18. Critical Mathematics Education in Action: To Be or Not to Be¹

Paula Andrea Grawieski Civiero and Fátima Peres Zago de Oliveira

Society today presents a civilising equation where it is crucial to unveil and guide the imbricated relationship between what is technical and what is human. Therefore, the study of contemporary variables is central to the interpretation of this reality. By considering that critical mathematics education (CME) is the most developed approach to treating such themes in mathematics classes, we present a landscape of investigation developed with high-school students, step by step, based on reflective didactic transposition (RDT) of a Scientific Initiation (SI) project in high school. SI enables the investigation of contemporary themes that, in turn, approach the concerns of CME by fostering questions, autonomy, decision-making and a critical interpretation of reality. This proposal evidenced the approximation of SI concepts with landscapes of investigation, just as it was possible to perceive the urgency of the imbrication between the different milieus of learning and the contemporary variables of this complex civilising equation.

To be or not to be, that is the question: will it be more noble.

In our spirit suffering stones and arrows

With which Fortuna, enraged, targets us,

Or rebel against a sea of provocations

And in struggle to put an end to them? Die … sleep: no more.

(Hamlet—William Shakespeare)

During the last decades, technoscientific development has been reaching unthinking levels, which, in turn, has boosted a new civilising behaviour.² This reveals that people need to consume and possess, rather than be. An organised society is susceptible to the commands disseminated by those who dominate the technoscientific apparatuses that, in turn, are treated as instruments of power, and not as a vehicle for human development.

Thus, there is a civilising equation—a metaphor, utilised by Bazzo (2019)—which could be a tool “to bring together the most different variables that arise at all times in a civilisation that is vulnerable to the most accelerated mutations in its daily behaviour” (p. 21). The tool could be even more significant given the implications that these issues have on society. In other words, these issues give us the urge to “provide reflections and changes in our ways of working knowledge in such serious times of human problems” (p. 20). The civilising equation has both more technical and more human contemporary variables. It is aimed at the overlapping of the variables, so that the result of the equation at least guarantees the principles of human dignity.³ The current social, economic and political variables are considered essential elements for the analysis and interpretation of reality. Some examples are environmental issues, the immigration process, social inequalities, the hybrid crisis, the atomic bomb, global warming, chemical wars, biological wars, pandemics—such as coronavirus—and so many other variables that compose the civilising equation.

We understand that mathematics is especially relevant in knowledge processing, and operates in the process of globalisation,⁴ i.e., it interferes in several aspects that integrate with society. We admit that globalisation refers to all aspects of life and that, depending on how it is questioned and operationalised, it may or may not be beneficial. Therefore, globalisation “has to do with the construction, codification, and distribution of knowledge that turns into goods for sale” (Skovsmose, 2014, p. 130). This way, mathematical knowledge is involved as part of the foundations of society, making it necessary to question its position in this laborious civilising equation (Civiero and Bazzo, 2020).

In this context we delegate some power to critical mathematics education (CME) by considering that it can contribute to the formation of critical individuals by promoting reflection on this process. Bazzo (2019) considers us to experience a civilising equation whose variables need to be discussed in schools. Also, Civiero (2016) shows that CME is today the most developed approach to dealing with contemporary variables in mathematics classrooms. The author discusses the importance of these questions being part of the training of mathematics teachers.

Based on the exposed understandings, we advocate the construction of landscapes of investigation⁵ in mathematics classrooms to provide students and teachers with the opportunity to investigate themes that may foster reflection on contemporary issues. We identified as a possibility for the development of landscapes of investigation the Scientific Initiation that forms part of the curriculum in high school, at the Instituto Federal Catarinense—Rio do Sul Campus, Santa Catarina, Brazil.⁶ It is a space for the development and alteration of perspectives on scientific activity, which provides undergraduates with an initiation into research in basic and higher education. It is also a path to intellectual independence, creativity, curiosity and autonomy, and can be a means to sharpen the students’ critical awareness. According to Freire (1974):

the critical consciousness is characterised by delving into the interpretation of problems; substituting causal principles for magical explanations; testing one’s “findings”; and by opening to review; by attempting to avoid distortion when perceiving problems, and preventing preconceived notions when analysing them; by refusing to transfer responsibility; rejecting passive positions; by offering sound argumentation; by practicing dialogue rather than controversy; by receiving well the new for reasons beyond mere novelty and by having the common sense not to reject the old just because it is old—by accepting what is valid in both the old and the new (p. 15).

As an example, we present here part of Civiero’s research (2009). At each step of the experience, we use the students’ speeches, interspersed with theory, to highlight the constitution of the landscape of investigation.

Finally, we show that the landscape of investigation leads students and teachers to experience actions based on mathematics. In this way, we realise the imbrications between mathematical knowledge and the contemporary variables of this complex civilising equation.

1. Scientific Initiation in High School

Both Scientific Initiation and CME are possible places for the discussion of contemporary issues, and they contribute to the critical education of the students. Scientific Initiation is a fundamental space for research. Therefore, we contend that research is:

the search, the study, the knowledge, the explanation, and the understanding of the world that surrounds it, motivated by actions of the subject that makes science. This demonstrates that it is not enough to fulfil the requirements of the system, it is also necessary to reduce the gap among areas of knowledge and between the technical and the human (Oliveira 2017, p. 32, our translation).

