Mathematics education in the time of COVID-19: a public health emergency exacerbated by misinterpretation of data

Abstract

The COVID-19 public health emergency has been characterized by an abundance of data in the form of numbers and charts. Although these data are readily available, there have been challenges associated with their interpretation—exacerbated by generally low numeracy rates. Consequently, people may underestimate the speed at which the disease spreads and the ensuing risk to themselves and others, resulting in a lack of compliance with non-pharmaceutical interventions. This article discusses misconceptions associated with the popular terms, metrics and graphs commonly used to describe this pandemic. We suggest and illustrate how mathematical literacy is necessary to understand and convince people of the necessity for various restrictions and lifestyle changes. As the pandemic progresses and in anticipation of any future outbreaks, it is important to rethink the teaching of these concepts so as to equip learners with the proper tools for informed decision making—now and in the future.

1 Introduction

COVID-19 has not only brought with it a ‘new normal,’ but introduced a whole new vocabulary. As a result of public health campaigns, augmented by figures and graphs, terms like flattening the curve, exponential spread, fatality rate etc. are now part of our everyday language when describing the risk and scale of contagion. Despite their widespread usage and vital role—not only in understanding the extent and the seriousness of the pandemic but also in its containment, insufficient effort has been devoted by public health officials to providing a clear explanation of these terms, figures and graphs to the general public. Instead, these have been thrust upon a mathematically naive public, with assumptions that they have the background necessary to interpret their significance.

Given that 617 million children and adolescents worldwide—6 of 10—are not reaching minimum proficiency levels in reading and mathematics (UNESCO, 2017), it should come as no surprise that these data and their underlying concepts are subject to misinterpretations, thus causing people to underestimate or otherwise misinterpret the risks posed by COVID-19. This may have detrimental effects; perceived risk or susceptibility is generally considered an important condition for behavioural change and engaging in preventive action (Kittel et al., 2021).

This article describes some of the misconceptions associated with the popular terms, metrics and graphs commonly used to describe this pandemic. Concepts include proportional reasoning, fractions, percentages, rates and ratios, exponential growth and the graphical representation of data. For each concept, we explain its use in describing the pandemic and illustrate both the potential for, as well as, examples of misinterpretation.

2 Fractions, percentages, rates and ratios—Putting the pieces of information together

With all the raw numbers bandied about and often no reference points, it is easy to feel confused about their meanings and context. Interpreting the numbers defining the COVID-19 outbreak requires an understanding of proportional reasoning in the forms of fractions, rates and ratios where numbers are considered in relative terms, rather than absolute terms. However, proportional reasoning is one of ‘the most difficult to teach, the most mathematically complex, the most cognitively challenging, the most essential to success in higher mathematics and science’ topics in the mathematics curriculum (Lannon, 2007; Gläser & Riegler, 2015) and consequently presents a challenge—even for many adults (Tourniaire & Pulos, 1985).

Perhaps this was why during the early stages of the pandemic, many believed that COVID-19 was less fatal than seasonal influenza (Lewis, 2020). While the data comparing the number of deaths may have been true at that point in time, they conveniently ignored the timeframe over which the data was reported and actually compared COVID-19 deaths over a few months with estimated influenza deaths over the period of a year.

But even something as well defined as the metrics used to track the course of the pandemic are subject to misinterpretations. Two of the metrics popularly used in the media are the case fatality rate (CFR) and the test positivity rate (TPR) as defined in Table 1. These serve different purposes. The CFR is an estimate for the disease lethality, while the TPR gives an indication of how widespread infection is in the area where the testing is occurring.

Though these definitions appear straightforward enough, there are underlying factors (apart from limited testing resources) to consider when it comes to testing. Specifically, symptomatic people are more likely to be tested than asymptomatic people. Also, not everyone with mild symptoms similar to that of a cold will go in for voluntary testing—some of these may well have COVID-19 but it will not have been detected. In particular, the severely ill at home and those in hospital will be tested for COVID-19 according to national policies/guidelines—deaths in COVID-19 are more likely to occur amongst these subgroups. Thus if the tested subpopulation is limited to those who are symptomatic (especially those in hospital who are really ill and are more likely to die), this may lead to a systematic bias and there will be a greater CFR than if we tested generally in the population. This can therefore give rise to widely varying estimates of CFR (Rajgor et al., 2020) so that this metric may be misleading when comparing disease lethality across regions.

