importance are qualitative methods, which are also applicable to thestudy of (historical) documents. Given the primary importance of cultural history,anthropology provides relevant methodological resources as well.

Some Elements of a Historico-Systematic Analysis

Frame of the Present Study

In view of the extraordinarily broad range of topics within our historical field, thenumber of states and cultures throughout history where mathematics was taughtandthe different levels of school systems we had to narrow our focus. The focus chosenis institutionalized forms of teaching and learning – in types of schools equivalent toprimary and secondary levels. And teacher education is included for these types ofschools, which at least partially belonged and belongs to higher education. Theteaching of mathematics at universities in general has not been included, however,since it is more tied to the history of mathematics proper and thus provides otherresearch questions, and needs different methodologies. On the other hand, thetransition from informal to modern institutionalized teaching allows insight intosystematic dimensions and has therefore been considered, too.

Functions and Dependences

Despite the focus on modern times for the studies to be presented in our topic group,it is necessary to unravel categories permitting systematic analyses to assess what isalready known about the overall development of mathematical instruction.

In fact, institutionalized teaching is almost identical to public instruction organizedby the state. This already shows the structural importance of the state for understand-ing the history of mathematics instruction. But its role is even more important.

Compared with the several thousand years of teaching mathematics, the fewcenturies of institutionalized teaching in modern times constitute a remarkably shortperiod. And secondary schools present the historically youngest element in the struc-tures of teaching. The first types of schools since Antiquity can be understood as

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professional schools, training specialists for the different administrative services of therespective state.4

When first realizations of state-independent forms of liberal education becamepracticed, they could be judged as forms of higher education. Later on, there existeduniversities on the one hand, and rudimentary schooling of the primary educationtype on the other. Genuine secondary schools emerged last, between the sixteenthand the nineteenth centuries, and typically as forms of public instruction.

The crucial point in all the different moments of the long history of mathematicseducation proves to be the rank attributed to mathematics within the respective set ofsocial and cultural values. And this rank was intimately related to the function exertedby mathematics instruction.

One can assert that mathematics enjoyed an unquestioned, firm and central posi-tion as key discipline in the historically first forms of systematic instruction: in thescribal schools of Mesopotamian Sumer and of Egypt, together with the second keydiscipline of language.

On the other hand, this instruction clearly was oriented professionally towards theconcrete needs of the state, for its highly developed system of administration. Thehigh rank attributed to mathematics instruction hence derived from this administra-tive, i.e. professional, function.

In later societies and cultures, where higher social classes became established,enjoying and using leisure time for learning, there emerged certain forms of liberal (orgeneral) education. Remarkably enough, apparently the only civilization among thesesocieties according mathematics a high rank for this function of education wasAncient Greece, with the well-known high value in Plato’s Academy. Within theRoman civilization, however, rhetoric was the leading discipline.

In the other states and cultures, even in the other Axial Age Civilizations (a keyterm in sociology for investigating the first monotheist religions as breakthroughs incivilization), mathematics was only accorded a minor propaedeutic function.5

In fact, it is well known that, in Ancient China, mathematics belonged to theknowledge necessary in the branches of state administration, and that it constitutedtherefore one of the examination subjects for entering the civil service. On the otherhand, it is likewise known that Confucianism, the Chinese ‘official’ philosophy, didnot value mathematics.

And within the Hindu civilization in India, mathematics was only ranked to exerta propaedeutic function, as auxiliary knowledge for the religiously valued astronomy.

4 Cf. Høyrup, Jens. “Influences of institutionalized mathematics teaching on the developmentand organization of mathematical thought in the pre-modern period. Investigations in an aspect ofthe anthropology of mathematics.” Studien zum Zusammenhang von Wissenschaft und Bildung.Materialien und Studien des Instituts für Didaktik der Mathematik der Universität Bielefeld 20 (1980):1–137.

5 Cf. Eisenstadt S. N., ed. The Origins and Diversity of Aaxial Aage Civilizations. Albany: StateUniversity of New York Press, 1986. I am grateful to Erika Hültenschmidt who drew my attentionto this sociological theory.