In line with this notion of research, according to Bazin (1983) and Oliveira et al. (2013), Scientific Initiation is a path of intellectual independence. As a scientific activity, it does not occur outside a social context.

Hence, we understand Scientific Initiation as a collaborative space of authorship experience, as the “search for the understanding in which the human being lives” (Oliveira, 2017, p. 32). Its insertion in basic education is pertinent, as Scientific Initiation offers scientific and technological education that contributes to the formation of the individual by fostering curiosity, creativity, authorship, decision-making and interpretation of reality through an initiation into research.

Despite its importance, Scientific Initiation in high school, which is aimed at students aged between fifteen and eighteen, is recent in Brazil. According to Oliveira (2017), it can currently be classified into three modalities: as an institutional program (since 1986), as public policy (since 2003) and as a component of the curriculum (since 2001).

Regardless of the modality, Scientific Initiation in high school must be distanced from technical rationality. That is, it must not focus on crystallised techniques and methodologies, such as imitation, repetition and reproduction, because:

high school Scientific Initiation can articulate and integrate diverse knowledge, theory and practice and teaching, research, and extension. Dialogicity, problematisation, critical reflection, and collaboration are the basis for the development of people’s autonomy. Based on these potentials, Scientific Initiation in high school is not just a space for methodological learning or research initiation focussed on training researchers concerned with an object of study that is alien to reality, society, and the civilising process (Oliveira, 2017, p. 147, our translation).

Oliveira, Civiero and Bazzo (2019) uphold the idea of Scientific Initiation as a component of the curriculum, a space in which the project selected for transposition was developed. For the authors, Scientific Initiation in high school “is a possibility to deal with contemporary issues and bring knowledge from different areas to the student’s reality and, therefore, to bring reflective and critical discussions” (p. 469).

We advocate this potential of the Scientific Initiation. Therefore, we have the modality of Scientific Initiation as a curriculum component at the basis of this study, as it is an environment that guides the permanent reconstruction of knowledge. This modality takes place at the Instituto Federal Catarinense (IFC), Rio do Sul Campus,⁷ locus of this study.

2. Scientific Initiation as a Curriculum Component in High School

In 2001, the Scientific Initiation Project started at IFC—Rio do Sul Campus as a project that was part of the diverse element of the high school curriculum matrix, with a two-hour workload per week. One or two teachers manage this workload, where students are offered training on epistemological foundations of science, as well as aspects of research methodology, and the writing of projects and reports.

The insertion of the Scientific Initiation Project into the curriculum allows for the production of knowledge and the articulation of different areas of knowledge, minimising the boundaries between the curriculum components (Scheller et al., 2015). Therefore, there are structural elements that permeate the organisation of Scientific Initiation from the beginning, as shown in the chart below.

Fig. 1. Structuring elements of the curriculum component Scientific Initiation. High school, IFC, Rio do Sul campus, 2001–2019.

With a view to the organisation in Figure 1, we realise that, at first, the teaching plans support the discussion of topics that instigate students’ critical reflection on the world, leading them to perceive themselves as subjects in and of the world. Through talks and deconstructions of myths and taboos about science, technology and scientists, students start the project, which involves choosing a topic under the guidance of a campus professor.

Scientific Initiation as a curriculum component allows all students to participate, to establish dialogical teaching and learning relationships in a process of dodiscence. “Teaching, learning, and researching deal with these two moments of the gnoseological cycle: the one in which the existing knowledge is taught and learned, and the one in which the production of knowledge that does not exist is yet to be worked” (Freire, 1996, p. 28).

Thus, Scientific Initiation is a space aimed at instigating the student’s enjoyment of—and inclination to—learn. In short, it is a place that may encourage students to be curious, reflective, to argue, to seek answers and to carry out the process of building knowledge in a critical way, which is not a tradition in education and society. In other words, Scientific Initiation is:

an educational process capable of equipping the student for critical reading of the social practice in which he lives is the means that will make the school democratic. I understand that a democratic school takes the student to be a transforming subject of their reality; a critical look is not enough; the student must be inserted into the project, think, and plan necessary changes, believe in them, and put them into action. For this process to become effective, it is necessary to assume democratic attitudes when restructuring the didactic procedures (Civiero, 2009, p. 53, our translation).

This educational process that constitutes Scientific Initiation approaches the landscapes of investigation in problematisation and knowledge production. Besides, Scientific Initiation intertwines with scientific and technological education; therefore, it is a space for discussion about the variables of the civilising equation, so that the practice of Scientific Initiation:

[…] needs, in its conduction and supervision process, a dialogical practice that problematises, that questions, that criticises knowledge, that values the other, that integrates, that instigates autonomy, and that takes care of life as the greatest social good, being essential the training of guiding teachers and/or researchers. The understanding and practice of scientific initiation need to go beyond the reproduction only of issues already posed “culturally” for research and teaching, such as, for example, bureaucracy, elitisation, selective character, training, focus on the method, and reproduction of technical rationality. To impact upon humanising formation, it is necessary to have as a main pact the critical and reflective search to understand the world in which we live, established by a collaborative environment permeated by problematising dialogicity that relates science and technology and the civilising process, the advisor and the student (Oliveira, 2017, p. 275–276, our translation).

