Population growth presents a unique opportunity to make the connection between mathematical and biological reasoning. The objective of this article is to introduce a method of teaching population growth that allows students to utilize mathematical reasoning to derive population growth models from authentic populations through active learning and firsthand experiences. To accomplish this, we designed a lab in which students grow and count populations of Drosophila over the course of 12 weeks, modifying abiotic and biotic limiting factors. Using the data, students derive exponential and logistic growth equations, through mathematical reasoning patterns that allow them to understand the purpose of these models, and hypothesize relationships between various factors and population growth. We gathered student attitudinal data and found that students perceived the lab as more effective, better at preparing them for lecture, and more engaging than the previous lab used. Through this active and inquiry-based method of teaching, students are more involved and engaged in both mathematical and biological reasoning processes.

## Introduction

One of the greatest obstacles for biology teachers is moving beyond the traditional methods of passive instruction and superficial understanding (Tunnicliffe & Ueckert, 2007). The goal, instead, is to reconnect the students and allow them to experience the living world through active, firsthand experiences (Wood, 2009). In addition, the integration of mathematical models within biological topics can present a challenge for students (Brent, 2004; Gross, 2004; Hoy, 2004). We found that population growth was a challenging subject to teach because of the difficulty of witnessing it firsthand and its heavy reliance on mathematical modeling. However, we also found it to be the perfect opportunity to integrate mathematical and biological reasoning, making the connection in students’ minds between the two seemingly divided subjects.

With the globally documented decline in mathematical skills among undergraduates (Mulhern & Wylie, 2004), the National Science Foundation has issued a call to incorporate mathematical reasoning into applicable contexts for students (Feser et al., 2013). In fact, quantitative reasoning is one of the core competencies of Vision and Change in Undergraduate Biology Education (AAAS, 2011). Unfortunately, researchers have documented a deficiency of quantitative approaches in most introductory biology curricula (Gross, 2000; Bialek & Botstein, 2004). When mathematics is taught in biology classes, it is typically done in isolation from the biological content (e.g., Eastwood et al., 2011; Llamas et al., 2012). Yet even modest improvements to the quantitative components in a laboratory-based course can significantly improve student attitudes (Goldstein & Flynn, 2011).

To combat this issue, we have incorporated mathematical reasoning patterns into several of our introductory biology labs, guiding students through the derivation of mathematical models when the subject matter permits (e.g., Hardy-Weinberg equation, genetic probability, and population growth). The novelty of our current approach is that it not only exposes students to authentic population growth, but also integrates a constructivist mathematical approach that allows students to construct the mathematical models of population growth for themselves.

Our previous population growth lab used a more direct and artificial method involving penny flipping to simulate population growth. Instructors noticed that students were disinterested and disengaged, finding the lab very tedious and difficult. In addition, students struggled to apply lessons learned in the lab to real-world biological situations. Furthermore, they struggled with the mathematical models involved. In order to address these problems, we developed a new, guided-inquiry design for teaching population growth that allows students to construct concepts of the biological phenomenon of population growth and of the mathematical model used to describe it.

Many educators have developed lesson plans to teach population growth using authentic organisms. For example, Baumgartner et al. (2015) used phytoplankton to demonstrate population growth. McCormick (2009) outlined a method to demonstrate exponential growth using bell pepper plants. Oswald and Kwiatkowski (2011) used Euglena. Street and Laubach (2013) created a computer simulation of population growth that allows students to manipulate the components of the equation to see their effects. However, to our knowledge, none have provided a method combining an authentic population growth experience with a constructivist approach to student derivation of the population growth equations such that students understand, at a fundamental level, the reason for and use of the mathematical models. Below, we describe the implementation of our new teaching approach, the supplies and preparations needed, and student attitudinal data about the newly reformed lab.

## Learning Goals & Objectives

This lab has been designed for undergraduate students in introductory biology but could easily be adapted to high school. It can be performed in a traditional lab setting or in a more lecture-based setting. The growth phase of the fruit flies requires 10–12 weeks; however, it requires only five minutes, once a week, to tally population sizes. The actual derivation and synthesis takes approximately two 50-minute class periods, or one two-hour lab period.

