Students learning the skills of science benefit from opportunities to move between the scientific problems and questions they confront and the mathematical tools available to answer the questions and solve the problems. Indeed, students learn science best when they are actively engaged in pursuing answers to authentic and relevant questions. We present an activity teachers can use in the classroom to introduce the concepts of species richness and diversity. We break down the history and logic behind the two primary statistical tools ecologists use to quantify species diversity: Simpson's and Shannon's diversity indices. With hypothetical data, we show how students can learn about and practice the calculations. We then describe an activity where students collect authentic ecological data with pitfall traps while learning some arthropod systematics and practicing their newly acquired quantitative reasoning skills, all within the context of edge effect ecology and habitat conservation. The entire activity reinforces for students how interesting and helpful mathematical models and quantitative reasoning in science can be for understanding biological phenomena, but also for generating more questions, and for designing additional data-collection techniques and experiments.

## Introduction

The May 2018 issue of The American Biology Teacher (vol. 80, no. 5) includes three articles that emphasize teaching the concepts of species richness and biodiversity (Davis-Berg & Jordan, 2018; Doup, 2018; Minteer et al., 2018). The present article extends that work by having students collect authentic ecological field data and utilize diversity indices to quantify biodiversity. Species diversity (biodiversity) is often confused with richness (the number of species in an area; Spellerberg & Fedor, 2003). However, species diversity is actually a function of both the richness and evenness of the species within a community (Delang & Li, 2013). Ecologists have developed mathematical tools to quantify diversity and other ecological phenomena, as well as borrowing tools from other fields. Here, we – a biology teacher (Strode) and two of his former students (Prinster and Hoskins) – provide a theoretical context for how the tools for quantifying diversity were derived. We then explain how diversity indices can be taught in the classroom. Finally, we demonstrate how students can collect authentic data in the field while learning some arthropod systematics and practicing their quantitative reasoning skills, all within the context of edge effect ecology and habitat conservation.

The Next Generation Science Standards (NGSS), the AP Biology Course and Exam Description (AP Bio CED), and Vision and Change all call for life science teachers to incorporate into their curricula as many opportunities as possible for students to practice quantitative reasoning (i.e., quantitative analysis or mathematical thinking; AAAS, 2011; NGSS Lead States, 2013; College Board, 2015). Though often loosely referred to as numeracy, quantitative reasoning has been defined by Thompson (1993) to include not only counting and measuring, but also the act of comparing quantities, determining differences between quantities, and analyzing those differences. The NGSS extend Thompson's framework to include using mathematics to make quantitative predictions (NGSS Lead States, 2013). Mayes and Myers (2014) have expounded on the relevance of reasoning practice through their description of quantitative reasoning within context (called QR-C). Mayes and Myers define QR-C as “mathematics and statistics applied in real-life, authentic situations that impact an individual's life as a constructive, concerned, and reflective citizen.” Indeed, a primary and critical role of the science teacher is to provide students with as many opportunities as possible to practice QR-C in a scientific context. The pitfall trap activity we describe below is a cheap, simple, and weeks-long investigation that provides an opportunity for students to engage in QR-C while they are exposed to several ecological concepts and two data analysis techniques.

### Curriculum Connections

Within the topic of Conservation and Biodiversity in the IB Biology course guide (International Baccalaureate Organization, 2014), students are expected to understand the biogeographic variables that affect biodiversity, including edge effect, and be able to analyze and compare “the biodiversity of two local communities using Simpson's reciprocal index of diversity” (p. 131). Although the AP Bio CED does not mention edge effect specifically, students are expected to understand that species composition and diversity are used to measure and describe community structure. A study of the edge effect phenomenon serves well to fulfill this objective (College Board, 2015). Moreover, Learning Objective 4.12 in the AP Bio CED suggests that teachers should provide students with an opportunity to apply quantitative reasoning within the context of an ecological phenomenon such as edge effect: “The student is able to apply mathematical routines to quantities that describe communities composed of populations of organisms that interact in complex ways” (p. 87). An investigation of edge effect on diversity also allows students to employ several of the science practices described in the AP Bio CED. Thus, the edge effect study described here not only emphasizes an important focus on QR-C, but also fulfills learning objectives in the IB Biology course guide, the AP Bio CED, and likely most any college general biology lab course.

### What Is Edge Effect?

Habitat edges are abrupt transition zones where communities from adjacent and distinct habitats meet and interact (Holland & Risser, 1991). Edge effect theory, first popularized by Eugene P. Odum in the first edition of his widely used textbook Fundamentals of Ecology (Odum, 1953), states that due to the impact of biotic and abiotic factors from multiple habitats, species diversity is expected to spike at an edge along a transect that moves from one habitat to another (Magura et al., 2001). Edge characteristics can also create distinct microclimates that provide valuable resources to insects and other small animals (Lövei & Sunderland, 1996), which can contribute to overall ecosystem function.