In this way, we recognise that the Scientific Initiation works developed in the IFC—Rio do Sul Campus are landscapes that encourage authorship and critical reflection on knowledge. Thus, it can be transposed into the classroom and constitute landscapes of investigation. Therefore, the first author selected Scientific Initiation projects and leveraged them into proposals of landscapes of investigation in mathematics classes. The author has reported this experience in her Master’s dissertation (Civiero, 2009). She considers this a Reflective Didactical Transposition (RDT) process. The RDT process considers Didactic Transposition (Chevallard, 1991), but criticises and adds to this theory the concern about reflection on mathematical contents and the real context. In other words, the knowledge to be taught is adapted to provoke reflections on reality. Thus, RDT intends to transpose the knowledge to be taught in a landscape for investigation in order to instigate the subject’s questioning of their own reality. In the process, student participation is promoted, and discussions and decision-making are provoked. Thus, the knowledge developed in the Scientific Initiation Project was transposed to mathematics classes, according to the CME perspective. For RDT, “it is not enough to transfer knowledge; it is necessary to instigate reflections on mathematical subjects linked to reality” (Civiero, 2009, p. 49). Below, we present the experience lived in one of the landscapes.⁸

3. Experiencing a Landscape of Investigation

Landscapes of investigation are milieus of learning built in the classroom. They allow for investigation, where students are invited to make discoveries in a process full of questions, curiosities, explanations of perspectives and critical reflection. Therefore,

the important point is that the landscapes of investigation are not explored based on a previous list of exercises. On the contrary, explorations take place through a “learning guide,” in which students can point out directions, ask questions, ask for help, make decisions, etc (Skovsmose, 2001b, p. 64).

In this context, we carried out the RDT with two classes of first-grade high-school students at the IFC—Rio do Sul Campus. When developing the activity, reactions and comments from the students were observed, as well as the teacher’s perceptions. The students’ names have been substituted with letters of the Roman alphabet to conceal their identities.

In a context of uncertainty, in a landscape of investigation, it is paramount that the students accept the invitation to participate. According to Skovsmose (2001), a landscape of investigation is constituted when students accept (and assume themselves as active participants in) the process of exploration and explanation.

[A] landscape of investigation is one that invites students to ask questions and seek explanations. The invitation is symbolised by its “Yes, what happens if …?” In this way, students get involved in the exploration process. The teacher question: “Why is this?” represents a challenge, and the students’ “Yes, why is this…?” indicates that they are facing the challenge, and that they are looking for explanations (Skovsmose, 2008, p. 21).

Therefore, we sought to encourage the investigation to arouse students’ curiosity. We presented the material attractively, so that the activity was not seen as a command but rather as a different way to learn mathematics, to prioritise the reality in which the students were inserted. In this way, the teacher’s role in this space is that of an inquirer and mediator, to avoid defined and unquestionable concepts.

At first, students were invited to investigate windrow behaviour through composting, based on the work “Composting From Various Organic Wastes”, developed in the Scientific Initiation Project (2006/2007). The windrows are structured with a base of vegetable dry matter and layers are interspersed with organic matter. The system works with passive aeration, ensuring the thermophilic composting process. Figures 2 and 3 below illustrate the windrows.

The students who attended the Technical Course in Agroecology or Agriculture integrated with high school⁹ showed interest. Thus, raising concerns about the environment as one of the contemporary variables in this globalised world is part of humanising education, as well as professional education.

4. From the Landscape to the Investigation

In the first stage of transposition (RDT) from the elements of work developed in the Scientific Initiation Project, students reflected on the importance of the project and the social relevance inherent to the theme. They also raised some hypotheses about its implementation. According to the authors of the Scientific Initiation work,

the compost is very important for agriculture because it follows the concepts of agroecology, as it is a way of nourishing plants with all the macros and micronutrients that it needs without any type of external inputs that can harm nature (Battisti, Campos, and Souza, 2007, our translation).

The students began asking a series of questions that involved the subject: “What is organic matter? What is compost? What are the benefits of composting in the soil?” Such questions prompted the search for explanations. At the same time, students began to ask new questions: “How do you make compost?” (Student C). “What types of materials can you use to make the compost?” (Student J). “How do I prepare a compost pile?” (Student A). “How do I maintain this compost?” (Student P). “How do you check the maturity of the compost?” (Student K). “What are the stages of composting?” (Student B). In this way, the initial curiosity, which can be called naïve (Freire, 1996), was instated as they accepted the invitation.

