Transitions from school mathematics to university mathematics – an international variable for educational policies

Abstract

The approach of coloniality to analyse the impact of political-cultural power structures on mathematics teaching is currently being discussed in mathematics education, especially in Latin America. In particular, one affirms there a hierarchy between the mathematical knowledge that is to be taught in the different educational levels, and is lamenting the hierarchy: as an effect of coloniality and as an expression of a global, Eurocentric hegemony. However, such relationships between the different content parts of the educational structure have been established in each country in a specific way, that is, according to the forms in which the educational system had emerged in that country. Whether hierarchies are implied has to be investigated for each case separately. The core of the differences is constituted by the structural functions achieved in Modern Times by the former Artes Faculty of Medieval Times. It depended on these functions which requirements were instituted for the transition from secondary level to higher education and which authority had to decide whether the requirements were met. In particular, the paper will discuss how the different paths for the transition from secondary to higher education have also shaped different structures of school mathematics and to what extent power mechanisms and hierarchies can be proven in them.

1. Introduction

The two–antithetical–concepts of coloniality and decoloniality were recently developed by Catherine E. Walsh (Ecuador) and Walter D. Mignolo (USA) in order to analyse ongoing Eurocentric practices in dealing with former colonial countries, as well as to promote local, different practices, referred to as “decolonial attitudes” (Walsh, 2018, p. 17). ¹ It is a radical critique of a “colonial power matrix” exercised by European powers since about 1500 – including US policy in more recent periods – and imposing occidental modes on other cultures and societies. A characteristic description of the decoloniality programme has been given by Walsh.

“Decoloniality denotes ways of thinking, knowing, being, and doing that began with, but also precede, the colonial enterprise and invasion. It implies the recognition and undoing of the hierarchical structures of race, gender, heteropatriarchy, and class that continue to control life, knowledge, spirituality, and thought, structures that are clearly intertwined with and constitutive of global capitalism and Western modernity. […] Decoloniality, in this sense, is not a static condition, an individual attribute, or a lineal point of arrival or enlightenment. Instead, decoloniality seeks to make visible, open up, and advance radically distinct perspectives and positionalities that displace Western rationality as the only framework and possibility of existence, analysis, and thought” (Walsh, 2018, p. ii).

Criticisms of Western notions and concepts of rationality and modernity are by no means new. They already arose in the debate about “Orientalism”, as coined by Edward Said in his much-debated book, first published in 1978, which describes the Orient as a construction of the Occident, of the West (Said, 1978). Likewise, in his provocative trilogy Black Athena, Martin Bernal (1987–2003) showed that the Occident is also a construction; his extensive and novel analyses were dedicated to reveal that the origins of modern science are not deducible just from ancient Greece but can be rooted in Africa.

However, these new concepts of coloniality/decoloniality appear to be very radically all-embracing, especially with their questioning of rationality.

A reception and adaptation of this approach to mathematics education has been published, in 2019, by two Brazilian researchers in mathematics education, Victor Giraldo and Filipe Fernandes. The title of this paper Caravelas à vista!–Caravels in Sight!–is programmatic: it aims to reverse the perspective. The traditional historiography uses to speak of the “discovery” of America–as if the continent had not been populated before the arrival of the colonists. Thus, the traditional report of the cry “Land in Sight!” on the Portuguese ship approaching what later became Brazil is here reversed to the cry of the indigenous people remarking the ships nearing to their coast.

In this adaptation of the general concept of coloniality/decoloniality to mathematics education, the focus of the analysis is the institution university. They are denouncing:

[…] an epistemic model propagated by the euro-USA-centric modernity, with the University being the locus that embodies disciplinary thinking and establishes an arboreal organisation of its structures (Giraldo & Fernandes, 2019, p. 474; transl. G.S.).

There, the University not only impedes interdisciplinarity:

In this view, the University intensely marks the boundaries between the fields of knowledge, and initiatives to transgress them are discouraged (ibid.),

but the University is also declared to be an agent of coloniality, acting for establishing hierarchies:

Here, it seems to us, an important trait of coloniality is operating: a hierarchisation of knowledge directed towards the hierarchisation of social bodies; an organisation of knowledge that places the pedagogical not as maturation – even if treated at the end of training –, but as qualifying only when mathematical knowledge, superior, precedes and determines it (ibid., p. 475).

This criticism, which is related to the traditional form of structuring mathematics teacher education in Brazil, ² relies on a highly problematic, even stronger and more fundamental criticism:

Thus, being a teacher would be epistemologically and subjectively conditioned to being a mathematician: there cannot be a teacher of mathematics who is unaware of the mathematics practiced in scientific-academic spaces (ibid., p. 475).

What is preoccupying with this criticism is that it implies the claim that one can become a mathematics teacher in school being unaware of present-day mathematics. In fact, it is assumed in this conception that the teachers could build up their own separate “teacher-training mathematics” as resistance against the Eurocentric university mathematics. Mathematics is even decried “as a Eurocentric epistemic field” (Fernandes et al., 2022, p. 13).

A basic problem in practically all approaches to coloniality and decoloniality is that universal statements do assume a global validity. Already the criticism of Euro-centrism is undifferentiated – even now in its extension to a Euro-USA-centrism – as if all European colonial powers had pursued the same political, social and cultural agenda.

Actually, a broader investigation of these colonial practices regarding mathematics teaching is still needed. I have given a short analysis of mathematics teaching in various colonial systems: of the Portuguese colony Mozambique, of North Africa in the colonialism era – Egypt before becoming rules by Great Britain, Tunisia before becoming a French colony and Morocco as French and as Spanish protectorate -, and of British India (Schubring, 2021, pp. 1459 ff.).

