Vol.:(0123456789)1 3

ZDM (2020) 52:395–406 https://doi.org/10.1007/s11858-019-01110-3

ORIGINAL ARTICLE

Numeracy in Youth and Adult Basic Education: syntactic, semantic, and pragmatic dimensions of a discursive practice

Maria da Conceição Ferreira Reis Fonseca1

Accepted: 18 November 2019 / Published online: 3 December 2019 © FIZ Karlsruhe 2019

AbstractThe diversity of vulnerability conditions, that have prevented children and adolescents from exercising their right to school education, also produces a diversity of cultural references of the public that comes to Basic Education programs for Youth and Adults in developing countries. This diversity often forges appropriation processes of numeracy practices that defy the rationality assumed by hegemonic mathematics and confront us with the insufficiency of not only the syntactic approach, but also the semantic resources in school mathematics teaching. Focusing on events occurring in classrooms of different programs that I have been accompanying in Brazil, I identify and analyze tensions, disputes, and complementarities among the signification efforts being referenced in syntactic, semantic or pragmatic dimensions of language games that constitute the discursive interactions in these contexts. My analysis is based on theoretical tools of language studies that invest in reflec-tions on pragmatics and, as in Wittgenstein’s later works, assume that meaning is determined by use, allowing scholars to approach linguistic signification as a social phenomenon and numeracy practices as discursive practices of social subjects in interaction. The findings provide evidence to further discuss the complexity of the classroom, to which the vulnerability of the students’ living conditions adds decisive elements, challenges and possibilities.

Keywords Youth and Adult Basic Education · Appropriation of School Numeracy Practices · Diversity of forms of life in the classroom · Pragmatics of language · Meaning and signification

1 Research context

In Brazil, as in most Latin American countries, the expres-sion “Youth and Adult Education” (YAE) is mainly associ-ated with basic school education for young people and adults who, as children or adolescents, have been excluded from the school system or never had access to it. This exclusion is usually caused by the incompatibility between what school demands and offers (which all children and adolescents should be able to meet and enjoy) and the early need to take on heavy responsibilities for paid or domestic work (which obliges children and adolescents to abandon not only school, but also childhood) (Di Pierro 2005). Thus, as pointed out by Gal et al. (2020) in the paper that opens this issue, their life conditions configure a framework of social vulnerability,

which precedes and triggers restrictions on the access to school education, and is aggravated and reproduced by such restrictions. Moreover, to the precariousness of the eco-nomic conditions of these subjects and their families are added restrictions of rights (Haddad 2006). These are related to cultural or infrastructure issues and adequacy of the pub-lic system, reaching mainly girls (often prevented from stud-ying even when families’ economic conditions would allow it), black populations, indigenous peoples, families of rural workers, and even urban groups originating in or belonging to those previous groups, who live in the outskirts of cities.

Among the motivations that take adults back to school are the desire or need to access new occupations or to meet new demands of the same occupations, and the search for better conditions in the social instances in which they participate or seek to participate. Younger people, in turn, often return without having actually left: although Basic Education was universalized in Brazil in the twenty-first century, learning conditions have not yet been democratized, and thus, school failure leads them to exceed the age considered ‘appropriate’ to study in ‘regular’ classes, and the school system transfers

* Maria da Conceição Ferreira Reis Fonseca mcfrfon@gmail.com; mcfrf@fae.ufmg.br

1 Faculty of Education, Universidade Federal de Minas Gerais, 6627 Antônio Carlos Avenue, Pampulha, Belo Horizonte, Minas Gerais 31270-901, Brazil

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them to YAE classes (Catelli Jr. et al. 2015). However, embedded in these and other motivations are both the pursuit of rights and the will to know—the search for knowledge.

Alongside these motivations, traces of the vulnerability contexts also configure these people’s relations regarding school knowledge—in particular, mathematical knowledge—in this new opportunity, and the dispositions and obstacles to access it. Researchers and educators refer to, analyze, and confront the configuration and effects of these marks from different perspectives1 and, in a way, it is this very exercise of identification, analysis, and confrontation attempt that has been developed by the Group of Studies on Numeracy (GEN—“Grupo de Estudos sobre Numeramento” in Portu-guese) of the Federal University of Minas Gerais—Brazil.

In this exercise, however, GEN’s researchers try to avoid interpreting the relationships of teaching and learning school mathematics as actions to face the precariousness of the mathematical skills of students in YAE programs. We do so in an attempt to address them in the pedagogical and epistemological fertility of the tensioning that the diversity of the subjects in these programs produces on the rationality that presides over the numeracy practices in school—which is often averse or inadequate to numeracy practices in other situations of their social lives. Such tensioning is strongly felt in the discursive interactions established in YAE math-ematics classes; thus, our analysis intends to show how this emerges when confronted not only by the insufficiency of the syntactic approach (that conceives, presents and uses mathematics only as a set of rules governing the behavior of a system), but also of the semantic resources (concerned with meaning, understood as the relationship between sig-nifiers—such as mathematical signs, symbols or even con-cepts—and what they stand for in ‘reality’, their denotation), usually employed in teaching to increase the possibilities of signification of mathematical knowledge.

2 Signification processes in YAE mathematics approach

The lack of specific training to work in YAE and the absence of a systematic reflection on the ways of appropriating social practices developed by its subjects lead to the reproduction of the same pedagogical strategies that have been used (not always successfully) while working with students who not only had their relation with knowledge formatted by the school approach (by taking part in it since early childhood), but also who belong to social groups identified with the same Cartesian matrix rationality that shapes school numeracy practices.

This search for favoring the signification processes in the school approach to mathematics through the semantization of concepts and procedures has been justified and grounded by theories that find considerable acceptance in the field of mathematics education, still echoing, with smaller or larger accommodations and distortions, contributions of Piaget-ian constructivism (Oliveira et al., 2015). This semantic approach is adopted as a reaction to the markedly syntactic approaches that have characterized (and often still charac-terize) the school way of conceiving, presenting, using, and evaluating such knowledge, especially when it comes to mathematical knowledge.