The learning outcomes for this lab are that students will be able to

• Synthesize models of population growth based on authentic population data

• Compare and contrast exponential and logistic growth

• Hypothesize the biotic and abiotic limiting factors that influence population growth

• Hypothesize the factors that change exponential growth patterns into logistic growth

## Materials

Students work in groups of four. Each student within the group makes a unique culture: one large jar with normal flies, one large jar with vestigial-winged flies, one small jar with normal flies, and one small jar with vestigial-winged flies. Instructor and student guides are available in the Online Supplemental Materials. We purchased all materials from Carolina Biological Supply Company (the product number is given in parentheses after each item for easy ordering).

• Drosophila: wild type (no. 172100) and vestigial chromosome 2 mutants (no. 171460)

• FlyNap Anesthetic Kit (no. 173010)

• Drosophila Culture Vials (1.25 × 4.00 inches; no. 173120) and Culture Bottles (240 mL; no. 173135)

• Foam plugs (no. 173122)

• Medium-Formula 4-24 Instant Drosophila (we prefer the blue, as it makes the larva easier to see; no. 173210)

• Different colored permanent markers

## Experimental Setup

On the first day of class, students are given a homework assignment to determine the difference between a male and a female fruit fly (Drosophila melanogaster). Students are told they will be required to distinguish between the sexes in the first laboratory session. During this first session, students prepare the Drosophila medium and place it into the bottom of the jar, anesthetize the flies, determine their sexes, and place one male and one female in each jar.

At the end of the experimental setup, students are given time to discuss the different setups and make predictions about what they might see from the different jar types and fly types. Students are asked to perform background research on fruit flies; specifically, they are instructed to investigate information in the scientific literature regarding life expectancy, life cycles and stages, and other factors that may influence their population's growth. They will use what they learn to evaluate and reconsider their initial predictions. In addition, they must come up with a practical way to count the number of flies each week.

## Weekly Procedure

At the beginning of the following week, the instructor discusses with the students a practical way to count flies, based on student suggestions. We have found that the easiest way to count is to use a different colored marker each week to mark new puparia that remain adhered to the jar (Figure 1). After being guided to a feasible counting method, students will count the number of Drosophila once a week over the course of 10–12 weeks and record the number by jar size and fly type (Table 1; see Online Supplemental Materials). After approximately 10–12 weeks (depending on jar size and ambient temperature), the population will crash, having reached its carrying capacity.

Figure 1.

Culture jars with population counts marked in different colors.

Figure 1.

Culture jars with population counts marked in different colors.

Table 1.
Typical table in which students record idealized numbers (“Expected Numbers”) and experimental numbers of flies for each treatment condition.
WeekExpected NumbersLarge Jar, NormalLarge Jar, VestigialSmall Jar, NormalSmall Jar, Vestigial
0
1
2
3     …etc.
WeekExpected NumbersLarge Jar, NormalLarge Jar, VestigialSmall Jar, NormalSmall Jar, Vestigial
0
1
2
3     …etc.

## Culminating Procedure

Once the students have recorded their data for 10–12 weeks, they are ready to derive the exponential, or idealistic, population growth equation. To do so, students are asked to calculate the idealized population sizes over the 10–12 weeks. In order to do this, they must refer to their prior research on growth rates of flies. Students will need to discuss with their group and come to an agreement on a typical growth rate. In their experiment, they did not take fly death into account (as this is rather difficult to do, given the setup); thus, in these idealized numbers, they will ignore fly death. (For the instructor: the time required to produce offspring is approximately one week, and the average breeding pair produces 10 offspring per reproductive cycle.) Students record idealized, or expected, numbers along with the experimental numbers for their four group members (i.e., large jar, normal flies; small jar, normal flies; large jar, vestigial flies; small jar, vestigial flies) in a table (Table 1; also see Online Supplemental Materials: full printable data tables are included in the Student Guide; idealized numbers are included in the Instructor Guide).

