In this article, we describe how to utilize differently colored M&M candies to represent species within a simulated biological community, and obtain population and diversity estimates utilizing Lincoln-Petersen and Shannon-Weaver methods, respectively. Through use of this activity, our students gain a better understanding of mathematical applications in biological research, and are exposed to basic census and community analysis techniques utilized by practicing biologists. Additionally, this activity may be utilized in various instructional situations where, otherwise, it might prove impractical to take students on a fieldtrip to allow practice of these procedures.

## Introduction

As is well known, biological sciences, medicine, and every STEM-related field require that students and practitioners have a good understanding of what the National Council of Education and the Disciplines (NCED) and the Woodrow Wilson National Fellowship Foundation (WWNF) called quantitative literacy or numeracy. The Next Generation Science Standards (NGSS) point out that there is a direct correlation between mathematics and success in college course work and likelihood of successful graduation (NGSS, 2013). Some examples of applications of quantitative literacy and numeracy in biological sciences and medicine include descriptive statistics, computer-based mathematical/statistical programs for mapping genomes, assessing laboratory experiments and clinical trials, conducting ecological studies, comparing risks, and determining rates of change—to list only a few (Steen, 2001). Additionally, almost every major public issue depends upon data, projections, inferences, and systematic thinking and methods related to quantitative literacy (Steen, 2001). As stated within Appendix C in the NGSS, scientists must be able to integrate content knowledge and practice in their work, but emphasis upon this integration is not always practiced in science instruction (NGSS, 2013).

Researchers have demonstrated that appropriate use of manipulatives broadens students' understanding of mathematical concepts, transforms mathematical problem solving into a social activity, increases student engagement, and decreases student anxiety (Jungck et al., 2010). When teaching basic ecological principles and systems at the high school, community college, or university level, it is an excellent practice to involve students with living organisms in a “field-based” context or laboratory (Brady & Brady, 2009). However, many factors may preclude this approach, including the location of the school (urban, suburban, or rural), school policies and procedures, and budgets for travel and insurance as may be required for field trip activities. Additionally, online course delivery formats usually preclude field-based activities that may be relevant, group-based, and ecological. As a result, some educators have developed classroom or laboratory activities using live organisms in order to model ecological and biological community principles (Whiteley et al., 2007). However, factors such as approval of these types of activities through animal care and use committees and/or the size (enrollment) of the lecture or lab classes may place limits on these in-class activities.

## Methods

To help dispel the idea that biology is a numeracy-light discipline and to facilitate integration of knowledge and application, we present two activities utilizing M&M candies that allow biology students to conduct quantitative sampling and estimation techniques in a manner that is both rigorous and accessible. Utilizing our activity, community college students learn the conditions and use of the Lincoln-Petersen population estimate and Shannon-Weaver diversity index. The mathematical algorithms associated with these basic, ecological census techniques are currently utilized by practicing biologists. Our exercise is based upon the biological content areas of ecology, population biology, and diversity. In turn, these content areas of biological sciences are an underpinning for students' understanding of natural selection and evolutionary processes—the central, unifying theme(s) of modern biology.

Use of the packaged M&M candies provides some advantages in a large classroom setting or when use of live specimens is not possible. One advantage is that the M&M candies can be easily explained as a “closed population” because the candies are already, literally, enclosed within a bag. Also, the candies are of several different colors, making them distinct for the counting/marking purposes necessary to obtain data utilized in calculation of the Lincoln-Petersen population estimate and Shannon-Weaver diversity index. Another advantage is that students are able to literally see the entire, simulated community and the differently colored species within this community. This allows visualization of the mathematical results of their sampling activities, as explained below. An additional advantage, if your students do not have food allergies, is the experimental subjects are edible and, if handled with care during the experiments, may be presented to the students at the end of the lesson as a reward.

We explain to students that scientists, researchers, and consultants need to know the various types and species of organisms present in natural communities and ecosystems. Additionally, to make informed decisions related to the well-being of the natural community or particular species within that community, scientists also need to know (or be able to approximate) the population size of the various species. It would be helpful if, prior to conducting the activities, students were able to distinguish between populations, communities, and ecosystems.

