Abstract
This article presents a study describing the agency of non-human entities within a mathematics education practice involving a teacher and seven high school students from a Brazilian public school. Data collected through observations were analyzed using Actor-Network Theory, mainly drawing on Bruno Latour's sociological insights. The analysis suggested that viewing mathematics education as a complex network of interconnected human and non-human actors allows us to understand how non-human agents contribute to performing a mathematics task. This study provided valuable insights into the role of these agents, revealing whether they act. Elements such as computers, calculators, the task itself, and the computer lab did not operate as mere tools but as active agents that influenced decisions, generated conflicts, and redirected the learning process in mathematics. The analysis also suggests that these non-human and human actors take on different roles, sometimes as intermediaries when their influence goes unnoticed, and sometimes as mediators when they actively intervene and alter the course of action. Furthermore, the study highlights the instability of hybrid association networks, showing how students’ leadership and strategies are constantly reshaped through interactions with various material objects.
The World is Full of Things, But That is What Makes it so Fascinating
From reading Alrø and Skovsmose (2004), we assume that mathematics education practices
Alrø and Skovsmose (2004) discuss mathematics education practices in terms of and in the context of possible learning environments for carrying out exercises or mathematical inquiries. Although they recognize the importance of the type of task carried out, the characterization of a learning environment mainly concerns the kinds of established social relationships and forms of communication. We might challenge this characterization as being excessively anthropocentric (the human being as the centre and measure of all things) since it only partially considers the material dimension (the type of task carried out in each environment; other objects are missed).
Imagine a mathematics lesson without books, pencils, erasers, tables, chairs, or, in some more favored socio-economic realities, computers, tablets, etc.! In short, the world is full of things, but that makes it fascinating. You might ask: Are they saying that objects also play an active part in maths education? Yes, of course… but calm down! Don't be alarmed! Throughout the text, it will become clearer what we mean when we say that things are also participants and protagonists in this context. For now, remember that we are challenging the old anthropocentric ideal (Simon, 2010).
This idea of the active participation of objects in mathematics education practices may at first sound strange, but it's nothing new! In philosophy and the social sciences, the protagonism of objects has been studied through different theoretical lenses. For example, to support this idea, Michel Serres uses the common figure of boys playing ball, arguing that this artefact is not just a thing but an agent capable of creating relationships between the players. It (the ball) not only traces relationships but “follows its trajectory, the team creates itself, gets to know itself, presents itself. Yes, active, the ball plays (…)” (Serres, 1993, p. 45).
Can we say that objects also take an active part in carrying out a mathematical task? We hope that, by the end of this article, we will have produced a satisfactory understanding of this possible question. We will not delve into this now, but we must clarify that we intend not to create a reading path driven by suspense. However, we need to lay out our theoretical foundation before outlining an understanding with statements that, without proper contextualization, could sound misplaced.
In this study, we draw on the principles of the “material turn,” an ontological movement that emphasizes the importance of objects, treating them symmetrically with humans. Specifically, we focus on the conceptual contributions of Bruno Latour, the French philosopher and sociologist who established Actor-Network Theory 2 . We will produce a momentary description (we are not interested in conclusive explanations) based on the following research question: How do objects such as pencils, paper, rulers, textbooks, calculators, and computers participate in mathematics education practices? (Here, we intuitively use “ participation,” with a conceptual definition to be provided later.)
We recognize that research in Mathematics Education has explored the role of objects. For instance, the human-with-media construct (Borba & Villarreal, 2005) highlights the mutual influence between humans and technological devices, emphasising how these media shape human thought. Similarly, the notion of embodied cognition (Núñez et al., 1999) views the mind and body as inseparable and interacting with objects in the environment. This perspective supports analyses of how bodily gestures, e.g., touching tablet screens, contribute to teaching and learning processes in mathematics (Bairral, 2019, 2020).
Although these analyses highlight objects, they do not assume an ontological symmetry between humans and non-humans (such as physical and digital objects, animals, plants, microorganisms, etc.). However, Latourian sociology is grounded in generalized symmetry, which seeks to support non-hierarchical and non-deterministic approaches to human-non-human relations. By challenging analytical asymmetry, it rejects the notion that only humans are active agents in the unfolding of practices.
However, you may already be wondering: how can the approach of non-humans in mathematical education practices truly contribute to teaching and learning mathematics? Sinclair and Freitas (2019) seek to answer this question by highlighting the mutual influences between humans and the various types of non-humans present in a mathematics classroom. The authors argue that hybrids emerge from the associations established between them that are not merely the sum of their parts.
In this sense, they assume these hybrids make learning and teaching mathematics possible. Therefore, a mathematics education practice cannot be reduced to isolated actions of teachers and students, but should be understood as an inherent coupling between them and the artifacts with which they are associated. In other words, the associations formed shape how mathematics is learned and taught, whether through space organization, using manipulable materials, digital technologies, or other elements, thus shaping how the discipline is experienced.
