Introducing the study

The studies on orality and writing that expanded significantly abroad, during the 1970s, ended up constituting a field of studies of an interdisciplinary nature in which themes related to the possible cognitive, sociocultural, linguistic and political effects of writing (^{BATISTA, 2000}). From this, the numerous researches that have been carried out in the scope of writing indicate that its domain causes changes in the state of knowledge of a certain society or cultural group.

If we understand orality as a practice used by different cultural groups, it is easy to highlight the existence of tensions between oral and written practices, given that writing is usually legitimized and configured as hegemonic in modern society, while orality is often marginalized and seen as invalid (^{WEBER, 2002}). ^{Bunzen (2014}, p.14), referring to written language, affirms that there is an imposed challenge to understand it not only as “[...] a linguistic (psycho) point of view, but it is also historical, anthropological and cultural one, taking into account power relations”. The idea is putting the legitimacy of writing under suspicion, creating conditions of possibility for the different manifestations of language, among them, in its oral form.

In his doctoral thesis, ^{Souza (2008}) argues that writing produces truths and manufactures certain types of subject “[...] as people lacking a specific reasoning mode, typical of those who master writing technology” (^{SOUZA, 2008}, p. 233). Thus, we have evidence that writing has great power in the constitution of the modern subject, since it is associated with both scientific progress and academic culture. In this way, orality is moved to the “magical world of voices” (^{CERTEAU, 1994}, p. 224), giving an air of inferiority to speech. Given this fact, this article problematizes some written and oral mathematical practices of a group of colonists descended from Germans from a region of German colonization in the Vale do Rio dos Sinos², and aims to verify how such experiences were used and conceived by these subjects, both in the school period and in their work activities carried out in agriculture.

The empirical material consisted of statements taken from a master's thesis³, produced through semi-structured interviews conducted with three settlers with low schooling⁴. The school period of the three subjects interviewed by ^{Kroetz (2015}) occurred between the 1920s and 1930s, concomitantly with the time when agricultural activities represented the families' only means of subsistence. Furthermore, this period was characterized by a peculiar political movement, the Nationalization Campaign, a set of measures adopted by the Getulist government during the Estado Novo, whose main objective was to reduce the influence of foreign immigrant communities in Brazil, forcing their integration with the Brazilian population (^{SEYFERTH, 1990}).

Since the subjects' narratives are directed to a period of the past, oral history was adopted as a research methodology; since it is possible to concatenate aspects of the historical scenarios linked to past school practices and understands the reflexes nowadays (^{GARNICA, 2004}).

About the Oral History

Oral History (OH) is traditionally used in the field of History and Social Sciences, and aims, according to ^{Silva and Souza (2007}), the valorization of oral narratives as study material. In this perspective, the use of OH began to be considered as historians understood that historical documents were susceptible to adulteration (^{GAERTNER; BARALDI, 2008}). However, it is worth noting that using this approach is not about admitting the existence of a true story. In view of this, ^{Garnica (2005}) emphasizes important elements brought by orality, such as dynamism, record corrections and the chance of transforming objects of study into subjects, which by other instruments of data collection might not be accessible. According to Silva and Souza (2007, p. 142):

Oral narratives are seen by oral history as sources from which it becomes possible to get closer to the meanings attributed to the realities experienced by the narrator, since it seeks (in most cases) to preserve, in an almost literal presentation of the narratives collected through interviews, the legitimacy of the narrator.

The procedures required by OH include:

The pre-selection of deponents, probing of possible documents that deal with the topic of the interviews to be recorded, recorded interviews that will constitute the base document of the research, instances of transforming the oral document into writing: literal transcription, textualization and transcreation [...]; ‘Legitimation’ [...] and, finally, a moment of analysis, whose character varies according to the purposes of each research (GARNICA, 2007, THOMPSON, 1998 apud ^{SILVA, SOUZA, 2007}, p. 144).

According to ^{Gaertner and Baraldi (2008}), the reports obtained through interviews constitute individual trajectories and must be understood as such. Also, the researcher is required to have ethical principles in relation to the interviewees that must be respected, since memories can be revived marked by emotions and facts that were hidden. Still, another important aspect is that OH is conditioned to the interviewee's memory, and this is selective, being able to create multiple versions of past events (^{GAERTNER; BARALDI, 2008}). It is also noteworthy that, in addition to the materials obtained through the interviews, the researcher can use other research sources.