However, some students opposed the activity, claiming that they did not perceive a relationship between the activity and mathematics. This resistance may be due to the exercise paradigm. Student C’s question, “What does this have to do with the math class?” is an example of such an objection. They felt a little uncomfortable as they had to move from a passive to an active position. Resistance strengthened the premise of the ideology of certainty, imbued with a culture that supports the power of containing the ultimate argument attributed to mathematics. For Borba and Skovsmose (1997, p. 133), “students should, therefore, be persuaded against ideas such as a mathematical argument is the end of the story; a mathematical argument is superior by its very nature; the numbers say this and this.” The only methodological option was to try to detach from the exercise paradigm. We started to discuss how urgent it was to use a different approach with the mathematical contents in order to analyse that reality.

After reading the proposition, the students delved into the theoretical foundations and were free to research more elements, which led them to become collaboratively involved. Classes started to gain momentum; students were not waiting for ready-made answers and investigated them when necessary, manifesting the criticism of the initial curiosity (Freire, 1996, 2006). Throughout the activity, the teacher felt she was in the risk zone, because, in a landscape of investigation, uncertainties are part of the process. Uncertainties also manifested in the students: “Teacher, when will mathematics appear?” (Student A) or: “This class is different, what does the teacher want?” (Student J).

After that, students were motivated to recognise the materials and methods used by the Scientific Initiation Project in the process. According to Figure 2:

Fig. 2. Adapted from Battisti et al. (2007).

Photos of the experiment were used to detail each step, to facilitate the understanding of the practice (Fig. 3 and Fig. 4).

Fig. 3. Taken from Battisti, Campos and Souza (2007).

Fig. 4. Taken from Battisti, Campos and Souza (2007).

After getting to know the compost production process, the next step was to start data analysis. At first, the teacher prompted the students to make estimates: “How do you think the composting occurred? How did the windrows behave? Which decomposed more quickly? How did this happen?” After many conjectures, Table 1 was created with the data.

Table 1. Behaviour of windrows with different residues for composting, Rio do Sul. Taken from Battisti et al. (2007).

Based on the data, students were encouraged to analyse the results. Again, the teacher acted as an inquirer, triggering questions such as: “What data are listed in the title of the chart? Does the title tell us what is being presented? Tell us when and where the experiment took place. What can you see in the height variation of the windrows? What was the starting height?”

During the discussions, these observations emerged: “Look at the windmill with organic material, it always reduces equally. Did this really happen, teacher?” (Student C).

We emphasise that the way communication develops between students and the teacher can influence the learning (Alrø and Skovsmose, 2002). The teacher plays a vital role in supervising this dialogue, as discussed by Milani (2017) and Milani et al. (2017).

In this process, the teacher instigated the students’ understanding of the behaviour of the windrows, aiming to show them the difference between the windrows and the organic material (waste) and the fibrous windrow (straw).

The windrow of fibrous material decreased faster due to disruption and varying the decrease, while the others gradually decreased by 1.2 cm per week. The windrows were made alternately with fibrous material and organic waste, and at the end, each windrow was 30 cm high. The different windrows gradually lowered around 1.2 cm each week, and in February, their height stabilised at 12 cm. The windrow that had only straw fell faster, varying its decrease due to decomposition and de-structuring of the compost, but ended up the same height as the others (Civiero, 2009, p. 83, our translation).

Student A observed: “Now I am aware of mathematics that appears in this data.” Following this speech, other students spoke, willing to establish mathematical relationships and to represent them graphically. The students interacted and wanted to be involved in the discussions. The commitment and willingness were different from most maths classes the teacher had experienced before.

5. Mathematics-Based Actions

Figure 5 presents the behaviour of the data graphically, but also gives the mathematical models that have best adapted to the data.

Fig. 5. Evolution of windrow height for composting. Taken from Battisti, Campos and Souza (2007).

This stage of RDT is in line with Skovsmose (2001b) regarding the three aspects of teaching and learning that the social argument of democratisation must present.

1) The material has to do with a real mathematical model; 2) The model has to do with important social activities in society; 3) The material develops an understanding of the mathematical content of the model, but this more technical knowledge is not a goal. The goal is to develop an insight into the hypotheses integrated into the model and thus develop an understanding of the processes (for example, decision processes) in society (Skovsmose, 2001b, pp. 43–44).

The developed activity converges with the three aspects. As for the latter, it was necessary to instigate discussions on the importance of models in a highly technological society. When analysing the model and its relationships, it is essential to check estimates and approximations. This allows the identification of an object of reflective knowledge, distinct from the object of technological knowledge.

The teacher asked questions: “What do they represent? What kind of curves appeared? What do the coefficients mean in the function?” Students were asked to define the mathematical model that best adapted to the curve.

Thereby, students became curious about mathematical models and started a process of mathematical discovery. In this case, specific mathematical knowledge and its concepts were essential to explain the reality. This speech can express some of the reactions: “Now I understand where the teacher wants to go. The project is full of mathematics” (Student D).

The students realised that in the windrow formed by various residues, the height variation was gradual, exactly 1.2 cm per week. They soon identified this number as the angular coefficient of the linear function, advancing to the concept of rate of change, which is made explicit in the statements: “The windrow is always lowering equally” (Student G). “Look at the table; each week, the windrow lowered 1.2 cm, it’s the same number that appears in the function” (Student A).