For example, in the study by Ng and Ferrara (2020), the portable 3D printer acted as a central non-human agent, reshaping classroom interactions. The device sparked students’ curiosity by making it easier to produce geometric objects. It redirected the lesson toward hands-on investigations of solid properties, such as prisms and cross-sections. This association showed how artifacts can co-construct learning environments based on real-world problems.
Thus, a possible answer to the above question is that we can expand our understanding of actions related to teaching and learning mathematics by turning a more attentive gaze to non-humans and recognizing them as active agents in mathematical education practices. We assume that this shift in perspective can contribute to rethinking mathematics education practices, allowing us to trace how objects agentially influence (we will later discuss the concept of agency) teachers and students (and vice versa) in these practices, as they emerge from the hybrid associations between humans and artifacts mobilized in either formal or informal educational contexts.
This approach to understanding mathematics education practices engages in dialogue with principles from other sociomaterial perspectives, such as cultural-historical activity theory and complexity theory (Fenwick, 2010). These frameworks also reject the classical modern dichotomies, particularly the division between subject and object. According to Fenwick, although these approaches present significant differences, they are complementary, as each highlights distinct dimensions of the sociomaterial in educational practices. Together, they contribute to broader understandings of the constitution of subjectivities, the production and circulation of knowledge, and the ongoing configuration and reconfiguration of these practices.
By highlighting the participation of non-humans in these practices, the title of this paper might suggest a defense of technological determinism in human actions. However, this differs significantly from our perspective, as we aim to overcome the ontological dichotomies between human beings and the objects surrounding them. In other words, objects are neither neutral instruments nor independent autonomous forces. Instead, humans and objects act as mediating, hybrid agents (Latour, 1994). We acknowledge that action occurs through interactions with humans, but since human action is commonly assumed, we are tracing the participation of objects.
ANT has been criticized as an apolitical sociological approach detached from power relations. However, this is not a fair criticism, since, for example, the Latourian perspective in sociology addresses the notion of power in terms of the capacity to establish associations. To illustrate, if we looked at a guide to commercial flights in Brazil, the airports of Guarulhos and Congonhas would appear more powerful, as they concentrate most of the country's flights.
In this sense, power is fluid, distributed across the entire network of associations, shaping relationships, as discussed in the previous two paragraphs. The notion of network, therefore, also works as a quality control for research narratives, encouraging the most detailed description possible. This quality depends on the researcher's ability to observe the actors’ behavior.
In the next section, we will look at the notion of agency and others related to it so that we can present the aim of the research in a clearer form. We will detail the methodological approach, the data analysis, and the discussion, and finally, we will present the final considerations.
Things Are Not Simple Tools
Inspired by Latour (1991), we emphasize that recognizing non-humans as capable of influencing the flow of a mathematics education practice is not an attempt to anthropomorphize them. Instead, it aims to shift the perspective from viewing them as mere tools with a useful function to acknowledging them as active protagonists in mathematics education practices. We thus assume that a mathematics education practice comprises a set of heterogeneous actors in complex associations. An actor is “anything that modifies a situation by making a difference” (Latour, 2007, p. 108). From the ontological perspective of Actor-Network Theory, humans (teachers, students, etc.) and non-humans (objects present in a classroom) are actors.
Humans alone, without objects, could not constitute mathematics education practices because even a mental calculation presupposes a series of non-human elements such as signs (consensual symbols). The reverse is also true: even artificial intelligence cannot operate without human input or the involvement of another artefact, as it requires a command to perform a calculation or any other action. We have listed some examples of non-humans that go beyond those mentioned. In Latourian sociology, non-humans encompass everything unrelated to the human biological substrate (whether material or immaterial). Still, we will not go into this discussion because it is broad and beyond our scope, although it has implications for different research fields, such as Communication.
Rather than paying attention to what changes in a student's or a teacher's knowledge during a practice in mathematics education, we focus on the associations established between these actors and material objects. This allows us to “expand different possibilities of relations” (Ryu, 2019, p. 138) among them. Is knowledge unimportant? We would not deny its importance; however, what actively sustains and validates what we recognize as knowledge? A simplified answer might be: “maintaining links among the actors” (Ryu, 2019, p. 138).
Latour (2007) describes this network of actors, humans, and non-humans engaged in mutual interactions as a fragile structure. Despite its delicate “ties,” the network tends to maintain active connections through the agency of its members. The author argues that agency is intrinsically linked to the notion of action, which is not limited to humans but distributed throughout the entire network. That implies that action is always the result of reciprocal interactions, in which these two categories of actors are interconnected, so it is impossible to say where it begins and ends since it “spreads” throughout the network.
Thus, a plausible question when faced with an actant is: does it make a difference to the course of action of another agent? (Latour, 2007). Thus, an actor's capacity for agency indicates that it produces movement and transformation. In other words, if it does something to someone or something, it can cause disturbances in the network or even disintegrate it.
We find examples of material agency in Delacour (2022) in his discussion of using manipulable didactic materials within mathematics education practices. In his analysis, the author does not treat these actors as mere tools for teaching a mathematical concept but as actants capable of mobilising children's interest in investigation, ultimately leading to the learning of mathematical concepts. Thus, in the studied case, the manipulable material “left traces and influenced both the teacher and the children” (p. 4), meaning that the non-humans prompted human actors to take action.