A similar reasoning process can explain a varying test positivity rate in a population. So for example, if you limit testing to those with severe symptoms and those in hospital on any one day, then the test positivity rate on that day will be high compared to if there was large-scale testing of a population.

In spite of our familiarity with fractions from the first year of elementary school and their use in everyday life, fractions are still a source of confusion for many (Gabriel et al., 2013). Even public figures are not exempt—for example increased testing in the United States was blamed for ‘creating’ more cases resulting in an increasing test positivity rate (Fricker, 2020). Some responsibility may perhaps be allocated to whole number bias—where natural number comparisons are applied to fractions, while ignoring the relationship between the numerator and denominator (Alonso-Díaz et al., 2018) so that for example, though 10 is greater than 5, 1/10 is actually less than 1/5. According to the definition of test positivity rate (Table 1) and the equivalence properties of fractions, increasing the denominator (i.e. the number of people being tested) only increases the test positivity rate if the additional people tested are more infected than the original group (e.g. if firstly only people with symptoms are tested but then this is increased to include people who want to check if they have the virus in case they infect others). As a matter of fact, the increasing test positivity rate at that time was validated by a corresponding increase in hospitalizations and deaths.

These rates and other ratios have generally been represented as percentages in media reports—perhaps because people are already familiar with their use in everyday life from consumer arithmetic. Yet despite this, when used to compare quantities or quantify changes, there are challenges, not only in interpretation but in language (Schield, 2000). There are two main ways of quantifying changes using percentages– as percentage points where one percent value is subtracted from another and as a percentage change of the initial value. From an update article (Smith, 2020)—‘the number of Vietnamese people wearing face masks in public has risen 26 percentage points, from 59% on 23 March to 85% by 30 March… Mexico, which has seen a rapid 23 percentage increase from 39% being very or fairly scared of catching the disease last week to 62% this week.’ Despite being from the same article, the terms percentage points and percentage increase are used as synonyms. While this may appear to be merely an issue with semantics, the percentage increase in fear is actually 59%, which is significantly more than the 23% mentioned. This may then necessitate some form of intervention and illustrates the importance of a proper understanding of these concepts.

3 Exponential growth

The greatest shortcoming of the human race is our inability to understand the exponential function (Bartlett, 2004).

Exponential growth is used to describe the unconstrained spread of an infectious disease. Despite its widespread use in this pandemic, exponential growth is a difficult concept to understand (Levy & Tasoff, 2017; Schonger & Sele, 2020) with many people confusing it with linear growth (Wagenaar & Sagaria, 1975). In linear growth, the numbers infected increase by the same amount each day. Whereas in both cases the numbers are initially small and remain small (but increasing) for a time, exponential growth quickly ‘blows up’ and outpaces linear growth as demonstrated when another popular term—the basic reproduction number (R₀) is considered.

The basic reproduction number (R₀) refers to the transmission potential of a disease i.e. how many healthy people one diseased person can infect in the absence of any interventions in a susceptible, non-immune population. If this number can be brought below one, then the spread will slow and the disease will eventually die out. The higher the value of R₀, the faster an epidemic will progress. R₀ can vary by country or region, R₀ in Japan using data near the start of the pandemic was estimated to be 2.6 (Kuniya, 2020), whereas in Wuhan, R₀ was estimated to be between 1.4 and 6.5 (Liu et al., 2020).

To demonstrate its growth potential, if we assume a value of R₀ of 2, the number of infected individuals doubles each generation. This means that a single infected person can infect two people (generation 1) so that by generation 7 we have 128 new cases with a total of 255 infected—all this resulting from a single infected person. In contrast, with linear growth, by generation 7, we would have 15 new cases with a total of 64 infected.

It is easy to underestimate this rapid, exponential growth—a mistake known as ‘exponential growth bias’ (Lammers et al., 2020; Banerjee et al., 2021). Thus, people may misjudge the speed at which the disease spreads, the resulting risk to themselves and the importance of non-pharmaceutical interventions (Lammers et al., 2020; Banerjee et al., 2021). This is what is alarming about infections that spread exponentially and leads to surprise by the general public when the change in the number of new cases increases rapidly and suddenly.

With COVID-19, there may be ‘invisible’ spread since infectivity begins before the end of the incubation period, i.e. before becoming symptomatic. Also, depending on the severity of the disease, it can be spread for up to two weeks after infection. This means that there are no clearly defined generations in time—the daily number of new infections can come from different generations. This is why the number of new infections may vary so much on a daily basis and may lead to people ‘letting down their guard’.