Our technique involves an application of the Lincoln-Petersen method of population estimation and calculation of a Shannon-Weaver diversity index value to our classes utilizing plain, chocolate M&Ms as our different species within a hypothetical community. Each different color of M&M represents a different species of animal. For example, the teacher might explain that, due to difficulties of travel, equipment, and time, we will use each different color of M&M to represent a different species of mammal that we could not ordinarily study. The blue M&M candies might be Meadow Voles, the green M&Ms might be Least Shrews, the yellow M&Ms might be Wolves, the brown M&Ms might be White-Tailed Deer, etc. The teacher may leave the composition of the simulated community up to the students. We explain that knowledge of the number of a particular species within a community is essential for a working biologist to make informed decisions regarding management and conservation of that species within its community.

We explain that the Lincoln-Petersen method is the most basic population estimation method and involves one session of capture, marking, and release of individuals of a particular species (or in our simulation, a particular color of M&M) within a study community, and a second, recapture session (Greenwood, 1996). In essence, the Lincoln-Petersen method estimates the population size of a particular species as the ratio or proportion of recaptured marked individuals compared to all individuals of that species (marked and unmarked) captured during the two sampling sessions. For the Lincoln-Petersen estimate to be accurate, the study population must be closed, at least during the study session. This means that no individuals die, leave, or are born within the populations and community during the study period. Also, all individuals of a particular species under study must have an equal probability of capture or census. Additionally, identifying marks used on captured individuals must remain distinguishable throughout the brief study period.

We point out to our students that the Lincoln-Petersen technique is most accurate for large sample sizes of a particular species, assuming that the overall population size of a species within a community is fairly large. Researchers have determined that, if the number of marked individuals obtained in the second sampling event is less than eight, the Lincoln-Petersen population estimate will be biased. Confidence limits calculated from Lincoln-Petersen estimates have been found to be reasonably accurate if the number of marked individuals found in the second sample is greater than 50 (Greenwood, 1996).

The Shannon-Weaver diversity index calculation requires that the researcher be able to identify different types of individuals within a community. In a biological context, this might mean that the researcher is able to identify and classify individuals to the species level, or at least, be able to confidently separate the community into different, distinct, groups.

Before beginning this portion of our activity, we explain to students that the Shannon-Weaver index is a simple, common way in which researchers can determine the richness and evenness of species' distributions within natural communities. The greater the number of species within a particular community (species richness) and the more evenly they are distributed throughout the community is a good measure of community stability. The Shannon-Weaver index is based upon information theory and is, essentially, a measure of uncertainty in a system/community. Calculated values of the Shannon-Weaver index will range from 0 to 4. A value of 0, or close to 0, indicates a high probability that your next result will be the same as the previous result. For instance, if you chose a blue M&M from the sample container, there is a high probability that your next choice will also be a blue M&M. However, if the value is the opposite—a value of 4 or close to 4—this means that there is a low probability of obtaining the same color of M&M in your next choice. In other words, when the calculated value is close to four, the community or system under study is much more diverse. In a biological context, this means that many different kinds of species would be found within this community. Also, a value closer to 4 would indicate that the individuals within the community are evenly distributed throughout this community (Smith, 1996). The most important requirement for calculation of the Shannon-Weaver index is that individual species or types of organisms within a community be distinguishable from each other, as are our differently colored M&Ms.

All of these aforementioned conditions are satisfied well by our simulated M&M community. Additionally, use of the candies would also allow instructors to impose effects of natural selection on the community (adding or removing M&M candies, for instance) in order to have students analyze these effects upon their resulting population and community diversity estimates. With this prior knowledge, students are prepared to engage in the activities as teams of two partners. A detailed description of the activities follows, with examples and worksheets in the  Appendix to this article.