The author recounts how the teacher's attempt to track the proposed task hindered the children's interaction with the non-human actors in the classroom. For Delacour (2022), this type of interaction is crucial, as children are constantly investigating the world through objects, which, in turn, helps them make sense of it and continue learning. In this episode, the teacher experiences the same tension Oliveira and Barbosa(2012) described regarding the fear caused by the initial experiences implementing inquiry-based mathematical tasks. As Alrø and Skovsmose (2004) remind us, such tasks may lead us into a “zone of risk,” where a simple question can redirect the entire activity's flow and challenge the teacher.
In Delacour (2022), the teacher enabled stronger interactions between the children and the objects by weakening the attempt to control the task's flow. Although the proposed mathematical concept (the study of geometric shapes) was not always the central focus of the children's activities, they “allowed their imagination and the materialities to guide what happened” (p. 12, our translation), leading them to explore other mathematical concepts. Some non-human actors demonstrated clear traces of their agency on/in the children, such as rods and the floor, which were fundamental in helping them understand how to construct geometric shapes by drawing them on the ground.
Considering non-human agency, Latour (2007) suggests that an actor can be either a mediator or an intermediary. An intermediary is an actor that does not produce transformations; it exists within the network but does not impact it. A mediator, on the other hand, is an actor that drives transformations within the network. It is important to highlight that this classification is not fixed; an actor may function as a mediator in one instance and an intermediary in another.
Ramey et al. (2024) conducted a study on adopting and implementing an educational innovation in teaching Natural Sciences and Mathematics. This innovation, focused on project-based learning tailored to students’ interests, was analyzed in the context of schools in the United States and Finland. Their research revealed that a specific non-human actor established associations with students, shaping their engagement with the proposed tasks, thereby acting as a mediator. For instance, with desktop computers, because each student was assigned a specific computer, they could not switch seats or sit closer to certain peers. This material contingency imposed new forms of association during the project's implementation. Had this non-human actor not influenced the network, it would have functioned merely as an intermediary.
Having established our epistemological assumptions, we are now ready to restate our research goal, this time in theoretical terms. This paper reports a study describing non-human agencies involving school students within a mathematics education practice. In the following section, we present the methodological approach.
A Journey to Describe How Non-Humans’ Agency Relates to Humans
Let's consider that qualitative research turns toward the individual or the collectivity of humans, i.e., their lived experiences, perceptions, and understandings, in short, toward how they organize their experiences. We conclude that it is an anthropocentric kind of research (St. Pierre, 2021). Since we are placing the ontological assumptions of humanist approaches in suspension, we need, according to that author, to consider paths that likewise bring non-humans into view so as not to produce epistemological noise in our research. This entails a shift to a posthumanist and therefore post-qualitative approach, requiring an adaptation of the instruments employed.
Thus, adopting the lens of Latourian sociology as our theoretical-methodological framework implies resisting the impulse to explain the processes observed, precisely because this approach proposes only to describe them. This means relinquishing the search for causal links for this or that action, since we already know that they “spread” through the continuous flow of the sociotechnical network. Describe… describe… and, when further questions arise, keep describing: this is the motto we infer from Latour's ironic advice to “just” follow the actors and “simply” say what they do.
Accordingly, our stance was to proceed without delineating our path in advance, without losing sight of our starting point, always following and describing the tangles of interactions between humans and non-humans wherever they led us (Latour, 2007). In this way, to describe the role of non-human actors in the observed mathematics education practices, we adopted the Teaching Experiment (Steffe & Thompson, 2000). This approach emerged within mathematics education research. However, we adapted it to Latourian presuppositions, aligning it with a posthumanist perspective (St. Pierre, 2021). This means that we remained attentive to the associations that emerged among the actors in the observed practices, avoiding reductionist explanations and bringing non-humans to the center of the analyses undertaken.
In practical terms, our field observations did not follow any observational protocol but assumed an eminently exploratory character. We did not impose pre-established categories; instead, we paid attention to the “performances” of the actors as the sociotechnical network under observation gradually took shape before us, and to how the actors enrolled one another. This is because, as we have already stated, we were not seeking explanations but to describe the momentary flow of the network—in other words, seeking a “narrative image” that did not—and does not—aspire to be definitive.
By ethical standards, the human actors involved (a teacher and seven first-year high school students from a public school in a Brazilian city) signed informed consent forms. These documents outlined the research objectives and the ethical treatment of data. While participants were allowed to choose pseudonyms, they declined to do so. Legal guardians also signed the consent forms. The group was selected based on participants’ willingness and availability to attend the teaching episodes scheduled in the afternoons.
Three sessions were conducted in a school-provided computer lab, each lasting one hour and forty minutes. This duration was sufficient to complete the task selected by the teacher from a supplementary textbook designed to encourage interdisciplinary connections between mathematics and the social sciences. Specifically, the task addressed Quantities and Measurements, geometric magnitudes, monetary systems, and financial education, contextualized through a discussion on the social impact of community gardens.