I will mention here in particular two cases of colonialism.

1.1 An extreme case of colonialism

In the case of India, one can distinguish three periods of colonial teaching of mathematics. The first two are marked by the governing of even more extended parts of India by the EIC, the East India Company, founded in 1600 to trade in the Indian Ocean region The EIC had expanded enormously in India since about the middle of the 18^th century, and beyond its original commercial function also now acting politically as colonial agent. Originally, it had not intended to organise education there and had sent administrative staff from Britain.

It was at first British settlers who founded colleges, from the 1780s–parallel to the already traditionally existing schools for the Hindu and Muslim population, where teaching was effected in Sanskrit and in Arabic. At the colleges created by the EIC from 1805, called “Oriental colleges”, teaching should be done in the local language, thus in general in Arabic or Sanskrit.

The second period was initiated due to a revealing brake of colonial policy, marked by the so-called Macaulyan-Minute of 1835: it ridiculed the local languages as “pure and rood” and required that the colleges should teach in English. At these anglicised Oriental colleges, a class of local colonial officers should be formed. Yet, during both these two periods, mathematics textbooks there were based on books used at British military schools and at the proper EIC colleges in Britain preparing for the colonial service: these textbooks were more modern than those used at the English universities – for instance A Course in Mathematics by Charles Hutton (Aggarwal, 2006, pp. 18, 27).

However, after the rebellion of 1857, the British Crown dissolved the EIC and took over the direct control of the former EIC colonies in India, transforming it to the British Empire of India. From now on, teachers sent to the English-Oriental colleges were increasingly graduates of English universities. Given the dominance there of English editions of Euclid for teaching mathematics, teaching mathematics in British India became ever more based on Euclid editions as textbooks (see Schubring, 2021, p. 1462; Aggarwal, 2006, pp. 243). The case of British India shows thus, furthermore, that the basic mathematics transmitted during Imperialism was of a different kind and level than that transmitted by other European powers. ³

1.2 Teacher education in Brazil–a case of coloniality?

I will analyse here the alleged hierarchy in Brazilian mathematics teacher education structures, as an example how detailed investigations can relativise or differentiate global affirmations.

The major issue of the coloniality-criticism in Brazil is the affirmation of a global “knowledge hierarchy between university and school” (Matos, 2019, p. 20). And their focus is to outline that mathematics teacher education in Brazil had been, from its beginnings, subject to such a hierarchy.

Giraldo and Fernandes base their affirmation of hierarchy upon a historical sketch of the history of teacher education in Brazil. To understand the history of teacher education in Brazil, one needs to know that from 1808, when Brazil suddenly turned from a colony, exploited for its raw materials, to the mainland of the Portuguese Empire, there existed no teacher formation at all within the system of higher education then eventually established. This system of higher education was modelled according to the French system, established in the Napoleonic era, providing separate professionalising faculties – hence without an integrating Philosophy Faculty (see Schubring, 2002a). The only institution where academic mathematics was taught was the Military Academy, for civil and military engineers. Those who became teachers at the colleges, established as secondary education, used to be persons who had studied as engineers or were even graduates of that Academy, or of polytechnic schools founded later on.

The decisive change occurred only in the 1930s when genuine universities were founded in Brazil, no more as mere aggregations of separate faculties, but with a Faculty of Philosophy as an integrating kernel. And the declared intention of these new institutions was to eventually establish teacher education for the secondary schools. The first such university was the Universidade de São Paulo (USP), in 1934. From the first creation on, this intention was followed at various places, founding universities with a Philosophical Faculty; the next one was in 1935 the Universidade do Distrito Federal (UDF), in Rio de Janeiro.

Regarding this first real university, Giraldo & Fernandes maintain:

Together with the hegemonic narrative, which converts the mathematics course at the USP into the cry Terra à Vista! regarding the history of training those who teach mathematics in Brazil, different studies point out that the main function of the first courses was to prepare mathematicians, with the intention of training teachers professionally being subordinated to the training of scientists (Giraldo & Fernandes, 2019, p. 473).

While they speak here of “different studies” without referencing them, they give a reference in their parallel paper, published in English. Presenting there “this hegemonic narrative about Mathematics teachers’ education in Brazil” that it “reproduces traces of coloniality, strongly affecting our contemporary teachers’ education programs”, the “first trace” being:

Particularly in Mathematics teachers’ education, the knowledge on didactic-pedagogical aspects of teaching was subordinated to the mathematical scientific-academic knowledge. In a study on the 80^th anniversary of Mathematics Undergraduate Program at USP, Gomes (2016) states that “the main role of the program was the education of mathematicians, pushing aside the goal of professional education of Mathematics teachers to be subordinated to the education of scientists” (p. 429, our translation) (Fernandes et al., 2022, p. 16).

Actually, it proves that neither Gomes nor the other authors she is quoting there had verified the original course structure established at the USP in 1934. Researching the statutes and practices of the first years, one remarks that the opinio comunis is mixing up different degrees, for differing academic levels, and mixing up different periods of degree regulations.

To comment these claims, one has to remind, first of all, a basic result from the research upon the emergence of research into pure mathematics. It had been the restructuring of the Philosophical Faculty in Prussia, as an integral element of the neo-humanist reform of education from 1810, that the Faculty had obtained a proper function, namely training teachers for the reformed secondary schools (Gymnasien), that the Faculty professors had been enabled to specialise professionally in their disciplines – and to inspire their students with a scientific spirit (see Schubring, 2002a, p. 55).