4 Understanding and interpreting the graphs—When a picture may not be worth a thousand words

While a noteworthy feature of this pandemic is the rapid and easy access to information via social media and other channels, it is often difficult to visualize and see trends and patterns when the information is presented as raw numbers. A useful way to illustrate and interpret the variation, patterns, and trends within data is by using a graph. Graphs have been used not only to highlight the change in the epidemic-related data (such as new cases) over time but also in displaying proportional information such as the percentage of people vaccinated for example. It is unfortunate that the misuse of graphs by those in authority (Engledowl & Weiland, 2021) and as propaganda (Pullman, 2021)—deliberate or otherwise—has occurred. Sampling bias, truncated axis distortion, irregularity in axis scaling—not using a consistent scale on the x-axis (Garzón-Guerrero et al., 2020; Engledowl & Weiland, 2021; Kwon et al., 2021) have all contributed to the misinterpretation of graphs. This may have dire consequences on the course of the pandemic.

Despite some familiarity with graphs from school, newspapers or online articles (mainly histograms, bar charts, pie charts and line graphs), people often have difficulty in interpreting graphs—a challenge originating from their school days (Woolnough, 2000). There is also the temptation to ‘view graphs as representing literal pictures of situations rather than abstract quantitative information’ (Glazer, 2011). To make matters worse, some of the graphs used to describe the pandemic in the media are not part of the general mathematics curriculum (Kwon et al., 2021). This includes the choropleth map—a thematically shaded map using differences in brightness— to show the total number of confirmed COVID-19 deaths per million people in different countries.

Even those that are familiar—such as line graphs—may be presented in an unaccustomed format using a non-linear (for example a logarithmic) scale to show cumulative cases. Unlike a linear scale, on a logarithmic scale the numbers on the y-axis don’t move up in equal increments but instead each interval increases by a set factor say of 10. This means that a graph using a linear scale cannot be compared to one using a logarithmic scale. For example, Fig. 1 shows the same cumulative data displayed using a linear scale and a logarithmic scale. Since most people are unfamiliar with the intricacies of the logarithmic scale, they may therefore not interpret these graphs correctly (Romano et al., 2020; Ryan & Evers, 2020).

With such a bewildering array of options available and the ability to go online and generate customized graphs, it is important to be able to not only choose the appropriate variable, but to identify the most suitable graph/chart to fit the data and the questions asked. For example, the daily number of new infections presented as a histogram will reveal meaningful trends at a glance. It may be used to determine how fast the epidemic is growing, if it is decreasing or stabilizing, but does not give information of the full extent or the pandemic. For that, a cumulative graph (as shown in Fig. 1) is more useful. However, this type of graph should be used cautiously as with the count of infected people numbering into the millions, the upward rise of the graph may give the appearance that despite mitigation efforts, the situation is getting worse.

This pandemic has served to highlight the discrepancy between the graph literacy taught in schools and that needed for real life (Kwon et al., 2021). Though a picture may be worth a thousand words, the abundance of graphs used in media reports and online may be a source of confusion to the public. When graphs are used in conjunction with data to justify measures such as lockdowns, a lack of understanding may lead to non-compliance and further spread of the disease (Romano et al., 2020; Ryan & Evers, 2020; Banerjee et al., 2021).

5 An unexpected boon—Online mathematics tools

An unanticipated benefit of the COVID-19 pandemic was a large shift to an online learning environment and the resulting increased use of technology to help students grasp mathematical concepts. With their open source software and straightforward interfaces, online and app-based digital visualization tools, such as Desmos and GeoGebra, are powerful partners to educators, especially in the teaching of graphs. Similar to graphing calculators, they can generate graphical representations of data, relate these representations to their algebraic equations and link these visual representations to physical phenomena. These tools are useful in connecting mathematical concepts to real life problems and experiences and may thus play an important role in increasing students’ interest in mathematics (Vos, 2018).

During the COVID-19 pandemic, graphs such as Fig. 1 featured prominently in the media—providing an excellent opportunity for educators to improve graph literacy. With the ease of access to COVID-19 data repositories, such graphs may be created by students using graphing tools. Figure 2, generated with the graphing calculator in Desmos, illustrates the number of daily cumulative cases worldwide using a linear and a logarithmic scale.