Human-created edges have been increasing for decades as a result of habitat destruction and fragmentation (Saunders et al., 1991; Haddad et al., 2015), and the resulting edge effect can negatively influence species survivorship and community stability. Fragmentation increases the proportion of edge to habitat area and can separate or isolate existing communities, disrupting gene flow and genetic diversity (Sarre et al., 1995). While edges and the interior habitats they outline tend to have distinct species assemblages (Saunders et al., 1991), survivorship of habitat generalists is often favored due to increased adaptability (Elton, 1958; Gibbs & Stanton, 2001). Thus, the study of edges has become progressively more important to the field of conservation biology, particularly with regard to the preservation of biodiversity.

The goal of the activities described below is to authentically and effectively expose biology students to a multitude of ecological phenomena and quantitative reasoning opportunities in the context of habitat edges and edge effect.

## Teaching Diversity Indices

### Quantitative Reasoning in Context (QR-C)

Nearly 100 years ago, scientists began applying mathematical models in an attempt to describe and predict community interactions such as predator–prey population dynamics (see Lotka, 1920). By the middle of the 20th century, ecologists began to tackle the problem of describing and comparing community characteristics like diversity, especially in the context of increasing human disturbance to ecosystems. Inventing new mathematical models and adapting existing models was necessary, and Simpson's and Shannon's diversity indices emerged as useful quantitative solutions in the context of species diversity.

The species richness of a community is simply a count of the total number of different species found in an area and is an easy concept for most students to grasp. However, as teachers present the concept of species diversity to students, they must keep in mind that species richness in a community can easily be mistaken as an indication of species diversity, but richness does not account for the fact that the individuals among species in most communities are distributed unevenly. Indeed, communities can be dominated by individuals from just a few species while all other species are rare. Therefore, species diversity indices account for both richness and evenness.

Have students consider the data in Table 1, which compares two hypothetical communities of wood warblers, a type of songbird. While both communities have the same species richness (i.e., 10 wood warbler species), the individuals in community A are more evenly distributed among the 10 species than the individuals in community B. In fact, the individual yellow-rumped warblers in community B occupy a greater proportion of the community than all other species combined. Ask students about which community is more diverse and why. To elicit ideas, teachers can ask students to also think about and discuss in what ways the two communities might function differently – for example, how competition for food might look different in each community. Finally, before presenting the diversity indices described below, tap into students' quantitative reasoning skills by asking student groups to see if they can come up with a way to use math to describe both species richness and evenness in the two communities. In large classes, at least one group is likely to suggest calculating the proportion of the total number of individuals in each community occupied by individuals within each species. Calculating proportions is the first step in calculating diversity indices.

Table 1.
Distribution of individuals among 10 species of wood warbler in two hypothetical communities.
SpeciesCommunity ACommunity B
TotalTotal
Yellow-rumped warbler 38 128
Black-throated green warbler 30 32
Palm warbler 24
Magnolia warbler 24
Blackburnian warbler 18
Nashville warbler 18
Yellow warbler 16
Orange-crowned warbler
Wilson's warbler
SpeciesCommunity ACommunity B
TotalTotal
Yellow-rumped warbler 38 128
Black-throated green warbler 30 32
Palm warbler 24
Magnolia warbler 24
Blackburnian warbler 18
Nashville warbler 18
Yellow warbler 16
Orange-crowned warbler
Wilson's warbler

### Diversity Calculation with Simpson's Lambda

Simpson (1949) developed a way to quantify diversity (ƛ or D; hereafter D) in a population, as shown by Equation 1:

$D=∑i=1sni(ni−1)N(N−1)$

where N is the total number of individuals in a community of S groups (species) and n is the total number of individuals of a species in the community.

Present the Simpson model to students and explain that Equation 1 is known as the unbiased estimate of Simpson's diversity in a community. In statistical terms, D is the probability of randomly sampling an individual (without replacement) from a community of species i two times in a row from N possible species. We subtract 1 from both ni and N because once the first individual is removed from the community and identified, there is one less individual in the community when the second sample is taken. Putting it more simply for students, Simpson's diversity index can be described as the probability that two individuals, chosen one after the other at random from the community, will belong to the same species.

Since subtracting 1 from ni and N when samples are large has little effect on Simpson's D, students are pleased to learn about Equation 2 (which is an approximation of Equation 1):

$D=∑i=1Spi2$

where pi equals the proportion of individuals of the ith species (i = any of the possible species) to the total number of individuals of all species combined, or the proportional abundance.