The same pattern is confirmed by the creation of these new universities in Brazil. Originally, the new Philosophy Faculty was structured to provide two degrees: the licenciado (sometimes also called Professor Secundário), the graduate of the teacher formation course, and the doutorado degree, as the following level for scientific production. In fact, the first of the yearly reports of the Faculdade de Filosofia, Ciências e Letras (FFCL), underlined for its first year 1934–1935 the primacy of teacher training in a scientific spirit:

That is why the creation and functioning of the Faculty of Philosophy, Sciences and Letters at the University of São Paulo, in 1934, established with the aim of giving teaching a scientific nature and making it possible to prepare teachers for secondary education (Anuário, 1934–35, p. 215; transl. G.S.)

And in 1936, the director of the FFCL declared in his discourse for the first conferring of degrees:

If the Faculty of Philosophy, Sciences and Letters could decide between its two purposes–that of training teachers for our secondary education and that of starting scientific production – it would naturally be the first one which would become its predilection from the beginning. Before we will have scientists, it is essential that we provide a basis with a body of selected secondary teachers (Anuário, 1937, p. 186; transl. G.S.).

All the first graduates of the FFCL obtained the licenciado degree; for mathematics it was five graduates, becoming mathematics teachers (ibid., p. 190). Their disciplinary studies occurred together with studies on pedagogy and psychology, in a parallel school, resp. faculty. The Statutes of the FFCL provided for a doctor's degree: students could enter these courses after obtaining the licenciado degree. The doctorate was a course after the teaching course – not a parallel course. The first “scientists”, i.e. doutorandos, graduated quite a period later, in 1952. The first one in mathematics was Chaim Hönig (1926–2018) (Cavalari & Nobre, 2019).

A drastic change of this degree structure occurred in 1939, by an intervention of the Federal government into the functioning of the universities so far established by the governments of federal states – and this during one of the dictatorship periods of the president Getúlio Vargas. Then, in 1939, the president dissolved the UDF and transferred its parts into the 1920 founded Universidade do Brasil, upgrading thus – what had been so far a mere sum of faculties – now to a real university, with a Philosophical Faculty. This Faculty obtained now the ambitious name of Faculdade Nacional de Filosofia (FNFi), implying that its structure should be the model for all the universities of Brazil. The Federal government decreed its Statutes, and stipulated that the FNFi had to confer two degrees.

Actually, with this decree-law of 4 April 1939, its Art. 48 imposed the bachelor degree as the principal degree of the FNFi. Its Art. 49 simply degraded the licenciado to be an extended form of the bacharelado: After the three years of disciplinary studies, a student having obtained the bachelor degree, could obtain the “diploma de licenciado” in this discipline, after an additional year of pedagogical-didactical studies (Decreto-lei 1.190). This decree established thus the formula “3 + 1”, characterising for decades the practice of teacher training in Brazil. The decree gives no reasons at all, in particular not for the preference given to the bachalerado degree and to the splitted form of teacher training. I am still researching for reasons; probably, the education minister Gustavo Capanema was conceiving of the Philosophy Faculty in the traditional forms of polytechnic schools, where one was conferring the bachalerado degree.

For this structure and during the period of its validity, the quoted hierarchy criticism can be applied, on the one hand. On the other hand, no coloniality is evidenced by this restructuration. It shows the intervention within one country, motivated maybe by the traditional forms of training engineers.

This the more, as the imposed degree remained dead letter for decades. Given that no careers other than the “magistério” were accessible for graduates, the licenciado remained the principal access to a profession for those studying mathematics. For instance, then at the USP, the yearly report for 1952 reports about the number of graduates (Figure 1):

Figure 1.

Anuário, 1952, p. 47.

This very low number of those who accepted the new degree imposed from above (Anuário, 1952, p. 47) shows that there were no real career perspectives outside of the education sector. This began to change only later on, when the country had become more industrialised. The reform of Brazilian universities of 1968 is connected with this change.

And it was in the 1970s that the strange formula “3 + 1” became abandoned. Due to the new structuring of university studies, according to the US-American model, into an undergraduate and a graduate programme, the undergraduate mathematics course became organised as two parallel courses: for the bachalerado and for the licenciatura. The Federal government is defining the general structure of these teacher education programmes, by the Diretrizes Curriculares Nacionais para a Formação Inicial de Professores para a Educação Básica (MEC, 2019). And in these National Curricular Guidelines for the contents to be studied in training teachers for public schools, it is declared that the definite criterion for these studies is the curriculum of the public schools. ⁴ Hence, the criterion for structuring the contents of teacher education in Brazil is the school curriculum – and not the standard of the mathematics Bachelor degree.

This case of Brazil shows changes in the values attributed by the society to teacher education. We remarked here a competition between valuing to increase generally the education, on the one hand, and valuing engineering qualifications for a part of the new generations. Such changes in social values are related to differing socio-political situations – occurring frequently in other countries, too. Depending upon social developments, changes in value systems are specific, and not signs for a coloniality.

1.3 Transitions from secondary to higher education

The focus of the coloniality critiques in Brazil is the global critique of “knowledge hierarchies between university and school” (Matos, 2019, p. 20). They are claiming to have “revealed characteristics of coloniality that reinforce hierarchies of knowledge based on the primacy of scientific knowledge as the basis for school and professional knowledge of teachers” (ibid., p. 154; transl. G.S.). Therefore, a hierarchy of academic mathematical knowledge over school mathematical knowledge is denounced there in general.