Students plot their experimental numbers versus the expected numbers, with week number on the x-axis and Drosophila count on the y-axis. Typical growth curves are shown in Figure 2. Students determine the slope using the idealized numbers during each time interval (Remember: slope is the change in y over the change in x; i.e., slope = Δy/Δx). Students will notice that a comparison of slopes shows that they increase with time. This increase indicates what students should already intuitively understand from the fruit fly experiment (i.e., the number of new individuals increases with each passing generation). The population soon “explodes” in terms of numbers of new individuals. At this point, students should revisit their previous predictions about the population growth after a week and a month. By comparing their experimental numbers to the idealized numbers, students will likely notice that the numbers match for the first few weeks of the experimental period.

Figure 2.

Plot space for Drosophila count vs. week, showing idealized numbers and two hypothetical populations.

Figure 2.

Plot space for Drosophila count vs. week, showing idealized numbers and two hypothetical populations.

From idealized numbers, students are asked to determine the rate at which the populations should grow and to write a generic formula to express the intrinsic growth rate for any population. Students are allowed to work on this individually and to discuss it with their group before discussing it as a class. A typical equation is shown in Equation 1. Keep in mind that in order to determine the initial population size for the week, students will need to subtract the new number of offspring counted from the total population size.

(1)

Other variations typically seen are included in the Instructor Guide (see Online Supplemental Materials). If done correctly, the intrinsic growth rate should be the same number from week to week (i.e., it does not change, despite the changing population size). If students agreed upon a typical growth rate for fruit flies, they should all arrive at the same r value. However, differences may occur between groups (based on prior research that students did) and should be discussed as a class.

Students are then asked to reflect upon how they came up with idealized numbers to fill in their table. From this, they derive the ideal change in fly population (N) over change in time (t) from the intrinsic growth rate and fly population (i.e., a model for predicting the idealized population size of any generic population), as shown in Equation 2.

$ΔNΔt=rN$
(2)

This is the exponential growth equation, which can be used for any growth rate and population size with no limiting factors or carrying capacity.

Students compare experimental numbers to idealized numbers and discuss differences observed. They hypothesize reasons why their data did not follow the ideal data (especially toward the end of the growth period). By doing so, students list limiting factors of Drosophila growth. Students’ attention is directed to the horizontal asymptote observed in their data and they are taught about environmental carrying capacity. Each student reports their system's carrying capacity and discusses differences observed in carrying capacities, data trends, and limiting factors between the different jar sizes and wing types. Additionally, students consider other limiting factors not addressed in these tests and predict how those factors would affect carrying capacities and data trends if included in future experiments. (The instructor may even choose to have them design experiments.)

From their experimental carrying capacity, students are asked to consider a population's “room to grow” (RTG) for any given week, or the percentage difference between the current population for the week and the carrying capacity (K). They are asked to write a mathematical expression describing the RTG and are given time to discuss this in their groups (for leading questions and tips, see Instructor Guide and Student Guide in the Online Supplemental Materials). They should come up with an expression similar to Equation 3.

$RTG=(K−N)K$
(3)

With the population's RTG equation, students modify the growth rate at each interval to better estimate population growth, as shown in Equation 4.

$ΔNΔt=K−NKrN$
(4)

This is the logistic growth equation, which can be used for any growth rate and population size with limiting factors and a carrying capacity.

Using the logistic growth equation and their experimental carrying capacity for each treatment (jar size and fly type), students add a logistic plot to their previous graph and discuss differences in plots between the exponential, the derived logistic, and the experimental values. Their discussion should include questions such as these: Which model better represents fruit fly population growth? When do populations grow exponentially? What causes logistic growth? Most students will notice similarities between their experimental data and the derived logistic growth predictions, which will help them construct the idea that growth in a contained environment with limiting factors can be understood best through a logistic growth model.

