Evolutionary Medicine

Background Knowledge & Content 

Exponential growth is a process “in which the rate of increase of a quantity is proportional to the present value of that quantity.”⁷ In other words, exponential growth occurs when a quantity grows by an ever-increasing rate over time. A linear growth pattern, in comparison, grows by a constant rate over time; for instance, an employee at a store makes a fixed rate of money per hour.

Virus replication is an example of exponential growth. A virus invades a cell; the virus then duplicates itself, creating 100 or more progeny particles within this host cell. The virus eventually spreads to other cells where the same replication process occurs. The speed with which virus pathogens grow within human cells while mutating to create new genotypes causes their rapid evolution, forcing us to constantly create new iterations of vaccines in an effort to combat the spread of viral diseases. However, the inability of our vaccine formulations to keep pace with virus evolution can further the virus’s continued spread of infectious disease. This inability of hosts, such as humans, to evolve at the same pace as their pathogens is a key example of evolutionary medicine, which applies evolution thinking to better understand how and why health and disease problems occur.

While scientists and mathematicians may be familiar with the concept of exponential growth in evolutionary medicine and disease, the vast majority of people are not; they misunderstand exponential functions and, instead, tend to assume a linearized pattern of growth. This misunderstanding, referred to as exponential growth bias, can have dire consequences, as seen with the COVID-19 pandemic.⁸

Previous studies of epidemiology show that the initial spread of infectious disease, including COVID-19 caused by SARS CoV-2, often follows an exponential growth pattern.⁹ Li et al documented the initial spread of COVID-19 in Wuhan, China by analyzing the first 425 confirmed cases of the disease. In this analysis, Li and his colleagues concluded that the rate of infection grew exponentially over the first month or so of its presence in humans.¹⁰  (While the R-value, or “the number of individuals that the average infected person will infect,” changes over the course of the spread,¹¹ it appears to remain somewhat constant at the onset of infectious disease).¹²

Despite this documented proof of exponential growth during the initial phase of viral infection, many did not understand or appreciate the rapidity with which the disease would spread. In the United States, for instance, many political figures downplayed the spread of COVID-19 due to their exponential growth bias, either failing to respond to the pandemic or outright refusing to follow guidelines set by the U.S. Centers for Disease Control (CDC).¹³ Exponential growth bias discouraged the general public from following non-pharmaceutical interventions, such as masking, social distancing, and quarantining; the lack of following these interventions had dire consequences on public health.¹⁴

Another virus pandemic is a very real possibility, and the potential to misunderstand its initial pattern of spread is quite high. Therefore, the need to inform the next generation about exponential growth bias and its consequences is essential. It is imperative that students learn the difference between exponential and linear growth, not only so that they can take appropriate action in the event that another pandemic occurs in their lifetime, but also so that they can be better informed about general trends that follow these patterns.

To facilitate student comprehension about both linear and exponential functions, I will utilize the Herbartian instructional model. This model focuses on “the creation and development of conceptual structures that would contribute to an individual’s development of character.” Johann Friedrich Herbart, a German philosopher, found both student interest and conceptual understanding to be foundational components of teaching.

There are two types of interest: (1) that “based on direct experiences with the natural world” and (2) that “based on social interactions.”¹⁵ In my unit, I plan to engage the first type of student interest through examination of a disease they are all too familiar with: COVID-19. Students will explore the exponential growth patterns associated with advent of the COVID-19 pandemic in Wuhan, China through hands-on activities, tables, and graphs. While the cause of the pandemic may or may not have been what is considered to be a natural occurrence, it spread naturally, therefore facilitating student learning by evoking memories of direct experiences that students had with the natural world.

The second type of student interest will be engaged through socialization, which will be imparted via collaborative learning (collaborative learning will be expounded upon in the next section, Teaching Strategies). Through this collaborative learning, students will work together to digest and discuss various depictions of exponential functions, including tables, graphs, and word problems. Furthermore, toward the end of the unit, students’ collaborative learning will facilitate their understanding of the difference between linear and exponential functions.