To examine how non-human agency unfolded during the mathematics education practice, we observed the sessions, took field notes in a research journal, and video-recorded the activities. Additionally, we collected students’ work produced during the sessions to identify possible instances of object agency. Data analysis occurred concurrently with data production, as insights from one episode informed focal points for observation in subsequent sessions.
In the following section, we present and discuss the analyzed data through the lens of Latourian conceptual frameworks.
Things Are So Interesting When They Come Into Play: A Narrative in Two Acts
Drawing on Sperling et al. (2022) and in light of the discussions/reflections presented, we aim to offer you, dear reader, a narrative of a mathematics education practice, seeking to describe the actors’ performances, their agency, and the network woven during the teaching experiment. Accordingly, the data produced are presented in two Acts, using the theater metaphor, for our level of analysis is that of a narrative description inspired by Latourian ethnography (Latour & Woolgar, 1986). Thus, we did not undertake coding/categorization or triangulation of the data, since our purpose is to trace hybrid associations. We emphasize that tracing here is understood, as indicated, as following the actors, describing the ties they establish, and tracking the effects they produce within the network.
Act I
The curtains open. Who are the actors in this narrative? Who was the first on stage? Some stayed in the “backstage,” while others promptly stood under the lights. Trying to identify them was our first step, but we realized that it would be better to characteriz them gradually as they “entered the scene.” This way, we sketched momentary perceptions of the network being formed, configured, and reconfigured throughout the teaching experiment.
The first actor we met was Computer Lab Number 4 (Figure 1), a space the school's management made available. It is one of six laboratories available at the institution. In this space, we found other actors: computers, a workbench, chairs, a table, a multimedia projector, air conditioning, good-quality internet, a whiteboard, and large glass windows that made using artificial lighting unnecessary during the observation period.
Computer laboratory number 4. Source: Research data.
When we arrived at Computer Lab Number 4, it was a bit “warm” and we could hardly switch on the air conditioning. Not even the technician in charge was successful, but after many attempts (he had already given up), we got it working ourselves. The space was ready to receive the students. We can say that this material apparatus (Computer Lab Number 4) was also the “stage” where the performances of the other actors could be observed during the teaching experiment, which is why we characterize it as an “actor-theatre”.
The students turned out to be punctual actors; they did not take long to arrive; they came in groups and relaxed as they settled in: some started interacting with the computers (games) (Figure 2), and the others kept looking at their mobile phones.
Students waiting for the teaching experiment to start. Source: Research data.
While they were distracted by the digital devices, another actress came on the scene 3 ,Simone, an experienced mathematics teacher with a Master's degree in Mathematics Education who has been teaching for 27 years and at this school for over 23 years. She arrived, opened her laptop, connected it to the multimedia device, and took the opportunity to make the final adjustments while waiting for two students who had committed to taking part in the teaching experiment to start the activities (these two did not come).
Teacher Simone begins to present the topic proposed for the mathematics task (community gardens) through an informal conversation and using a supporting actor, a PowerPoint presentation, and a brief explanation of the social importance of these initiatives. She stopped, put the presentation on one side, and asked the students if they could tell her where one of these gardens was in the city. No one could answer, and she took advantage of this lack of knowledge to problematize the discontinuation of a community garden in an area close to the school due to the construction of a highway by sharing memories from her childhood.
The teacher asks, “What does the word garden bring to mind?” Student actor 2, perhaps feeling the mathematical approach, says that geometry and other possible calculations related to planting and harvesting came to mind. At this point, the teacher avoids going deeper into the discussion from a mathematical perspective, and the conversation then talks about community gardens from a sustainability perspective. After problematizing the use of agrochemicals and ending the initial discussion to contextualize the proposed topic, the teacher suggests that community gardens could also be agents of social inclusion through “community empowerment” (in the sense that they enable people to achieve a better quality of life and mobilize the community to fight for more rights).
Next, teacher Simone organized the students into two groups (one with three students and the other with four, in such a way that our observations will focus on the latter group due to the limitation of having only one video camera) to read the text of the proposed task (Figure 3), which was extracted from the textbook entitled “Scenarios for Investigation - humanities and maths in context” (Paulussi; Grassmann, 2020, p. 84). At this point, we saw the entry of a new actor, the task taken from the textbook. The “A mathematical look at community gardens” task proposes a link between a social issue and mathematics. Its structure is as follows: a motivating text that raises questions about ethical, social, political, and mathematical aspects to come up with a proposal for creating a community garden, followed by a proposal to draw up three budgets that make it feasible to build a hypothetical garden or one that will be implemented.
First page of the mathematics task. Source: Research data.
The students read the motivating text, resuming the dialogue together; the teacher asked what they would consider necessary to build a community garden and asked them to list what they would need.
Student 1 made the following intervention: […] Knowing the spacing, knowing if the garden will need water transmission [irrigation], how the light will be, what area for the plants to be exposed, and seeing the geography of the place, right? Even the time the light hits.