In fact, however, many developments and structures in the field of education and science are not the expression of universal and global characteristics, but result from the characteristics of a specific society, an individual country or a cultural tradition. This shows the need for a concrete socio-historical analysis – it is afforded in each case for the transition between secondary school and higher education.

Actually, the reflections about this transition understand it as just one issue. A closer look reveals, however, that two different aspects are implied:

The relation between the mathematics school curriculum and “academic” mathematics, and

the regulations for the transition from secondary school to higher education, or – according to the practice in many countries – for the entrance into higher education.

We will discuss here both aspects.

1.4 Conceptions on relations between the mathematics curriculum and university mathematics

There is an excellent recent study, which has analysed the relationship between school mathematics and university mathematics (Scheiner & Bosch, 2023). It has identified two “extreme” positions in mathematics education for conceiving of this relationship, understanding it more generally to confront school disciplines and academic disciplines (ibid., p. 767). One of the extremes is identified as:

school mathematics and university mathematics are identified as a single entity, ‘mathematics’, with different elements selected as school content or university content (ibid.).

The other extreme being:

school mathematics and university mathematics can be considered independent fields of knowledge, where one does not influence the other and should therefore be considered as separate entities (ibid.).

The two authors have restricted themselves to study the first extreme. There, they have selected and analysed three approaches for this vision, as constituting the major and pioneering approaches. These are the approaches by Felix Klein (Germany, 1908), Lee S. Shulman (USA, 1986) and Yves Chevallard (France, 1985). They alert that Shulman's approach is not specific for mathematics, but applies to all school disciplines, and that each approach has to be analysed within its specific context:

the intellectual climate of the time and the region strongly influenced the way Klein, Shulman and Chevallard expressed themselves. (ibid., p. 768).

The approach of Shulman, a researcher on educational psychology, is well-known for his conception of pedagogical content knowledge. There, academic knowledge is assumed as given and stable, providing the content knowledge which novice teachers will transform “into pedagogical forms that were accessible for students” (ibid., p. 770). Academic or university mathematics constitutes the defining instance, without considering historical developments and changes. Chevallard is likewise well-known for his conception of transposition didactique. Academic mathematics is assumed as fixed and stable, too. He introduced a new instance for deriving from it the school mathematics: the professional group of mathematics educators, called noosphère by him.

Regarding the directionality between university mathematics and school mathematics, Scheiner and Bosch emphasise that Shulman had “explicitly asserted that the academic discipline precedes the school subject” (ibid., p. 772). And for Chevallard they state “that the knowledge taught in school is derived from a pole of scholarly knowledge and transposed to a seemingly subordinate pole in the classroom” (ibid.).

Felix Klein's approach was instigated by his intention to overcome the “double discontinuity” he had revealed as a major obstacle for mathematics teaching: the student, entering university, has to forget what he had learnt in school, and graduated in mathematics and returning to school as mathematics teacher, he has again to forget what he had acquired as modern mathematics being forced to practice again traditional school mathematics. Klein understood mathematics as a unity, “holistically, as constantly evolving and reforming through a process of elementarisation” (ibid., p. 769). This elementarisation has nothing to do with a simplified elementary mathematics as school knowledge. Rather, elementarisation means a process of restructuring mathematics, achieving a new, relatively coherent architecture based on elements as basic concepts, allowing to develop the conceptual structures – thanks to a ‘maturation’ process of newly established disciplines which had evolved independently, but where the elementarisation had revealed their underlying common elements (see Schubring, 2016) – like Klein's Erlanger Programm of 1872 had revealed group theory as the common element to all the geometric disciplines which had developed in particular during the 19^th century, providing thus an elementarisation of geometry for his time.

It is therefore most significant that Klein, while understanding the entire mathematics holistically, did not understand school mathematics as an “elementary” form of academic mathematics. Rather, he attributed schools an entire independence in deciding about the contents of their mathematics curriculum:

The normal process of development […] of a science is the following: higher and more complicated parts become gradually more elementary, due to the increase in the capacity to understand the concepts and to the simplification of their exposition (“law of historical shifting”). It constitutes the task of the school to verify, in view of the requirements of general education, whether the introduction of elementarised concepts into the syllabus is necessary or not. (Klein, 1907, p. 90; transl. quoted from Schubring, 2019, p. 175).

As a matter of fact, Klein is the only one among these three approaches who was aware of the necessary socio-cultural context of schools, which would be decisive for the curricular decisions:

“School mathematics, however, will again have to select from this elementary mathematics what best corresponds to the aim of secondary schools, namely: to provide a general basis for understanding our present-day culture. (Klein, 1907, p. 111; transl. quoted from Scheiner & Bosch, 2023, p. 769).

As Scheiner and Bosch emphasise, school mathematics has according to Klein to be in an “epistemic” agreement with the presently restructured state of mathematics (Scheiner & Bosch, 2023, p. 772) ⁵ – like “functional thinking” had been that epistemic element in Klein's time.

Surely, Klein was not a mathematics educator – mathematics education as a discipline did not even exist then–and he was not developing curricula; he was rather delineating general visions. But it is likewise significant that he had not required immediate impacts of elementarisation processes within mathematics upon school mathematics. Instead, he had considered that there can occur a certain “hysteresis” for such impacts.

Klein does not propose to treat the latest scientific results; rather, he allows proper choices according to criteria of the school system, yielding a certain “hysteresis” behind the recent, not yet elementarised state (Schubring, 2019, p. 167 f.).

