1 Introduction

The use of graphs in different types of media has become a fundamental tool to facilitate communication to the general public regarding the COVID-19 pandemic (Kwon et al., 2021). During this pandemic, the media have become important allies of health agencies in disseminating to the public information about preventive and control measures (da Silva et al., 2021; Parada & Zambrano, 2020). Despite the fact that, for the most part, research on graph studies has been carried out in the field of journalism (e.g., Dick, 2015), it is necessary, in agreement with Kwon et al. (2021), that research in mathematics education also pay attention to the way in which graphs are used in the media. In that way, it would be possible to provide theoretical guidelines for the teaching of mathematics that would enable citizens to be capable of understanding the information delivered in this type of graphs.

One of the central mathematical insights for understanding the information associated with the COVID-19 virus, which is provided through graphs, is that of exponential behaviour. Indeed, there is research that shows that this type of behaviour makes it possible to explain, through the use of graphs, different types of phenomena linked to this virus. To mention some of them, there is the work carried out by Tejera-Vaquerizo et al. (2020), in which exponential behaviour allows the estimation of the hypothetical impact on survival that the increase in the size of squamous cell carcinomas and melanomas would have as a result of an indefinite confinement given by COVID-19. On the one hand, through exponential behaviour it is possible to make predictions about the number of infections and deaths caused by this virus. For example, Medina et al. (2020) made some predictions about the number of confirmed cases and deaths associated with COVID-19 in Cuba, based on logistic and exponential models. On the other hand, Rojas et al. (2021) used logistic and exponential models to predict the initial behaviour of COVID-19 in Costa Rica.

Despite the importance of studying the mathematical knowledge associated with exponential behaviour to understand phenomena related to COVID-19 through the use of different types of graphs, several studies show that this type of knowledge is not usually present in the teaching and learning processes associated with the exponential function. Just to mention some of the research, we have the work done by Heyd-Metzuyanim et al. (2021), Casanova and Concha (2020), Banerjee et al. (2021), Aguilar and Castañeda (2021) and Kwon et al. (2021) who posit the existence of a discrepancy between the skills needed to interpret pandemic graphs presented in the media and those developed in school in their countries (Israel, South Korea, and Chile, respectively). In the particular case of Aguilar and Castañeda (2021), they identified and characterised the mathematical knowledge and skills needed to interpret COVID-19 reports, based on eight constituent components of mathematical competence called mathematical competencies, in the sense of Niss and Højgaard (2019).

Particularly, Heyd-Metzuyanim et al. (2021) reported the low ability of students from Israel to interpret the mathematical aspects of news about COVID-19. Banerjee et al. (2021) reported a generalised tendency of the population to linearise or underestimate the dizzying growth of exponential functions when evaluating them intuitively, as in the case of the interpretation of graphs with logarithmic scales.

Kwon et al. (2021) and Aguilar and Castañeda (2021) pointed out that the graphs used in real-world news stories are often beyond the scope of mathematical skills developed in schools. In that regard, Kwon et al. (2021) referred to the existence in the media of a variety of composite graphs that group heterogeneous data, as well as the graphing of differences through the use of different colours. This practice is in contrast to the idealised graphs presented in school mathematics, where artificial and refined situations are shown. Along the same lines, Casanova and Concha (2020) pointed out the great difference between the graphs presented in the press and in an official school textbook in Chile. These authors indicated that the graphs shown in the media usually start from a reference point other than \((\mathrm{0,0})\), presented in both linear and logarithmic scales; there is overlapping of different curves in the same graph and in many circumstances, the pandemic graphs do not express an exponential function. In contrast, all the graphs shown in the school textbooks were centred at the point \((\mathrm{0,0})\) used only linear scales, and no graphs were shown that come from a real context, but rather the phenomenon was always idealised to an exponential function made explicit and treated algebraically. In summary, and in the particular case of Chile, the teaching and learning processes regarding the exponential function concept traditionally focus on the study of different properties of the exponential function, seen as a mathematical object.

In this regard, the Socioepistemological Theory of Mathematics Education (STME) is a theory that aims to promote the learning of mathematics by positioning human activity in the construction of mathematical knowledge (Cantoral, 2013, 2019). To do this, the theory proposes the need to carry out in teaching a “decentring of the mathematical object” (materialized in definitions, techniques, algorithms, or rules) to attend, as a priority, to the analysis of the practices that precede and accompany its production (Cantoral & Farfán, 1998). As an example, Reyes-Gasperini (2016) explained how the practices of comparing, equivalence and commensuration allow the construction of mathematical knowledge associated with proportionality. Cantoral (2013) pointed out that the use of variational practices such as comparison, seriation and prediction allow students to construct mathematical knowledge for predictive purposes. However, these practices are not usually used in the teaching and learning processes involving the exponential function to ensure that students build mathematical knowledge to predict different types of phenomena, such as those related to the COVID-19 pandemic. According to OECD (2016), these processes are desirable to promote in the teaching of mathematics.