Student B reflected: “This number is constant in the organic material windrow.” Student J added: “Ah! That is why the graph is a line.” Student D immediately asked about the meaning of the correlation index: “Teacher, what does this R² represent?” Students were encouraged to investigate this issue.

Meanwhile, Student C asked: “What about the windrow made of straw? It is different, so how do we identify the rate of change if it was not constant?” They realised that these windrows’ behaviour was different, with the points not adapting to the linear shape, which would need a different mathematical analysis.

The students were curious about the functions that the Excel software presented. Student E asked: “Teacher, what do the charts have in common with these functions that Excel listed?” This question encouraged other students to speak out of curiosity. They were ready to start another stage of the investigation.

First, they understood that the Excel program had adjusted the curve according to the data that related the two quantities—that is, a set of coordinates (x, y). When asked which quantities were being related, Student A replied: “Of course, we are relating time (weeks) and height (cm).” Therefore, the variables x and y were revealed: variable x was the weekly variation, and the variable y was the windrow height variation. The students studied the mathematical concepts with interest and asked questions such as: What is that for? In traditional classes, this would usually become obsolete because this assumption had already been established. This issue disappeared as mathematical concepts were developed from the need generated in the context the students were inserted into.

In this phase of the explanation, we needed to refer to pure mathematics. Therefore, the students stopped to appropriate specific mathematical knowledge in order to understand the project. For the first windrow, which showed a linear decrease, they needed to study the function of first degree, recognising its main characteristics and rules. To exercise the content of linear systems, which emerged from the need to adjust curves, they used activities referring to semi-reality. Such activities make up an important part of the list of educational possibilities; however,

[s]olving exercises referring to a semi-reality is a very complex competence and based on a well-specified contract between the teacher and the students. Some of the principles of this agreement are as follows: the exercise’s wording fully describes semi-reality; no other information is essential for solving the exercise; more information is totally irrelevant; the only purpose of presenting the exercise is to solve it (Skovsmose, 2008, p. 25).

When carrying out the activities proposed, some students showed satisfaction, which they expressed in some comments: “Now, yes. The class became mathematics again” (Student C), which means that the student may still be framed by the exercise paradigm. However, this feeling was not shared by others, who said: “Wow, we did it again without knowing for what” (Student H). This student showed that they were uncomfortable with decontextualised exercises. And finally: “Of course not. We need to learn to calculate to understand the windrows” (Student M). We understand that this student realised that specific mathematical knowledge is needed to interpret reality. The student’s initial curiosity was criticised. But then, as they deepened their knowledge, it became an epistemological curiosity (Freire, 1996).

During this traditional class, the students behaved formally, i.e., they reproduced the activities. However, this stage was full of meaning, and the students had a goal. Even so, the teacher needed to create space for dialogue, through active listening. According to Milani, Civiero, Soares, and Lima (2017, p. 240): “When the teacher tries to perform active listening, he starts the movement of dialogue that seeks to understand what the student says. This movement is not simple and immediate, as it is a change of posture, in the epistemological, methodological, and political sense.”

In this process, it is possible to perceive students’ involvement in different milieus of learning,¹⁰ which highlights the potential of the landscape developed. Thus, they move between different milieus all the time, according to necessity. References to pure mathematics, in the context presented, are totally related to semi-reality and real-life. The landscapes of investigation concerning real life emerge naturally from a contemporary variable. Thus, this merging of milieus is not forced, like the grafts, but is a requirement for its development.

At the beginning of the following class, students were asked: “What did you see when observing the behaviour of elephant grass windrows and other waste? Why was it represented graphically by a line? How can you relate the data and the linear function presented by the Excel program?” After discussing, the students realised that the windrow height varies according to the passage of time (weeks). That is, in the function, it accompanies the x, which is representing the time in weeks. They also concluded that the parameter b of the function was representing the initial windrow height. To conclude, much dialogue was needed. Students went on debating, and, from time to time, the teacher intervened with a challenge to help them understand the mathematical relationships. The definitions became evident in students’ statements: “Well, if the table shows that the windrow has always lowered by 1.2 and that each collection was made weekly, then, just by multiplying the week by 1.2, and we will know the windrow height” (Student H). “However, you can’t forget that it has to be 1.2, because the windrow is lowering and a function y = -1.2x + 30 that expresses the windrow height according to the weeks is obtained” (Student C). “Interesting, then, that is why the graph is a straight line, week after week, the windrow decreases equally” (Student J). “Yes, the variation is always the same. I also noticed that the line is decreasing, which is logical because the windrow is lowering” (Student B). “Teacher, number thirty appeared in the function, is it the initial windrow height, or just a coincidence?” (Student G). “It is obvious, you see, now I understand. The graph is showing exactly the behaviour of the windrow. It starts with 30 cm and lowers by 1.2 cm, which must be represented by 1.2 cm because it is going down, i.e., it is decreasing” (Student D).

During the dialogue, there was active listening, and the students tried to emphasise the appropriate nomenclature. For example, when they said that the windrow was lowering, they were taught to use the term “decreasing”, which was related to the position of the line. Gradually, they adapted language and mathematical symbology.