In this study, oral sources are used, based on interviews provided by three subjects, settlers, residents of the city of Santa Maria do Herval, Rio Grande do Sul. The statements analyzed here are an excerpt from the work of ^{Kroetz (2015}), followed by the methodological procedures suggested for Oral History, seeking contributions regarding the understanding and negotiation of mathematical meanings for this cultural group (^{GARNICA, 2005}).

About the oral and written practices

R⁵: And what did you do to control what you earned?

IC: The control was all with me, but I didn't need to write anything, I knew everything in my head.

R: Didn't you need to write anything down on paper? IC: Nothing, I remembered things.

IB: We used to do the math more in our heads, for selling, for buying; we knew that, the father taught all of this.

R: And how did you know it was worth the same [exchanges]? IB: Ahh, that was in the “eye”, right? In the “head”. IA: Yes, then if it is decreasing and I need more, I borrow one, we learned this at school. But I always had my bills ready in my head before going on sale (market)

IA: We were very quiet at school, one sitting behind the other, straight, almost without moving, I could only look at the board and write.

IB: The teachers wrote on the board and that’s all, I had to copy it. The children studied, if we could continue to study. First we learned einmaleins, 1 x 2, you know? Then ABC. First the father taught this and he said: how do you want to learn the rest if you don't know this?

The excerpts above were taken from ^{Kroetz's dissertation (2015}). In her studies, where the author aims to identify and understand the processes of generation, organization and dissemination of the knowledge of colonists who descend from Germans living in the Vale do Rio dos Sinos region, she verifies that the generation of the knowledge of this group occurred with their parents and acquaintances, and that the language games⁶ used in agriculture were different from those imposed by the school. One of the differences pointed out is that if writing used to predominate at school, for example, in the subjects' experience, orality prevailed.

While in the colonists' work activities, they affirm that they do not need to write anything down, know everything in their heads, it is easy to see that orality was a practical way of solving their daily situations. However, in the school period, the same subjects emphasize that mathematics was only considered on paper. When pointing out the rigidity of the teachers, who used disciplinary mechanisms to shape the students, the colonists affirm that the contents were written on the board and that it was necessary to set up the calculation and put the figures under each other in mathematics classes⁷, which can be seen in the following passage: IA: We suppose to add like this [the calculation], one under the other, you know? … Adding and subtraction, as I learned at school. R: Were the calculations always under each other? IA: Yes, then if I were subtracting and I needed to “borrow” one, we learned this at school.

It is clear, then, that the school was responsible for the control and use of time with techniques that branched out into a hierarchy of knowledge, organizing internships through graduated tests, which based on the writings of ^{Foucault (1987}, p. 135), “[...] they include exercises of increasing difficulty; qualifying individuals according to the way they go through these series”. One of the procedures of this technique is the exercise, understood by Foucault (1987) as a technique that imposes several repetitive and intensified activities that obey a hierarchy, it was marked by writing.

One of the materials used by the colonists to do school exercises in the past was tafel and griffel. According to the colonists interviewed, the tasks were reproduced on slates made of smooth slate stones. They wrote on its surface with a pointed object made of stone, which they called the griffel. They used a piece of cloth as an eraser, but in most cases, they rubbed it with their own hands. For ^{Souza (2007}, p. 83), the slate and the blackboard "[...] served as instruments for the schooled child to learn the first movements of correct writing." The subjects stand out: “We used a tafel, a little board, and we wrote with a nail” and “We had tafel⁸and a griffel⁹. If it fell on the ground it would break. We dipped a pen in an ink, I could only dip a little, if I put it too much it would still smudge. ” The tafel consisted of a stone or wooden surface used as a support for writing, it used to be efficient in assisting the teacher in his teaching and making students copy what was necessary.

According to the narratives of the interviewed subjects, the exercises were copied from the board using a tafel and a griffel (figure 1), two artifacts with a significant symbology regarding the school period of these subjects.

Source: It was extracted from ^{Kroetz (2015}).