The logic behind Equation 2 is that we know that ni/N equals the proportional abundance (pi) of the ith species. Since (ni − 1) and (N − 1) are nearly equal to ni and N, respectively, the unbiased estimate of D from Equation 2 is not notably different from pi2. Therefore, we most often use Equation 2 for simplicity, and that equation is indeed easier for students to set up in a spreadsheet.

Note to IB teachers: The IB Biology course guide (2014) describes Simpson's reciprocal index of diversity. Some ecologists do use this statistic, $D′=1−D$, as their diversity index, but it is rarely found in the literature. This reciprocal index is formally called the Gini-Simpson index and returns the probability that two individuals chosen at random from the community will belong to two different species. The higher the $D′$, the more diverse the community.

### Diversity Calculation with Shannon's H′

The Simpson index is mathematically simple for students to grasp; however, the Shannon index is the most common diversity index found in the ecology literature. Students may find the mathematical model for the Shannon index more interesting than Simpson's D and the story of its invention intriguing.

After World War II, Claude Shannon published a landmark paper titled “A Mathematical Theory of Communication” (Shannon, 1948) in which he simultaneously explained his wartime discoveries from investigating cryptography and founded the field of information theory (Tellenbach et al., 2009). Among the seminal concepts that Shannon outlined in this paper was a mathematical expression for optimizing encrypted messages and quantifying information uncertainty. This expression came to be known as Shannon's H, and it quickly found applications in a variety of academic fields, one of which was ecology. Specifically, ecologists adapted Shannon's H to a version known as Shannon's H′ or the Shannon index. The Shannon index is a measurement of species diversity for a given area that accounts for the total number of species and the evenness of the distribution of individuals among those species. The general expression for the Shannon index is shown in Equation 3 (with symbols following those used in Equations 1 and 2):

$H′=∑i=1S−pi(lnpi)$

Explain to students that the simplest conceptual explanation for how the Shannon index is applied in ecology is that it represents the average “uncertainty” in predicting to what species an individual chosen at random from a collection of S species and N individuals will belong. Precise discussion of the nature of mathematical uncertainty is beyond the scope of most biology classrooms, and thus it is sufficient to define uncertainty for students as the amount of information needed to predict a given outcome. For example, the amount of uncertainty associated with predicting whether a fair coin will land on heads or tails is less than the uncertainty associated with predicting the number that will result from the roll of a fair die. Students should understand that the Shannon index increases with either a greater number of species or more “evenness” in the number of individuals among all species, or a combination of both. Increasing the number of species is analogous to increasing the number of sides on a die. Similarly, decreasing the “evenness” in the number of individuals among all species is analogous to weighting the die or altering the faces so that certain numbers are more represented than others (i.e., more dominant). From this analogy, students might develop the intuition that a maximum Shannon value would be an ecosystem with an infinite number of species and with each species represented evenly, much as the most uncertain die would be a die with an infinite number of faces and an equal likelihood of landing on any particular face.

### Student Practice

The calculations for Simpson's and Shannon's diversity indices for the hypothetical data in Table 1 are shown in Table 2. Have students use Equation 2 and the data in Table 1 to calculate Simpson's D for communities A and B. Students should find that community A has a Simpson's D of 0.1218 while for community B, D = 0.4412. Ask students for ideas about how to interpret D. In brief, if all individuals are of one species, then D = 1 and the probability that two individuals selected at random from the community will be the same species is also 1. As the proportion of the individuals of a single species in a community approaches 1 (i.e., one species dominates the others with respect to number of individuals), D approaches 1.0 and there is low diversity in the area. This phenomenon can be seen with community B (D = 0.4412) in Table 2. In other words, the probability is high of randomly selecting a yellow-rumped warbler as both the first and second individuals from the community. However, if an area has high species richness and all species are represented nearly equally, as can be seen with community A (D = 0.1218) in Table 2, then D will be far less than 1.0, indicating that there is high species diversity in the area and that two individuals chosen at random from the sample will likely not belong to the same species.