This ‘hysteresis’ is a second category indicated by Klein for characterising school mathematics, besides his ‘historical shifting’. Requiring a certain maturity for new mathematical theories before being able to enter in a restructuration process, Klein was quite sceptical about the fresh set theory entering into the school curriculum at his time (see ibid., pp. 176 ff.).

It is to be regretted that Scheiner and Bosch have not included the other extreme into their study. Its main representative is the French André Chervel, a researcher on the history of education. It is important for assessing his approach that it was not developed for mathematics, or the sciences, but for the humanities – in particular for the native language, history and geography, religion, and even philosophy – where there do not exist such clear-cut differences between school knowledge and academic knowledge. Therefore, the focus of Chervel's approach

is the socialising function of school and hence, in particular, of school disciplines. The school culture and school subjects are thus characterised by autonomies: it is believed that school disciplines enjoy autonomy with respect to the other disciplines (Chervel, 1988, p. 73), while the example of mathematics shows that the set of all disciplines influence strongly the status, the level and the views of school mathematics (Schubring, 2005). Moreover, Chervel emphasises the generative nature of the school, which results in creating, due to its character understood as relatively autonomous, school disciplines (Schubring, 2019, p. 171).

Chervel is emphatic in emphasising the autonomy of school disciplines, independent even of the academic disciplines, his key concept being the school culture:

The teaching contents are conceived as sui generis entities. specific to the school class, independent to a certain extent of any cultural reality outside the school, and enjoying an organisation, an intimate economy and an efficiency which they seem to owe to nothing other than to themselves, that is to say to their own history (Chervel, 1988, p. 64; transl. G.S.). ⁶

Chervel's conception of the independence of school disciplines, being analysable only from the point of view of school culture, has been applied in particular in Brazil as a conception for studying the history of mathematics teaching (see Moreira, 2004, p. 16).

Resuming these four approaches, one remarks that for three of them mathematics constitutes a fixed and stable block, without a historical evolution of its elements which would affect the relationships with school mathematics. It is only Felix Klein's approach, which is conceiving of this historical variance, on the one hand, and, on the other hand, conceding schools the independence to act in a certain epistemic correspondence.

Besides the socio-cultural differences of the theoretical approaches in mathematics education, one has to be aware that the construction of a mathematics curriculum always occurs within a specific country, based likewise on the specific socio-cultural framework of that country. This issue will be discussed in the section following the analysis of the transition regulations.

1.5 Determinants for the transition from secondary school to higher education

Since educational structures in Western Europe in Modern Times – as being decisive for what would become transmitted in later times to other regions and continents – were prefigured by the forms of universities as they emerged in Medieval Times, I have at first to evoke basic features of these first institutions for higher learning.

The universities in the Middle Ages formed the core of the institutionalised structures, or more generally: what is now called higher education. There was practically no distinction yet between secondary and higher education, as evidenced by the university names of the time, such as “Archiginasio” and “Hohes Studium” (see Figure 2). Admission to the Faculty of Arts could already take place at the age of 11 or 12.

Figure 2.

Entrance to the main building of the University of Padua (Italy), dating from the Middle Ages: “Gymnasium omnium Disciplinarum”.

The later differentiation into primary, secondary and university education, with the regulations for transition from one to the next, occurred differently in the nation states that emerged since early modernity. It was transformations of the Faculty of Arts, which provide the key for understanding the differing patterns arising for the transition from secondary schools to universities.

The first decisive transformation was the splitting of this Faculty into a secondary school, assuming the teaching of the younger students admitted so far to the Faculty, and the remainder of the Faculty, providing now more focused lectures. There were two reasons for this splitting, occurring since the fifteenth century:

as students of the Faculty, the adolescents were academic citizens and thus exempted from public order in the university towns; they were hence permanent trouble makers in the urban life. Universities began, hence, to establish colleges with a sequence of yearly, ascending school forms and providing systematically organised and structured teaching;

a second root were schools established by municipalities; these also became soon organised in this structure of ascending yearly school forms (see Schubring, 2014, p. 131).

The second decisive transformation occurred as a consequence of the Protestant Reformation and the ensuing Catholic Counter-Reformation. The essential difference of the educational structure as it became established thereafter manifested itself in the different fate of the former Artes Faculty.

While in Protestant, especially in Lutheran territories, the Artes Faculty was able to achieve a more independent position – with greater developments for its disciplines – and this rise became evidenced by its renaming as “Philosophical Faculty”, in Catholic territories the faculty did not only remain in its subordinate function, it was even largely substituted by the secondary school-like colleges. This radical functional change in the Catholic territories was essentially a consequence of the work of the Jesuit order, which in the first century of their impact for the Counter-Reformation, from about 1550, was in fact the only force to build up a higher Catholic educational system. The structures of the two principal forms, the Protestant-Lutheran one and the Catholic-Jesuit one, are visualised in Table 1.

Table 1.

The structures of the Protestant (Lutheran) and the Catholic-Jesuit education systems. ¹¹

Since the Jesuit colleges had not the right to confer academic degrees, the former Artes Faculty was only maintained as the examination board for conferring the degree for entering one of the (formerly: higher or professional) faculties. By this structure, the former Artes Faculty was “absorbed” by the college.