Figure 1: Tafel and Griffel 

The griffel can be considered as an artifact used to write on the taffel what the teacher wrote on the board, as evidenced by interviewee A: “We were very quiet at school, we used to sit one behind the other, in lines, almost without moving, I could only look at the board and copy it”. It is noticed the existence of a superiority attributed to the written language in the school period of these subjects, different from the practices carried out in agriculture, where orality prevailed. This valorization of writing over orality, if we take ^{Foucault's writings (1987}, p. 157) in supervising and punishment, places the subjects in a kind of “[…] network of written notes [that involve] documents that they capture and fix them”. For ^{Souza (2008}), it is this written practice that strengthens and prioritizes the repetition and training of exercises, as well as the use of different forms of disposition on paper. Such practices would be considered as essential so that the subjects could develop their mathematical thinking.

There is still the impression that teachers, in the school period of these subjects, generalized the way of thinking emphasizing a hierarchy - naturalized by modern teaching - to the ways of teaching, which is noticeable when the colonists say that “First we learned einmaleins¹⁰, 1 x 2, do you know? Then, ABC. First the father taught this and said: how do you want to learn the rest if you don't know this? Let's say, 8 x 4 what is the result, 32, then we wrote it on the tafel. If you didn't know that, how can you go on? ” (Interviewee B).

The interviewees' speeches confirm a certain rationality regarding the learning of Mathematics. Fernanda ^{Wanderer (2007}), when analyzing the speeches about school and school mathematics of a group of colonists of German descent who attended a rural school in the municipality of Estrela, during the period of the Nationalization Campaign, highlights, in this matter, it was “[...] constituted by rules that say the importance of decorating the multiplication table” (^{WANDERER, 2007}, p. 173). The multiplication tables and armed accounts characterized the teaching of mathematics during the school period of the subjects, of a mathematics that had no meaning in the social scene in which the subjects were inserted, agriculture.

Emmanuel ^{Lizcano (2004}) is going to propose a change in the way we refer to Mathematics, pointing out that we have the custom of “[...] giving it as a suggestion (that is, placing it under us, as fixed ground) and since then, look at popular practices, in particular, at popular ways of counting, measuring calculating [...] ”(^{LIZCANO, 2004}, p. 125). From this school mathematics - this recontextualization of academic mathematics -, we tend to analyze local practices, always using more advanced mathematics as a reference. Based on this, Lizcano (2004, p. 125) points out that the others’ mathematical practices are delegitimized because they are less like the mathematics we learn at school: “But what happens if we invert our eyes? What do we see if instead of looking at popular practices from “mathematics”, we look at mathematics from popular practices? ”

Unlike mathematics taught at school, where it was necessary to present calculus in written form, oral mathematics practiced by the colonists interviewed was permeated by language games marked by estimation, proportional thinking and orality. This can be seen below:

IC: That's right. Then, when I got older, we got fertilizer from Banco do Brasil, and they gave us a 40% discount. R: And is it a good discount? IC: Ahh, it is. It's almost half off. R: And what would you do to know that? IC: Look, 100 out of 40, right? So it is percent I think. R: Could you give me an example? IC: Let's suppose, you bought it for R$ 1,000.00, you will earn R $ 400.00. That's a good discount. Imagine ... it is almost a half of the value.

IA: More or less, for each 20kg of lard we used about 4kg of soda. R: And if I only had 2.5 kg of soda, how much lard would I need? IA: About 12kg and 500 grams. R: How do you know that? IA: If I need 4 kg of soda for 20 kg of lard, then for 1 kg of soda I need 5 kg of lard, right? R: Yes. IA: So if it is for 1 kg of soda I need 5 kg of lard, for 2 kg of soda I need 10 kg of lard. And I'll do the rest later. R: What rest? IA: The decimal numbers. If for 1kg of soda I need 5kg of lard, for half a kilo I need 2.5kg of lard, right? So it plus 10 is 12.5kg.

However, even though these colonists used oral mathematical practices to deal with business and various activities in the past, the writing used at school was a mechanism exercised by those who dominated it: teachers. This is also noticeable when the colonists affirm that nowadays business is done on paper, but in the past they could do business by word of mouth.