Table 2.
Distribution of individuals among 10 species of wood warbler in two hypothetical communities (see Table 1), with calculations for proportion, Simpson's index, and Shannon's index.
Community ACommunity B
SpeciesTotalProportionSimpson's
pi2
Shannon's
pi(ln pi)
TotalProportionSimpson's
pi2
Shannon's
pi(ln pi)
Yellow-rumped warbler 38 0.19 0.0361 0.3155 128 0.64 0.4096 0.2856
Black-throated green warbler 30 0.15 0.0225 0.2846 32 0.16 0.0256 0.2932
Palm warbler 24 0.12 0.0144 0.2544 0.04 0.0016 0.1288
Magnolia warbler 24 0.12 0.0144 0.2544 0.04 0.0016 0.1288
Canada warbler 20 0.10 0.0100 0.2303 0.03 0.0009 0.1052
Blackburnian warbler 18 0.09 0.0081 0.2167 0.03 0.0009 0.1052
Nashville warbler 18 0.09 0.0081 0.2167 0.02 0.0004 0.0782
Yellow warbler 16 0.08 0.0064 0.2021 0.02 0.0004 0.0782
Orange-crowned warbler 0.03 0.0009 0.1052 0.01 0.0001 0.0461
Wilson's warbler 0.03 0.0009 0.1052 0.01 0.0001 0.0461
Total 0.1218 2.185 Total 0.4412 1.295
Community ACommunity B
SpeciesTotalProportionSimpson's
pi2
Shannon's
pi(ln pi)
TotalProportionSimpson's
pi2
Shannon's
pi(ln pi)
Yellow-rumped warbler 38 0.19 0.0361 0.3155 128 0.64 0.4096 0.2856
Black-throated green warbler 30 0.15 0.0225 0.2846 32 0.16 0.0256 0.2932
Palm warbler 24 0.12 0.0144 0.2544 0.04 0.0016 0.1288
Magnolia warbler 24 0.12 0.0144 0.2544 0.04 0.0016 0.1288
Canada warbler 20 0.10 0.0100 0.2303 0.03 0.0009 0.1052
Blackburnian warbler 18 0.09 0.0081 0.2167 0.03 0.0009 0.1052
Nashville warbler 18 0.09 0.0081 0.2167 0.02 0.0004 0.0782
Yellow warbler 16 0.08 0.0064 0.2021 0.02 0.0004 0.0782
Orange-crowned warbler 0.03 0.0009 0.1052 0.01 0.0001 0.0461
Wilson's warbler 0.03 0.0009 0.1052 0.01 0.0001 0.0461
Total 0.1218 2.185 Total 0.4412 1.295

Next, have students use Equation 3 and the data in Table 1 to calculate Shannon's H′ for communities A and B. Students should find that community A has a Shannon's H′ of 2.185 while for community B, H′ = 1.295 (Table 2). Community A has more evenness among the 10 species compared to community B and thus has higher species diversity. Some students may notice that in community B, the dominant yellow-rumped warbler (p = 0.64) has a −pi(ln pi) value (0.2856) that is nearly identical to the −pi(ln pi) value (0.2932) for the black-throated green warbler, even though p = 0.16 for the black-throated green warbler.

Lastly, ask students to study Figure 1: a graph of −pi(ln pi) that illustrates the uncertainty contribution of a single species i as a function of pi (the proportional abundance). Help students recognize that Figure 1 shows why the Shannon index reaches a maximum uncertainty value as the distribution of individuals among species in a community becomes more even. Using Figure 1A, explain to students that by using natural log (ln), Claude Shannon was able to create a weighted index where the maximum contribution of a “species” is not in the center of the distribution. For example, a species with pi = 0.25 contributes more to the overall diversity index than one with pi = 0.75. Indeed, the maximum value that any one species (species “II”; Figure 1A) can contribute to the overall diversity index is just under 0.4 (0.3679), and rare species (species “I”) contribute as little to the diversity index of a community as do highly dominant species (species “III”). In fact, the more dominant one species is in a community, the more relatively rare all other species must be. However, students should understand that rare species can make significant contributions to ecosystem functioning (Lyons et al., 2005). For perspective, rare species are defined by ecologists as comprising less than 5% of the maximum observed percentage of counts (animals), 1% of the maximum percentage cover (plants) (Mouillot et al., 2013), or 1–5% ecosystem biomass (Lyons et al., 2005). It is important to note here that for all practical purposes, Shannon's H′ is only an estimate of species diversity, because for communities that are high in species richness, surveys are likely not to “capture” all species, especially those that are rare.

Figure 1.

(A) The Shannon diversity index mathematical model for species in a community occupying any proportion of the total number of individuals of all species. The maximum value a species (II) can contribute to Shannon's H′ is 0.3679. Rare species (I) and dominant species (III) can contribute equally to Shannon's H′. (B) An unbounded graph of −x(ln x) as a function of x, demonstrating that the Shannon index is defined on the interval from 0 to 1 due to the input being the proportion pi.

Figure 1.

