A “Rule-of-Five” Framework for Models and Modeling to Unify Mathematicians and Biologists and Improve Student Learning

Figures & data

Figure 1. Depiction of the blind men and the elephant parable, which illustrates the relationship between reality (the small circle) and different types of model representations (the large circle) [48], discussed fully in Section 3.1. Elephant images used in this illustration are Creative Commons public domain from Pixabay.com.

Figure 2. “Rule-of-five” framework for models and modeling. Each box is a model representation (defined in Table 1). Each arrow is an activity in the modeling process (defined in Table A1).

Table 1. The five types of model representations, corresponding to Figure 2 with their definitions and classroom examples

Type of representation	Classroom examples
Experiential: Direct experiences that are concrete rather than abstract, such as virtual laboratories and animations, kinesthetic experiences and manipulations of physical models, actual scientific experiments, case studies, and simulation case studies	A video of bacterial growth; beanbag biology [30]; virtual laboratories (e.g., SimBio [35], the BUGBOX-predator virtual laboratory [35]) Physical model of the structure of DNA Experiment to measure bacterial growth in the laboratory An in-class activity where students make decisions based on a dice roll to shake hands and transmit a disease
Verbal: Hypotheses used to design experiments, predictions, assumptions used to construct mathematical models, simple descriptions of observations, qualitative experimental data	Hypothesis: A mutation in a particular gene will reduce the rate of bacterial growth because the mutation impairs DNA replication Prediction: On average, global temperature will increase Assumption: We assume that the population is well mixed Simple descriptions of observations: The rate of increase is decreasing; we observe far more of the blue flower type then the purple flower type Qualitative data: Spiciness ratings by tasters of chili peppers
Numerical: Data sets collected from model-based simulations, data calculated from other data (derived data), experimentally-collected quantitative data	Numbers of infected individuals calculated from a Symbolic epidemic model Derived data: Low density growth rate and carrying capacity calculated from plotting relative growth rate versus population for logistic growth Measured population counts from experiments
Visual: Graphs, schematics	A graph of relative growth rate versus population A schematic of an epidemic model; stock-and-flow diagrams Data visualizations (e.g., histograms, scatter plots, infographics, etc.)
Symbolic: Formal mathematical constructs such as formulas, equations, algorithms, parameters, and state variables	Discrete difference equation for geometric growth xn + 1 = λxn and continuous differential equation for, exponential growth $\mathit{dP}/\mathit{dt}=\mathit{rP}.$ If in HWE, p = frequency of one allele, p^2 = frequency of homozygotes for that allele State variable: P(t) = population at time t (in years) Equation from a linear regression Equation from a probability distribution

Table A1. Modeling activities that correspond to the arrows in Figure 2 with a description and an example.

Arrow	Classroom Example
1. Exp→Ver: Crafting a formal scientific hypothesis (which implies mechanism) based on observations; predictions; collecting of qualitative experimental data.	Students make a hypothesis based on their observations of bacterial population growth A population satisfying the assumptions of HWE will maintain the same allele frequencies generation after generation
2. Ver→Exp: Designing an experiment	Based on a hypothesis about bacterial growth rates, make a prediction about growth rates under different conditions. Design an appropriate experiment to support or refute the prediction Based on a hypothesis that a population is in HWE, plan an experiment to test for HWE.
3. Exp→Num: Collecting data	Collecting quantitative data from the Experiential simulations/animations Sampling populations over time in a field study Measuring a culture of bacteria over time through spectroscopy
4. Num→Exp: Feeding experimental data into a simulation; using experimental or simulated data to make a physical model; performing model validation against new experiments	Using coordinates derived from X-ray crystallography to build a physical model of DNA Using data from a numerical simulation to create an animation of a population growing Running a longer experiment to test when an exponential growth model of bacteria fails to match the data
5. Ver→Num: Estimating; approximating	Finding estimates for parameters in a logistic model from the literature Back-of-the-envelope calculations or reasoning that allow students to test a Verbal prediction
6. Num→Ver: Describing patterns and trends in the data; using data to refine hypotheses	Looking for trends in a data set that may indicate the presence of a carrying capacity for the population Interpreting the results of a statistical test for HWE.
7. Sym→Ver: Interpreting/analyzing a mathematical model.	$\frac{\mathit{dP}}{\mathit{dt}}=\mathit{rP}-\frac{r{P}^2}{K}$ is the rate of change of population size, positively affected by a net positive intrinsic birth-death rate and negatively affected by intraspecific competition over resources Interpreting p and q in the HWE model as frequency of allele A and frequency of allele a
8. Ver→Sym: Mathematizing	The converse of arrow 7, going from the explanation to the Symbolic representation Activities traditionally associated with mathematical modeling, i.e., formalizing the language of the Verbal description by assigning parameters such as carrying capacity and intrinsic growth rate, state variables such as population size and writing formulas and equations such as, $\frac{\mathit{dP}}{\mathit{dt}}=\mathit{rP}-\frac{r{P}^2}{K}$.
9. Num→Sym: Statistical modeling	Fitting a logistic model to a data set Testing the experimentally measured data of genotype frequencies to the HWE null statistical model
10. Sym→Num: Simulating data	Simulate data from the logistic differential equation using an ODE solver and a particular parameter set Performing in silico experiments
11. Vis→Ver: Interpreting visualizations of processes or results	Going from a “stock-and-flow/box-and-arrow” schematic representation of a model to the Verbal description, a hypothesis, or prediction to test Interpretation of a graph of logistic growth, identifying the lag phase (early exponential), the log phase (greatest rate of growth), and the stationary phase (saturation) of the logistic growth rate
12. Ver→Vis: Sketching a graph or schematic that illustrates a hypothesis or observation	Sketching a graph of a population that starts off growing exponentially, but then has a carrying capacity Drawing a schematic of what processes might be involved in a logistically growing population
13. Num→Vis: Graphing, visualizing data	Traditional plot of population data versus time or rate of growth versus population size Creating appropriate infographics for a “big data” set
14. Vis→Num: Interpolating data points from a graph, using a visual modeling program such as STELLA, Insightmaker, or Simulink	Interpolating between given data points Estimating data points or parameter values, such as carrying capacity, from a graph (a.k.a., “Reverse engineering”)
15. Sym→Vis: Graphing or drawing a schematic of a process described by a formal mathematical model	Traditional graphing of the logistic growth curve Drawing a stock-and-flow diagram from seeing SIR model equations
16. Vis→Sym: Modeling based on qualitative features of a graph or processes laid out in a schematic	The equation for logistic growth of a population based on a graph showing the carrying capacity, the initial population size, and the time at which the population is at half the carrying capacity or the doubling time Build the Symbolic representation of a model based on the schematic of processes involved, such as net exponential growth and death due to competition
17. Sym→Exp: Programming an animated simulation	Programming in NetLogo a simulation illustrating the logistic growth of bacteria in a virtual Petri dish Creating a manipulative beanbag biology experiment to explore algebraic relationships of the HWE equation
18. Exp→Sym: Writing mathematical equations directly based on observations (experienced modelers may not need to pass through additional Verbal or Visual boxes)	Recognizing logistic growth is at play based on an experiential activity and immediately writing the resulting equations
19. Exp→Vis: Drawing a schematic or cartoon based on experiential observations	Drawing a sketch of DNA from an animation or a physical model Drawing a process schematic or concept map of processes at play when observing the growth of a bacterial cell culture
20. Vis→Exp: Experimenting with manipulatives, assembling a physical model that replicates a process schematic, or animating a sketch or cartoon of a process	Constructing a physical model of DNA from a picture Perform an experiment to replicate graphical or schematic results

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