Based on the above, and from a socioepistemological perspective, it is evident that there is a problem present in the teaching and learning processes involving the exponential function in Chile. This problem consists of being focused on mathematical objects, leaving aside variational practices that contribute to the making, by students, of predictions associated with phenomena of exponential behaviour linked to the COVID-19 pandemic. In order to provide elements to model a social construction of mathematical knowledge that guides the design of school situations, that allow students to predict phenomena associated with COVID-19 through the use of exponential behaviour graphs and the variational practices of comparison, seriation and prediction, the following research questions were posed: (1) What types of exponential behaviour graphs are used in the Chilean media to inform the population about the COVID-19 pandemic? (2) What types of variational practices are present in the use of this type of graphs by the media?

In the first part of this paper we address the first question, where the thematic analysis (Braun & Clarke, 2006; Braun et al., 2019) allowed the description of exponential behaviour graphs used by the media to inform the population about the COVID-19 pandemic. It is important to note that this first part was not framed in any particular theory, and it was necessary to carry out the second part of this research, which addressed the second question, which was framed within the socioepistemological research line called Variational Thinking and Language (Spanish acronym: Pylvar), in which the ways of thinking, arguing, organizing and mathematically communicating change processes are studied, in order to analyse variational practices (Arrieta & Díaz, 2015; Cantoral, 2019; Farfán, 2012).

The structure of this paper is as follows: in Sect. 2 we present the theoretical framework that supports the research (specifically, the second part of this research). In Sect. 3 we present the methodology that made it possible to respond to the research objectives. In Sect. 4 we present the research results, showing a description of the type of exponential behaviour graphs that a certain newscast used to inform the population about the COVID-19 pandemic in Chile and the use of variational practices that emerge from this type of graphs. In Sect. 5 we present a discussion of variational practices present in this type of graphs that can be used by the educational community for the construction of a reference framework that contributes to a decentring of the exponential function as a mathematical object. Finally, the conclusions of this research are presented in Sect. 6.

2 Theoretical framework

The STME is a theoretical framework cultivated by researchers from different Latin American countries since the 1990s (Cantoral, 2019). It aims at social and educational transformation, which is sought through a change in the way mathematics is conceived and taught. As a starting point, this theory considers that mathematics is an element and part of a culture that ‘lives’ outside the classroom but is then ‘recreated’ inside it. In this way, mathematics.

is used in different settings, let us say that it ‘lives’ throughout the most basic actions of all human activities: construction of houses, planting and weaving, development of protocols for the use of drugs or toxics, making recipes, designing wine tanks, calculating medical dosages, making mathematical conjectures explicit, coordinating movements of a pilot landing on a difficult track, mathematizing biological phenomena, making decisions for financial investments, interpreting public opinion, simulating continuous flows, bartering in traditional markets, studying the consolidation of saturated fine soil, controlling temperature, narrating, comparing, transforming, estimating, adjusting, distributing, representing, constructing, interpreting, and justifying, among many others (Cantoral et al., 2018, p.79).

For STME, practices precede and accompany the constitution of mathematical knowledge. Details of the socioepistemological study of the practices that precede and accompany the production of mathematical knowledge were given by Cantoral et al. (2018) and Cantoral (2019). One of the research lines developed in STME is Pylvar, whose purpose is to study the variational practices that give life to the mathematics of variation and change. Examples of these variational practices are comparison and seriation. These practices can be characterized, following Cantoral (2013), as follows:

  • Comparison: It refers to the study of differences between states, identifying which is greater and which is lesser. It is a comparison of two moments with the purpose of identifying and quantifying the change. It is linked to the question, how much does it change?

  • Seriation: It refers to the simultaneous comparison of more than two moments (three or more cases) and their ordering, to establish qualitative descriptions of the type of change. It is then a matter of organizing and establishing a logical order to a finite set of comparisons that makes it possible to determine the stable nature of the change (Caballero, 2018). It is linked to the question. how do the variables change?

Figure 1 exemplifies the way in which the analysis of the variational practices of comparison and seriation is carried out.

Fig. 1
figure 1

Techniques for the analysis of the variational practices of comparison and seriation (Caballero, 2018)

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Caballero (2018) reported that comparison and seriation can be inferred through an analysis of heights and tangents. Figure 2 shows how both analyses allow researchers to account for change and variation in the plotted phenomenon.