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We can therefore not only state two conflicting functions of mathematics instructionthroughout history, a professional function and a propaedeutic function, where thelatter might evolve to the higher form of liberal or general education of the mind. But,moreover, it now becomes clear also why sociology of religion is relevant to the historyof mathematics instruction.

Evidently, the role exerted by knowledge in society is one of the favorite subjects ofsociology. One has to be aware, however, that in all civilizations the state functionedin tight union with some official religion (at least until the historically late processesof secularization and the separation between state and church). Within this classicalcoalition, it was the task of religion to define and to regulate the social appreciationof knowledge, in particular the hierarchy of the different fields, or disciplines, ofknowledge.

It is remarkable how many of the official state religions in history, even of the AxialAge Civilizations, did not fully accept mathematics, and attributed it a minor rank intheir hierarchy of knowledge. While it was legitimized for some religions, or forreligiously welcome professional goals, it was subordinated in the conceptions ofknowledge and hence in the forms of liberal education, too.

To mention some examples, besides the cases of China and India:

● Within the Islamic civilization, some applied arithmetic was legitimized by theproblems of inheritance as emphasized by the qoran, but mathematics was either ina marginal or endangered position, as ‘foreign’ or ‘ancient’ science, or as anauxiliary propaedeutic subject for the madrasa, which were maintained for religiousends.6

● Within the likewise religiously organized universities of the Christian West,elementary astronomical knowledge was welcome as ‘computus’, for calculatingthe Christian calendar, but the famous ‘Quadrivium’ meant elementary subjects ofsecondary relevance.7

● Kastanis’s contribution shows instructively how strongly the Orthodox ChristianChurch opposed a modernization of learning and a higher rank for mathematicsinstruction at the end of the eighteenth and the early nineteenth century.

It is revealing that the first movement in favor of according mathematics a high rankas knowledge and as a teaching subject in Europe originated outside the teachingsystems, and that means external to church and religion, and likewise independent ofa state and its goals: it was the movement of Humanism during the Renaissance inEurope, which claimed to revive the knowledge values of Ancient Greece.

And it is likewise revealing that the teaching systems did not implement suchreforms by themselves – as evidenced so many times later! – but that pressure fromoutside was necessary to achieve reform. Actually, it was now the state that, in thecourse of the process of emerging national states, began to change character, assuming

6 Cf. the contribution by Abdeljaouad in this volume.7 Cf. Schöner, Christoph. Mathematik und Astronomie an der Universität Ingolstadt im 15. und 16.

Jahrhundert. Berlin: Duncker & Humblot, 1994.

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some functions for interior policy hitherto exerted by the Church, making the sover-eign decree the establishment of teaching disciplines as proclaimed by Humanism,against the resistance of the traditional corporations.8

Again, after Christianity in Western Europe split into Protestant and Catholicfaiths and Churches, one can observe the relevance of sociology of religion, and inparticular of Max Weber’s claim of a Protestant Ethic and of the Merton thesis: it wasthe Protestant states that maintained and pursued the Humanist reforms of learningand accorded mathematics a firm position as an increasingly independent discipline,whereas the Jesuit teaching system in the countries of Counter-Reformationessentially re-established a minor, even marginal status for mathematics. In fact,according to the Jesuit interpretation of Aristotle’s philosophy, mathematics made noassertions about things themselves, but only about some accidental properties, so thatmathematics was not the instance for them to assure the necessary ‘certitudo’ ofknowledge.9

One of the roots of the modern public system of schooling, together with its notionof liberal education elevating mathematics to the rank of one of the major disciplines,thus originates in this secularization of education brought about by the ProtestantEthic.

The other root extends from the major strand of modern philosophy, fromRationalism in France, developed by Descartes with the purpose of establishing thecertainty of mathematics against the complex of Jesuit interpretation of Aristotelianideas mentioned above. Rationalism was continued in the nineteenth century byPositivism, which accorded mathematics the leading position among the sciences.