### Materials Needed

1. Chocolate M&M candies in “Fun-Sized” bags.

2. Appropriate plastic containers, with reclosing lids, for mixing the candies. (We use rectangular, 14 × 10 × 7 cm containers of 710 ml [24 oz] volume.) This size allows sufficient space for mixing and distribution of the candies and more random selection as students reach into containers to draw out individual M&Ms.

3. Calculation worksheets (see examples in the  Appendix).

4. Pencils and calculators (if allowed and appropriate).

5. Non-toxic markers (Crayola markers work well) if the instructors do not wish students to scratch the candies to mark them.

## Procedure: Lincoln-Petersen Estimate Calculation

The formula for a corrected Lincoln-Petersen estimate calculation is:

$N=(MS1+1)×(nS2+1)÷(MS2+1)−1$

Wherein: N = population estimate obtained from sampling efforts on two occasions.

MS1 = all individuals of a particular species marked in Sample 1.

nS2 = all individuals of a particular species (marked and unmarked) obtained in a second sample.

MS2 = all individuals of a particular species that were found to be marked in the second sample.

Adding values of 1 to the sample sizes and subtracting 1 from the total population estimate helps to correct for some bias in the basic Lincoln-Petersen calculation.

Each color of M&M will represent a particular species population within the total community. Plain chocolate M&Ms come in colors of blue, brown, green, orange, red, and yellow. The Mars Corporation currently packages M&M candies according to weight; therefore, proportions of colors in M&M candy packages are no longer standardized and tend to vary. However, the last reported proportions given by the Mars Corporation for M&M milk chocolate candies were 24% blue, 20% orange, 16% green, 14% yellow, 12% red, and 12% brown.

### Obtaining the First Population Estimate

1. Distribute Fun-Sized bags of M&Ms to the class, and have students work in pairs to complete the Lincoln-Petersen and Shannon-Weaver estimates.

2. Give each student team (pair) one plastic container (with lid).

3. Student pairs will place their M&Ms into the container and count the total number of M&Ms and total numbers of each color in the container. For the Lincoln-Petersen calculations, each color represents a different species of M&M. For example, each color may be a different species of mammal, butterfly, bird, or whatever the students wish to imagine the candies to represent.

4. Students record the resulting counts of the differently colored M&Ms on the provided worksheet (see student worksheet and example worksheet in the  Appendix).

5. Explain to the students that, by placing the M&Ms into a container, we have created a closed population where no individuals will be exiting the population, reproducing, or dying between sample events (if you can keep the students from eating the M&Ms!). At this point, the instructor might remind students that closed populations, short durations of time between sampling events, and no significant changes within the population between samplings are essential conditions that must be satisfied to obtain accurate population estimates using the Lincoln-Petersen method.

6. Students determine which color of M&M occurs in the highest number(s). This number will then be used as the sample size of subsequent sampling events (see student worksheet and example worksheet in the  Appendix).

7. For example, if blue M&Ms are in the greatest number in the population, blue M&Ms will be marked if any are randomly picked from the container in the first sampling event (MS1). Explain to students that, in a field survey, researchers might use tags, dyes, florescent powders, or other types of marking/identifying methods to designate individuals captured in the first sample. Also, remind students that the population estimate will be more accurate if large sample sizes are taken.

8. Return all M&Ms to the container and gently stir or shake the container (with the lid on, if shaken) to randomly distribute the candies in the container. Remove the lid and have one (or both) students close or avert their eyes and randomly draw out the same number of M&Ms as the most numerous color (blue, in our example). Make sure students draw out the M&Ms one at a time. This will assure equal probabilities of capture or census. (Remind students that this is another required condition of the Lincoln-Petersen estimation method).

9. If any of the M&Ms of the highest number (blue, in our example) are found in the sample, these candies will be marked by gently scratching their surface. A light fingernail scratch mark should be sufficient to identify them later. Alternatively, if the teacher desires, a non-toxic marker (such as a Crayola marker) may be used to mark the candies instead of scratching. Be sure to record the number of designated M&M species (blue, in our example; see accompanying worksheet) captured in this first sample on the provided worksheet. This is MS1.