Fig. 2
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Techniques for the variational analysis of graphs (Caballero, 2018)

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In addition to comparison and seriation, a third fundamental practice for the variational analysis of graphs is prediction. According to Cantoral (2013), prediction is a fundamental practice for the development of human thinking linked to the study of variation and change. Pandemic is an example of this, as the mathematical modelling processes of phenomena associated with COVID-19 are traditionally oriented towards predicting the behaviour of an epidemic curve to avoid hospital saturation (see Fig. 3).

Fig. 3
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A “flattening the curve” graph (Heyd-Metzuyanim et al., 2021)

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At a local level, prediction operates by anticipating the behaviour of a variable in the near future using state information and its variations. This can be done by qualitative analysis of the behaviour of change, through comparison and seriation of heights or tangents (see Fig. 2). At a global level, prediction requires the recognition of a mathematical model that describes the behaviour of the phenomenon, i.e., a model that accounts for the variables that change and how they change. In short, in a process of decentring the mathematical object, variational practices allow researchers to study the process of social construction of mathematical knowledge through the analysis of its use situated in specific reference practices.

3 Methodology

In order to respond to the research questions, the methodology used was qualitative. A descriptive methodology was used to answer the first question and an inferential methodology was used to answer the second question. Both are detailed in Sect. 3.3.

3.1 Chilean context

As reported by the Chilean Ministry of Health (MINSAL, 2022), the first case of COVID-19 in Chile was reported on March 3rd, 2020. It was a person who had returned from his vacation in Southeast Asia, who tested positive for COVID-19 after analysis of his sample performed at the laboratory of the Hospital Guillermo Grant Benavente in Concepción, Chile. On March 16th, 2020, the President of Chile reported that the country had entered Phase 4, meaning that there was viral circulation and community spread of the disease. The number of infected had increased from 75 to 155 cases in just 24 h and continued to grow with an exponential trend in the following weeks and months (MINSAL, 2022).

By April 2nd, 2020, there were more than 3,400 infected persons and 18 deaths in Chile (MINSAL, 2022). At the end of April 2020, the infection curve began to increase and marked the beginning of the so-called “first wave”, a period of time in which the Chilean press focused entirely on informing the population about the evolution of the pandemic through the use of various types of graphs. As of April 30th, 2020, the most affected regions of Chile by the pandemic were the capital and the main cities of this country. According to reports from the Ministry of Health, at that time, there were 16,023 confirmed cases in the country—although 53.5% had recovered (8580)—and 227 deaths were recorded, of which almost 70% were people over 70 years old, and the figure rose to over 85% for those over 60. On June 14th, the Ministry of Health announced the highest number of infections reported for a single day: 6938 cases. In terms of deaths, this was the day on which, according to epidemiological reports, more people died: 195 deaths. Finally, the so-called “first wave” in Chile lasted until the end of July 2020 (MINSAL, 2022). Due to the exponential behaviour of the COVID-19 infection curve that existed from the beginning of the pandemic in Chile until the end of the first wave, and the large number of graphs that Chilean newscasts used to keep the population informed about this phenomenon, it was of interest for this research to focus the study within this particular period of time and thus be able to respond better to the research objectives.

3.2 Data collection

In order to respond to the research objectives, the data were collected from what was observed in the Chilean television newscast called Meganoticias Prime, broadcast by the open channel Mega. The decision to choose a television newscast was mainly due to two reasons: (1) this type of newscast informs almost the entire public through television, (2) television newscasts, in an attempt better to explain facts related to the COVID-19 pandemic, make use of a great diversity of graphs to inform the population about the number of COVID-19 infections in Chile. The decision to collect data from the television newscast Meganoticias Prime, was due to the fact that this news program, during 2019, completed 60 consecutive months leading in ratings, according to information obtained from its web page (Meganoticias Prime, 2019).

The period of data collection was from March 3rd, 2020 (beginning of the pandemic in Chile) to June 12th, 2020. Data were collected from two sources. The first of them entailed graphs used by the newscasts to inform the population about the number of COVID-19 infections in Chile. The newscasts used a total of 71 graphs for this purpose, and the screen time dedicated to showing these graphs, was approximately 177 min. The second data collection was based on interviews conducted by the journalist with different specialists, in which they talked about issues related to the COVID-19 pandemic.

3.3 Data analysis

Due to the different natures of the research questions, different data analyses were performed, which are described below.

To identify the types of exponential behaviour graphs used by the television newscast Meganoticias Prime (research question 1), a thematic analysis was used (Braun & Clarke, 2006; Braun et al., 2019), as it was an exploratory, descriptive and interpretative approach that allowed the identification of themes that permitted us describe the types of exponential behaviour graphs that the newscast used to inform the population about the COVID-19 pandemic in Chile. In order better to analyse the information underlying the data obtained from the newscast studied, the thematic analysis was combined with documentary analysis techniques (Rojas, 2011). The combination of these analyses was carried out separately by two groups, each of them formed by specialists in mathematics education, who were part of the technical staff of this research. Once each group had performed their respective analyses, they met to discuss together the results of each group and, in this way, to recognize similarities and discrepancies in the analysis process carried out by each of them. This process had the aim of giving a better description of the types of exponential behaviour graphs that the newscast used to inform the population about the number of COVID-19 infections in Chile.

