One Fish, Two Fish, Many Fish, Few Fish

4 min readMar 18, 2021

Emily Morgan Liljestrand

If you exit left out of the elevator on the basement floor of MSU’s Natural Resources building, and walk past the conference room with green walls, past Dr. Brian Roth’s wet lab, and even past Jill Cruth and Jim Schneider’s offices, there’s a perpendicular sparse hallway. In the last room on the left of that oft overlooked hallway lies the FW storage room. The contents of the lockers are lab-specific and, because of the diverse research interests of our department, can run the gamut from broken centrifuges to electrofishing nets to camera traps. But twice a year I wiggle my way around all that equipment to retrieve my favorite bit of “research” supplies — two decorated cardboard boxes, some pipe cleaners, and about 500 small plastic fish.

Outreach supplies drudged up from the basement and into the light.

These supplies were purchased back in 2019 by the FW Graduate Student Organization (GSO) using funds earmarked for outreach activities. I specifically requested them to use at the Graduate Women in Science (GWIS) Girls Math and Science Day. This annual event invites mid-Michigan 3rd-5th graders to explore the diversity of science. The specific discipline I wanted to introduce them to was “quantitative fisheries,” and I just needed to design an appropriate lesson plan that would appeal to my audience.
I knew the average 3rd-5th grader would not be familiar with “quantitative fisheries” and I could not blame them. It wasn’t even until after my undergraduate degree that I learned about how you could input fisheries data into computer models that then output statistics like abundance and population death rate. Though I love the diversity of mathematical models in fisheries, I wouldn’t be able to cover everything in the 30-minute time limit. Then it hit me, why not teach them Lincoln-Petersen!

A “Lincoln-Petersen” estimator is a population tagging model commonly used in quantitative fisheries where a sample of individuals are captured, tagged, and released. Scientists use the number of tagged fish that are recaptured to estimate the number of fish in a closed population. I decided an approximate activity would make for a great hands-on opportunity where students could practice their multiplication and division skills while simulating real experimental protocol!

The aforementioned boxes served as our fictional “lakes.” I filled each with an unknown population (N) of plastic fish. I invited the participants to work as a team, “fishing” out small portions of the population (a total of n individuals), then “marking” them with segments of cut up pipe cleaner before returning them to their lake. After “mingling” for a bit with the rest of the fish (i.e., shaking the box really hard), the population was re-sampled. The ratio of tagged fish (t) to total fish (T) in the second sample is roughly equal to the total number of tagged fish to the total population, i.e.,

And since we knew t, T, and n, we could use some simple algebraic finagling to predict N. The students eagerly demonstrated their long division skills by solving the equation on the whiteboard, often to the fifth decimal place! The lesson ended when I revealed the real number of fish in each lake and pointed out how much harder it would have been to count them all by hand, rather than using a handful of “tags” and pinch of math.

From left to right, Mimi Gong, Samantha Strandmark, Emily Liljestrand (Author), and Lydia Wassink, just after completing the last of four workshop sessions.

The activity was not without its struggles. It took some trial and error to determine the best number of fish to have in the lake and how many to sample. Tag too few fish and you could seriously bias the estimates. For example, if your second sample has no tagged fish in it, the equation works out to N=∞! I’ve gotten some great feedback from my amazing assistants, all of whom are graduate students in the Department of Fisheries and Wildlife. They’ve pointed out how I could adjust my language, using phrases like “number of fish” instead of “population size,” and to pose discussion questions that relate the activity back to broader ideas like conservation. I look forward to using this constructive criticism to keep making my outreach even better.

I anticipate that Girls Math and Science Day won’t happen again until 2022 at the earliest. Though the majority of my research items have moved to my home office, I’m a little pleased whenever I remember that stuffy locker in a storage room at the end of an oft overlooked hallway in the basement of the Natural Resources building. And when the world gets a little less dangerous, I’m sure there will still be eager and bright junior scientists ready to learn how to use math to count fish.


Emily Liljestrand is a Ph.D. candidate with Dr. James Bence. She is interested in using novel statistical and modeling methods to improve fisheries stock assessment. She can be reached at or her personal website




Spotlight Magazine is the official student-run annual publication from Michigan State University’s Fish and Wildlife graduate students.