(A) The Shannon diversity index mathematical model for species in a community occupying any proportion of the total number of individuals of all species. The maximum value a species (II) can contribute to Shannon's H′ is 0.3679. Rare species (I) and dominant species (III) can contribute equally to Shannon's H′. (B) An unbounded graph of −x(ln x) as a function of x, demonstrating that the Shannon index is defined on the interval from 0 to 1 due to the input being the proportion pi.

## Applying the QR-C of Diversity Indices with Real Data: The Student Edge Effect Study

The QR-C approach involves moving back and forth between the realms of science and mathematics (L. S. Mead et al., unpublished data). In our case described here, students move between the science of ecology and edge effect and the mathematics of diversity indices. The entire data-collection activity spans several weeks, with sampling occurring once per week. We use the other classroom days to discuss various additional ecological concepts in the curriculum. As a culminating activity, student groups write condensed scientific papers (more to follow). Throughout each component of the activity, the primary goal is for students to utilize their QR-C skills in the context of a realistic, hypothesis-driven investigation. We encourage teachers to borrow from the information we provide below and create their own data-based engagement activity that is specific to their local ecology.

### Study Area and Sampling Design

The field activity described here takes place annually in a “natural” unmowed hillside that slopes down to a small lake (Figure 2). The hillside is ~30 m wide and runs the length of the east side of the lake and the west side of the school. Between the unmowed slope and the school is a mowed lawn 10–15 m in width. Where the two areas meet is a hard tall-grass to mowed-grass edge. The student goal for the last several years has been to investigate edge effect theory by testing one of its central hypotheses: edges have characteristics of both converging habitats and thus attract species from both habitats. Students then make the prediction that when arthropods are collected at the edge of the unmowed area and within its interior, they will observe greater arthropod species richness and species diversity in the edges than in the interior. Sampling within the mowed-grass habitat is not possible as a third treatment because mowing and other activities are uncontrolled.

Figure 2.

Rough placement of the edge (E) and unmowed interior (I) pitfall traps (white circles) within the study area.

Figure 2.

Rough placement of the edge (E) and unmowed interior (I) pitfall traps (white circles) within the study area.

Student groups set up two 40 m transects within the grassy area west of the school. One transect is placed along the habitat edge where the mowed and unmowed area meet and the other runs parallel to the edge transect 10 m within the unmowed area (Figure 2).

Students collect arthropod samples along each transect by using pitfall traps. Pitfall traps are animal sampling devices, long used by ecologists to passively collect population and community data on target species when individuals fall into them. Students use a pitfall trap method similar to that described in Magura et al. (2001). Each year, student teams dig holes every 5 m along each transect. The holes are big enough to fit two 470 mL (16 oz.) plastic cups (Figure 3). The first cup is used as a soil stabilizer so that the inner cup can be removed and brought inside for analysis and refilling. The inner cup is filled with a 5% ethylene glycol (antifreeze) solution to mitigate evaporation and overnight freezing. The solution is also 0.1% detergent to remove surface tension. One class section installs the edge pitfall traps and the other class section installs traps along the interior transects.

Figure 3.

An edge pitfall trap installed by a student team.

Figure 3.

An edge pitfall trap installed by a student team.

### Data Collection

Each student team (eight teams per class section) collects data three times from a single pitfall trap over a three-week period in October. Trapped arthropod samples are removed with tweezers from the solution and placed on white paper towel for identification. Students use two online methods to identify their samples: Google Images and the American Museum of Natural History dichotomous keys for arthropods (https://www.amnh.org/learn/biodiversity_counts/ident_help/Text_Keys/text_keys_index.htm); teacher guidance is sometimes helpful. Having students accurately identify all of the individuals they collect to the species level would be an overreach in this context, given students' lack of experience with systematics. Thus, while students try to identify many individuals to species, they ultimately use the family taxon as a proxy for “species” in their richness and diversity calculations. Data sharing with other groups across both class sections is accomplished with a shared Google speadsheet where students record data from their weekly collections (Figure 4).

Figure 4.

Example of a student pitfall trap sample.

Figure 4.

Example of a student pitfall trap sample.

Each lab group then analyzes all the shared data after the three-week collection period as if they performed the entire study themselves. Once students have tabulated and pooled all of their data, they calculate species richness and use both Simpson's and Shannon's indices to calculate diversity.

### Data Analysis

Table 3 shows actual student-processed data from the collection seasons of 2016 and 2017. Notice that species richness and diversity are different between the two years among both habitats sampled. Also notice that only the species diversity indices from 2017 support the hypothesis that edges attract species from both converging habitats; the 2017 collection reveals slightly more diversity at the edge (D = 0.25; H′ = 1.91) than in the interior (D = 0.32; H′ = 1.71).