Student actress 3 now feels interested in taking part in the conversation: […] My father has a farm, and he needs to irrigate it; he has a pump (…) because, without water, the plants would die.
The teacher questions the idea of necessary mathematical knowledge, highlighting the importance of intuitive and culturally constructed knowledge and how the different areas of knowledge are integrated to solve a problem: […] You thought about geographical issues, biological issues, he talked about maths […], about 20 cm depth. The knowledge that can be popular is common sense knowledge, used in everyday life by people living in that locality […]. I don't need an engineer […] to plant a vegetable garden. The people who live there […].
Student actor 6 spoke, but we could not capture his words because of the low volume. However, we infer that it was an attempt to reinforce the importance of traditional community knowledge. Student 2 makes an apparent counterpoint to the teacher's speech, and this is a possible speech by his colleague, emphasising the relative ease of access to information that we have nowadays: […] To start a community garden today, a normal person can research and delve into a certain topic by researching the right sources and trying to learn mathematical formulas […] to build the garden.
The students react to the text of the mathematical task, and the format of a more in-depth discussion shifts the dialogue again, now to the economic issue, to the prices of organic products. Student 1 assumes this results from the clash between family and large-scale farming. For student 2, the latter is “crushing” the former. However, they do not discuss this, and the teacher invites them to consider possibly planning a vegetable garden. This was her “cue” to address the need for planning to build a community garden. She informs them that, at the next meeting, they will begin to work out a budget for a hypothetical community garden as a group.
After these conversations and guidance on the task, the meeting draws close. Then, the curtains close. End of Act I.
When watching a play, an audience member quickly identifies the characters. So, a first question would be: which role or roles are played by the actors in the first Act? We know that each can “act” as a mediator or intermediary and change roles in the middle of the play. According to Latour (2007), when everything is in place and there are no mistakes or errors, some actors go unnoticed; they are there but don't draw attention to themselves.
Laboratory Number 4 is home to several other actors who, in this first episode, played the role of intermediaries, such as the computers, chairs, and table. As supporting actors, there is not much to report about them, except that everything went according to the plan: the computers worked, the chairs, the bench, and the projector were ready for use, and we could even forget about them. They are part of the network, but they have not produced any transformation in it.
Lab Number 4, however, did not go unnoticed because, as the stage for this mathematics education practice, it acted as a way for the students to distribute themselves around the space to carry out the task. This type of user-space interaction has been analysed by Sharif (2022), emphasising the impact of this type of material apparatus on the daily lives of human beings. He uses the corridor of a university department as an example, enabling a variety of associations, such as casual, unavoidable, etc. He also discusses the possibility of a physical space like this cooperating with collaboration between students (collaborative associations), thus strengthening their bonds.
We agree with Sharif's idea (2022) because we observed that, when divided into groups, the students spontaneously sought, even with a particular spatial limitation, a distribution in circles that favored dialogue between them. That made it possible, for example, for one of the groups to take turns reading the text of the task involving all the members so that they could listen to each other. In the author's terms, we could say that Laboratory Number 4, as we call it, is an “actor-theatre” and can maintain different networks of associations. The air conditioning malfunction was a clear example of this, illustrating Latour's thesis that errors reveal associations. This defect acted as a mediator, “behaving” unexpectedly and exposing the complexity of the socio-technical network in which we were embedded.
This space (Laboratory Number 4) is part of an institution that establishes a set of rules for its use, and different employees work there, such as the one who came to help us. From that space, we began to glimpse the network (equipment, institution, norms, people, etc.) forming before us so that the Teaching Experiment could be carried out. In other words, the network could expand indefinitely. Still, to avoid getting lost in this “entanglement,” we restricted ourselves (applying the metaphor of the theatre) to what was happening on the stage of Laboratory Number 4.
The task, structured differently from a typical exercise and incorporating a text for discussion, sparked the introduction of various sub-themes during the dialogues. These included topics like social justice and related ideas, such as building a vegetable garden at the school and addressing space conflicts with cars in the parking lot. For this reason, we can describe Lab 4 and the mathematics task as mediating actors. This characterization highlights their agency and how their influence is distributed across the network of actors.
Sayes (2014) reminds us that the concepts of agency, actions, and network are almost equivalent; in other words, they are presented by Bruno Latour as spiralling and necessarily intertwined. The author suggests that resistance to the controversial notion of non-human agency might diminish if people recognised Latour's body of work does not aim to construct a substantive theory of agency. Instead, it provides methodological pathways for acknowledging the importance of including objects in sociological analysis. To reiterate, we do not intend to attribute intentionality to non-humans, but they can alter the course of action and influence the trajectory of a mathematics education practice. Thus, from this perspective, ignoring them would be a significant oversight.
You, dear reader, may wonder: what does recognising the agency of the mathematical task for the students and the teacher in this Teaching Experiment add to the analysis? What contributions does describing how it raises such ethical questions make to Mathematics Education? We agree that these questions are not unreasonable because anthropocentric analyses are still prevalent in our field of research. Still, to address them, we need to change the focus. We must say that human intentional agency is a type of action that does not exclude others. According to Sayes (2014), intentionality, subjectivity, and free will are often dissociated from agency.