10. Make sure all of the M&Ms are replaced into the provided container. Place the lid on the container, and again, gently shake the container or mix them.

11. Follow the sampling procedure describe in Step 8, have students obtain a second sample of M&Ms, and record the sampling result as nS2 on the worksheet.

12. Explain to the students that these particular steps are important because they allow for a random sample, wherein each individual has an equal probability of capture. This is one of the essential conditions that must be satisfied for an accurate population estimate utilizing the Lincoln-Petersen method.

13. Count the number of marked individuals obtained in the second sample; in our example, you would count the number of marked (scratched) blue M&Ms obtained in the second sample. This count gives you the value of (MS2). Record this value on the worksheet (see student worksheet and example worksheet).

14. Students will use their counts from the samples and complete a Lincoln-Petersen estimation of the M&Ms by utilizing the formula given in the worksheet (found in the  Appendix).

15. Students should be reminded (if necessary) to add 1 to their counts and subtract 1 from the obtained population estimate to account for sampling bias.

### Obtaining a Second Population Estimate

1. Next, have students utilize the color of M&Ms occurring in the lowest number (green, in our example found in the  Appendix) as the number of initially captured marked and released individuals in an initial sample (MS1).

2. Have the student return all M&Ms into their container and gently shake the container to randomly distribute the M&Ms.

3. Students (with their eyes closed and drawing M&Ms one at a time) will then draw out the same number of M&Ms as in Step 8. For instance, if students sampled 10 M&Ms in their first MS1 calculation, they will once more randomly draw out 10 M&Ms for the second initial sampling event and population estimate for the color of M&Ms in fewest number (green, in our example). If any of the randomly selected M&Ms are of the least common color, students will scratch them for marking purposes, then return all M&Ms to the container. Record the number of least common M&Ms obtained for MS1 on the provided worksheet.

4. Utilizing this sampling methodology assures that sample sizes are equivalent for the two population estimates. This will allow for comparison and discussion of sample size effects upon the Lincoln-Petersen population estimate.

5. Students will again gently shake or stir the candies in the container, and then randomly draw out the designated number as described in Step 8. Count the number of marked M&Ms (MS2) of the second, random sample nS2 (as described in Step 11, above).

6. As before, the number of marked green M&Ms in the second sample represents MS2. Record this value on the worksheet.

7. Using the provided formula on the worksheet, calculate a population estimate for the green M&M species in the community (as in Step 14).

8. Students will then compare their population estimates to the actual number of M&M species of both colors (those in highest [blue] and in lowest [green] numbers) in their simulated community, and discuss how accurate their estimates are compared to the actual counts of M&M candies.

### Follow-up Activities

A few questions for students to consider are:

• How close were your population estimates to the actual number of M&M species in the community?

• Were your estimates of the populations higher or lower than the actual population number in either or both estimates?

• If your population estimates were not very close to the actual numbers of the M&M species, what might be some factors that would account for the difference?

• Assume that all the conditions required for the Lincoln-Petersen index are demonstrated for a population of animals that you are studying. What could you infer about the population numbers of a particular species if you captured, marked, and returned several individuals during your first sampling period, but obtained no marked individuals after the second sampling session?

• Could you suggest a way (or ways) to obtain a more accurate population estimate?

After the activity is completed, instructors should discuss the results and students' conclusions concerning their obtained population estimates. It is reasonable to assume—because of the relatively small total number of M&M candies in the sample communities—that almost all of the population size estimates obtained would be biased.

One way to improve our estimates would be to combine more student groups' M&Ms and thereby obtain the larger sample sizes necessary for more accurate estimates. Also, as a possible extension of this activity, the instructor might point out that a repeated sampling method, such as the Schnabel method, will likely give more accurate results than the more common Lincoln-Petersen method. However, in a real-world application, it may not be possible for a researcher to obtain multiple samples in some locations or ecological/environmental settings. For instance, it might be undesirable to disturb the environment or species of concern on more than two occasions if the researcher is working within a sensitive environment or with a threatened or endangered species.