The narratives elucidate a certain tension between these subjects’ oral and written mathematical practices, as in the school period of the colonists, the practices adopted were focused on writing, and in agriculture the colonists used techniques based on orality. When subjects opt for orality for possible business and agricultural activities, they have no need for written mathematics, and do not need to set up the account and place one under the other, as they do at school. In addition to mathematical knowledge, it is perceived other knowledge implicit in these people’s actions, knowledge probably not stimulated at school, since the formal education provided to them was based on the learning of mathematics, literacy through Portuguese and the teaching of religion.

The 21^st century school is not much different from the schools attended by the colonists interviewed. Oral practices in mathematics classes tend not to be considered, and if the student does not present a specific or “appropriate” type of reasoning from the teacher's point of view, he/she often becomes the target of corrective pedagogical interventions. The school does not meet the expectations of the students, since for some cultural groups, this school mathematics, coming from academic mathematics, does not make sense. The colonists did not need the paper and the pen to solve various situations that occurred in their ways of life, and in this sense it is worth mentioning that oral mathematical practices are distinguished from writing not only by dispensing with the registration of writing, “[...] but also because they are parameterized by other values ​​and intentions (such as pragmatism in the option for the agile production of an approximate response at the expense of the meticulous search for precision) ”(^{SOUZA, 2008}, p. 251).

The tensions between orality and writing, based on an ethnomathematical approach, are the target of studies by Gelsa ^{Knijnik (2007}). The author points out that, at school, written mathematical practices have more value than oral practices, a study carried out by the author in a settlement of the Movimento Sem Terra, MST, in Rio Grande do Sul. In her studies, Knijnik (2007, p. 36) denounces the “[...] disappearance of oral mathematics as a cultural practice in educational processes.” In the same direction, ^{Souza (2008}, p. 234) points out that the truths constructed about written mathematics produce certain types of subjects, “[...] those who dominate writing technology and are therefore allowed to 'practice 'school mathematics and those who still need to master this technology to be inserted in school mathematical practices.”

Thus, the domain of writing appears as a device of power, as it determines access to a type of knowledge. In Microphysics of Power (2014), Foucault describes the devices as material operators of power, as a heterogeneous set that encompasses statements, laws, institutions and discourses, where the device is the network that forms between these elements (^{FOUCAULT, 2014}). Among these elements, there is a set of games that allows the formation of a new field of rationality, that is, it is from them that the asymmetries of relationships become visible. Thus, the relationship between knowledge and truth is also evident, as mastering writing implies a power that affects the maintenance of a regime of truth - in this case, the mastery of writing to learn official, true Mathematics. This mathematics would reinforce the effects of this power, making us believe that without knowing how to write it would not be possible to know.

In this scenario the French anthropologist Florence ^{Weber’s studies (2002}) on the diversity of the natives’ reasoning and the economic practices of families in her country can be referenced, where the author states that such reasoning is not always explicit, because the natives have a plurality of reference systems, rules and objectives, using different rationales that only make sense in what the author will call the social scene. Weber (2002) highlights that it is this plurality of measurement and calculation systems adapted to different uses that leads to the conclusion, after articulating several studies carried out in different social scenes, that the units of measurement adopted depend on the context of the practice, and that it is the interaction that gives meaning to the transaction that always occurs in a given scenario.

Regarding purchases and the different businesses established in agriculture, the colonists affirm that nowadays, if nothing is on paper, nothing else is worth it, and that in the past business was done by word, that is, nothing was signed, only agreed. We are so used to the legal and institutional transaction system that we end up forgetting all the other ways of doing business. ^{Weber (2002}, p, 162-163) points out that this fact is strange for us, because “[...] far from the legal guarantees offered by contracts and commercial law, which are accompanied by written evidence and signatures, they take us to the universe in which the word given and belonging to the group are sufficient.” The guarantee and the trust that the colonists place in each other makes the orality also marked by values ​​that they consider essential in their community, such as the credibility and trust established among the members of their group.