As we have already said, it is important to ask: who or what is at work? Who or what is producing transformations in the socio-technical network? Making other actors act differently? Thus, it is clear that we are not trying to compare human and non-human agencies but to generate uncertainty about the course of action (vagueness of where it starts from) and about the very definition of action. For White (2019), recognising this type of agency can enrich theoretical frameworks for analysing classroom tasks, considering the complexities involved in their development and the effective presence of a given object in a mathematics education practice.
In short, the task and Laboratory Number 4 were two actors who, in this first Act, proved to be prodigious in changing the course of action of the students and the teacher. They experienced an unusual mathematics lesson on social justice, ethics, and political debates. Still, mathematics was present, and when the discussions seemed to end, they were fed back by important points from the task text.
Act II
After the first break, the curtains open for the second act. The teacher reviews the discussions held in the previous episode and details the work proposal for that episode: to draw up a budget for the construction of a community garden, the students would survey the availability of sites close to their homes or the school where it could be built; they would demarcate the area by distributing the land according to the specifications of the task (distribution of the vegetable beds, establishing circulation spaces between them, etc.).
The task detailed how the students should interact with the computer and the website, from how to use the mouse to how to use certain icons to establish the perimeter and area of the chosen location. The suggested website link had only begun to be spelled out by the teacher, and Student 6 already indicated that he had it open and was ready to work.
We witnessed a change of role: if, at the first meeting, Student 6 was shy and not very participative, now, in front of the computer, he takes on the role of a leader, even if only for a short time (Figure 4).
Student 6 in association with the computer and leading the group. Source: Research data.
If Student 2 had concentrated most of the associations previously, Student 6 now becomes a node with greater density (a greater number of associations within a network). Relationships have changed. Everyone in group 01 sits around the expert who emerges from the interaction with the machine, so two associated actants star in the scene: a human (the student) and a non-human (the computer).
The students begin to analyse satellite images to choose a suitable location (Figure 5). Their first attempt is frustrated, as they are informed that the chosen area corresponds to an environmental reserve: Student 2: […] parks can't…, dams can't… Student 5: Then there's nowhere! Student 6: […] go up (moves the mouse), here is a farm, here is a football pitch… here (pointed) you can't, because it's a slope…
Site chosen by the group to build the hypothetical vegetable garden. Source: Research data.
A discussion unfolded until they decided on a suitable location: an abandoned plot of land next to a shopping mall in the city, close to Student 6's house. Once the discussion was settled and the location chosen (there was no information available on the Internet for the team to check whether the land belonged to the public authorities or to the shopping centre itself), they began to delimit the space intended for the construction of the community garden, following the commands of the task. A new discussion began about the geometric shape of the community garden. Student 6 came up with a very complex layout, and his momentary hesitation in successfully demarcating the area for the garden caused Student 2 to reposition himself and claim his position as leader of the group: Student 2: […] We're going to have to delimit this space and draw (….) I think it's better if we measure in yards (in the meantime, attention was drawn to the fact that the unit of measurement, metres, would make their work easier). Let's draw a rectangle inside this space: the vegetable garden. This space here (the previous drawing) is not a rectangle. Student 5: We'll take that part in the middle (pointing to the screen again and demarcating the area that the rectangle should delimit)(Figure 6). They again discuss the unit of measurement for length and area.
Delimitation of the space earmarked for building the community garden. Source: Research data.
Student 5: (speaking to Student 6) Just follow the line […], is it already saved? Then just redo the measurement. Give me a sheet of paper and a pencil to write down these measurements (Student 5 hands him these objects).
Teacher: […] We first want to think about the garden's shape and the type of crops we're going to plant there. Are you going to plant coriander? Are you going to plant spring onions? What are you going to plant?
(The group's concern is correctly establishing perimeter and area measurements.)
Student 2: The terrain is uneven (then uses a mobile phone to use the calculator function) […] 96 m here, 250.9 here. (The students record these calculations, as shown in Figure 7).
At this point, new actors, such as paper, pencil, mobile phone calculator, and Excel spreadsheet, enter the stage in association with Student 2. Initially, the last actor preferred to abandon the computer to carry out some calculations with a calculator and then check them by hand using a pencil and paper. Teacher: […] You have to think about how long and wide you are to define each vegetable bed's area. Student 2: It's already written down. Teacher: How many beds will you have? Student 2: Now we just have to divide 693.9 m. (The students were confusing the notions of perimeter and area.) Student 5: We have to see how many beds there will be. Student 2: We have to see how many things we're going to plant. Student 5: Ten beds. Student 2: (using the mobile phone calculator) 693.6 divided by 10. Student 5: That's 69, 36. Student 2: In this case, it's 6.936m Student 5: Divided by 10? Student 2: Yes, it was divided by 10. Student 6: It would be too big
Paper and pencil actors entering the stage. Source: Research data.