6. Different Approaches

Next, the students chose their resolution method: either the Simple Linear Regression or the Least Square Method. They found the values of the coefficients a and b, whose meanings and values they had previously recognised, and checked the mathematical algorithm. The variables x and y were already known. For example,

y = ax + b, wherein:

They used different methods to solve the problem. The purpose of fostering the study of different methods to determine parameters a and b is consistent with their desire to create decision situations. The students can decide which path they want to follow when developing the work. This freedom to choose the method also occurs in Scientific Initiation projects.

Knowing the parameters, we proceeded to interpret the model. For this, we had to observe the usual correlation coefficient and the determination coefficient. After an investigation of the mathematical devices needed to solve this situation, the students used the correlation coefficient formula:

This calculation requires students to understand that a perfect correlation is one that approaches 1, which shows a perfect match between the data and the function. Student K interrupted, saying: “So, the result is always 1?” To answer the question, we argued that “the ideal would be 1, which would indicate a 100% correlation. However, this is difficult to achieve in practice because many factors determine the relationships between variables in real life. In our case, it did occur because the variation was constant, and all points were aligned, but it is not always so. Usually, some points are off the line, so the correlation index will not be exactly 1.”

After the discussions, we suggested that students apply the formula to understand the concepts discussed. Then, replacing the values obtained before, they found the following calculations:

When they saw a negative result, there was some anxiety. Student G immediately exclaimed: “It’s wrong, look, it is negative!” Then, we discussed what this negative result would mean and that it would be interesting to explore its concept before excluding it. For this, they found that the coefficient can vary from -1 to 1, indicating that there is a strong relationship between the variables. What is explicit in the model as weeks go by, (x), the windrow is decreasing. That is, it is decomposing and, thus, reducing in size, which also happens with the graph. In the same way, we proceeded to the calculation and analysis of r² and the rate of change.

At this point, it is possible to infer that this dynamic occurred according to Skovsmose (2008, p. 13), who considers “that a new critical mathematical education must seek educational possibilities (and not propagate ready-made answers)”. Postman and Weigartner (1969), who discuss the importance of changing attitudes, moving from a school of answers to that of questions.

7. The Model: The Importance of Critical Reflection

Asked to write a report on the main mathematical characteristics evidenced in the investigation, the teacher drew attention to one more detail: “Can we consider the function indefinitely?” Soon, Student C said: “It is absurd!”, along with a colleague, Student J who, until then, had not spoken but was aware of everything: “It stopped decomposing.” Student A also intervened: “If we only consider the function, the windrow height will begin to decrease indefinitely after the fifteenth week, which is absurd, because the height of the windrow has stagnated.” Student G then expressed his curiosity: “And now, what do we do? How does mathematics explain this?”

After they spoke, we explained that it is necessary to indicate the domain of this function. Some students realised that the image (y) was a consequence of the domain (x) and reached a conclusion, expressing the answers as follows:

During the RDT development in classes, we also highlighted the discussion about the importance of adjusting the curves. Different actual situations can present problems that require solutions and decisions that can be solved by a mathematical formulation. Thus, a mathematical model is represented by symbols and mathematical relationships that seek to translate, in some way, a phenomenon of reality. In this perspective, when proposing a model, we must keep in mind that it comes from approximations made to try and understand a phenomenon better. Thus, these approaches are not always consistent with reality, but they portray aspects of the situation analysed. Therefore, it is necessary to criticise the model in all its dimensions.

We highlighted that several observations could be made regarding mathematics in action and thus justify how mathematics can operate in technologies, production, management schemes and decision-making. As part of the laborious civilising equation, it can change social and cultural behaviours.

We discussed how society is technologised and how much mathematics helps to shape this society, which overlaps with Skovsmose’s (2008, p. 112) third concern about mathematics in action, when he states that it “is a paradigmatic space to discuss structures of knowledge and power in today’s society.”

In this debate, we emphasise the accelerated change in the civilising equation led by the Fourth Industrial Revolution. According to Schwab (2017), this behaviour announces a 4.0 Revolution, characterised by the transition towards new systems that overcome the digital revolution. Consequently, the power relations underlying the processes of commercialisation and industrialisation are closely linked to technoscientific development, which, in turn, is conditioned by mathematical algorithms.

We also observed that mathematics provides the possibility of hypothetical reasoning, i.e., it can analyse the consequences of an imaginary landscape. On the other hand, mathematics can also help to construct (true or false) justifications to legitimise some decisions and actions. The students raised the issue of the electoral season when we look at the polls. Depending on their organisation, they do not always represent reality but are used to influence and pressure us to choose a specific candidate. In 2020, we could discuss statistical data from the COVID-19 pandemic. For example, the case fatality rate (CFR) depends on the number of confirmed cases, and for that, it depends on the number of tests that are carried out. Therefore, it is extremely challenging to make accurate estimates of the actual death risk.

The students were frightened by the power of mathematisation, which could be seen by the speech: “Wow! Mathematics has power over everything; it gives me a shiver. Knowing that everything is calculated in advance, and callously” (Student J).