In accordance with the provisions of Braun and Clarke (2006), the thematic analysis consisted of six stages: (1) familiarization with the data, (2) generation of initial codes, (3) search for themes, (4) review of themes, (5) definition and naming of themes, and (6) report generation. The first stage consisted of reading all the information obtained from the data collected, allowing a first understanding of the type of exponential behaviour graphs used in the newscast. In the second stage, a descriptive coding was carried out in order to recognize certain codes of a more general nature, which made it possible to generate potential themes. Regarding the codes, it is important to note that they identify a characteristic of the data (semantic or latent content) that appears interesting to researchers, and refer to the most basic segment, or element, of the raw data or information that can be meaningfully evaluated in relation to the phenomenon (Braun & Clarke, 2006). In this sense, codes differ from themes, as the latter are usually broader (Braun & Clarke, 2006). Following the example given by Braun and Clarke (2006), Table 1 shows some examples of how codes were assigned to short segments of data obtained from phrases pointed out by news reporters while showing graphs of exponential behaviour associated with the COVID-19 pandemic. Table 2 shows the codes obtained after the analysis of the exponential behaviour graphs used by the Meganoticias Prime newscast. In the third stage, a refinement of the coding process was carried out. Stage 4 brought together the analyses of all code associations made by the experts, who worked in groups and in parallel, and subsequently, discrepant codes were agreed upon. In this sense, stage 4 included a review of stages 1, 2 and 3, which made it possible to generate a first version of the themes. Stage 5 focused mainly on generating the definitions of the themes to faithfully represent the data. Finally, stage 6 consisted of generating the themes that allowed us to describe the types of exponential behaviour graphs that the newscast used to inform the population about the COVID-19 pandemic in Chile.

The inference of the variational practices present in the use of exponential behaviour graphs used in Meganoticias Prime newscasts (research question 2) was carried out in three moments:

Table 1 Example of assigning codes to short segments of data from the Meganoticias Prime newscast
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Table 2 Codes resulting from the thematic analysis of the data obtained from the newscast Meganoticias Prime
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Table 3 Inference of variational practices
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  • First moment: The graphs identified after answering the first research question were analysed, based on an analysis of the variational reference system (Caballero, 2018). This analysis made it possible to identify the following aspects in these graphs: (a) the variables involved; (b) the reference points where the graphs began and ended; (c) the unit of measurement used in the graphs, and (d) the timing proposed in the graphs, i.e., the marking of two or more states that allowed comparisons and seriations to be made.

  • Second moment: The oral explanations made by the journalists when using the graphs during the broadcasting of the news programme were analysed. For this, an analysis of the use of graphs was used (Torres-Corrales & Montiel-Espinosa, 2021). In this analysis, the following were identified in the journalists' explanations: (a) what is done or said? and (b) how is it done or said? It should be noted that both questions refer to observable and explicit aspects of human activity.

  • Third moment: For Cantoral (2019) practices are not observable, but inferred. At this point, the variational practices present in the use of the graphs used by the journalists were inferred. The inference process was carried out through the observable (moments 1 and 2), mediated by the definitions of the variational practices made in the theoretical framework and using Table 3, which operated as an analysis protocol and allowed, from the observable, the inference of the use of variational practices.

The results of this research are presented in the next section. They are shown separately for each research question.

4 Results

The results made it possible to identify the types of exponential behaviour graphs used by the news programme studied (see Sect. 4.1) and to infer the types of variational practices present in the use of these graphs (see Sect. 4.2).

4.1 Description of exponential behaviour graphs

During the period studied, the Chilean newscast used 71 graphs to inform the population about the COVID-19 pandemic in Chile. The codes shown in Table 2 contributed to the identification of three topics that allowed us to describe the type of exponential behaviour graphs that the television newscast used to inform the population about what happened with the COVID-19 pandemic in Chile. It is important to note that this description of graphs was made based on what was obtained from the thematic analysis and was not related to the theoretical framework used in this research. The three identified topics, which are also summarized in Fig. 4, are detailed below:

  • Cumulative frequency graphs. Graphs showing the evolution of cumulative values of the pandemic, both infected and deceased in the country (12 graphs).

  • Comparison graphs. Graphs comparing the number of infections or deaths between different countries, considering cumulative cases over time (17 graphs).

  • Simultaneity graphs. Graphs that simultaneously include the cumulative and daily cases of infection in the country (15 graphs).