A Synthesis of Recent Research

The processes of modernization and of transmission are intimately related. It isclearly observable in history that there was no universal and uniform growth of math-ematics and mathematics instruction on the various continents. Rather, there were afew centers of development from which knowledge was transmitted to other regions/other cultures and the centers that were strong in a given period did not remain stable,their strength being in vigor only during limited periods, whereafter other centersemerged, in general by transmission from one of the hitherto dominant centers.

There are numerous spectacular cases of emergence of mathematics in a hitherto‘underdeveloped’ country or culture by transmission:

● the transmission of Greek and Hellenistic mathematics to the culture of the Arabs;● translation of Euclid’s textbook into Chinese by Jesuits at the Chinese court, and

their adaptation of European astronomy to Chinese calendar-making;

8 Cf. ibid.9 Cf. Krayer, Albert. Mathematik im Studienplan der Jesuiten: die Vorlesung von Otto Cattenius an

der Universität Mainz (1610/11). Stuttgart: Steiner, 1991.

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● the Meiji Revolution in Japan in the 1860s. The Empire of the Rising Sun had beenclosed entirely against influences from abroad. After opening had been forced,Western European culture and science were introduced;

Less spectacular or less well-known cases are:

● Brazil, a Portuguese colony until 1808, was intentionally held in a state of under-development by the Portuguese government. When the Court shifted the centre ofPortuguese government to Brazil in 1808, it introduced a decisive modernization,in particular by transmitting French science and mathematics to the educationalsystem now established.

● Another case is Greece, which has been presented here by Kastanis and Kastanis.While at the end of the eighteenth century, during the Ottoman dominance, tradi-tional German mathematics and also some French mathematics was brought homeby Greeks having studied in these countries, an enormous movement of transmis-sion began after 1821, and following independence. For the system of publicinstruction now rapidly being established, modern mathematics textbooks – nowmainly from France and to a lesser extent from Germany – were translated, andused as textbooks in schools and universities.

● Yet another case has been presented by Alexander Karp for Russia: given the stateof general underdevelopment, the energetic introduction of an educational systemby Tsar Peter the Great meant a decisive modernization of the country. For a longtime, its evolution relied on the transmission of foreign science, and it took a longtime until significant national production in science began.

● A revealing case is presented by the USA: since colonization by British settlers,there was for almost two centuries no serious mathematics instruction. A changecame about early in the nineteenth century, with the founding of the militaryAcademy at West Point and the importing of French textbooks. Thereafter, thenational production of adapted textbooks began and permitted reasonable, yet notparticularly noteworthy mathematical instruction. A new change came about in thelast third of the nineteenth century, due to the university level – when the firstuniversities began to create graduate studies and to call professors from abroad forthese – for mathematics, predominantly professors from Germany. This rise wascomplemented by a great number of students going to Europe to study at thesources of modern science – for mathematics again mainly in Germany.

It was this direct transmission at the tertiary level that also raised the quality of math-ematics in secondary schools, and established a reasonable level of national researchin mathematics. The real ‘take-off’ of mathematics in the USA, which shifted thecentre from Germany to the USA, was due, however, to the later exodus of mathe-maticians who fled the dictatorships and persecutions by Fascist regimes in Europe.

Reviewing this continuing process of spreading mathematical culture from onecountry or region, from an erstwhile ‘metropolis’ to another country at the periphery,which might later become a new metropolis of learning, the question is what isprimary for establishing a mathematical culture in a country hitherto not ‘affected’:

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whether it happens by research, through some more or less isolated specialists orpractitioners, or whether it happens by teaching, and by disseminating textbooks.

It seems a bit difficult to answer this question, but the cases just described hint atthe primary importance of teaching, of instruction for establishing a mathematicalculture, which then becomes supportive of research.10

Reflecting on the notion of transmission, it can be noted that the knowledge trans-mitted in general does not remain unchanged upon being integrated into the newcontext of another culture. Rather, it is adapted to the specific traditions and valuesof the new context. That transmitted ideas are changed on reception into anothercultural context is well known in the humanities, but it also applies to mathematicsand mathematics education. A characteristic example was shown in Yamamoto’scontribution: The insistence of the German Treutlein on abolishing the strictseparation between plane geometry and solid geometry was eventually reduced in theJapanese reception to a methodological reform in dealing with solid geometry.