Table 3.
Species richness and Simpson's and Shannon's diversity indices for edge and interior community samples from three weekly collection periods in October, 2016 and 2017, at Fairview High School in Boulder, Colorado. Note that smaller values for Simpson's D and larger values for Shannon's H′ indicate greater species diversity. See Supplemental Material Tables S1 and S2 for raw data from these years, including the categories within which species were grouped.
20162017
EdgeInteriorEdgeInterior
Species richness 14 17 16 20
Simpson's D 0.21 0.11 0.25 0.32
Shannon's H′ 1.99 2.41 1.91 1.71
20162017
EdgeInteriorEdgeInterior
Species richness 14 17 16 20
Simpson's D 0.21 0.11 0.25 0.32
Shannon's H′ 1.99 2.41 1.91 1.71

Results like these provide a valuable opportunity for students to come up with additional hypotheses to explain unexpected results. For example, student groups from 2016 hypothesized that the mowed habitat presents arthropods with enough of a hazard that many species avoid the area, which may include the edge. If students are provided with data from previous years, they can be challenged to think about why data like these might vary from one year to the next. For example, it is possible that the richness and diversity differences we observed between years may simply be a function of the uncontrolled variable of weather, which can have a strong effect on the fall activity for many kinds of insects (Kingsolver, 1989). Indeed, in 2016, the October collection period occurred after several weeks of warm weather without rain, whereas the October collection period in 2017 followed a more seasonable September with occasional rain events and cooler temperatures that included a short snow event between the first and second sample collections.

In the 2018 collection season, we were able to add the mowed area to our pitfall trap samples. Students discovered that the species found in the mowed area were essentially a subsample of the interior (tall grass) area. They hypothesized that the tall grass could be the source of species occupying the edge and mowed areas, thereby challenging assumptions and conclusions made in earlier years.

### A Statistical Extension for Data Analysis

If students have already been exposed to and had practice with inferential statistical tests like Student's t-test, they may be interested in a more rigorous comparison of the Shannon index for the two habitats. In 1970, Hutcheson developed a version of the t-test to use with the Shannon model (Hutcheson, 1970), our Equation 4:

$t=H′a−H′bsH′a2+sH′b2$

where Ha is the overall Shannon index for one of the habitats and Hb is the overall Shannon index for the other habitat. The sample variance for each habitat is indicated by s2 and is calculated after determining H′ for each individual pitfall trap.

If students use the Hutcheson t-test to test the null statistical hypothesis that the Shannon index between two habitats is not different, it is important to mention and emphasize to students that their ability to make any sweeping claims is considerably limited not only by sample size but also by the size of their particular field site. Our site is long and narrow, spanning 30 m at its widest, and is close to the west wall of the school building. Edge effects can be measured up to 100 m into an intact habitat (Broadbent et al., 2008), so it is likely that our interior is functioning as all edge habitat. However, this limitation does not reduce the potential of this activity for guiding students through quantitative reasoning to reveal and reinforce interesting ecological phenomena, but also for generating more questions, and for designing additional data-collection techniques and experiments.

### Student Reports

From the results of each year's edge effect study, student groups write miniature scientific papers, complete with a title, abstract, introduction, methods, results, discussion, and literature cited section. Students are provided with online links to several freely accessible published papers from which they gather information to aid in the writing of their own papers. A sample of these papers can be found in Table 4. The published papers also function as mentor texts from which students model the scientific practice of communicating through technical writing. Students also use Google Scholar to search for other relevant papers to help them with their reports.