We tried to sharpen the discussion by emphasising that mathematical models are not always constructed from a socially just perspective. Encouraging critical mathematics is “integrating students’ lives, knowledge, and cultures; having students learn important mathematics and about their world; and supporting them to act on the injustices they perceive and experience” (Gutstein, 2012, p. 65). Therefore, to paraphrase Skovsmose (2008, p. 118), we emphasised that “mathematics should be a theme for reflection and criticism in all its forms of action.”

In this context, we sought to show how critical education can be guided toward emancipation. These discussions were based on Skovsmose (2008, p. 94), when he mentions critical citizenship: “[…] it can ‘challenge’ the constituted authority. It carries with it the opposition to any decision considered unquestionable.” We highlighted the relevance of knowing how society is managed and how situations are planned, often mathematically, with its algorithms.

Consequently, this educational possibility was planned, aiming to bring about changes to problematise the need to criticise the social system. During the discussions, the significance of dialogue between the teacher and the students became evident. However, we agree with Skovsmose’s concerns when he states that:

the important question now is how well mathematics education can prepare [students] for critical citizenship. I do not see that such preparation is related to the school’s mathematical tradition. I do not even see it linked to the intimate nature of mathematics. It has to do with a possible function of mathematics education (Skovsmose, 2008, p. 95).

We argue that discussions of this type can help mathematics education set critical citizenship in motion. In this way, students can experience actions based on mathematics to realise how relevant reflections are. One of the students commented in between discussions in class: “We cannot accept everything as finished, full stop; we need to understand the process to be able to accept it or not” (Student F).

We concluded the study of the function of the first degree with this analysis. However, we ended the class by encouraging students to observe the behaviour of the windrow made up of fibrous material, whose model refers to a quadratic function. Nevertheless, we will talk about this development at another time, or it can be seen in Civiero (2009).

For Civiero and Sant’Ana (2013, p. 695), the landscapes of investigation from Scientific Initiation constitute a “critical reflexive approach that can relate teaching to the act of questioning and making decisions, establishing a link with life in society and mathematics.”

In this context, we advocate the relevance of mathematics teachers’ scientific and technological literacy, so that they are prepared to promote RDT through landscapes of investigation about contemporary variables. In this regard, Civiero, Fronza, Oliveira, Schwertl and Bazzo (2017, p. 2673), state that:

for the teacher to develop mathematical concepts related to reality, he needs to recognise it, read the news critically, expand his list of readings, understand, make decisions, evaluate and criticise social, political, economic, scientific, and technological issues. Recognise the dynamics and complexity of the educational world imbricated with reality outside it, acting cooperatively and collaboratively (our translation).

In this way, we emphasise that the teacher must assume a critical epistemological conception to foster a critical posture in his students, making them capable of analysing and making decisions that can interfere in reality and, consequently, in people’s quality of life.

8. Additional Considerations

Hamlet’s dilemma: “To be or not to be, that is the question”, one of the most famous phrases in world literature, may seem complex, but it is actually very simple. “To be or not to be” is about acting, decision-making and positioning oneself or not in the face of events. In this perspective, the phrase led us to question the possibilities of CME in today’s society.

We live in a civilising equation on a different scale, scope and complexity from any that has ever occurred before. The challenges for those who aspire to social justice because of the equation are increasingly complex. Therefore, we must urgently enter all possible spaces to show that a better world is possible, a world where all people have equal and inalienable rights as the foundation of freedom, justice, peace and social development.

For this purpose, the description of the Reflective Didactic Transposition (RDT) of a Scientific Initiation project for mathematics classes provided the development of a landscape of investigation, which, in turn, encouraged reflection and criticism. The development of this approach evidenced the approximation of Scientific Initiation conceptions with the landscapes of investigation. Therefore, when proposing activities from the perspective of the CME, we provide discussions about the variables of the civilising equation. It is a way to equate the many elements of the interwoven relationship between technical aspects and human issues.

When exploring the behaviour of the composting windrows (the theme of the Scientific Initiation Project), our objective can be an environmental study, one of the essential variables that reflect fundamentally on the existence of the Earth. We invited students to get involved in the production of the mathematical model and reflect on how the results are related to the criteria used and how they can be used in society. We emphasise that mathematics plays an essential role in social issues. Therefore, we reiterate the importance of looking at the context and appropriating it for reflection and action to foster collective decision-making, given the needs established in the process.

It was also possible to highlight the need for urgent imbrication of the variables, the scientific initiation, and the different milieus of learning. When looking for different milieus, one can articulate the dimensions of the methodological specificities of pure mathematics, semi-reality and reality. In this context, the landscapes of investigation with real-life references—environmental variables related to the professional course—took shape. Its content was essential for the interpretation of life through mathematics, which is found in Skovsmose’s (2005, p. 96) words: “In this way, they experienced what actions based on Mathematics can mean and realized the importance of reflection.”