However, while we cannot deny the advances in the pos-sibilities of signification that semantic approaches eventually provide, our analysis of mathematical education practices in (less or more institutionalized) YAE projects—observed in the GEN research program on the appropriation of school numeracy practices by different cultural groups—cannot help but make us reflect on its limits. The work hypothesis of this reflection thus associates the diversity of forms of life that coexist in the contexts of YAE—forged, among other factors, in the experiences in different conditions of vulner-abilities—to the greater incidence of cases which show the insufficiency of both syntactic and semantic approaches to address the signification of knowledge and the appropriation of school practices by these students. The notion of forms of life introduces, in this study, references to the Philosophical Investigations of Wittgenstein (1986). More than the biologi-cal dimension, this notion involves, especially, the cultural dimension of possibilities of living: language establishes itself in the form of life (see P.I. Sects. 19, 23, 24).2

It was the empirical material gathered in our research program that led us to observe that this diversity called into question the rationality that, in general, supports the seman-tic approach, suggesting a perspective of analysis that, to a certain extent, paraphrases the strategy adopted by Condé (2004) in the discussion on the conception of pragmatics present in the Philosophical Investigations. Condé proposes to show that “a conception of rationality based predomi-nantly on semantics can no longer adequately position itself in the face of important philosophical questions, such as the crisis of reason” (p. 52). This and other strategies are undertaken by Condé to show that “the Wittgensteinian conception of the pragmatics of language establishes a new model of rationality that enables us to deal with contempo-rary philosophical issues much more effectively” (p. 46).

1 See, for example, the studies gathered by Powell and Frankenstein (1997) or by Yasukawa et al. (2018b).

2 In this paper, the excerpts from Wittgenstein’s Philosophical Inves-tigations are quoted as they are in the works of their scholars: referred to only by the initials P.I. §, followed by their paragraph number, when it is from the first part of the work.

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With considerably more modest intentions, our analysis assumes that, in school contexts of mathematics teaching and learning, understood as efforts to promote the appropria-tion of numeracy practices, argumentation “based predomi-nantly on semantics” cannot “position itself adequately” regarding important pedagogical issues interposed by the multiplicity of uses of mathematical ideas, expressions, and procedures caused by the diversity of subjects that character-izes YAE classrooms.

Therefore, the analytical exercise I propose here seeks to highlight the pragmatic character of the arguments which, in addition to those that appeal to the syntax and semantics of concepts and procedures, are elaborated and considered by YAE students in their processes of appropriation of school numeracy practices.

The option for the pragmatics of language offered by Wittgenstein’s Philosophical Investigations, in the midst of diverse conceptions and currents of pragmatism, is in agree-ment with the concept of language that substantiate that work, which is constituted from the notion of signification as determined by use (Hallett, 1967). This notion is particularly interesting in the development of my argument regarding the insufficiency of the semantic approach in mathematics teach-ing strategies, especially in YAE, although one may identify advances on the semantic perspective when it is compared to an exclusively syntactic approach. In my analysis, I am interested in pointing out how these dimensions coexist and are also pragmatically mobilized by the subjects in the pro-cesses of appropriation of school numeracy practices, thus establishing instances of signification based on use.

3 Methodological concerns on researching YAE numeracy practices

The observation of several YAE groups within GEN’s research program enabled us to produce a substantial set of situations concerning the appropriation of numeracy prac-tices by students and teachers doing course work or in train-ing. I gathered the empirical material of 18 studies produced from an ethnographic perspective, which sought to devote special attention to the sociocultural dimension of behav-ior and communication (Green et al. 2001). This material includes experiences of YAE developed in municipal, state, and federal public systems or in projects promoted by, or in partnership with, social movements, in urban contexts, rural education, and indigenous education. The material articulates a rich repertoire of events that were carefully, meticulously, and ethically narrated based on transcriptions of audio and video recordings, on the analysis of collected and/or photographed artifacts, and on the rewriting of notes in the field journals written during the observations of these experiences.

I organized this repertoire in a database gathering 108 events, in which researchers or teachers would be able to select events related to a special group of students, or a certain level of schooling, or dealing with a specific math-ematical concept or field. My whole research project focuses on the mobilization of syntactic, semantic, and pragmatic dimensions in the language games that are established in numeracy events. The Wittgensteinian notion of language games to approach the interactions in YAE classroom seemed fruitful as it involves not only expressions but also the activities to which these expressions are connected (see P.I. Sect. 7).

In my first study, I selected those events forged while students had to solve mathematical problems and, in doing so, they speculated on how to perform arithmetic opera-tions.3 Based on these events’ narratives published in the research reports, I had to construct a new narrative for each one, searching other relevant information about the subjects and production conditions in those reports. Although my approach is quite different from the analysis originally devel-oped by the researchers responsible for the fieldwork, it pre-served the ethical agreements established with the subjects when participating in those investigations, and has allowed the Group to enhance the efforts undertaken by these sub-jects—students, teachers, researchers—in understanding their interpersonal relations and their relations with school knowledge, as well as its appropriation processes configured in the discursive interactions that constitute the educational practices analyzed.

The variety of contexts in which these studies were devel-oped reflects the diversity of YAE provision in Brazil, each one destined for specific groups and educational purposes. This diversity was forged in attempts (less or more success-ful) to suit different sociocultural groups, whose members, subject to different restrictions, could exercise their right to Basic Education only as adults. Many of these YAE provi-sion modes are today threatened by the lack of definition of their directions and restrictions in the educational actions of inclusion and empowerment for low-income classes dictated by a paradigm shift in educational policies, due to the recent rise to power in Brazil of a national project that hardly iden-tifies with those classes’ yearnings and rights.

The discursive practices established in these contexts will not, however, be immune to the conquests and contradictions forged in the diversity of conditions that compose them, the forms of life that coexist in them, and the educational projects developed in such contexts. Possibilities and restrictions that

3 I am currently analyzing events in which students are called upon to deal with negative numbers to discuss the relations between syn-tactic, semantic and pragmatic aspects involved in the need to expand numeric fields.