With regard to the second notion within this first dimension of ongoing research,i.e. modernization, one must be aware that it concerns changes within a given educa-tional system: while transmission induces changes adopted from foreign systems,modernization occurs within a system already in function.

Even then, a scrutiny of primary schools and secondary schools as a subsystem ofthe entire educational system of a country seems to reveal that these school systemsnever produced reforms from within, but rather showed considerable inertia, tendingmore toward conservatism than to modernization. Impulses for modernization eithercame from the tertiary level of education, or from applied professions, or from someenlightened, far-sighted personalities exerting leadership by charisma.

A good example of this inertia is the German school system during the second halfof the nineteenth century. Although it was quite evident that the traditionalmathematics curriculum with its dominance of Euclidean geometry was no longerappropriate for the industrially and technologically advanced country, no initiativesin favor of modernization came from the mathematics teachers, not even from theteachers in the realist secondary schools, or from the association of mathematics andscience teachers founded in 1891. In this case, the impulse for change was given byan enlightened personality in the tertiary system: the university professor Felix Klein.

In reality, modernization cannot only happen within the same state but can also bebrought about by transmission from foreign instances.

In fact, mathematics apparently constitutes the only school discipline to experiencetwo prolonged movements of modernization based on international transmission ofreform concepts: the movement of the years after 1908, and the ‘modern mathematics’movement.

10 See Schubring, Gert. “Production Mathématique, Enseignement et Communication.Remarques sur la note de Bruno Belhoste, ‘Pour une réévaluation du rôle de l’enseignement dansl’histoire des mathématiques’ parue dans la RHM 4 (1998): 289–304.” Revue d’histoire desmathématiques 7 (2001): 295–305.

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The first movement of reforms was launched in 1908 by the International Commis-sion on Mathematics Instruction (IMUK/CIEM). Felix Klein, the first president ofIMUK, had understood that the school curricula for mathematics needed to beprofoundly modernized. Following his advice, the main reform agenda became tointroduce the concept of ‘functional thinking’ as a basic notion pervading the entiremathematical curriculum. This main agenda should have as its main impacts:

● the use of graphical elements in the teaching of geometry;● reconciling rigor and intuition;● the fusion of hitherto separated branches of school mathematics, such as plane and

solid geometry;● the introduction of the basic notions of differential and integral calculus.

Evidently, the reception of these reform ideas did not occur uniformly in the variouscountries, but they adapted them to their own cultural and social traditions. Ingeneral, one can state, however, that in a considerable number of countries definitiveeffects of modernization of curricula and teaching were achieved. For instance, as adirect effect, the first all-Russian mathematics teacher congresses discussed reformagendas for Russia based on the ICMI agenda (see the contribution by AlexanderKarp).

There were even direct effects when the movement reached the periphery with somedelay: a telling case is that of Brazil with its reforms of 1929–1932, in the genuine spiritof Felix Klein.11

The modern mathematics movement was launched after the Sputnik shock of1957. This international movement, much broader than the first, clearly alsooriginated from outside, from the OECD – hence from an economic and social driveto develop the industrialized countries in the West in competition with the socialistsystem. There are numerous studies of this movement and also internationalevaluations; the definitive historical study has not yet been done. But Bjarnadóttir’scontribution, in particular, shows that this movement – despite all its shortcomings,and despite its short duration – had a lasting effect:

● It set off an enormous dynamic of reform together with a remarkably broad partic-ipation of teachers.

● It resulted in a lasting modernization of the curricula for secondary schools.● It abolished the separation of the primary schools, and integrated them into a

consequent curriculum aimed at developing mathematical competence for all.● It affected not only the industrialized countries of the West, as originally intended

but a large number of underdeveloped countries, as well.