Although perhaps not the most ideal method to reinforce or teach ecological principles and/or techniques to students, we have found that our students have responded favorably to this technique and seem to obtain better conceptualization and understanding of the underlying mathematical and ecological concepts related to estimation of biological populations.

Because students have already counted the differently colored M&M species within their communities, a natural extension of our activity is to have students calculate a Shannon-Weaver diversity index number for their M&M communities and discuss the resulting value in terms of species richness and evenness. Because the numbers of the differently colored M&M species have already been determined, it is relatively easy (with calculators having a “natural log” function key).

### Procedure: Shannon-Weaver Diversity Index

The formula for calculation of the Shannon-Weaver diversity index is:

$H=Σ(pi)(−ln pi)$

Wherein: H = Calculated Shannon-Weaver index value

Σ = total sum

pi = proportion or average of a particular species obtained in a sample

-lnpi = inverse (opposite) of the natural log of the proportion for a particular species in the sample

Student teams will enter their previous, total counts of the M&M species in the provided worksheet and complete calculations for the Shannon-Weaver diversity index for their simulated community (see student worksheet and calculation examples worksheet in the  Appendix). The teacher might remind the students, as mentioned earlier, that this index is a measure of uncertainty and, the closer the obtained value is to 0, the more certain you become of sampling the same species (color) of M&M from the community each time that a sample is taken. The closer the value is to 4, the less likely you would obtain the same species of M&M candy from differing samples.

Extending our activity by having students calculate diversity indices for their M&M communities provides greater depth, practice, and reasoning into the biological and ecological concepts. Students learn techniques that researchers utilize in real-world estimates of biological community structure and abundance. The M&M methodology might also be incorporated into lessons and demonstrations concerning adaptation, cryptic or non-cryptic coloration, and basic genetics lessons. Instructors can design many lessons incorporating biological and mathematical concepts and critical thinking skills with the M&M model, therefore, in a manner of speaking, the lessons are already “in the bag”!

## References

References
2009
).
Using field experiences to study the land-use legacy
.
American Biology Teacher
,
71
(
7
),
419
423
.
Greenwood, J. J. D. (
1996
). Basic Techniques. In W. J. Sutherland (Ed.),
Ecological Census Techniques a Handbook
(
11
110
).
Cambridge
:
Cambridge University Press
.
Jungck, J. R., Gaff, H. & Weisstein, A. E. (
2010
).
Mathematical manipulative models: In defense of “Beanbag Biology.”
CBE Life Science Education
,
9
,
201
211
.
. (
2013
).
Next Generation Science Standards: For states by states
.
Washington, DC
:
.
Smith, R. L. (
1996
).
Ecology and Field Biology
(5th ed.).
New York
:
HarperCollins College Publishers
.
Steen, L. A. (Ed.). (
2001
).
Mathematics and Democracy: The case for quantitative literacy
.
National Council of Education and the Disciplines (NCED)
.
Woodrow Wilson National Fellowship Foundation
.
Whiteley, A. R., Woolf, J., Kennedy, K., Oberbillig, D., & Brewer, C. (
2007
).
Classroom mark-recapture with crickets
.
American Biology Teacher
,
69
(
5
):
292
297
.

### Appendix: Lincoln Petersen Estimation and Shannon-Weaver Calculation Student Worksheet

A particular color of M&M will represent a distinct species within a hypothetical community. Plain Chocolate M&Ms come in colors of blue, brown, green, orange, red, and yellow.

Please count the number of M&Ms of each color in your sample (combine two Fun-Sized bags of Chocolate M&Ms) and record the total numbers of each color below:

 Blue = Brown = Green = Orange = Red = Yellow =

Total Number of M&Ms in sample (the total population) =

#### Lincoln Petersen Estimation Calculation

The formula for a corrected Lincoln-Petersen Estimate Calculation is:

$N=(MS1+1)×(nS2+1)÷(MS2+1)−1$

Wherein: N = population estimate obtained from sampling efforts on two occasions

MS1 = all individuals of a particular species marked in Sample 1

nS2 = all individuals of a particular species [includes marked and unmarked] obtained in a second sample

MS2 = all individuals of a particular species that were found to be marked in the second sample

Adding values of 1 to the sample sizes and subtracting 1 from the total population estimate helps to correct for some bias in the basic Lincoln-Petersen calculation.