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constitute this diversity and are also constituted by it are woven in the meshes of language, producing discourses and being produced by them. However, even if plural, these discursive practices are established in language games that are governed by rules dictated by the intentionalities, values and power rela-tions that are instituted in the social group and in these games:

[What is true and what is false] is what human beings say that is true and false; and they agree in the lan-guage they use. That is not agreement in opinions but in form of life. (Wittgenstein, P.I. §241)

Therefore, the diversity of contexts, forms of life, and educational projects needs to be considered, in the prop-osition of an investigation of appropriation processes of numeracy practices and in their analytical procedures, as a structuring condition of the complexity and multiplicity of discursive practices in classrooms, especially discursive practices in which one can identify the relations concerning knowledge, procedures, criteria and representations that we learn to relate to what we call mathematics and which, in general, involve dealing with quantification, measurement, organization and appreciation of spaces and forms, stand-ardization, classification, and ordering.

In the survey of literature conducted by Yasukawa et al. (2018a), they identified “Literacy as social practice” (LSP)—a theoretical school of thought developed in the context of what became known as “New Literacy Studies” (NLS), which grew out of the work of Brian Street (2003)—as one of the influences on numeracy studies which take it as social practice. GEN’s studies undeniably enjoy the influence of LSP, as well as the legacy of ethnomathematics, as pointed out by Geiger et al. (2015). However, when they call those discursive practices “numeracy practices”, they do not use the ideas of NLS to draw an analogy to numeracy, as proposed by Baker (1998). They include numeracy practices among the literacy practices because they understand that, in a grapho-centric and quanticratic society (Fonseca 2015), numeracy practices compose the ways of using written language and are constituted by them. This inclusion is necessary not only because mathematical representations appear in written texts, or because our cultural heritage has given us written ways of doing mathematics, but because written culture itself, which permeates and constitutes the mathematical practices of the graphocentric society, is also permeated by principles based on the same rationality that forges or parametrizes these prac-tices and which is reinforced by them.

Considering the discursive nature of numeracy practices, I searched for theoretical tools that allow me to discuss mod-els of rationality that permeate and/or are confronted when syntactic, semantic, and pragmatic arguments are mobilized and hierarchized in the language games in which learners and educators from diverse contexts of YAE participate. For this reason, I turned to language studies that focus on pragmatics,

in particular, the contributions of Wittgenstein’s later work and the studies that have developed from it in philosophy and sociology. These contributions have been mobilized in some studies on mathematics education, as developed by Knijnik and Wanderer (2010, 2018), to discuss philosophical and pedagogical limits of modern European rationality to which mathematics taught in schools is subjected.

If considering the instances of signification based on use seems relevant in any context of mathematics education, my work as researcher, educator, and teacher trainer shows me that such concern is crucial while working with students whose life journeys make them wary of the rationality that prevails in school practices.4 In particular, in YAE class-rooms, various scopes and levels of vulnerability to which students have been or continue to be subjected inevitably permeate the interactions between subjects, sometimes more implicitly, other times more explicitly. These forms of vulnerability tend to emerge quite dramatically, not only because of their greater occurrence and diversity in the lives of adult people excluded from school education, but also because of the repercussions of the ways in which each of the personal histories was lived by the subjects who, no longer children, have dealt with their living conditions, or having undergone them for a longer time, drawing from them many of their relations with the world and with other people, as well as with institutions, learning and knowledge.

4 Syntactic, semantics, and pragmatics dimensions of school numeracy practices: discursive positions and power relationships

To substantiate my discussion, I present, initially, three events that took place in different mathematics night classes of the same YAE group, in a middle school that works as an outreach project of a public university in a large Brazilian city. They compose the empirical material of Cibelle Lima’s master’s thesis (2012), in which they are narrated5 and ana-lyzed in a debate on the influence exerted by people’s previ-ous schooling when they return to study as adults, focusing on the ways they deal with school knowledge and didactic resources used in this new educational context. The analysis I propose, however, seeks to identify, in the appropriation

4 It should be noted that the prominence I give to the pragmatic char-acter of the notion of use is not only unidentified with, but can also often be contrasted with, a didactic perspective of “mathematics use-fulness”.5 The narratives presented here have received some adaptations to favor the understanding of the interaction situation, since, in the origi-nal research reports, the events were presented with another argumen-tative intention.

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of numeracy practices by those adults, tensions, disputes and complementarities between efforts of signification that are referenced in syntactic, semantic and pragmatic dimen-sions of language games that institute those practices, while permeating and configuring the interactions established in YAE’s school mathematics learning contexts.

This set of events, however, will be confronted with another event, taken from the rich empirical material of Maria Celeste de Souza’s doctoral thesis (2008). Looking at the relationships between gender relations and different mathematics, Souza’s research analyzes the production of games of truth in numeracy practices performed by women and men who work in an Association of Paper and Recy-clable Material Collectors. This research has inspired many of the reflections and papers we have co-written, contem-plating, mainly, how gender relations are established in numeracy practices and by them—in other words, how the discursive webs suppose, challenge, or reinforce gender inequalities by devising feminine mathematical practices and masculine mathematical practices, differentiated in their constitution, legitimation, and social worth.

As it is inevitable to take into account the discussion regarding the gender relations that permeate and delineate numeracy practices, in this paper I try to point out how these relations and other different conditions and dimensions of vulnerability define the mobilization of the pragmatic dimen-sion on the process of shaping the signification modes carried out by YAE students, provoked by interdictions and diversi-fied demands in their relation with mathematical knowledge.

All protagonists of these scenes are adults of different ages, all over 18 years old, in accordance with the norms set by the university’s project and the association of recyclable material collectors. Adulthood, however, is not defined by age limits, but by the conditions and responsibilities of adult life, especially regarding work and the need to support them-selves and their families.

4.1 “It’s called practimetics”

Night of May 26th, 2011, in a classroom at a university, at nights occupied by a YAE project.