If instructors desire, data can be collected and combined between semesters for a deeper understanding. Because we have only recently implemented this reformed lab, we have not, as of yet, tried that strategy. However, combining data from semester to semester would allow students to make more definitive conclusions about the effects of jar size and wing type on Drosophila reproduction rates.

## Assessment

To determine student perceptions of the reformed lab, we collected attitudinal data from two semesters – teaching assistants (TAs) taught the lab in one semester using the original lab and in the other semester using the reformed lab. Both control (fall 2013) and test (fall 2014) data came from the introductory biology course for majors. In fall 2013, 71 students participated in the original lab (15–24 students per TA-led lab section). In fall 2014, 164 students participated in the reformed lab (again, 15–24 students per TA-led lab section). All TAs were trained by one of the authors to conduct the lab according to the Instructor Guide. This course is taught every semester and contains mostly freshmen, with some sophomores and juniors who recently switched majors. After completing the lab (original or reformed), we asked students to rate the lab according to their agreement with the following statements, using a Likert scale (1 = strongly disagree to 6 = strongly agree):

• This lab was effective in teaching me the concepts of population growth.

• This lab effectively prepared me for the information we covered in lecture. [Note: our class is designed after the 5E learning cycle model, such that labs precede lecture.]

• This lab effectively prepared me for the quiz.

• This lab was engaging (i.e., I was on task a majority of the time).

We compared the results of the original lab to those of the reformed lab using nonparametric Mann-Whitney U-tests. Students who participated in the reformed lab reported that they felt it had been more effective at teaching them population growth (U = 6897.5, P = 0.016), that it had better prepared them for lecture (U = 7109.0, P = 0.005), and that they had felt significantly more engaged (U = 7122.5, P = 0.005) than those who had completed the original lab. While there was a slight increase in the reported level of preparedness for the quiz, the results were not statistically significant (U = 6423.0, P = 0.192; see Figure 3).

Figure 3.

Response rates on a six-point Likert scale for the original and reformed labs. Students considered the reformed lab more effective, more engaging, and better at preparing them for upcoming lectures. Asterisk indicates a significant difference between treatments.

Figure 3.

Response rates on a six-point Likert scale for the original and reformed labs. Students considered the reformed lab more effective, more engaging, and better at preparing them for upcoming lectures. Asterisk indicates a significant difference between treatments.

## Conclusion

We created an inquiry-based and authentic population growth lab to help students be more involved and engaged in the learning process and to construct mathematical functions that apply directly to authentic organismal populations. Our results show that students felt they learned more, were more engaged, and were better prepared for lecture with this reform. We believe these improvements are largely attributable to two factors. First, students were given an authentic, hands-on interaction with population growth. They were able to witness growth trends firsthand, with a biological example rather than a simplified simulation, and they did so through a guided inquiry-based approach. This approach is successful in teaching science concepts and increasing scientific reasoning skills (Howard & Miskowski, 2005; Spiro & Knisely, 2008; Rissing & Cogan, 2009; Minner et al., 2010). In addition, the constructivist approach, which allows students to construct their own understanding, led students to gain deeper conceptual understanding and ownership of their knowledge (Heiss et al., 1950; Renner et al., 1973; Minner et al., 2010).

Second, students were given the opportunity to derive the mathematical models for themselves in a constructivist fashion, allowing them to better understand the purpose and implementation of mathematical models in a biological framework. This contributes to the goals of Vision and Change to teach undergraduate students to apply quantitative skills to biological topics (AAAS, 2011), a skill that is in growing demand in the professional community (Feser et al., 2013). We believe that by introducing active learning and real-life biological examples as well as opportunities for mathematical reasoning, this pattern of reform could likewise improve engagement and preparation in other traditionally difficult areas of biology.

A Teaching Enhancement Grant from the College of Life Sciences, Brigham Young University, provided funding for the development of this lab. We wish to thank Dr. Riley Nelson for his invaluable suggestions about logistics, teaching assistants for putting in the effort to learn a new teaching style, and our students for being willing participants.

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