Besides these two principal structures, ⁷ there were two more educational structures in Western Europe: the less known Calvinist-reformed variant of the Protestant structure and the often not well understood English structure, based on the Anglican religion, as institutionalised by the Elizabethan Reform of 1570. Their structures are shown in Table 2: In the Calvinist or Reformed territories, the secondary school, sometimes called Pädagogium, was organised as part of higher education, as the preparatory stage before entering the Philosophical Faculty of the university. In the Netherlands, the so-called Latin schools, with this preparatory function, were by law understood as part of higher education. A law of 1863 introduced “secondary education” in the Netherlands, but by creating the HBS (Hogere Burgher school), a new realist school type, not providing access to higher education. Another law, of 1876, confirmed a decree of 1815, defining the Latin schools as part of higher education – only renaming them Gymnasia. This remained their official status, until a new law of 1963 (see Smid, 2022, pp. 100 ff.).

Table 2.

a) structure of the Calvinist-Reformed education system; b) corrected structure of the English education system, showing the colleges dominating all the teaching at the universities, different from the faculties like in the Protestant universities on the Continent.

The two English universities, Oxford and Cambridge, are known as collegiate Universities (see McConica 1968). Whereas colleges had become the function of secondary schools in the Catholic-Jesuit countries, the English universities had expanded the notion of ‘college’ to be the basic teaching unit of higher education–and no longer faculties. Thus, after frequenting a grammar school, a student would enter one of the colleges of a university. Only the final exam, for graduating, would be organised by a board obtaining to the original structure of faculties (see Schubring, 2002b, p. 368). Isaak Barrow, for example, entered a college of Cambridge University at the age of 13 years.

1.5.1 Transition by the Abitur

Within these various educational structures, it was for a long time understood as the right of the parents to decide when one of the sons should enter a university; at best, there was some rather formal exam by the rector for enrolling him. The first formal exam to conferring the right for entering university was introduced in Prussia, in 1788, a Protestant-Lutheran country. And this right was defined as enabling transition: as the final exam of the secondary school, tere called then Gymnasium or Gelehrtenschule. The final exam should verify whether the student had achieved the maturity for university studies, called Abitur, ⁸ or Matura. At first, this exam was not obligatory; during the feudal epoch, parents could not be restricted in their decisions for their sons. The Abitur exam became obligatory for the transition in Prussian, in 1834. Thus, it was the school which exerted the right for deciding upon the transition; universities had to accept Students provided with the Abitur for enrolling.

It has to be noted that this conception of Abitur implied that the exam exercises were defined locally, by the teachers of the respective discipline at each Gymnasium. This was an expression of the strong social position of the teachers, established by neohumanism in Prussia. This aspect is reinforced, when one knows that the Abitur was introduced also in Bavaria – one of the Catholic states in Germany, but in the form of a Zentralabitur, where the exercises were defined by central state agencies – due to distrusting the qualifications and the impartiality of the teachers (see Schubring, 1989, p. 284).

1.5.2 Transition by the Bac

In the traditionally Catholic countries of Western Europe, on the other hand, it was the universities that decided on admission to studying. This pattern had become established as a continuation of the traditional structure in the Jesuit system: the function of the remainder of the former Artes Faculty, reduced there to the entrance examination, was retained as a power privilege of the universities. In France, for example, the entrance exam is known by the short name “Bac” – for Baccalauréat, reminding the final exam of the Medieval Artes Faculty.

In Brazil, a traditionally Catholic country, each university organised annually a vestibular, an entrance examination, for admitting new students. A short time ago, this vestibular has been substituted for most of the universities by the ENEM – Exame Nacional do Ensino Médio – organised by the Ministry of Education, with the participation of universities for elaborating the exercises.

1.5.3 The British special way

Regarding the transition, Great Britain practices again a quite special way. There, a proper school type serves for preparing and for deciding about the transition to higher education. There, students have to undergo an intermediary education stage (Further Education) and sit for national exams that measure their progress, according to the so-called A-Levels.

Advanced Levels or A-Levels are subject-based qualifications that British students aged 16 or older must get if they plan to enter university or just want to gain better knowledge in a particular study area or profession they are aiming to into.

A-Level courses are provided by Sixth Form Colleges and Further Education Colleges. Essentially, these two institutions are the same except that Further Education Colleges operate separately from high schools, offer a wider range of courses and qualifications; moreover, they ensure a different learning environment compared to high schools.

A-Levels are divided into two stages:

AS Level – it is taken by students aged 16 to 17, and

A2 Level – this is taken by students aged 17 to 18

At the end of each stage, the students have to sit for a certain examination. To move in the upper stages of education, one must necessarily pass these examinations.

Resuming this variance of transition regulations, one remarks that there existed and exists no overall dominance of universities about secondary schools. The differences are expressions of differing cultural traditions – revealingly based originally on religious conceptualisations of education.

1.6 The emergence of school mathematics in relation to academic mathematics

Thus, even if the Gymnasia in Germany were able to establish themselves as independent compared to the universities, the question remains how school mathematics established itself in relation to academic mathematics–whether there was or is a power relationship and hierarchy by university mathematics regarding the determination of contents and concepts of school mathematics. I will discuss this question here for Germany, as a country where the transition is determined by the school. “School mathematics” could only be configured from the formation of public school systems, i.e. with regulations for curricula and examinations only after 1800.

And an essential constituent of school mathematics is that it cannot be determined solely or primarily through its relationship to academic mathematics, but that it is first and foremost an element in the concert of all school subjects–and the determination of the proportions of the individual school subjects within the whole and their relative attribution of values is the result of a complicated socio-cultural process of negotiation. Strangely, this embedding into determining the overall curriculum of the respective school type is often neglected in the history of mathematics teaching. And the texts by Brazilian authors presented at the beginning also do not see this necessary embedding of subject teaching and teacher training into the context of the school.