All the members of the group, even Student 5, accept the calculation proposed by Student 2 without question. Student 2: No, it'll be less than that, because we'll still have to calculate the circulation metres. It will end up being approximately 4 metres for each plantation. So that's 4.93 because we're left with two metres for circulation. Two metres is Gabriel's height. Student 5: Is it the length? Student 2: Teacher, it's difficult to calculate the length. Teacher: The length is fine. What about the width? Student 2: The width. Teacher: If you're 2 m long, you can put 3 m wide and get 6 m².
The teacher tries to correct the group's calculations with some explanations, and Student 2 redoes some calculations using the calculator, but is not convinced by the result presented by the device. Student 2: That's crazy! It's a hundred and a bit. That doesn't make sense! That calculator is a bit crazy. Student 5: Do you want to take mine? Student 2: The comma is coming here (points to the mobile phone screen, says something we can't make out). Then I'd understand. Let's go again. If you give the same answer, I'll get angry! You gave another answer (showing the mobile phone to Student 6)! It's 106! Crazy! [Grabs paper and pencil] I'll do the maths myself! Who remembers division involving numbers with commas? (Student 5 helps him, and he finds the same result as the first calculator.)
Soon after Student 2's initiative to record the perimeter measurements on paper, the teacher asked them to represent the terrain on paper to make it easier to measure the vegetable beds and other spaces to be built in the garden. Student 7 (Figure 8), who had kept quiet most of the time until then, became a key figure in the team's ability to fulfil the challenge proposed by the teacher. The other students in the group said they found it difficult to make such a record:
Student 7 drawing the floor plan of the community garden. Source: Research data.
This moment allowed new actors to take centre stage in this mathematics education practice: the paper, the pencil, and the ruler, all interacting with Student 7. As the spotlight shifts to them, their association results in the drawing shown in Figure 9.
Student 7 drawing the floor plan of the community garden. Source: Research data.
In addition to the beds for planting the vegetables, this drawing also shows a warehouse and a toilet because, in their discussions, they raised the issue of the quality of the working environment that would be provided for the workers who would join the project. Student 7, to draw the picture on paper with a pencil and ruler, had to calculate the areas of the vegetable beds and the other spaces stipulated by the group. This way, we can see that the non-humans interacting with the students were responsible for correcting the team's course. However, you will not see these measurements because the concern was correctly recording the perimeter.
With the garden floor plan ready, they had to draw up three budgets containing all the materials they thought they would need to set up the community garden. The task already proposed a model for tabulating the data organised into the following categories: tools and objects; soil, sand, and fertiliser; seeds and seedlings; and other materials.
The teacher instructed the students to check the prices in online shops, and the students chose to record them in an Excel spreadsheet and check them on three well-known sales websites. Student 5: Why are we putting 20kg
4
Because it's 17.00 reais! So, I don't know. Let's put 1.00 real each. Student 2: If you have 20 kg, put the amount in a bag. Student 5: But we need to record the unit value. Student 2: So that's right. In the end, we add up what we spent on everything and add the total amount. [Addressing Student 6 again] If you don't know how to use (…) [quoting the name of the spreadsheet editor], go to (…) [quoting the name of the text editor], which is much simpler.
As they return to working with the computer, a new actor enters the scene: the word processor. The students abandon the idea of tabulating the data in the spreadsheet editor, feeling more comfortable using the former software. However, the spreadsheet editor ends up mediating a new association. Student 5 questions whether the tabulation method proposed by the task is the easiest or most suitable, but they ultimately agree to follow the task's instructions, and the price consultation progresses. After some time and considerable effort searching the internet, they conclude their search, producing the resulting spreadsheet shown in Figure 10.
Budgets drawn up by the students. Source: Research data.
The episode ends with the students presenting their work and sharing their setbacks: “I would never have imagined that we could study maths with the theme of community gardens” was repeated by some of them. Making comparisons between the budgets and questioning the criteria for choice (cost/benefit), they end up concluding that the third budget, although not the cheapest, is the one that manages to combine good brands at a low cost.
So, once the task has been completed and presented, Act II ends and the curtains close, the end of the show… of a mathematics education practice in Laboratory Number 4.
In Act II, we observed some changes in the roles of both human and non-human actors, who began to take on the position of mediators, prompting transformations in the network. As Latour (2007) points out, these roles are not fixed. First, we see the case of Student 6 in association with the interactive maps website, where the materiality and its features (such as the icons for calculating area and perimeter) facilitated this shift. The ease with which the student interacted with the tool led to an unexpected outcome: taking on the group's leadership.
The stability of the network was thus put to the test. As Law (1987) argues, it is important to recognize that it is difficult to keep actors in place because they test the limits of the vehicles that connect them. Engagement and consensus are required to prevent the network from disintegrating. Previously, much of the associations centred around the task and Student 2. However, a new group leader emerged from the interaction between Student 6 and the computer, albeit briefly acting as a new mediator. According to Law (1987), networks maintain cohesion (the balance of links) to avoid dissociation. As we observed, when Student 6 hesitated in their association with the maps site, unable to draw a geometric shape acceptable to the group, Student 2 reclaimed their leadership position. Interestingly, it was the same non-human actor that facilitated this shift.
