What good is modeling? Introducing biology students to theory

This paper describes a graduate-level course designed to help empirically-focused biology students overcome mathematical barriers and better understand theoretical biology through evidence-based teaching strategies, ultimately fostering a more productive dialogue between theory and empirical science.

The Gist

The Big Picture: Why Do We Need a "Theory Class"?

Imagine biology as a massive, bustling construction site.

• The Empiricists are the Bricklayers. They are out there mixing cement, laying bricks, and measuring walls. They know exactly how the building feels, looks, and stands up in the real world.
• The Theorists are the Architects. They sit in the office with blueprints, math, and models. They aren't touching the bricks, but they are figuring out why the building stands up, predicting where the next wall should go, and checking if the design will collapse before anyone lays a single brick.

The Problem: The Bricklayers and the Architects speak different languages. The Bricklayers often think the Architects are just playing with numbers that don't matter. The Architects often think the Bricklayers are too busy to understand the big picture. This paper argues that biology students (the future Bricklayers) are being taught to ignore the Architects, and that's a mistake.

The Three Walls Blocking the Door

The authors say there are three main reasons why biology students are afraid of or confused by mathematical models:

1. The "Math Phobia" Wall: Many biologists haven't used algebra or calculus since high school. When they see a theory paper, it looks like a foreign language. They think, "I can't read this, so I won't try."
2. The "Lost in Translation" Wall: Over time, brilliant ideas from math models get passed down as simple stories. Imagine a complex architectural blueprint being turned into a rumor: "Oh, the building needs a wide base." The next generation hears the rumor and thinks it's obvious common sense. They forget that a mathematician actually proved that a narrow base would collapse. The "magic" of the math is lost, and the value of the model disappears.
3. The "Physics vs. Biology" Wall: In physics, models are like crystal balls. You build a model to predict exactly what will happen tomorrow (e.g., "The ball will land here"). In biology, models are often more like stress tests. You build a model to ask: "Is this idea even possible?" or "Is this idea impossible?"
   • Example: If a biologist says, "Maybe birds fly because they have magic feathers," a model can prove that magic feathers are impossible without needing to catch a single bird. The model didn't predict the future; it just ruled out a bad idea.

The Solution: A New Kind of Class

The authors created a special graduate course to fix this. Instead of teaching students how to become mathematicians, they teach them how to read the blueprints.

Think of the course like a tour guide for a museum of complex ideas. You don't need to be an artist to appreciate a painting; you just need to know what to look for.

How the class works (The "Secret Sauce"):

• Backwards Design: The teachers start with the end goal: "Can you read a theory paper and explain what it means?" Then they build the whole semester around that.
• Active Learning: Instead of sitting in a lecture hall listening to a professor talk, students are put in small groups. They are given a paper and a set of questions. They have to figure it out together, like detectives solving a mystery.
• "Just-in-Time" Teaching: If a student gets stuck on a specific math concept (like a differential equation), the teacher explains it right then and there, only as much as is needed to understand the paper. It's like learning how to change a tire only when you have a flat, not memorizing the whole mechanics of a car engine.
• The "Black Box" Trick: Students are taught to treat the math like a black box machine. They don't need to know how the gears inside turn; they just need to know what goes in (inputs) and what comes out (outputs).
  • Input: "We assume bees want to save energy."
  • Output: "Therefore, bees should visit flowers in a specific pattern."
  • Result: The student understands the insight without needing to solve the equation.

The Toolkit: What They Actually Read

The class reads famous, historical papers. Here is how the authors explain them using analogies:

• Hardy-Weinberg (The "Reality Check"): People used to think dominant genes would take over the world. A simple math model proved that was wrong. It's like realizing that just because a loud person speaks up, it doesn't mean everyone else will stop talking.
• Maynard Smith & Price (The "Game Theory"): Why don't animals kill each other in every fight? A model showed that being a "Hawk" (always fight) is actually a bad strategy compared to being a "Dove" (back down sometimes). It's like realizing that in a game of rock-paper-scissors, always throwing Rock is a losing strategy.
• Hubbell (The "Neutral Theory"): This paper asked: "Do we need complex reasons for why some trees are common and others are rare?" The model showed that sometimes, it's just random chance (like flipping a coin). It proved that a "boring" model could explain complex data, saving scientists from inventing fake, complicated reasons.

The Takeaway: Why This Matters

The paper concludes that science is a team sport. If the Bricklayers (empiricists) don't understand the Architects (theorists), they might build a wall in the wrong place, or they might miss a structural flaw.

By teaching students to read models, we aren't trying to turn biologists into mathematicians. We are giving them a superpower: the ability to look at a complex idea, strip away the scary math, and see the core truth underneath.

In short: Theory isn't about doing the math; it's about understanding the logic of life. And once you understand the logic, you can build a better science.

Technical Summary

Based on the provided text, here is a detailed technical summary of the paper "What good is modeling? Introducing biology students to theory" by Anna Dornhaus and Joanna Masel.