Table 4.
Four key published papers available for free online, with summaries, that students can use in the writing of their own reports of their edge effect study. The TinyURL links will take students directly to a pdf of each paper.
AuthorsTitleSummaryTinyURL
Broadbent et al. (2008)  Forest fragmentation and edge effects from deforestation and selective logging in the Brazilian Amazon The authors quantify and summarize fragmentation in the Brazilian Amazon and find that 53% of forest habitat is located within 2 km of an edge. Students will benefit from the descriptions of 146 edge effects that are also presented. https://tinyurl.com/y9efsl57
Magura et al. (2001)  Forest edge and diversity: carabids along forest–grassland transects The authors used pitfall traps to collect carabid beetles and used Shannon's H′ to show higher diversity at an edge compared to a forest interior. Several abiotic factors were shown to be associated with the edge effect. Students can use this paper for their methodology and as a comparison to their own results. https://tinyurl.com/y89ck3d2
Mouillot et al. (2013)  Rare species support vulnerable functions in high-diversity ecosystems In the context of conservation and extinction, the authors discuss what it means for a species to be rare in its habitat and the ecological functions maintained by rare species. Most of the species students collect will likely be rare, and this paper puts their results in a greater ecological context. https://tinyurl.com/zskotmg
Saunders et al. (1991)  Biological consequences of ecosystem fragmentation: a review The authors argue that research on fragmentation should focus on the impacts fragmentation has on natural systems and that managing these systems should focus on conservation. Students can use this paper to build their own case for doing research on the effect of edges on species diversity, even in their own “backyards.” https://tinyurl.com/y9ucgaqs
AuthorsTitleSummaryTinyURL
Broadbent et al. (2008)  Forest fragmentation and edge effects from deforestation and selective logging in the Brazilian Amazon The authors quantify and summarize fragmentation in the Brazilian Amazon and find that 53% of forest habitat is located within 2 km of an edge. Students will benefit from the descriptions of 146 edge effects that are also presented. https://tinyurl.com/y9efsl57
Magura et al. (2001)  Forest edge and diversity: carabids along forest–grassland transects The authors used pitfall traps to collect carabid beetles and used Shannon's H′ to show higher diversity at an edge compared to a forest interior. Several abiotic factors were shown to be associated with the edge effect. Students can use this paper for their methodology and as a comparison to their own results. https://tinyurl.com/y89ck3d2
Mouillot et al. (2013)  Rare species support vulnerable functions in high-diversity ecosystems In the context of conservation and extinction, the authors discuss what it means for a species to be rare in its habitat and the ecological functions maintained by rare species. Most of the species students collect will likely be rare, and this paper puts their results in a greater ecological context. https://tinyurl.com/zskotmg
Saunders et al. (1991)  Biological consequences of ecosystem fragmentation: a review The authors argue that research on fragmentation should focus on the impacts fragmentation has on natural systems and that managing these systems should focus on conservation. Students can use this paper to build their own case for doing research on the effect of edges on species diversity, even in their own “backyards.” https://tinyurl.com/y9ucgaqs

## Conclusions

From the activity we describe here, one can see the great potential for students to practice quantitative reasoning within the context of a simple, yet data-rich, investigation during a curricular unit on ecology and conservation. Students learn about and practice mathematical techniques for quantifying diversity and come away with a clear understanding of the difference between species richness and species diversity. Students also learn and practice a common and useful field technique for collecting information on arthropod populations in the context of community ecology and habitat conservation. Additionally, the activity reinforces for students how interesting and helpful mathematical models in science can be.

We encourage teachers to attempt some version of this activity with their students to bring more QR-C into the curriculum and to send students literally into the field. Whether the activity includes a much more ambitious data-collection investigation than we have described here or is a “dry lab” model of an ecosystem, students inevitably benefit from opportunities to move between the scientific problems and questions we present to them and the mathematical tools available to answer those questions and solve the problems. Again, in the words of Mayes and Myers (2014), it should be a primary goal of all science teachers to include as a central curricular focus “mathematics and statistics applied in real-life, authentic situations that impact an individual's life as a constructive, concerned, and reflective citizen.”

The authors would like to thank E. Schultheis, S. Bergman, A. Martin, and R. Langendorf for their extremely helpful comments and suggestions on earlier versions of the manuscript. The manuscript was also improved by the comments of two anonymous reviewers. Finally, we would like to acknowledge the hard work and dedication of P. K. Strode's AP/IB Biology students of 2016, 2017, and 2018, as well as V. J. Haggans (2018) for providing some additional clarifying insight into the weighted nature of the Shannon model.