During the exercise correction, the students showed difficulties with the question, due to a decimal number operation. The resolution of the question depended on the operation 35,10a divided by 2. When solving on the blackboard, the teacher Edna wanted to “remove the comma”, but the students reacted, asking: “Leave the comma there”. Edna responded to the class’ request and did as suggested by the students: in this type of operation, the students said they left the comma figured in the numeral, but operated the division as if the numbers in the dividend and the divisor were integers (temporarily “forgetting” the comma); in the end they counted how many decimal places the numerals in the dividend and the divisor had, and proceeded “as in multiplication”, mark-ing the comma in the quotient so that the numeral had as many decimal places as the sum of the decimal places of the dividend and the divisor.

Observing the procedure adopted by the students, the teacher asksb:Teacher: Does it always work?Class: It works.Teacher: If I have, for example, here, another number. If I have a

comma here in this number? [She showed the divisor 2].Class: You count three places..Teacher: How do we divide when there is a comma? We don’t take

zero, we add zero. Since there are two places after the comma [She pointed to numeral 35,10 written on the board], I put two zeros here [She wrote the numeral 200 ‘completing’ the 2 of the divisor with two zeros].

At that moment, many students start speaking simultaneously, say-ing that they did not know this way and that they learned “forget-ting the comma”.

Antônio: Oh Edna, in multiplication, don’t we forget the comma and then count the places? In the division, too!

Teacher: I’ll take this up in the next class. I’ll discuss this here.The subject, however, did not come up on the next class.a In Brazil, the comma is used to separate the integer part from the decimal part in the numeralb The dialogues were audio recorded. The personal names were changed to pseudonyms

On the night of June 16th, the class was taught by trainee Gustavo, who had also attended all mathematics classes since the beginning of the semester. (…)

At the end of this class, the trainee-teacher, the student Antônio, and the researcher talked about the “divisions with comma” again. Antônio said that he always made the division without considering the comma and then just put it back afterwards. He also said that he learned to do so many years ago, when he did not have a calcula-tor yet. The researcher asked him to do the following operation 5,76:2,4. He divided it in his way and said that the result was 0.024, because, according to him, there were three places after the comma. Gustavo told him not to do the division and just reflect on the result.

Trainee-teacher: If I divide five something by two something, I must find two something, right?

Antônio: Yes… [He seems to agree initially, but then goes back to the operation made on the board and reacts] No! The result is way smaller.

Trainee-teacher: Then, later, do this counta on the calculator.Antônio: I always did it this way. When I studied, I did not have a

calculator, and I’ll tell you, I don’t know how to do it in the calcula-tor, I don’t know how to deal with it, I always make mistakes; but every time I do it this way [he pointed to the operation he had made himself on the board] I always succeed.

The conversation was interrupted because of the bus schedule…a In Brazil, when it is an informal situation, with colloquial language, we usually refer to arithmetic operations as simply “counts”

On the night of June 28th, the teacher Edna wrote the following problem on the board:    One room has an area equal to 46.8 m2.    I want to cover it with square-shaped pieces of slate whose side

measures 60 cm.    How many pieces will I spend?

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The class had some time to solve it. While correcting the exercise, the teacher turned 60 cm to 0.6 meters and found the area of a piece of slate (0.36 m2). She wrote on the board:    To know how many pieces of 0.36 m2 fit inside my living room,    we will do the following count.

Teacher Edna wrote the division (46.8 divided by 0.36) and performed it as the students said they used to do (“forgetting the comma”), finding 0.013.

Teacher: Check out the weird thing that’s going to happen here now. How many pieces fit in my living room? None. Look, the result was less than one: zero comma zero thirteen pieces. Is that okay? No. This right here is wrong. Look at the count the way you do, what happens to it? On this count here, I found that not even one single piece fits in the room. How can this happen? My room has forty-six comma eight square meters. A small piece that is not one square meter, it is less than one square meter, it has a zero comma thirty-six. How can’t it fit more than one? More than one has to fit there! So this count here that you’re doing does not work when there’s a comma here, see, in the denominator. It does not work, why? So what do I have to do? I have to do it the way Gustavo taught you. I thought you had seen it, but I guess you did not see that kind of count here.

The students asked how they should do it.Teacher: Let’s erase everything! [The teacher erases the operation

she has written and begins to do it again] We will match the places. Let’s walk with two commas, and we will have four hundred and sixty-eight divided by three point six. Four hundred and sixty-eight comma zero is the same as four hundred and sixty-eight. Then I’ll walk with the comma again. Then it will be four thousand six hun-dred and eighty divided by thirty-six.

Then she made the operation 4680 divided by 36, finding 130. João, however, intervened to explain how they would do the division in a practical situation:

João: Oh Edna, you know these counts? We make them directly… It’s called ‘practimetics’.

Teacher: Wow! How so?João: That zero that you moved down, we climb back with it. Sorted

out.a In Brazil, when it is an informal situation, with colloquial language, we usually refer to arithmetic operations as simply “counts”

4.1.1 Syntactic versus semantic approach

The sequence of interactions above shows, at first glance, the dispute between ways of performing a calculation: students position themselves in defense of an algorithm whose rules they claim to recall from their school memories. On the other hand, the educators try to propose another procedure, placing under suspicion the effectiveness of the algorithm adopted and defended by the students. Although they suppose some conception of what it means for “a count to be right”, the arguments on one side and the other do not initially focus on the semantics of the procedure, that is to say, the meaning of “forgetting the comma”, “adding decimals”, “counting deci-mal places” or “walking with commas”; the issue in dispute is the confidence in the effectiveness of the procedure and its universality (“Does it always work?”; “It works.”).

Valuing effectiveness, however, presupposes a crite-rion for evaluating correctness. It is in the election of this criterion that the students’ choice effectively differs from the educators’ (the teacher and the trainee), reaffirming Wittgenstein’s proposition about the decisive interference of the forms of life in the agreement on what is false or true (P.I. Sect. 241). Students opt for a syntactic criterion: the procedure is correct when following the rules. It is the trainee, however, when he returns to the question in the sec-ond scene, who first seeks to present a semantic criterion: the procedure is correct when it produces a response which is compatible with what it means to divide (in this case, divide two numbers, two measures, two quantities of the same nature—because the example given refers to division as in the idea of measure and not the idea of sharing, a clas-sification that, by itself, mobilizes a semantic interpretation of division).