The conceptual understanding component of Herbart’s philosophy deals with the coherence of ideas; new learning should be directly related to students’ prior knowledge. The COVID-19 pandemic, along with evoking student interest, will directly connect students’ learning of exponential functions to an event they all experienced and can relate to.

These foundational components of teaching, interest and conceptual understanding, are reflected throughout the Herbartian instructional model’s four phases: Preparation, Presentation, Generalization, and Application. This unit will consist of two cycles of the Herbartian model up to the Generalization phase; the first cycle will cover exponential growth and the second cycle will cover exponential decay. Students will be assessed on both types of exponential functions in the Application phase.

Preparation & Presentation

The Preparation and Presentation phases will both occur during the first lesson of my unit.

In the Preparation phase, the teacher brings prior knowledge to the forefront of the students’ learning experience; the practice of activating prior knowledge has been found to be beneficial to students’ comprehension of new, unfamiliar mathematical concepts.¹⁶ In the exponential growth cycle of the Preparation phase, students will be shown a video that splices news clips together from the onset of the COVID-19 pandemic.¹⁷ In the exponential decay cycle of this phase, students will be shown a news clip that discusses the need for the general public to get the COVID-19 vaccine.¹⁸ These clips will evoke personal memories that students had during their experience with the pandemic; the more personally meaningful and relevant the content is for students, the more likely they are to engage in learning.¹⁹

The Presentation phase focuses on connecting prior learnings to new learnings. During the Presentation phase of the exponential growth cycle of this unit, students will be introduced to exponential functions, particularly exponential growth, through a class simulation illustrating the initial spread of COVID-19. In the Presentation phase of the exponential decay cycle, students will visualize the effects of vaccines on the spread of COVID-19 through tree diagrams. An epidemic that students have great familiarity with, COVID-19 will, as seen in the Preparation phase, evoke prior knowledge and facilitate students’ connection to new learning about exponential functions. Students will collaborate to graph and analyze the data gathered and observed during both the simulation and visualization activities respectively, thereby introducing them to the mathematical application of exponential functions.

Generalization

During the Generalization phase, the teacher clarifies and facilitates development of students’ conceptual understanding. The Generalization phase of this unit will cover exponential functions through explicit (direct) instruction (which is discussed further in the Teaching Strategies section of this unit). Explicit instruction will more clearly define the following mathematical formulas for students:

• Exponential Growth and Decay Formula: f(x) = ab^x, where a is the initial (starting) quantity, b is the multiplicatory rate of change, and x is the amount of time passed (In this formula, exponential growth is represented by a b-value greater than 1; exponential decay is represent4ed by a b-value between 0 and 1).

• Exponential Growth Formula (with Percent): f(x) = a(1 + r)^x, where a is the initial (starting) quantity, r is the rate of growth (which has to be changed from a percent into a decimal or fraction before being substituted in the formula), and x is the amount of time passed.

• Exponential Decay Formula (with Percent): f(x) = a(1 – r)^x, where a is the initial (starting) quantity, r is the rate of decay (which has to be changed from a percent into a decimal or fraction before being substituted in the formula), and x is the amount of time passed.

Throughout the generalization phase of this unit (during both cycles of the Herbartian model), students will practice identifying exponential growth and decay rates through tables, graphs, and word problems and will translate these mediums into equations and evaluate said equations using the above formulas. Furthermore, they will compare and contrast exponential and linear functions in an effort to alleviate exponential growth bias within the student population. Students will answer questions about specific data points and general data trends within these functions, so that they can more successfully transfer this understanding to the Application phase.

Application

In the fourth and final phase, the Application phase, students demonstrate their newfound comprehension through application of concepts to new contexts. In this phase, students will research graphs depicting real-life scenarios in which exponential growth or decay is occurring. Students will have to analyze these graphs to determine what their exponential patterns mean and will communicate this meaning and its importance through a presentation to their classmates. In addition to this presentation, students will complete a summative assessment to show their overall comprehension of exponential (and linear) functions.