## References

References
AAAS
(
2011
).
Vision and Change in Undergraduate Biology Education: A Call to Action
.
Washington, DC
:
AAAS
. http://visionandchange.org/files/2013/11/aaas-VISchange-web1113.pdf.
Broadbent, E.N., Asner, G.P., Keller, M., Knapp, D.E., Oliveira, P.J. & Silva, J.N. (
2008
).
Forest fragmentation and edge effects from deforestation and selective logging in the Brazilian Amazon
.
Biological Conservation
,
141
,
1745
1757
.
College Board
(
2015
).
AP Biology Course and Exam Description
.
New York, NY
:
The College Board
.
Davis-Berg, E.C. & Jordan, D.R. (
2018
).
The innovative use of Mathematica to teach biodiversity
.
American Biology Teacher
,
80
,
372
378
.
Delang, C.O. & Li, W.M. (
2013
). Species richness and diversity.
In Ecological Succession on Fallowed Shifting Cultivation Fields
.
Dordrecht, The Netherlands
:
Springer
.
Doup, M.L. (
2018
).
Using an outdoor activity on local plant biodiversity to teach conservation ecology and promote environmentally responsible behaviors
.
American Biology Teacher
,
80
,
359
364
.
Elton, C.S. (
1958
).
The Ecology of Invasions by Animals and Plants
.
Chicago, IL
:
University of Chicago Press
.
Gibbs, J.P. & Stanton, E.J. (
2001
).
Habitat fragmentation and arthropod community change: carrion beetles, phoretic mites, and flies
.
Ecological Applications
,
11
,
79
85
.
Haddad, N.M., Brudvig, L.A., Clobert, J., Davies, K.F., Gonzalez, A., Holt, R.D., et al (
2015
).
Habitat fragmentation and its lasting impact on Earth's ecosystems
.
,
1
,
e1500052
.
Holland, M.M. & Risser, P.G. (
1991
). The role of landscape boundaries in the management and restoration of changing environments: introduction. In
Ecotones
(pp.
1
7
).
Boston, MA
:
Springer
.
Hutcheson, K. (
1970
).
A test for comparing diversities based on the Shannon formula
.
Journal of Theoretical Biology
,
29
,
151
154
.
International Baccalaureate Organization
(
2014
).
Diploma Programme Biology Guide
.
Cardiff, Wales
:
International Baccalaureate Organization
.
Kingsolver, J.G. (
1989
).
Weather and the population dynamics of insects: integrating physiological and population ecology
.
Physiological Zoology
,
62
,
314
334
.
Lotka, A.J. (
1920
).
Analytical note on certain rhythmic relations in organic systems
.
Proceedings of the National Academy of Sciences USA
,
6
,
410
415
.
Lövei, G.L. & Sunderland, K.D. (
1996
).
Ecology and behavior of ground beetles (Coleoptera: Carabidae)
.
Annual Review of Entomology
,
41
,
231
256
.
Lyons, K.G., Brigham, C.A., Traut, B.H. & Schwartz, M.W. (
2005
).
Rare species and ecosystem functioning
.
Conservation Biology
,
19
,
1019
1024
.
Magura, T., Tóthmérész, B. & Molnár, T. (
2001
).
Forest edge and diversity: carabids along forest-grassland transects
.
Biodiversity and Conservation
,
10
,
287
300
.
Mayes, R. & Myers, J. (
2014
). Quantitative reasoning: changing practice in science and mathematics. In
Quantitative Reasoning in the Context of Energy and Environment
(pp.
1
35
).
Rotterdam, The Netherlands
:
Sense
.
Minteer, C.R., Smith, M.C., Lake, E.C. & Pokorny, E. (
2018
).
Teaching complex ecological concepts through a demonstration garden: biodiversity, invasive species, and conservation in practice
.
American Biology Teacher
,
80
,
346
352
.
Mouillot, D., Bellwood, D.R., Baraloto, C., Chave, J., Galzin, R., Harmelin-Vivien, M., et al (
2013
).
Rare species support vulnerable functions in high-diversity ecosystems
.
PLoS Biology
,
11
,
e1001569
.
(
2013
).
Next Generation Science Standards: For States, By States
. http://www.nextgenerationscience.org/next-generation-science-standards.
Odum, E.P. (
1953
).
Fundamentals of Ecology
.
:
W.B. Saunders
.
Sarre, S., Smith, G.T. & Meyers, J.A. (
1995
).
Persistence of two species of gecko (Oedura reticulata and Gehyra variegata) in remnant habitat
.
Biological Conservation
,
71
,
25
33
.
Saunders, D.A., Hobbs, R.J. & Margules, C.R. (
1991
).
Biological consequences of ecosystem fragmentation: a review
.
Conservation Biology
,
5
,
18
32
.
Shannon, C.E. (
1948
).
A mathematical theory of communication
.
Bell System Technical Journal
,
27
,
379
423
, 623–656.
Simpson, E.H. (
1949
).
Measurement of diversity
.
Nature
,
163
,
688
.
Spellerberg, I.F. & Fedor, P.J. (
2003
).
A tribute to Claude Shannon (1916–2001) and a plea for more rigorous use of species richness, species diversity and the ‘Shannon-Wiener’ index
.
Global Ecology and Biogeography
,
12
,
177
179
.
Tellenbach, B., Burkhart, M., Sornette, D. & Maillart, T. (
2009
). Beyond Shannon: characterizing Internet traffic with generalized entropy metrics. In
International Conference on Passive and Active Network Measurement
(pp.
239
248
).
Berlin, Germany
:
Springer
.
Thompson, P.W. (
1993
).
Quantitative reasoning, complexity, and additive structures
.
Educational Studies in Mathematics
,
25
,
165
208
.