The graph with the linear scale shows the absolute number of cases over time. Where the number of new cases is increasing, the cumulative curve is concave up. Where the number of new cases is decreasing, the cumulative curve is concave down. This is easily seen in Fig. 2 (left hand panel): the number of cases is generally rising until about day 15, after this it starts to decrease. Where the number of new cases is staying the same, the cumulative curve is linear and therefore the slope is constant.

If we plot the logarithm of the number of cumulative cases along the Y axis against time (t) on the X axis, this gives a semi-log graph. Though its appearance may be similar to the previous graph, this graph gives a different sort of information. Suppose on any arbitrary day, the number of cases is |${y}_1$| and on the following day, the number of cases is |${y}_2$|⁠. From the semi-log graph, the slope of the line connecting these two points is |$\frac{\ \mathit{\lg}{y}_2-{lgy}_1}{1}=\mathit{\lg}\frac{y_2}{y_1}$|⁠. By definition, |$\mathrm{growth}\ \mathrm{rate}=\frac{y_{2}-y_{1}}{y_1}=\frac{{y}_2}{{y}_1}-1$| which can be estimated by |$\frac{{y}_2}{{y}_1}$|⁠. This implies that the slope of the graph with the logarithmic scale can give an idea of how quickly the number of cases is changing day by day or how the rate of growth is changing—i.e. whether it is slowing down (flatter) or accelerating (steeper). Figure 2 (right hand panel) shows this very nicely: the growth rate of new cases generally increases up to Day 7. After that the growth rate slows down and the curve begins to plateau, which means the relative change in cumulative infections on a daily basis has reduced significantly. This is the key difference between the two curves and highlights the importance of looking at the logarithmic curve during the spread of any infectious disease.

The difference between the two ways of displaying the same data can usually be viewed as being highly abstract, but this pandemic allowed the difference to become more imaginable. Thus in teaching about linear and logarithmic scales, we need to emphasize that both graphs in Fig. 2 tell the same story—the number of new cases is decreasing after a certain time. However, logarithmic scales can highlight the relative rates of change in a way that linear scales do not and are useful when assessing whether control measures are helping to curb the spread of infection.

6 Conclusion—A call to action

A common complaint by mathematics students past and present is the subject's applicability to real life. With the abundance of numbers detailing the course of this pandemic in the form of totals, fractions, percentages and graphs on a daily basis, this is certainly no longer the case and the examples provided here may provide fodder for activities to bridge this gap (Griffiths, 2011). Projects involving these topics may be used not only to create mathematically literate individuals but also to generate interest and excitement in mathematics.

Unfortunately, although information (as well as other health-related data) is readily available, there may be difficulties associated with its interpretation due to its novelty, exacerbated by generally low numeracy rates in the population (UNESCO, 2017). While a challenge at the best of times, the misinterpretation of numbers and figures in the time of COVID-19 may have serious consequences. It may result in misplaced optimism and complacency and a lack of support for non-pharmaceutical interventions. For example, when people perceive the exponential growth of COVID-19 as linear, they do not see the value of social distancing (Lammers et al., 2020). By the same token, the use of a logarithmic scale leads to both an underestimation of disease severity, as well as difficulty in predicting the trajectory of the pandemic and a resulting unwillingness to adopt precautionary measures (Romano et al., 2020; Ryan & Evers, 2020).

For this reason, the way the data is presented and interpreted is of vital importance. In order to make sound health decisions, individuals need to understand this information so as to weigh the pros and cons of their actions. Inevitably, this may have a detrimental impact on how people maintain their health and make informed medical decisions (Reyna et al., 2009). We speculate that at a time when people feel powerless in the face of COVID-19, with their freedoms curtailed, an explanation of the rationale behind certain actions—backed up by data—may lead to better compliance.

As the pandemic progresses and in anticipation of any future outbreaks, it is therefore important for educators to improve their instructional planning and pedagogical practices by using all available resources—real and virtual. Not only is this important in this ‘new normal’ but also its benefits will transcend this pandemic.

References

Terence Seemungal did his professional training in the UK, is certified in Chest and Internal Medicine and is a member of The Royal College of Physicians of London. He did his PhD at the University of London and his current interests are chronic lung disease and the epidemiology of lung disease. He is currently professor of medicine at The University of The West Indies. Professor of Medicine, Faculty of Medical Sciences, The University of The West Indies St. Augustine Campus, Trinidad and Tobago. Email: [email protected].

Dr Joanna Sooknanan did her PhD in mathematical modelling at The University of The West Indies. She is an Independent Researcher and a secondary school teacher with an interest in showcasing the practical applications of mathematics. Email: [email protected].