In fact, the term “limits” was very influential in Germany in the 19th century. From the 1830s, the analytical orientation of the Tralles-Süvern curriculum of 1810/16 in Prussia came under pressure (see Schubring, 1991, pp. 60 ff.). In the absence of a state-decreed curriculum, the requirements of the Abitur exam became increasingly decisive for what should be taught during the years of the Gymnasium. And there was a contradiction in the reform programmes, which turned out to be decisive: the Abitur edict of 1812 had required much less knowledge than the curriculum of 1810 – only up to the intermediate level, probably as an expression of the awareness that in those first years of the reform the requirements of the curriculum could generally not be met. But when, in 1834, an updated Abitur regulation was issued for the now consolidated system of Prussian Gymnasia, the requirements for mathematics had essentially not been improved (see Schubring, 1991, pp. 64 ff.).

As a result, in a December 1834 decree, the ministry excluded from the mathematics curriculum two subjects favoured by mathematics teachers: conics and spherical trigonometry, arguing that they were beyond the limits of the Abitur edict. Because conic sections were the paradigmatic subject for analytic geometry in school, their exclusion led to the emphasis on synthetic geometry. It became a catchphrase for the school authorities to demand that mathematics teachers not exceed the “limits” of the Abitur examination. The exclusion of these “higher” subjects was brought about by pressure from philologists who wanted to reduce the new character of mathematics as one of the major disciplines. However, the mathematics teachers in Prussia continued to argue for a long time that the content of the exams does not have to determine the content of the curriculum.

But the situation in the other German states was much more problematic, as the example of the state Kur-Hessen (Hessen-Kassel), one of the minor German states, shows. There, a decree of February 28, 1843, restricted considerably the Gymnasium curriculum, although mathematics constituted there, too, one of the major disciplines:

that the teaching of mathematics in high schools, in terms of its outer scope, is only extended to the equations of the first degree, while the equations of the second degree are omitted, and that the teaching of plane trigonometry is also limited to the elements, and accordingly the requirements in mathematics in the Matura examinations are also reduced (quoted from Schubring, 2012, p. 531).

These limits were decreed due to twofold pressure: on the one hand, from philology teachers who wanted to restrict the effect of the Abitur exam in mathematics upon the final mark, and, on the other hand, also from a mathematics teacher. This teacher, Ernst Wilhelm Grebe (1804–1874) exerted a double function. He was at the same time mathematics teacher at the Gymnasium Marburg and lecturer in mathematics at the University of Marburg, the only university of this minor state. Since the philosophical faculty in Marburg had not yet been reformed in the Prussian sense, the mathematics lectures there were still encyclopedic and propaedeutic. Grebe showed that his main interest was that mathematics lessons at Gymnasia should not “anticipate” lecture content that was relevant to him for his university lectures. Grebe proudly reported that he had achieved the elimination of conic sections from his high school curriculum (ibid.).

After Grebe's regional influence, Adolph Tellkampf (1798–1869) played an influential role in defining school mathematics, now for German mathematics teachers as a whole. Tellkampf had become Privatdozent, in 1822, at the not yet neo-humanist University of Göttingen, the University of the Kingdom of Hanover, but was unable to continue lecturing there because of a tuberculosis crisis. After his recovery, he first worked at the Prussian Gymnasium in Hamm until he became director of a new secondary school in Hanover in 1835. Even so, he was still striving to return to a university career. Tellkampf developed into a spokesman for German mathematics teachers, propagating the notion of limits between school mathematics and university mathematics. His schoolbook, published in 1829, expressed the intended preparatory function of school mathematics “for the actual study of science” already by its title “Preschool of Mathematics” – Vorschule der Mathematik. In his Schulprogramm of 1827, he devoted himself intensively to discussing the limits: limits not for exam knowledge, but limits of school mathematics to the science of mathematics.

In 1864, a meeting of German mathematics teachers took place, for the first time, within the traditional annual philologists’ meeting. Since it took place in Hanover, Tellkampf became the chairman of this section. The main point of its agenda was to establish a definition of school mathematics, actually in a defensive context – the role of mathematics again questioned by philology teachers. The definition adopted was formulated by Tellkampf:

that mathematics instruction in Gymnasia has to be limited to the domain of lower mathematics and that the part of science, which is based on the concept of the variable (higher mathematics) has to be completely excluded.

and emphasising in the same vein the predominance of synthetic traditional approaches over analytic approaches:

Furthermore, geometry, including plane trigonometry and stereometry, must be the predominant subject of that teaching (quoted from: Folkerts & Schubring, 2020, p. 225; transl. G.S.). ⁹

The evolution of school mathematics in Germany during the 19^th century thus shows, besides initiatives by some individual lecturers to establish a definite priority for university mathematics, a decisive role of pressure of other school disciplines. The relative weight of each discipline in this joint set of school knowledge expresses the values attributed to the cotemporaneous socio-cultural situation – thus, a hierarchy exerted by the educational system in a country.

This situation in Germany during the 19^th century reveals the deeper meaning of the Meran Programm of 1905. Klein's extensive work to modernise mathematics education is often understood as a reform of teacher education to close the “double gap” between school and university. However, the deeper meaning of his reform concepts was, with the demand that mathematics teaching in secondary schools should be based on functional thinking, to abolish this defensive encapsulation of school mathematics in a static and traditionally elementary level and to harmonise it with the science being in development. School mathematics was freed from hierarchical dominance by Klein's reforms and integrated into a constructive coordination of elementary mathematics and higher mathematics.