Finally, we argue that the development of landscapes of investigation is fundamental to highlight mathematics linked to technological and human issues that constitute the civilising equation. To this end, it is essential to provide discussions inherent to reality to appropriate specific mathematical knowledge imbricated with other areas of knowledge, therefore applying them to promote a society where the principles of human dignity are guaranteed, and social justice prevails.

References

Alrø, H. & Skovsmose, O. (2002). Dialogue and learning in mathematics education: Intention, reflection, critique. Kluwer Academic Publishers

Battisti, G., Campos P. C. de & Souza, W. O. (2007). Compostagem a partir de diversos resíduos orgânicos. Relatório de iniciação científica. EAFRS: Rio do Sul.

Bazin, M. J. (1983). O que é Iniciação Científica. Revista do Ensino de Física, 5(1), 81–88.

Bazzo, W. A. (2016). Ponto de ruptura civilizatória: A pertinência de uma educação “desobediente”. Revista CTS, 33(11), 73–91.

Bazzo, W. A. (2019). De técnico e de humano: questões contemporâneas. Ed. da UFSC.

Borba, M. & Skovsmose, O. (1997). The ideology of certainty. For the Learning of Mathematics, 17(3), 17–23.

Chevallard, Y. (1991). La Transposition didatique: du savoir savant au savoir enseigné. La Pensée Sauvage.

Civiero, P. A. G. (2009). Transposição Didática Reflexiva. Master’s thesis. Universidade Federal do Rio Grande do Sul, Porto Alegre.

Civiero, P. A. G. & Sant’Ana, M. F. (2013). Learning roadmaps from the reflexive Didactic Transposition. Bolema, 27(46), 681–696.

Civiero, P. A. G. (2016). Educação matemática crítica e as implicações sociais da ciência e da tecnologia no processo civilizatório contemporâneo: Embates para formação de professores de matemática. Doctoral dissertation. Universidade Federal de Santa Catarina, Florianópolis.

Civiero, P. A. G., Fronza, K. R. K., Oliveira, F. P. Z., Schwertl, S. L. & Bazzo, W. A (2017). Alfabetização Científica e Tecnológica no Currículo da Formação de Professores de Matemática. Ensenanza de Las Ciencias, Extra, 2669–2674.

Civiero, P. A. G. & Bazzo, W. A. (2020). A equação civilizatória e a pertinência de uma educação insubordinada. International Journal for Research in Mathematics Education, 10(1), 76–94.

Civiero, P. A. G & Oliveira, F. P. Z. de. (2020). Landscapes of investigation and scientific initiation: Possibilities in civilizatory Equation. Acta Scientiae (Canoas), 22(5), 165–185.

Freire, P. (1974). Education for critical consciousness. British Library.

Freire, P. (1996). Pedagogia da autonomia: saberes necessários à prática educativa. Paz e Terra (Coleção Leitura).

Freire, P. (2006). À sombra desta mangueira. Olho dágua.

Gutstein, E. (2012). Reflections on teaching and learning mathematics for social justice in urban schools. In: Teaching mathematics for social justice: Conversations with educators. Reston: NCTM, 63–78.

Milani, R. (2017). “Sim, eu ouvi o que eles disseram”: o diálogo como movimento de ir até onde o outro está. Bolema, 31(57), 35–52.

Milani, R., Civiero, P. A. G., Soares, D. A. & Lima, A. S. de. (2017). O Diálogo nos Ambientes de Aprendizagem nas Aulas de Matemática. RPEM, Campo Mourão, Pr. 6(12), 221–245.

Oliveira, F. P. Z., Civiero, P. A. G., Fronza, K. R. K., & Mulinari, G. (2013). Iniciação Científica para Quê? Enseñanza de las Ciencias, Extra, 2764–2768.

Oliveira, F. P. Z. de. (2017). Pactos e impactos da Iniciação Científica na formação dos estudantes do Ensino Médio. Doctoral dissertation. Universidade Federal de Santa Catarina, Florianópolis.

Oliveira, F. P. Z., Civiero, P. A. G & Bazzo, W. A. (2019). A iniciação científica na formação de estudantes do ensino médio. Revista Debates em Educação, 11(24), 453–473.

Postman, N. & Weingartner, C. (1969). Teaching as a subversive activity. Delacorte Press.

Scheller, M., Civiero, P. A. G. & Oliveira, F. P. Z. (2015). Pedagogical actions of reflective mathematical modelling. In G. A. Stillman, W. Blum & M. S. Biembengut (Eds), Mathematical modelling in education research and practice: Cultural, social and cognitive influences (pp. 397–406). Springer.

Schwab, K. (2017). The fourth industrial revolution. Crown Business.

Shakespeare, W. (2009). Hamlet. Cambridge University Press.

Skovsmose, O. (2001a). Landscapes of investigation. ZDM―Mathematics Education, 33(4), 123–132.

Skovsmose, O. (2001b). Educação matemática crítica: A questão da democracia. Papirus.

Skovsmose, O. (2005). Travelling through education: Uncertainty, mathematics, responsability. Sense Publishers.

Skovsmose, O. (2008). Desafios da reflexão em educação matemática crítica. Papirus.

Skovsmose, O. (2014). Critique as uncertainty. Information Age Publishing.