However, Law (1987) reminds us that maintaining the network is constantly tested because forces operate to disarticulate it. The disagreements among the students and the underlying competition for leadership are examples of such tensions. To sustain the network, new actors can be integrated. This was the case for the paper, pencil, and cell phone calculator brought to the centre stage to reaffirm Student 2's leadership role. They performed a perimeter calculation, mistaking it for the area, and when dividing the garden plots, realized the calculation from the first calculator could not be correct, as the measurement was too large. To resolve this, they turned to another calculator and obtained a new result, but were still surprised and decided to calculate it manually. A dispute arose within the group, but was quickly subdued by Student 2's leadership profile. We can say that these two devices mediated the discussions and the student's shift in strategy.
White (2019) describes a similar situation involving graphing calculators (capable of sketching function graphs from their equations) during a task comparing the graphs of two quadratic functions. Rather than performing their predictable role of simply producing graphical representations, the calculators acted unexpectedly: “Certain aspects of the graphs they produced added confusion, rather than clarity, to many students’ efforts to interpret the relationships between the two functions” (p. 15). This confusion, arising from the difficulty in accurately determining whether the two representations were identical, created an opportunity for enriched classroom discussions.
Similarly to what Delacour (2022) reports, the teacher also seeks to maintain control over how the task unfolds, including how students interact with non-human actors. She encourages them to create a ground plan to facilitate the calculation of the areas of the designated spaces. In doing so, she brings new actors into the scene: paper, pencil, and ruler. Their introduction mediates Student 7's shift from the role of an intermediary (supporting actor) to that of a mediator (protagonist), as she was the only one who claimed to be able to create the representation. This allowed her to interact with these objects and brought the ruler onto the stage.
One might argue that the student uses these objects as tools to create a drawing. However, note that only by introducing these new actors could we see this student act and demonstrate her skills. Before, she remained silent, agreeing with everything proposed, i.e., a typical intermediary. As Latour (2000) explains, what alters the movement, in this case, the drawing and the objects, does not possess causal force. “Whether it concerns the dominant subject or the causal object, what is set into motion never fails to transform the action, thus giving rise neither to the utensil-object nor the reified subject” (p. 191). In Latour's terms, this implies that we must direct our attention to the links that make action possible to the plurality of what acts.
When examining what sustains these links (associations) (Ryu, 2019), focusing on the network nodes with greater density is essential to describe how actors influence one another. During the task, we observed several phases of network stabilization, though not in the same sequence proposed by the author. First, the allocated space provided an adequate environment for completing the task. This was followed by discussions prompted by the task itself and the active engagement of the students. The discussions and the competition for leadership tested the network's stability, but introducing new actors reinforced it. Ultimately, the network stabilized, and the task was concluded by presenting a budget for a community garden, accompanied by reflections on the social relevance of sustainability.
Do Things Also Perform Exploratory Tasks? Rehearsing a Possible Understanding
In this article, we presented a study to describe the agency of non-human objects in a mathematics education practice with school students. During the proposed mathematical task, we observed that non-human entities acted as agents towards human actors, contributing to the stabilization of the emerging network, and acted as active participants in this mathematics education practice.
We also noticed that this practice can be described as a heterogeneous network, where a collective of human and non-human actors operates to remain interconnected, stabilising the network constantly configured, deconfigured, and reconfigured. This implies that teaching and learning mathematics, whether in or outside school, occurs within dynamic networks where new actors are integrated and others are excluded. Perhaps this perspective can help us rethink mathematics education practices without seeking a single entity to blame for students’ learning difficulties.
Recognising the participation of non-human actors in the course of action does not imply determinism; we are not suggesting that they act in place of humans. As previously stated, we do not aim to transform objects into causes whose effects are human actions. Instead, drawing on Latourian ideas and applying them to the field of Mathematics Education, we propose that, in these practices, non-human actors are not merely tools used or appropriated by humans; they are actants towards humans.
Thus, in their way, albeit differently from humans, non-human actors are indeed involved in developing the task assigned by the teacher, Simone. Considering mathematics education practices as networks, we realize that non-human entities can function as high-density nodes mediating numerous actors’ actions. Since action is dispersed within the network, it becomes impossible to pinpoint where it begins, leading us to abandon the notion of a detached, self-contained human or non-human actor (outside the network). Inspired by Latour (2000), by giving up this isolated agent that does not exist, we also let go of the notion of society as solely centred on human beings in our analyses. Instead, we adopt the concept of a heterogeneous collective, which comprises complex, networked associations of human and non-human actors.
Footnotes
Informed Consent
Author 1 obtained informed consent from all participants involved in the study.
Contributorship
The first author conducted the research for this study and was responsible for participant recruitment, data production and processing, and drafting the initial version of the manuscript. The second author contributed through critical readings, ideas, and suggestions, and worked on the final writing of the article. Both authors read and approved the final version of the manuscript.
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The first author received a Social Demand scholarship from Coordination for the Improvement of Higher Education Personnel, Brazil.
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.