Fig. 4
figure 4

Representatives of exponential behaviour graphical types shown by the Chilean newscast

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In order to infer variational practices present in the use of exponential behaviour graphs by the television newscast Meganoticias Prime, an analysis of the use of variational practices in the graphs described above follows.

4.2 Use of variational practices in exponential behaviour graphs

The inference of the use of variational practices in the exponential behaviour graphs that the Chilean newscast used to inform the population about the COVID-19 pandemic in Chile is divided into two parts. In the first part, the uses of the variational practices of comparison and seriation were inferred, while in the second part the uses of the variational practices of prediction were inferred.

4.2.1 Variational practices of comparison and seriation

In this first section, an inference about the presence of the variational practices of comparison and seriation in the use of the exponential behaviour graphs described in Sect. 4.1 is presented.

The graphs in Figs. 5 and 6 show an evolution of the cumulative cases of deaths from March 15th to May 31st, 2020, where the data for March 31st, April 30th and May 31st are indicated. In the explanation, the journalist pointed out the following:

(May 31st, 2020) Journalist: Day 90 of the pandemic and the virus gives no respite, this Sunday a discouraging record was set once again. Undoubtedly, the month of May has been the crudest and most lethal of all this time, if between March and April there were a total of 227 deaths, that figure almost quadrupled and only in the month of May there were 827 deaths, reaching 1054 people who have lost their lives by or with COVID-19 (Fig. 5).

Fig. 5
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Evolution of cumulative cases of deaths as of May 31st, 2020

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Fig. 6
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Graph of cumulative cases as of May 31st, 2020

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Regarding the graph of deceased persons, the journalist pointed out the following:

(May 31st, 2020) Journalist: This is the graph of deaths from March 15 when the pandemic began until May 31, marked by the end of the month. The rise in the last four weeks is evident. May has been the worst month in the middle of COVID-19, registering 78.46% of deaths.

Regarding the variational reference system, we identified the following:

  • Relationship between variables: the cumulative number of deaths relative to the elapsed time of the pandemic.

  • Reference values: The graph begins on day 0 of the pandemic in Chile (15/03/2020) where there are no deaths.

  • Unit of measure: both the number of deaths and the time (represented in weeks but plotted in days) grow similarly to an arithmetic progression.

  • Temporization: in May there are 827 accumulated deaths compared to 215 accumulated in April or 227 accumulated in March or April.

In turn, regarding the use of this graph made in the newscast, we identified the following:

  • What is done or said? In the explanation, growth intervals of the variable are contrasted, considering March–April (227 deaths) and May (827 deaths) as states (how much it changes).

  • How is it done or said? The amount accumulated in May (1054) is taken and the amount accumulated in March–April (227 deceased) is subtracted, obtaining the 827 deceased. Then the percentage of 827 to 1054 is calculated, obtaining a result of 78.46%.

Based on this analysis, and using Table 3, the use of the following variational practices can be inferred:

  • Comparison: States are compared to describe the growth of the variable in time intervals, considering March–April and May. To obtain the increase in May, a comparison is made by taking the May value and subtracting the March–April value (how much it changes).

  • Seriation: A seriation is performed between two or more states, by comparing the March–April and May increments, concluding that in May there is a quadrupling of the cases. In this way, the type of growth inferred from the graph (as it changes) is accounted for.

The presence of the practice of seriation by comparing increments can also be inferred from the use of the graph in Fig. 7. In this graph, the reporter noted that as of April 25th new cases did not exceed 600 [there were 552], but on May 25th new cases per day approach 5000 [there were 4895]. It should be noted that, in the other identified graphs, the explanation was not detailed or was only shown on screen with context information of the pandemic in the country.

Fig. 7
figure 7

Graph of new cases as of May 25th, 2020

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In terms of putting comparison and seriation practices to use in the analysis of exponential growth, we found the following scene in which a journalist interviews a doctor (see Fig. 8).

Fig. 8
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Graph of new cases as of April 9th, 2020

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The conversation was as follows:

(April 9th, 2020) Doctor: The values of the yellow bars (referring to the Fig. 8) are less than expected following the exponential projection.

(April 9th, 2020) Journalist: Are we going to enter a trend of over 400 cases per day or not necessarily?

(April 9th, 2020) Doctor: Probably. The fact that the value of the yellow bars (referring to Fig. 8) does not increase dramatically and fluctuates between 300 and 400 cases reflects that this is not the result of the normal spread of the disease, but is the result of human intervention due to the pandemic containment measures. The red curve is on an arithmetic scale and has not had a break with an exponential type of trend change.