1. Problem Statement

The authors identify a critical disconnect between empirical biologists (ecologists, evolutionary biologists, epidemiologists) and theoretical modelers. Despite the consensus that scientific progress requires integrating empirical data with theoretical frameworks, many biologists struggle to read and understand theory papers due to three primary barriers:

• Limited Mathematical Training: Many empiricists lack the technical skills to follow the mathematical derivations in theory papers.
• Historical Misrepresentation: Key insights derived from formal models are often passed down as "self-evident" verbal concepts, obscuring the utility of the models that generated them.
• Paradigm Mismatch: Biologists often apply a "physics paradigm" to biology, expecting models to generate specific, testable empirical predictions. However, in population biology, models are frequently used for "proof of principle" (showing a hypothesis is viable) or "disproof of principle" (showing a hypothesis is internally inconsistent), rather than for fitting data or generating specific predictions. This leads to "math phobia" and the misconception that models are inaccessible or unnecessary.

2. Methodology: The Course Design

To address these barriers, the authors developed a graduate-level course titled "Introduction to Modeling in Biology" at the University of Arizona. The course is designed for empirically focused PhD students with little prior mathematical training. The curriculum is built on three evidence-based pedagogical principles:

• Backwards Design: The course starts with the final assessment (a take-home exam where students analyze a theory paper) and designs all activities to practice the specific skills needed for that exam.
• Active Learning: Students engage in "think-pair-share" activities, analyzing papers in small groups. They are provided with reading guides (scaffolding questions) that direct them to identify the biological insight, the role of the model, and the inputs/outputs without getting bogged down in complex algebra.
• Just-in-Time Teaching: Mathematical concepts (e.g., differential equations, probability distributions, Markov chains) are introduced only when they become necessary to understand a specific paper.
  • Tool: The course utilizes Mathematica (a computational software) to allow students to manipulate equations symbolically and reproduce figures. This reinforces their mental representation of mathematical objects without requiring them to derive solutions manually.

3. Key Contributions and Curriculum Content

The course utilizes a curated set of historically significant papers (listed in Table 1) to demonstrate the diverse roles of theory in the scientific method. Key examples include:

• Hardy (1908): Used to demonstrate disproof of principle. The paper refuted the verbal theory that dominant alleles would inevitably increase in frequency, showing that dominance $\neq$ selection.
• Maynard Smith & Price (1973): Used to demonstrate proof of principle. The paper showed that ritualized animal contests could evolve via individual selection (Game Theory/ESS), overturning the prevailing group selection hypothesis.
• Luria & Delbrück (1943): Used to teach probability distributions. The paper distinguished between induced and spontaneous mutations by analyzing variance in bacterial resistance, highlighting the power of statistical distributions as evidence.
• Hubbell (1997): Used to discuss neutral theory. The paper proved that a simplified, "obviously wrong" model could predict complex ecological data, demonstrating that certain data types cannot distinguish between competing hypotheses.
• Fussman & Blasius (2005): Used to show sensitivity to model structure, illustrating how small changes in functional forms can drastically alter conclusions.

The "Black Box" Approach:
A central pedagogical strategy is teaching students to treat models as "black boxes." Students are trained to identify:

1. Inputs: What data or assumptions go into the model?
2. Outputs: What biological insights or predictions come out?
3. Classification: Is the model deterministic vs. stochastic? Analytical vs. numerical? Does it serve as proof of principle, disproof of principle, or a predictor?

4. Results and Outcomes

• Student Performance: The course successfully enabled students with minimal math backgrounds to extract biological insights from complex theory papers.
• Cross-Disciplinary Engagement: The course unexpectedly attracted theoreticians (e.g., from Applied Math) and undergraduates. Theoreticians reported gaining a better understanding of how their models interact with empirical science ("seeing the forest for the trees"), while empiricists overcame math anxiety.
• Skill Acquisition: Students learned to:
  • Skip non-essential mathematical details to grasp the core biological argument.
  • Classify models based on their function within the scientific method (e.g., distinguishing between forecasting and hypothesis testing).
  • Use computational tools (Mathematica) to visualize and manipulate mathematical objects.
• Feedback: Student feedback indicated that the "active learning" and "just-in-time" approach minimized "busy work" and maximized direct relevance to their research goals.

5. Significance and Conclusion

The paper argues that the integration of empirical and theoretical science is hindered not by a lack of interest, but by a lack of shared understanding regarding the nature of evidence and the role of models.

• Redefining Evidence: The course emphasizes that theoretical models provide a form of evidence distinct from empirical data. They can validate the logical consistency of a hypothesis or demonstrate its impossibility before costly empirical experiments are conducted.
• Curricular Reform: The authors propose that such courses should be a standard requirement in Ecology and Evolutionary Biology (EEB) programs.
• Broader Impact: By teaching students to critically evaluate the "process of science" and the specific utility of different modeling strategies, the approach fosters better communication between theoreticians and empiricists, ultimately accelerating scientific progress and critical thinking in biology.

In summary, the paper presents a successful pedagogical framework for demystifying mathematical modeling for biologists, shifting the focus from "doing the math" to "understanding the logic" of theoretical science.