11 Pitombeira de Carvalho, João Bosco, et al. “Uma coleção revolucionária.” História e EducaçãoMatemática 2, no. 2 (2002): 9–107, and id. “Euclides Roxo e as polêmicas sobre a modernizaçãodo ensino da matemática.” In Euclides Roxo e a modernização do ensino de matemática no Brasil, editedby Wagner Rodrigues Valente. São Paulo: Sociedade Brasileira de Educação Matemática, 2003:86–158.

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● It thus had, besides a democratizing effect within the countries concerned, at thesame time a globalizing effect.

The second dimension of current research concerns teaching practice and textbooks,and, in particular, teacher education. Evidently, all mathematical instruction dependson the teacher, and therefore the first object of all historical research is the mathemat-ics teachers. It is of particular interest for the period when mathematics instructionwas first institutionalized in a country to reveal who the first teachers were, how theyhad become qualified in mathematics, how had they been selected or hired for thisprofession, and how they were trained for their teaching profession.

And for the periods subsequent to this institutionalization it is of systematicimportance to know how mathematics teacher education for the existing school typeswas organized, and to which types of institutions it was attached. These topicsessentially belong to the educational history of the respective country, and may revealhow mathematics teacher education related to mathematics, already established as ascientific discipline, and what value was attributed to mathematics as a schooldiscipline in the period under investigation.

The point, hence, is to show whether a profession of mathematics teachers emergedin the given country, or whether mathematics instruction remained too marginal inthe school system to permit a separate discipline to become established among thegeneral teachers’ profession. A further issue is whether possibly different strataexisted within a profession of mathematics teachers, say, according to diverse schooltypes, imposing different views or epistemologies of mathematics in their instruction.Establishing such patterns, features and characteristics for the respective countrieswould provide a basis for comparing the structures in different countries, and fordistinguishing between general or common patterns and particular ones.

Another task to be tackled after these first tasks have been solved would be toestablish a prosopography of the mathematics teachers in that country, at least for thefirst periods of institutionalization of mathematics instruction. ‘Prosopography’ is atraditional method in history as a science to establish collective biographical data onspecific groups, and thus to reveal common structures in that group – say with regardto social origin, formation, professional orientations and shared beliefs.

A further task would be to study the professional life of these mathematics teachers,to unravel typical conflicts – say with colleagues of other disciplines, with the head-master, with the administration, with pupils and their parents – thus gaining accessto the historical reality, the everyday life of teaching.

Unfortunately, despite the systematic importance of these issues, one has to statehere one of the greatest deficits in research. There are only very few relevant studies.Maybe the gaps are due to the fact that the first issue, how the mathematics teacheremerged as a figure proper, in general requires extensive archival research, and thuscan be realized only by persons devoted to such research and methodologically qual-ified to carry it out. And the second issue, the history of institutionalized mathematicsteacher training, requires it to be situated within the respective history of education,and the history of mathematics, and makes for a quite complex research task, too.

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676 G. Schubring

Among the few studies one might mention, besides the research presented here byEileen Donoghue and some research in Brazil12 – both focusing on teacher educationin normal schools for primary schools and in universities for secondary, but mainlyonly for more recent periods in the early twentieth century – studies on theemergence of the profession of mathematics teachers in nineteenth-century Prussia,as part of the fundamental social and educational reforms realized there.13 Thesestudies reveal three successive generations of mathematics teachers as contributing tothat emergence:

● Since most of the Gymnasia existing at the time of reform already had somemathematics instruction, there were teachers who taught mathematics besidesother subjects as well. Some of these encyclopedically trained teachers continuedto teach mathematics.

● The second generation consisted of self-educated enthusiasts of mathematics whowere hired to meet the new high demand for mathematics teachers. Since theylacked systematic training, they were in general not helpful for implementingmathematics as a major new discipline in schools.