In the interaction with the trainee, however, the student Antônio reiterates his unwavering confidence in the proce-dure adopted by him, considering it an infallible procedure because it follows the right set of rules. Provoked by the trainee’s semantizing movement, he almost gives into the realization that the result should be “two something”. Nev-ertheless, between the variability of interpretation and the certainty of an algorithm (and above all, with airs of gen-erality, “in the same way as in multiplication”, therefore, in absolute agreement with the way in which he learned to conceive the grammar6 of school mathematics), he bets on obedience to the rules, without any concern for the adequacy of the response produced by the operation with a meaning that should be attributed to it—completely free of the “rep-resentation paradigm” (Condé 2004), which, in the evalu-ation of these students, does not seem to be what supports mathematical calculations.

In this sense, these students, who have experienced exclu-sion from school, conceive school knowledge, especially school mathematics, as a cultural product, and elaborate the-ses about its functioning based on hypotheses that they cul-tivate about the forms of life that produced it. The operative rules are presented to them as a social creation, proposed by one (other) social group and, therefore, “an arbitrary creation and, in this sense, […] an ‘invention’”. However, the rules cannot be completely arbitrary, since they have to maintain their “consistency with all other rules, that is, with grammar” (Condé 2004, p. 90).

Thus, when we inquire about the strength of the algo-rithmic procedure, in defiance of efforts of signification by

6 The notion of grammar proposed in Philosophical Investigations (Wittgenstein, 1986) comprises the dynamic and continuously flow-ing set of rules which establishes what is a correct or an incorrect use of language.

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means of semantics, we have to consider the set of values that the school experiences have brought students of Basic Education (those we focus on and many others) to associate with the mathematical knowledge in circulation there. It is these values that these students deem appropriate to mobi-lize in their discursive positions in the interactions that are established in this sphere of social life.

4.1.2 About grammar of school mathematics language games

“Then, later, do this count on the calculator”: this is the new attempt of the trainee-teacher, abandoning the project of a (semantic) signification of the operation and its result, and trusting the reliability of a result which would be produced by a machine.

The disdain that Antônio bequeaths to the teacher’s argu-ment is also forged in the different mobilization of values defined by different forms of life that coexist in classroom: the adult student, who claims that he has not learned to use the calculator in school, does not share the confidence of his teacher in the infallibility of the result produced by the calculation made on the machine (by him, who considers himself to be less skilled with the equipment): “I don’t know how to do it in the calculator, I don’t know how to deal with it, I always make mistakes”; (in relation to the result that I can produce when I do it by following the rules that I think to be correct, by means of which, “I always succeed”).

This may seem surprising at a time when access to calcu-lating machines is so widespread, especially among urban groups. However, it should be noted that Antônio refers to the success and the failure on doing mathematical operations at school. In the third event, João’s commentary suggests that, for these students, not only the procedures for perform-ing operations, in school situations or others, but also the criteria for evaluating the results adopted in these contexts may be different.

Wittgenstein’s pragmatic approach highlights that the dynamic and flowing rules which compose grammar are embedded in a social practice (Condé 2004). My analysis of the discursive positions assumed by these students considers the hypothesis they elaborate about the grammar that governs the language games of school mathematics. With this, I want to acknowledge that, when their positions tension the school numeracy practices, they use a knowledge (and not a non-knowledge) that authorizes them to reject both the procedure that the trainee and the teacher want to teach them, as well as the arguments that justify it.

This knowledge is made explicit in the procedure to adapt the operation’s result to the demands of the practical prob-lem that gave rise to it, presented by João in that new attempt of the teacher, 1 month after the first interlude, trying to propose another instance of signification as a criterion for

establishing the procedure and also for measuring its effec-tiveness. The division operation, inserted in a context of use, then, has an irrefutable argument to evaluate the correctness of its answer: the compatibility with the empirical solution.

João’s comment points, however, to a certain complexity in the consideration of the pragmatic dimension of the lan-guage games placed under analysis and in dispute in these interactions. This dimension is not explained by mere men-tion of a practical situation, but by the constitution of the situation as the “reference system” for signification (see P.I. Sect. 432) and thus for one to evaluate what is effective (and therefore correct) or not in the situation.

The mention made by João of another logic alerts us: the algorithm is a game that is played in school, because it conforms to a way of doing mathematics of a certain form of life: the school numeracy practices. In solving life’s prob-lems, another game is played; other practices are established, governed by other rules: “It’s called ‘practimetics’”.

4.2 “It’s a hundred bale, guys!”

Morning of March 30th, in the shed of the Association of Recycla-ble Materials Collectors where, very early, before starting work, a project developed by the City Hall offers classes, taught by a single teacher: literacy classes for non-literate collectors, and elementary school classes for the literate ones.

That morning, however, the researcher herself took over the work, with the literate students proposing a workshop using problems related to the Association’s production and sales report. At one point, the discussion turned to the Association’s billing, with the biweekly sale of 2708 kg of PET bottles, sold at “forty-five cents a kilo.”

Some students (three men and one woman) explained what they did: “I have multiplied and added”, “I set the count”; “I multiplied the five and the four”, “I multiplied and gave the total”. Other two women present, who used different calculation strategies than those using the written school record did not enunciate their modes of resolution, although they had reached a result.

At the end of the workshop, when the researcher asked “how much money would that pay”, the woman and the three men who used the written method said that the number found when making the “count” (registered as: “121860”) was, “in cash, one hundred and twenty-one reaisa and eighty cents”. That would be the amount that the Association would receive with that sale.

Ana, who is not literate, and who did not perform the operations on paper and was “officially” not participating in the mathematical activity, then challenged this result and triggered the questions of other women collectorsb who had hitherto been silent. It also unleashes male laughter and criticism.