Thus, the group of colonists interviewed, when constituted as belonging to a certain social scenario where agriculture, manual labor, exchange of products and the exchange of work predominates with the neighbors, systems of measurement and oral calculation that are directly involved in its use. A clear example of this is the unit of measurement called quarter¹¹ by the settlers, a unit taught by their parents, neighbors and people in the community. When asked about what would become a quarter, the settlers stated:

EA: A quarter is about 7.5 kg. Eight quarters is a sack. We measured with quarters in the past, few people used kilos. R: What was it like? IA: We didn't use a scale, right? That's why we used quarters and made these exchanges. We used to take a piece of meat in our hands and said how much it was approximately.

IB: Two quarters is a whole gallon, half a gallon full of seeds equals one quarter. Four full gallons gives about 60kg, it depends on what gives a full potato bag. R: And with corn, and other seeds, was it the same thing? IB: yes, the same thing

R: So, was it your measure?

IC: Yes, and if we did it right, we didn't even need to weigh it because it was the same thing. The scales [...] we started to use them later. R: And did this measure work for everything? IC: Yes, but I had to put it right too, ok? They were gallons of kerosene because we didn't have electricity, right? Then we kept these cans to put beans, potatoes, and for measure. Everyone did it in the old days, so we did it too. Four whole gallons were a full bag.

This metric system presented by the colonists uses specific practices for weighing and quantity of grains to be planted. On the different reasoning of the natives, ^{Weber (2002}, p. 160) highlights that

[...] in the universe where the clock, scale and register are ubiquitous, their use is not imposed uniformly in all domains of practice or [...] in all universes of social relations, in all ' social scenes'. In addition, the use of measurement techniques is sometimes accompanied by conflicts over the legitimacy of the measure or the relevance of the tools. Forgetting it is giving in to an angelic vision of social relations and of relations between science and society.

A unidade de medida adotada pelos colonos, além de fazer parte de uma cena social que só faz sentido no modo de vida dos colonos, é caracterizada por regras do mundo da agricultura, e não possui significado em outras cenas sociais. ^{Weber (2002}, p. 160) vai provocar, pontuando: “de que vale uma medida ‘científica’ para explicar comportamentos se ela não é utilizada, ou seja, percebida, pelas pessoas concernidas?” Na vida dos sujeitos, nas transações comerciais, na agricultura e em todas as estratégias adotadas, as necessidades de matematizar eram outras, e iam muito além do que a escola insistia em lhes cobrar, muitas vezes, de modo rígido e valendo-se de técnicas disciplinares como a vigilância e a punição (^{FOUCAULT, 1987}). Além disso, as práticas orais superam a funcionalidade de registros escritos, onde certamente não fazia sentido recorrer à ciência, ao cálculo exato e formal, uma vez que ele não se apresentava produtivo em suas vivências.

The unit of measure adopted by the colonists, in addition to being part of a social scene that only makes sense in the colonists' way of life, is characterized by rules from the world of agriculture, and has no meaning in other social scenes. ^{Weber (2002}, p. 160) points out: “what is the use of a 'scientific' measure to explain behaviors if it is not used, that is, perceived by the people concerned?” In the subjects' lives, in commercial transactions, in agriculture and in all the strategies adopted, the needs to mathematize were different, and went far beyond what the school insisted on charging them, often, in a rigid way and using disciplinary techniques such as surveillance and punishment (^{FOUCAULT, 1987}). In addition, oral practices surpass the functionality of written records, where it certainly did not make sense to resort to science, to the exact and formal calculation, since he was not productive in his experiences.

In addition to orality and writing, ^{Weber (2002}, p. 168) presents the concept of economic calculation, conceiving it as a way of life, since “[...] practical rationalities are not always offered to observation under the form of explicit reasoning, lists of notes or measured activities, but they can remain automatic or unconscious ”. Many of these ways of mathematizing are unconscious, made by rules, and operate through behaviors and strategies. Some reasoning is even automatic and, given that, it makes no sense to resort to numerical calculation if there is an effective way of economic calculation to minimize expenses and forecast your budgets, for example.