The rarely remarked contribution of mod math to the relationship issue turns out as a characteristic realisation of Klein's harmonising conception of elementarising mathematics. It had been Klein's intention to also include mathematics in elementary school in the modernisation movement. However, this was not possible due to the socio-political situation then. Primary school knowledge continued to remain completely separate from mathematics, as its own terminology and knowledge orientation, focussing reckoning and spatial doctrine (Rechnen und Raumlehre). The “modern mathematics” movement of the 1960s and 1970s is generally understood just as a failure and an erroneous conception. In fact, however, as a “collateral” effect, it had a thorough restructuring function. Elementary school knowledge is no longer a domain separated from other kinds of knowledge and teacher training–all levels of education are harmonised as just one mathematics! At least in Germany, the entire mathematics curriculum for primary and secondary education is constructed as unfolding of few fundamental concepts.

1.7 A new kind of coloniality: PISA and school mathematics – detached from academic mathematics

A new dimension of global coloniality can be shown in particular by a recently published analysis of attempts by post-Soviet Eastern European countries to follow the requirements and results of international performance tests, revealing at the same time a new kind of hierarchy between secondary and higher education. One international agency, PISA (Program for International Student Achievement), is revealed to be widely accepted for determining the national mathematics curriculum (Karp, 2020).

PISA was introduced by the OECD in 1997. The OECD was established, originally, in 1947 as a Cold War agency to administer the US-funded Marshall Plan. The influence of the OECD-organised colloquium in France at Royaumont in 1959, initiating an international modernisation of mathematics, is well known – when the role of education for economic development in the West was becoming apparent. Since then, the OECD is actively involved in the education sector, too.

The first international comparative performance studies were carried out by the IEA (International Association for the Evaluation of Educational Achievement), an international association of national research institutions, state research agencies and scientists. Their reviews, aimed primarily at the mathematics education communities, have not garnered much attention since the 1970s. In 1995, however, their TIMSS – Third International Mathematics and Science Study – had an impact on educational policy for the first time, also due to the new technology of publishing videos of lesson observations in three countries. PISA adopted structures from the IEA studies, now primarily targeting governments; PISA explains the differences on its website:

Both test different things. TIMSS is curriculum-based whereas PISA assesses its application of skills to real-life problems. TIMSS focuses on content normally covered in class while PISA evaluates overarching content categories that go beyond curricula. PISA also emphasises the importance of the context in which students should be able to use their skills (schools, home and society). ¹⁰

The origin of the testing programme PISA resides in the Realistic Mathematics Education programme (RME), elaborated in the Netherlands in the early 1970s, becoming there the major educational conception for mathematics teaching, overcoming the less strong enthusiasm there for mod math. RME has been strongly influenced by Hans Freudenthal; the theoretical conception was best established by Adri Treffers in 1978 (Smid, 2022, p. 249). Of its five educational principles, two became particularly influential in the testing programme:

the starting point of the learning process must be located in meaningful, realistic problems;

children should develop a rich, well-structured complex of skills, knowledge and insight, not only within the framework of the content matter itself but also in connection with applications in the real world (ibid.).

In fact, the characteristic of the test exercises, elaborated strictly within the OECD, is this focus on revealing the applicability of mathematics in real-life situations.

The chapter on Poland, in the mentioned volume on Eastern Europe, gives a eulogy of the evolution of the Polish test results, understanding them as an objective indication for the effects of mathematics teaching there:

In the PISA 2000 study, Polish students obtained 470 points, which placed them on the 25th position among 41 countries which took part in the survey. Twelve years later, in 2012, the average mathematical result of Polish students was 518 points. This put them at 13th place among 65 countries and economies participating in the survey and fourth among European Union countries. Polish students have achieved a level of mathematical skills identical to Canadian students and statistically indistinguishable from Finnish students (Karpinski et al., 2020, p. 158).

PISA has thus induced a new international coloniality in national educational programs. PISA thus exercises hegemonic control over curricula; due to the conception of skills-based competencies, curricula are dominated by labour market criteria and economic values, marginalising specific goals for education and cognitive development. This orientation introduces a hierarchy – but not, as criticised at the beginning, due to the universities, but due to a short-sighted focus upon school knowledge. The continuation of school learning in higher education is thus interrupted. The gap produced has been sharply criticised several times by university mathematicians, for example in the Netherlands, Finland and also in Germany (see Zhao, 2020).

2. Conclusion

As we have seen there is not one single, clear-cut issue of transition. Rather, transition between school and university constitutes a complex, multi-faceted process which does not serve for simple, short-circuit-form judgments or coloniality criticisms. There are multi-dimensional developments implied, which depend on various variables of a historical character. As it has been shown, the less known but most decisive one has its roots in the religious split after the Protestant Reformation of 1517. And that the variables, resulting in concrete forms for the relation of school mathematics with academic mathematics and for the transition regulations from secondary to higher education are shaped by socio-cultural structures in the respective country. Theories of mathematics education are not necessarily helpful in conceptualising these relations – in particular when they do not conceive of the historical character of the relations between school mathematics and academic mathematics. Yet, theories of mathematics education contributed to establish mathematics teacher education as a proper field, mostly not just a derivative of university mathematics.

This is not to say that there does not exist coloniality today – quite to the contrary, as the example of PISA as an international domination of national school curricula shows, imposing a one-sided conception of school mathematics, compromising the transition to study mathematics.