Ana: No… How come? Two thousand kilos of PET, paid that? No.Researcher: How much does it pay?

Ana: It’s gotta be more. [laughter, truck and pressing noises]Researcher: Guys, listen, this is important. Ana is saying it’s not

only that.Clélia: Almost three thousand kilos of PET…Ana: Two thousand seven hundred… No, it can’t be. I don’t agree.

It has to be from a thousand and five hundred to…

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Sebastião: Whoooah… [male laughter]Researcher: Who else thinks the same? How many bales?Dirce: Celeste, that’s something like a hundred bales. It doesn’t pay

only that little, no.Researcher: Let’s imagine the situation Ana is talking about and

the bales that Dirce spoke about. Is it right or is it wrong?Ana: It’s wrong.Jairo: Whoah.Ana: No way, it’s gonna be a thousand and… it’s gonna be a thou-

sand and something… it can’t be. Two thousand kilo, forty-five cents a kilo, it can’t be that…

Researcher: In the count, it was one hundred twenty-one and eighty?

Ana: It’s wrong.Cora: When she said a hundred bale’, I didn’t agree… It’s a hun-

dred bale’, guys!Ana: I don’t know how to do the count, but it seems that it’s some-

thing between a thousand and a thousand and five hundred.Faced with the diversity of calculation strategies, the teacher, who

observed the activity, seeks to ‘order’ the procedures in an attempt to teach the students a correct way to carry out such an operation that could safely and indubitably reach the right answer.

Teacher: Guys, just get it: ten kilos, forty-five cents a kilo, is…? And if you multiply by a hundred, it will be forty-five. For a thou-sand, it’s going to be four hundred and fifty… That’s the math. To facilitate these large counts, you only have to find the value of one hundred [repeating the explanation by recording the numbers as she speaks] and add up to one thousand. For two thousand you just double it [She writes “double” on the blackboard]. And the seven hundred and eight? Two hundred [kilos] will give ninety [reais]. What about the seven hundred? How do I find the seven hundred? [silence of the group and the teacher continues] If one hundred is forty-five, seven hundred will be: one hundred, two hundred three hundred… seven hundred [She writes 45 seven times on the blackboard] Did you see how easily you can do math counts? You just keep adding up, ten at a time. It’s the value of seven hundred [kilos] of PET: it’s three hundred and fifteen…And the eight? If ten costs four and fifty… But we still have to calculate eight kilos. How much, people? Eight kilos? (…)Forty-five plus forty[-five] makes ninety… adding up… [a woman who attended the Elementary School class follows, repeating out loud]. Look at what we did: we did the ‘doubling system.’ That’s the mathemat-ics!

On the other side of the ‘classroom’, non-literate women collectors, who did not officially participate in mathematics class, continue to develop another method of calculating:

Ana: If it was fifty cents [a kilo], it would make… a thousand.Researcher: How is that, Ana?Ana: Two [thousand kilos] is… fifty cents [a kilo]…Voices [of women and men]: Forty!Ana: I’m saying here: at fifty cents a kilo… [silence]Cora: I said fifty. [If] it was fifty…Ana: Two thousand kilos would be… would be a thousand. If it was

fifty cents, it would make a thousand. Right. Two thousand [kilos], fifty cents [a kilo] would make one thousand [reais]. Half.

a Real (pl. Reais) is the Brazilian currencyb Among the other three women collectors who spoke at that time, two were not literate and one had 3 years of schooling, but did not attend classes in the Project. Two non-literate men did not speak up

4.2.1 Vulnerabilities, interdictions, and tensioning

In the analysis of this event, and confronting the analysis narrated in the previous subsection, it is essential to consider the different work conditions and the differences regarding the participants of those two schools.

The first events happen in a YAE project that works in the facilities of a university, in a course corresponding to the middle school, taught by several teachers with specific train-ing to teach classes in mathematics, Portuguese, geography, history, and sciences, who have access to courseware, equip-ment, and the infrastructure of the university. The school that appears in the latter event, reproduced above, takes place in the shed of an association of recyclable materials collectors, along with the kitchen, the association’s office, the material sorting belt, the press, the deposit for already ‘clean’ mate-rial and for what has just arrived, as well as other activities of the association. The classes in the shed are differentiated into those students who are literate and those who are not, but they happen simultaneously, taught by the same teacher, in that same space, in parallel with several other activities of the associates.

The subjects’ vulnerability conditions also differ. Although all of them fall within the group of those who—due to socioeconomic, cultural factors and/or motives related to the educational system’s precariousness—were deprived of the right to Basic Education while children or adoles-cents, the public who studies in the University project is more diversified in terms of occupations. Moreover, most of them have somehow improved their living conditions when compared to those of their family of origin, a situation that allowed or encouraged them, in adult life, to take the initia-tive to re-enter the school.

The other group, however, is made up exclusively of people who work as recyclable waste collectors, which involves separating what is usable from what is not, in the different volumes of garbage that arrive in trucks, coming from various origins, at the Association’s shed. All of them are members of the Association. In addition to ensuring slightly better working conditions and a way of communi-cating with the public authorities, the ‘suppliers’, and buyers of the recycled material separated, cleaned, and organized by them, belonging to this Association also helps them to achieve greater perceived dignity for this line of work and to reduce tensions among the workers themselves. However, it is an arduous activity and a low-paying position which is not socially valued (in spite of the adhesion of almost all sectors of society to the discourses on its importance to improve the cities’ environmental conditions). Many of the collectors dedicate themselves to it by lack of choice. The school is also a benefit won by the Association, but attending the classes, taught by a teacher of the Municipal Education

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Network in the workplace, requires arriving at the shed ear-lier, and postponing other work tasks.

Thus, the numeracy practices forged in this sequence of interactions, motivated by the discussion of a problem that is originally arithmetic, are configured in the midst of many interdictions (Souza and Fonseca 2010): to poor people who did not have access to school while children and who, when adults, can only carry out precarious work activities; to illiterate people, discriminated against in comparison with their colleagues with little schooling, but who ‘know how to read’; to women, whose opinions are systematically disowned and even ridiculed by male colleagues; to oral calculation in relation to written calculation; to approximate calculation in relation to precise calculation…Among these interdictions, some explicitly refer to the values and rules that configure the grammar of school mathematics; but all of them are associated with the same rationality that sup-ports hegemonic values and rules, in the society in which this school culture (Knijnik and Wanderer 2018) is inserted.