The doctor assumes that, due to the dynamics of the contagion of a respiratory disease, its graph should follow exponential behaviour. However, after sustaining that the daily increments oscillate between 300 and 400 cases, it implies that the red curve (referring to Fig. 8) does not express exponential growth. After the analysis of the variational reference system of the graph and the analysis of the use of the graph made by the doctor and the journalist, it is concluded that the doctor used the comparison and seriation of the yellow bars to conclude that the red graph shown on the screen is not an exponential curve.Footnote 1

Next, an inference is presented concerning the variational practice of prediction present in the use of the exponential behaviour graphs described in Sect. 4.1.

4.2.2 Variational practice of prediction

The following scene was an interview that the news reporter conducted with a physician specialized in epidemiology regarding pandemic graphs (see Fig. 9):

(March 26th, 2020) Journalist: In this curve we start with 156 and it goes up to 1306 [cases in Chile]. Is this a curve that was expected or not?

(March 26th, 2020) Doctor: The truth is that this trend is quite predictable and if you notice at the end something even flattened.

(March 26th, 2020) Journalist: You say here, between day 10 and 11.

(March 26th, 2020) Doctor: Yes, the mathematical predictions we were making were even higher than what we have today [...].

(March 26th, 2020) Journalist: Italy starts with 132 and on the 11 it had 2500 cases.

(March 26th, 2020) Doctor: Exactly, if you see what is important: The steepness of the curve. The steeper the curve means that the growth is faster, that the progression of the epidemic is faster. The Chilean curve looks flatter.

(March 26th, 2020) Journalist: Yes, of course, on days 7 and 8 there is a jump, and definitely from day 9 onwards it really advances [referring to the cases in Italy].

(March 26th, 2020) Doctor: Of course, this is very interesting, because the situation in Italy today is really a catastrophe. One could say that it is a matter of time to get there, but if you look at the time trend at day 11 at the same moment the curve for Chile is flatter.

Fig. 9
figure 9

Comparison of infection curves between Chile and Italy (March 26th, 2020)

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After analysing the variational reference system of the graph in Fig. 9, as well as the analysis of the use of the graph in the dialogue between the doctor and the journalist, the presence of predictive practice was inferred using Table 3. Indeed, the doctor interviewed noted the following:

  • He referred to the importance of seeing the slope of the curve to interpret elements related to the variation of the phenomenon. If such a slope looks flatter it means less growth for subsequent states, and if it is steeper it implies a faster growth of the spread of the pandemic.

  • He noted that between day 7 and 8 of the graph in Fig. 9 the slope of the curve in the case of Chile flattens out and in the case of Italy it steepens. This is considered interesting given that the situation in Italy at the date of the analysis is considered a catastrophe, but from the difference in slope we have that the growth trend in Chile is different from that of Italy, therefore we can anticipate that Chile will not reach the hospital saturation that occurred in Italy.

  • He argued that the growth of 156 cases up to 1306 in 11 days is a predictable trend, and that even the mathematical predictions anticipated a higher growth. To make these predictions, a mathematical model is used that assumes exponential growth of the contagion.

The doctor’s ideas refer to the expression “flattening the curve”, used on several occasions in the data analysed. While this idea is specific to mathematics or the experimental sciences, it acquired colloquial usage in non-academic circles of society. As evidenced in this report, understanding the flattening or non-flattening of the epidemic curve (see Fig. 10) requires the use of variational practices. Indeed, estimating the short-term behaviour of the pandemic requires the ability to interpret qualitatively, for example, different changes considering different increments.

Fig. 10
figure 10

Epidemic curve (Cantoral et al., 2020, p. 14)

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Thus, when comparing different increments of new cases (serialize), if these changes decrease, there will be a tendency towards flattening of the curve (preventing too many people from becoming infected at the same time). In mathematical terms, this consists of limiting the variation in an interval so that the values of the function are lower than those expected without intervention and the occurrence of the maximum value is delayed. In short, the aim of containment measures is to reduce the speed of growth and, consequently, to reduce the estimated short-term values of the pandemic phenomenon without containment measures to avoid hospital saturation.

5 Discussion

Several investigations have evidenced the difficulty that middle and high school students have in understanding pandemic graphs (Aguilar & Castañeda, 2021; Banerjee et al., 2021; Kwon et al., 2021). Casanova and Concha (2020) analysed the treatment of the exponential function in an official book of the Chilean educational system published in 2020, where an exponential growth modelling problem is shown in an epidemiology problem. In such a problem, an exponential algebraic expression \(f(t)\) indicating the number of people infected by a virus is posed. The treatment proposed in the activity is to perform specific calculations of the function for t equal to 1, 4 and 10 days, plot the function in GeoGebra, analyse the long-term trend, and indicate whether the graph of the function is increasing or decreasing (see Fig. 11).