When showing different ways of mathematizing, such as organizations, techniques and predictions of the natives, ^{Weber (2002}) highlights the need to forget about calculus only as something formal, in its abstract conception, as there are specific categories of classification and incorporation that make sense in some social scenes. Weber's writings (2002) contribute to the problems that have been carried out in relation to the supremacy of writing, helping to understand that there are other ways to mathematize and that these depend on a context, on a social scene that only makes sense to those who are inserted in it. Some cultural groups, for example, do not use writing, counting or numerical calculation, but use techniques of ordering and classifying categories that allow the inference of certain expenses. However, such techniques and practices are not legitimized because they do not use the rules of academic mathematics and are often not even seen as mathematics.

One of the ordinary forms of calculation found by ^{Weber (2002}) in a study with horticulturists in France is orality, used to pass the budgets whose calculations were performed in the head. Such oral calculation strategies, for ^{Knijnik (2004}, p. 233), are “[...] interdicted at school in the name of written algorithms”. Orality represents a cognitive operation adapted to the practical needs of this subject, where the techniques of measurement and calculation owe nothing to the formal system (^{WEBER, 2002}). Thus, the abstract conception of calculus is abandoned and the native categories of classification and incorporation are verified. ^{Souza (2008}, p. 235) also affirms that in the school environment “[...] the use of writing also functions as a mechanism to legitimize school mathematical practices, when imposing itself in this space, many times, as the only way, and always the most correct way, of doing mathematics ”.

The low valuation of mental calculation and orality is still a matter of study, as evidenced by ^{Boni, Savioli and Passos (2015}), when they demonstrate that one of the issues that lead to this is the teacher's own deficiencies in their initial training, emphasizing the importance of mental calculation because it is a more functional way to be used in everyday life. However, as seen in the school period of the colonists, an official assessment or verification by the teacher of the student's proficiency through orality becomes more complex (^{BONI, SAVIOLI, PASSOS, 2015}). This ends up producing tensions between these two mathematical practices, “[...] a mathematics that is produced in the urgencies of life, in which orality is triggered with greater frequency, efficiency and legitimacy, and another, written, which meets demands in which generality, formality and control are valued ”(^{SOUZA, 2008}, p. 235-236). Such situations can be evidenced when the colonists emphasize that to make lard, to take care of the business of selling potatoes and for planting, for example, they need to mathematize in another way, as such situations go beyond the teachings seen in the classroom, which they are usually guided by written mathematics that is sometimes out of context with real situations.

Final Thoughts

This study, by presenting some written and oral mathematical practices by a group of colonists descended from Germans from a region of German colonization in the Vale do Rio dos Sinos, found that oral mathematical practices stand out in agriculture and in the practical activities of the interviewed colonists, while mathematics characterized by writing is legitimized at school. There is supremacy of writing over orality, which can be easily verified in the daily life of schools from prescriptions of program-curricula.

If we analyze current mathematical education, it is easy to see that oral practices have no place in the school environment. ^{Knijnik (2007}, p. 37) denounces the disappearance of oral practices at school, as this disappearance produces "[...] social effects that go beyond school failure". Accordingly, the author still complements that

[...] oral mathematics, specifically that linked to processes involving operations of addition, subtraction, multiplication and division, assumes a more prominent role, precisely because such oral practices integrate, even nowadays, the repertoire of knowledge that a significant number young people and adults have (^{KNIJNIK, 2007}, p.37).

The tensions between the oral and the written produce a discourse that the written has a greater value than what is said. Such discourse, according to ^{Souza (2008}), is put into circulation by many subjects of the field of education, such as students, parents, teachers, public policies, managers and the community in general. The constant use of writing in the school life of the colonists interviewed makes it seen as the only way to teach, and the oral model of thought is used by the subjects in agriculture as they use, for each occasion, strategies that differ from the rules imposed by the school.

Unlike the school period, where mathematics is presented as a formal and rigorous discipline, marked only by written practices and armed accounts, when using certain knowledge in their contexts using orality, mathematics can be viewed in another way, not as a hard science, but as something that transcends the exact calculations and can be found in the ways of knowing and doing in different cultures, permeating different knowledge used for certain purposes, not establishing disciplinary limits, but worrying about solving the group's daily problems of settlers.

Using practices of orality can constitute one of the directions of mathematical education, since written practices are still strongly consolidated in school contexts, producing subjectivities that legitimize such a way of mathematizing (^{SOUZA, 2008}).