In Ana’s questioning, however, there is also a tension-ing on these interdictions. Ana tensions the serial school organization, when, despite not officially taking part in the mathematics class, because she is not literate yet, she identi-fies the inconsistency of the calculation made by her liter-ate colleagues. Ana tensions the greater effectiveness of the written calculation in relation to the oral calculation and denounces the lack of relevance of the search for precision when compared to readiness to check the (in)correctness of the result obtained by performing the algorithm, identified in the comparison of this result with the evaluation of the order of magnitude of the result obtained by an approximate calculation. The criticism of the result produced by written procedures—sustained and perhaps even suggested by expe-rience (“one hundred bale, people”), but elaborated by an oral calculation strategy—puts under suspicion writing itself (and school mathematics, almost always a written mathemat-ics) as an exclusive or privileged promoter of objectivity, deductive reasoning, logical ability, correction and expan-sion of knowledge.

Moreover, Ana tensions the interdiction of women, always questioned as protagonists of numeracy practices by the discourse of the greater masculine capacity for math-ematics (Walkerdine 2003; Souza and Fonseca 2010). Ana is not intimidated by the ‘male’ and ‘more advanced’ col-leagues who seek to embarrass her, in defense not only of a procedure and its result, but also of the conformation of the discursive game that guarantees them a privileged posi-tion (in relation to their female and non-literate colleague), established by their supposed higher aptitude and/or presum-ably higher knowledge to perform a more valued calculation procedure and achieve a result with greater credibility. Ana, however, consistently defends her assessment that the result produced by them would be wrong, because it is incoherent

with her memory records of selling one hundred bales of PET and with the result of their calculation procedure, that taking advantage of the proximity between 45 cents and a half of one real, allows, circumstantially, the inversion of the multiplying and multiplier roles, leading to an approximate value which should be between a half of two thousand and a half of three thousand: “I don’t know how to do the count, but it seems that it’s something between a thousand and a thousand and five hundred”.

4.2.2 Contexts and syntactic, semantic and pragmatic instigations

However, if the practice made available by the solution proposed by the teacher to the collectors uses semantic resources which support the decomposition of the multiplier and the understanding of multiplication as the repetition of additions, this practice is backed up by a chain of procedures and arguments which obey a syntax. Such syntax follows a Cartesian inspiration (see Descartes 1983) that suggests the division of difficulties into parts of reduced complexity, the orderly and hierarchical conduct of the steps, the generaliza-tion, and, finally, the success guaranteed by this way of pro-ceeding: “Did you see how easily you can do math counts?”

This procedure, besides the legitimacy conferred by the fact that it was presented by the teacher, is also narrated as, more than adequate, infallible, general, simple and natu-ral, because it is in accordance with the grammar of school mathematics: “That’s the mathematics!” This is how you play this language game!

Infallibility, simplicity, generality: these are the same values that the YAE students who appeared in the first case mobilized in defense of their division algorithm. By bringing both scenes to inspire this reflection, I want to insist on the concern, which we need to instill in the teachers we train, to understand the meshes of values woven in the discourses conveyed by the mathematics they teach and the way they do so. This concern presupposes study and disposition, but also curiosity and detachment to conceive and face the con-sequences of considering knowledge as a cultural production (Freire 1973).

It is from this perspective that I want to point out dif-ferences between the pragmatic actions undertaken in the scenes in those two contexts (a University project and an Association of Waste Collectors)—actions that determine the taking of specific positions by the subjects.

In the first case, the discussion is, one might say, meta-linguistic: a dispute between syntaxes, between modes of conducting the operation, in which the results produced matter less than the defense of a procedure, or rather, the semantic analysis of these results only matters to the extent that it could legitimize or challenge what has been achieved by these or other ways of performing such operations. The

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pragmatics of the interactional linguistic games in this case is configured in a confrontation of authorities and of what legitimates the procedures and the adequacy of their mobi-lization in each situation: the solution of a school task, the solution of a practical situation, the decision to use (or not) a calculation instrument…

In the collectors’ case, however, what is being discussed, even for didactic purposes, is the Association’s sales report, which is what calculates the collective earnings and organ-izes its distribution among members. Therefore, elaborat-ing, conferring, discussing, and using the report involves numeracy practices forged in the fierce tension of many struggles: struggle for subsistence; for the valorization of their work; for a fair distribution of the Association’s gains, and also the criteria for the definition of ‘fair’—to determine the associates’ remuneration based on production (defended by the men, who, by force, have established that only men can handle the separation of metal parts, which generates the most valuable material) versus determining the associ-ates’ remuneration based on hours worked (as claimed by women, who dedicate longer and more regular shifts to the Association work). Moreover, there is also the struggle for the very existence of the Association, always threatened by the questioning of individualist arguments or by an immi-nent reopening of the dumping ground.

Therefore, Ana does not intervene, defying so many inter-dictions, in the defense of a procedure. She intervenes in response to a semantic clamor, but with a markedly prag-matic intention and action: to correct the result presented by her colleagues, which had generated a cash amount incom-patible with the billing expected by the separation of almost three thousand kilograms of PET. As someone always will-ing to defend the association, Ana cannot help but ques-tion—even in a situation simulated by a school activity—the production of a result that would suggest that so much work to organize “about a hundred bales” of PET would only gen-erate a little more than one hundred reais.

Thus, the act of making explicit her (approximate, not generalizable) calculation procedure has the pragmatic func-tion of convincing the teacher and the researcher, but espe-cially her female and male colleagues of the Association (which were, then, not mere schoolmates, since Ana did not ‘participate in mathematics class’, but coworkers and com-rades in the struggle for the Association) that their approxi-mate result evidenced the error of the ‘exact result’ produced by the calculation written and announced by the literate col-lectors who participated in the activity of mathematics.