Fig. 11
figure 11

Source: Osorio et al., (2020, p.39)

Example of the treatment of the exponential function in an official textbook of the Chilean Ministry of Education.

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Casanova and Concha (2020) stated that in this and other problems in which exponential growth is studied in real-life phenomena (such as earthquakes, population growth and bacterial culture), the proposed treatment of the problems is the same. That is to say: A function \(f(t)\) is given explicitly, the calculation is proposed for specific values for the function, the analysis of its graph and the variation of parameters and the determination of whether the function is increasing or decreasing. Indeed, exponential growth is defined as the increasing character of a function of the type \(f(x)=a{b}^{x}\).

The first question posed in this research was: What types of exponential behaviour graphs are used in the Chilean media to inform the population about the COVID-19 pandemic? The results showed that the types of exponential behaviour graphs shown, in the particular case of the television newscast studied in this research, are summarised in three themes, namely, Cumulative frequency graphs, comparison graphs and simultaneity graphs. However, these types of graphs do not make the function explicit in its algebraic form, nor do they express exactly an exponential function, as the official textbook of the Ministry of Education in Chile does. In addition to this, there are huge differences between the graphs studied in this textbook and those shown in the media, such as the information shown in the aforementioned television newscasts, a phenomenon that is also demonstrated in other types of research (e.g., Casanova & Concha, 2020; Heyd-Metzuyanim et al., 2021; Kwon et al., 2021). In short, a significant distance was identified between the way exponential curve graphs are conceived and used at school and in the media that formed part of this research.

Regarding the latter, Aguilar and Castañeda (2021) proposed that interpreting the pandemic graphs shown in the media requires the use of mathematical modelling skills at an advanced level. The authors pointed out that this not only implies the ability to interpret models in specific contexts, recognize and obtain data, and identify characteristics of the graphs; it also requires the ability to engage with the model, recognize its nature, its conditions and its characteristics in order to estimate and explain behaviours. That is to say, it is required to interpret data, relationship between variables, estimate graphic behaviours, recognize the model in a variety of contexts and situations, transfer information between graphs and formulate interpretations of them (Aguilar & Castañeda, 2021).

The second question posed in this research was: What types of variational practices are present in the use of this type of graphs by the media? The results of this research showed the presence of the variational practices of comparison, seriation and prediction in the use of exponential behaviour graphs given in the particular case of the Chilean newscast Meganoticias Prime. Given the central, leading, and underlying character of these variational practices in the understanding of the pandemic, it is necessary to address this issue at the level of mathematics education. Regarding the need to incorporate the development of these variational practices in teaching Cantoral (2013) pointed out that to do this requires a necessary decentring of the teaching of mathematical objects towards a teaching oriented to the development of mathematical competences linked to variation and change. Thus, to develop mathematical modelling competence at an advanced level, in the sense of Aguilar and Castañeda (2021), the incorporation of variational practices in the classroom is required, so as to accompany conceptual evolution (construction of mathematical knowledge) with a pragmatic evolution (development of practices that precede, accompany and are substantial to such knowledge) (Reyes-Gasperini, 2016). The necessity for this development can be seen in the research of Rotem and Ayalon (2021) who reported that only 3 out of 87 students put into use variational ideas, in the sense of this research, to provide explanations for official reports of the pandemic spread in media in Israel.

Developing variational practices in schools implies, at a certain level, a reformulation of how school mathematics is conceived, organized and taught, within the framework of promoting the development of epistemologies of practices in mathematics teaching (Cantoral, 2019). In socioepistemology, this is called redesigning school mathematical discourse (Cantoral, 2013). In this regard, there are several experiences that redesign the teaching of precalculus and university calculus oriented to the development of practices, particularly in Mexico. They were carried out in parallel to processes of teacher professionalization in the subject, whose effectiveness was studied in detail in various investigations (Arrieta & Díaz, 2015; Cabrera & Martínez, 2022; Cantoral, 2013, 2019; Cantoral & Farfán, 1998; Cantoral et al., 2018).

Another point we make is that, in school, rather than evoking realistic situations forced to be treated with the underlying logic of school mathematics, real problems of the real world should be incorporated (Cantoral, 2013; Cordero, 2016; da Silva et al., 2021). Now, doing this implies not only bringing the real-world problem into the classroom, but also the way in which mathematical knowledge is viewed and used in such everyday environments (Espinoza et al., 2020). That is, in the case of exponential growth, it is required to incorporate into the school problems of real-life phenomena whose approach and treatment is in correspondence with the use of knowledge in everyday contexts. In addition, in phenomena where change and variation are present, we know that variational thinking and language are some of those elements that characterize the ways of thinking and using mathematics in everyday contexts (Cantoral et al., 2018).