● Only the third-generation teachers trained at the likewise reformed universitieswere able to realize the neo-humanist vision of mathematics as an integral part ofknowledge.14

In the same vein, one should mention studies by Harm Smid, who has undertakenlargely analogous studies for the Netherlands for the first half of the nineteenthcentury.15

In contrast to teacher education, textbooks constitute a preferred and well-studiedhistorical issue in many countries. These numerous studies have already presented aconsiderable amount of knowledge on historical development. This knowledgeshould be integrated in order to establish what constituted ‘school mathematics’during a given period for a given country: as a hierarchical unfolding of mathematicalcontents, and tied to certain teaching methods and epistemological views.

One must be aware that the analysis of historical textbooks also presents method-ological challenges: a purely internal analysis of just one textbook will not yieldsignificant results; it requires comparison – but with what? In general, there is noother canonical textbook that can be used as a standard, as a measuring unity.Rather, one needs several textbooks in order to better grasp the spirit of a certainperiod. And in the case where several editions of a school textbook exist with changesin the presentation, an internal analysis will again not be sufficient, as the changes

12 See Silva da Silva Dynnikov, Circe M. “A preparação pedagógica dos professores de Matemáti-ca da Faculdade de Filosofia Ciências e Letras-FFCL da USP.” Cadernos de Pesquisa em EducaçãoPPGE/UFES (Vitória/ES) 8, no. 15 (2002): 8–37.

13 Schubring, Die Entstehung des Mathematiklehrerberufs, 1983.14 Ibid.15 Smid, Harm J. Een onbekookte nieuwigheid?: Invoering, omvang, inhoud en betekenis van het

wiskundeonderwijs op de Franse en Latijnse scholen 1815–1863. Delft: Delft University Press, 1997.

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Paedagogica Historica 677

need not be the result of cognitive progress of the author, but may (also) be due tosocial pressure on mathematics teaching.16

For the last major issue in this second dimension, teaching practice, there are alsoa considerable number of historical studies. If not regarded in an isolated manner,they already provide links to the cultural, social and political factors, since the meth-ods much favored at a certain time in a given country – say laboratory method, childorientation, problem orientation, etc. – might be strictly rejected in a later period inthe same country, or simultaneously in another country. A hint at these diversitiesbecame visible already in the discussion of the TSG’s second day, when practice-oriented methods according to John Dewey’s pragmatism, in complete agreementwith the educational values in the USA, were shown to be suspect of impedinggeneralization of concepts, in different social and cultural contexts, in the perspectiveof mathematics for all.

Regarding the last dimension, the cultural, social and political functions of mathe-matics instruction, the diversity of realizations of mathematics instruction seems toreach a peak, but, on the other hand, the applicability of comparative methods forunraveling comparisons and common patterns has in principle been achievable in thebest manner since functioning of traditional societies is investigated by anthropology,and modernization processes and social conflicts brought about by them by sociology.

In the preceding articles in this issue, a number of points relevant for this dimensionhave already been made. And in the existing studies on national history of mathemat-ics instruction, one can often find a wealth of information on this dimension. Thisinformation quite often needs to be ‘excavated’, to be made explicit, and to be placedin relation to the general educational and political history of the country concerned.

Livia Giacardi’s contribution is an excellent example, which makes the relationsbetween the different actors and instances clearly explicit – those between mathema-ticians, mathematics teachers, cultural traditions and their influence on schoolstructure, and political movements and decision. The extraordinary feature of theItalian case is the split within the mathematical community.

Evidently, ‘excavating’ in this way the ‘nuggets’ from existing publicationsestablishing relations to cultural, social and political history, and if necessary,complementing this by archival research, needs devoted specialists. And achievinginternational comparisons requires international cooperation.

Perspectives for Future Research

As regards future work we are confident that this Symposium will have enhancedinternational cooperation, so that continuing communication and cooperation willproceed to further substantial results. As a first step, we are establishing an interna-tional network, based on a website.

16 Cf. Schubring, Gert. “On the methodology of analysing historical textbooks: Lacroix as text-book author.” For the Learning of Mathematics 7 (1987): 41–51, and id. Análise Histórica de Livros deMatemática. Notas de Aula. Campinas: Editora Autores Associados, 2003.

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