Moreover, if there is political boldness in defying so many interdictions in her enunciation, there is also, in the calcula-tion procedure that Ana adopts, an intellectual boldness that, regardless of the impossibility of generalization of her proce-dure—because it is favored by the circumstantial contingency of the proximity between the multiplying (R$ 0.45) and half

of the unit—, reverse the roles of the factors, admitting and intuitively mobilizing the commutative property of multipli-cation (which involves a certain conceptual sophistication) in order to calculate “2000 X 0,50” (two thousand times fifty cents, that is, fifty cents plus fifty cents plus fifty cents… two thousand times) as “½ X 2000” (a half times two thousand or a half of two thousand): “(It) would be one thousand. Half”.

5 Pedagogical developments

In this paper, the analysis of these scenes is restricted to what happened due to the relation of the subjects with school mathematics. However, in mobilizing other possible inter-pretations of this relationship, I seek to provoke a discus-sion about the development of a sensitivity regarding the specificities of the pedagogical work in schools that teach adult people in situations of vulnerability. Concern with the specificities of this audience is often (mis)understood as ‘condescension’ with their ‘faults’, as a screening of the content they would ‘be able to’ or ‘would be interested in’ learning, or even, as a submission of a collective educational project to individual purposes or conditions (Fonseca 2002).

My intention, however, is to argue that this concern should lead to a disciplined attention to the conceptual and procedural subtleties that permeate the discursive positions held by these students. With this care, I must try to under-stand the pragmatic intentions that determine these posi-tions, and the social and political action that the subjects exert when assuming them. Such understanding, in turn, requires greater familiarity with the mathematics one wants to teach, to understand it not as dogma but also as a cultural product, permeated by pragmatic intentions exactly as the numeracy practices of other forms of life. It is this articula-tion between theoretical consistency and intellectual humil-ity that will allow teachers to fear less and to enjoy more the complexity of the classroom, largely forged by the diversity of life histories that interact there and the multiple webs of language in which they are entangled.

In the interactional linguistics games established from the estrangement and the disputes between criteria and pro-cedures—which are referenced in different rationalities—Antônio, João, Ana, Cora, Jairo, Dirce, Sebastião, and Clélia forge practices: they call for everyday experiences, mobilize conceptions of mathematics, evaluate possibilities of solving school tasks and school tasks in mathematics, implement a methodology to control the execution of these tasks, and hypothesize about the fields of possibilities offered by the resources of school mathematics, fields that are challenged (or even disdained) by the demands of everyday situations, within which it is also legitimate to challenge the experience forged in the concreteness of daily life.

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The discursive position assumed by João, when he explic-itly exposes the pragmatics that governs the choice of dif-ferent calculation procedures to be adopted in different situ-ations, highlights the sociocultural character of the use of language, the signification processes that this use defines, and the language games that in this way are produced (and thus the sociocultural character of the various mathematics produced by different forms of life). In addition, he and his colleagues denounce the distance (which the school expe-rience itself taught them to establish) between numeracy practices in school and practices in other instances of social life—whose distortions teachers cannot overcome or man-age, without acknowledging and bringing this character into account in pedagogical action.

Similarly, Ana and other women collectors face a wide range of interdictions, standing up not only for a result or a cal-culation, but for a pragmatic way of understanding the relations between quantities (of recyclable material evaluated by weight, recyclable material packed in bales, money, hours worked…). Beyond the relations of proportionality, this understanding seeks in memory and experience, put at the service of their discursive intention, the parameters of reasonableness, the con-vincing arguments, the multiplication tactics, and the criteria to hierarchize: precision and approximation, written calculation and oral calculation, registry and signification, embarrassment and self-confidence, silence and enunciation.

By highlighting this pragmatic dimension of numeracy practices carried out by YAE students, the studies GEN have developed approaches to signification (of language and, therefore, of mathematics) as a social phenomenon. This perspective, in a way, contrasts with the semantic concep-tion of signification intended by many school approaches to mathematics whose intention is the restriction of meaning fostered by the illusion of the uniqueness and homogene-ity of understandings and by the ideal of standardization of procedures. The strangeness of school forms of life that becomes explicit in the numeracy practices I focus on, how-ever, reiterates the pillar of language pragmatics proposed by Wittgenstein: “The meaning of a word is its use in language” (P.I. Sect. 43). Signification is, therefore, a pragmatic action.

The determination of the meaning of a word by its use in language, thus, the institution of this use as a reference sys-tem for the signification process (configured in a relational way, in accordance with the situation) arouses my interest, especially in the analysis of classroom interactions in YAE, because it highlights the creative character of signification, a decisive aspect in GEN’s choice to name the learning pro-cesses witnessed in the classroom as “appropriation of prac-tices”. In the concept of use and signification as constituted in relationships, the character of repetition of signification is considered, since meaning emerges from the regularity of uses; but by assigning the institution of signification to the situation, this conception ties in with the possibilities of

uses and their various contexts, an infinite possibility of the creation of meanings, that is, it assumes that “the limits of language are the limits of the pragmatics of the language of a form of life” (Condé 2004: 48). This makes social practice a creative practice of those subjects in interaction.

It is in this sense that the analysis of events from diverse contexts of YAE in which multiple language games are forged seeks to highlight the insufficiency and tense comple-mentarity of the syntactic, semantic, and pragmatic dimen-sions of language. It is intended to be an opportunity and an effort to make explicit and reflect on the complexity of the classroom, to which the vulnerability of the subjects’ living conditions adds decisive elements, bringing to the school scene different forms of life, and tensioning or complying with the language games that take place in it.

My intention is that this opportunity and effort can con-tribute to awakening or reconstituting the responsibility and awareness of the challenge, as well as the confidence in the subjects’ potentialities and the charm of the possibilities that open up to the teaching action and diverse educational prac-tices developed in Youth and Adult Basic Education.

Acknowledgements This research was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq–Brazil) and Universidade Federal de Minas Gerais (UFMG–Brazil).

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