Use the formula above to obtain a Lincoln-Petersen Population Estimate of the M&M species in a particular area. Show your numbers and calculations in the space below.

Calculations:

#### Extension Activity: Shannon-Weaver Diversity Calculation—Student Calculation Worksheet

Shannon-Weaver Formula:

$H=Σ(pi)(−ln pi)$

pi (proportions) of M&M species (should add up to 1.0):

 Blue = Brown = Green = Orange = Red = Yellow =

-lnpi (inverse proportions; inverse of the ln, because ln gives a negative value):

 Blue = Brown = Green = Orange = Red = Yellow =

Calculations:

#### Activity Examples with Calculations:

Lincoln Petersen Estimate and Shannon-Weaver Calculation Student Worksheet

A particular color of M&M will represent a distinct species within a hypothetical community. Plain Chocolate M&Ms come in colors of blue, brown, green, orange, red, and yellow.

Please count the number of M&Ms of each color in your sample (combine two Fun-Sized bags of Chocolate M&Ms) and record the total numbers of each color below:

 Blue = 11 Brown = 3 Green = 1 Orange = 5 Red = 5 Yellow = 5

Total Number of M&Ms in sample (the total population) = 30

The formula for a corrected Lincoln-Petersen Estimate Calculation is:

$N=(MS1+1)×(nS2+1)÷(MS2+1)−1$

Wherein: N = population estimate obtained from sampling efforts on two occasions

MS1 = all individuals of a particular species marked in Sample 1

nS2 = all individuals of a particular species [includes marked and unmarked] obtained in a second sample

MS2 = all individuals of a particular species that were found to be marked in the second sample

Adding values of 1 to the sample sizes and subtracting 1 from the total population estimate helps to correct for some bias in the basic Lincoln-Petersen calculation.

Use the formula above to obtain a Lincoln-Petersen Population Estimate of the M&M species in a particular area. Show your numbers and calculations in the space below.

Calculations: Examples, numbers, and calculations presented below are from an actual sample and trial of the methods.

BlueM&Ms (in greatest quantity in this community) are used as (MS1) = 4 Blue caught, marked (scratched to identify), and released.

Second Sample size (nS2) is 11. Per directions, use the number of the largest quantity color.

Stirred/shook M&Ms in container and randomly drew out 11; obtained 3 Blue (marked recaptures) in second sample (MS2).

This estimate is slightly higher than the actual number of Blue M&Ms in our simulated community.

Green M&Ms (in smallest quantity) are (MS1) = 1 Green caught, marked (scratched to identify), and released.

Use initial sample size of 11 for this estimate (nS2).

Stirred/shook M&Ms again and randomly drew out 11; obtained one Green M&M as a recapture. (MS2) = 1

The estimate for our Green M&M species is very much higher than the actual number in the simulated community. Overestimates are common with the Shannon–Weaver index; especially with low sample size numbers.

#### Extension Activity: Shannon–Weaver diversity calculation—Example with Calculations

Shannon–Weaver formula:

$H=Σ(pi)(−ln pi)$

pi (proportions) of M&M species (should add up to 1.0):

 Blue = 11/30 = 0.367 Brown = 3/30 = 0.100 Green = 1/30 = 0.033 Orange = 5/30 = 0.167 Red = 5/30 = 0.167 Yellow = 5/30 = 0.167

-lnpi (inverse proportions; inverse of the ln, because ln gives a negative value):

 Blue = 11/30 = 1.002 Brown = 3/30 = 2.303 Green = 1/30 = 3.411 Orange = 5/30 = 1.790 Red = 5/30 = 1.790 Yellow = 5/30 = 1.790

Species evenness and richness is relatively low in the M&M community. Therefore, your chance of obtaining the same color (species) of M&M each time you sample this community would be pretty high.