In summary, the results of the present research suggest the need to explore a different treatment of the exponential and its growth in school. Indeed, based on such results, we argue for the need to favour in the population the development of thinking that puts into use variational practices to achieve a better understanding of a phenomenon such as the pandemic and its spread, and thus, guide the right decision making. In this regard, it has been shown how comparison and seriation allow for qualitative characteristics of change and provide elements to make predictions of the behaviour of a specific phenomenon, in this case, the pandemic.

As stated by Cantoral et al. (2020), this variational reading allows us to understand that it is the action, both individual and collective, that allows us to flatten the curve. It is a reading that makes us aware that we are not external observers of the phenomenon through the analysis of the graphs, but that we are part of the phenomenon itself (as illustrated in Fig. 12).

Fig. 12
figure 12

Public image of “flattening the curve”

Full size image

What was stated in the previous paragraph means that there is a relationship between our actions and the flattening of the epidemiological curve, between individual experience and the mathematization of the phenomenon. In effect, since the pandemic has a dynamic based on contagion, actions such as maintaining a healthy distance, using masks, or frequent hand washing, have an impact on the reduction of new cases and therefore on the reduction of the pandemic's growth rate (flattening of the curve).

Change and variation are immersed in all areas of life and can be found not only in school situations but also in professional situations and everyday experiences (Cantoral et al., 2018). Within the Pylvar framework, the interest in studying change and variation comes from an inherent human need to predict due to an inability to look ahead in time to observe future outcomes. In the particular case of this research, this study provides elements for the construction of reference frameworks that guide the design of school situations that allow students to reason and act in changing situations. They allow students to predict phenomena associated with COVID-19 through the use of exponential behaviour graphs from a specific media outlet and the implementation of the variational practices of comparison, seriation and prediction that emerge from the use of these graphs. These considerations are an argument for teaching exponential behaviour and variational practices in schools based on the results obtained in this research.

Socioepistemology models the social construction of mathematical knowledge, that is, it models the dynamics of knowledge put into use (Cantoral, 2019). Along these lines, the importance of this research for the mathematics education community lies in providing elements to model a social construction of mathematical knowledge through the located use of variational practices in the context of a reference practice given by the television newscast in question.

6 Conclusions

The pandemic highlighted the question of how mathematical information is socialized to the population through the media. In this paper, we have argued the need to put into use variational practices in both the effective communication and comprehension processes of such information from the press. In the particular case of the newscast Meganoticias Prime, an answer to the first research question is that the graphs used to inform the population about COVID-19 are of three types, namely, cumulative frequency graphs, simultaneity graphs and comparison graphs. Regarding the second research question posed, it can be noted that it was inferred that the variational practices of comparison, seriation and prediction were present in the use of such graphs by this particular newscast. From these results we conclude the need to incorporate in the teaching of mathematics an approach that is decentred from the treatment of the mathematical object exponential function (Cantoral, 2013). Such incorporation would give priority to a conceptualization of exponential growth that allows a greater link between what is taught in school and how knowledge is used in everyday environments, for example the case of pandemic graphs in this Chilean newscast.

To achieve this outcome, it is necessary to go beyond simulated contextualisations in order to generate true processes of contextualisation of knowledge in the mathematics classroom (Espinoza et al., 2020). In this regard, the development of Variational Thinking and Language has been a line of socioepistemological research of vast development, which provides diverse research, curricular analysis, mathematical techniques, didactic proposals and implementation in reforms at the levels of courses, educational institutions, and national educational systems.

School mathematics is at the service of conceptual progress of the disciplinary field and the development of social thinking that allows the understanding and use of knowledge, both of which converge in the formation of a critical citizenship. Studies on new phenomena in diverse scenarios would give mathematics education a new field of exploration, with the purpose of abandoning the comfort zone in which everything is explained by being placed in a theoretical system. Such a development would involve the acceptance that it must be placed in ‘storm centres’, so that chaos and uncertainty arrive equally in the school and in our lives. In this regard, notions such as change, growth, stability, contagion and flattening of a curve, when linked to real life phenomena, become didactic elements of high social value, both for the scientific and governmental spheres and for citizens in general.

6.1 In memory

To Ricardo Cantoral Uriza, co-author of this article, teacher and friend, who passed away in December 2021 in a fatal traffic accident in his beloved Mexico. Ricardo was co-founder of the Latin American Committee for Educational Mathematics (CLAME), the Latin American Meeting of Educational Mathematics (RELME) and the Latin American Journal of Research in Educational Mathematics (RELIME). He was also the promoter of several postgraduate programs in Educational Mathematics in Latin America, author of more than 150 research articles, trainer of 123 doctoral students and science teachers and founder of the Socioepistemological Theory of Educational Mathematics, a theory that is currently widely used and developed in Latin America. Ricardo, may your legacy and struggle for a better world last for a long time.