How does validating activity contribute to the modeling process?

Abstract

Contemporary scholars describe mathematical modeling as a transformation of a real-world problem to a mathematical problem and back again. This paper treats a critical issue in the modeling process: how modelers determine if the transformation from the real world to mathematics was carried out well. I present an empirically derived typology of validating activities explaining how validating functions to ensure a mathematical model will yield a reasonably accurate prediction. The typology arose from analysis of four engineering undergraduates’ production of 276 instances of validating. The nuances of validating suggest that creating and maintaining relationships between reality and mathematics is more complex than a transformation and that we should afford a more prominent role to validation in the modeling process.

Appendices

Appendix A: Sample solutions to the study tasks

In this section, I provide one potential solution to each of the tasks, paraphrased from the participants’ work. Participants collectively exhibited multiple approaches to each task. The potential solutions are given as finished products, written up for quick understanding by the reader.

• 1. Baker’s yeast is a type of fungus that reproduces through budding. Each cell reproduces once every 30 min. To grow yeast for baking bread, you have to proof it first—allow it to form a colony—in a bowl of warm water. Suppose that in a particular bowl, after 6 h, the surface of the water is covered with yeast cells. When was half of the surface covered? [Based on Mance’s solution]

Since each cell reproduces once every 30 min, the number of cells is doubling every 30 min. Counting backwards from t = 6 h, when the surface of the bowl is covered, half of the surface would be covered at t = 5.5 h. Participants’ mathematical approaches included drawings, tree diagrams, tables, curve fitting, exponential growth, linear growth, and proportions.

• 2. Estimate how many cells might be in an average-sized adult human body. [based on Torrhen’s solution]

Approximate a cell as a cube with side length 50 μm. Then the average adult male is

$$ {\displaystyle \begin{array}{c}\frac{1.776\kern0.5em \mathrm{m}}{50\kern0.5em \mu \mathrm{m}}=35,520\kern0.5em \mathrm{cells}\kern0.5em \mathrm{high}\\ {}\frac{\left(10\kern0.5em \mathrm{in}\right)\cdot \left(.0254\frac{\mathrm{m}}{\mathrm{in}}\right)}{50\kern0.5em \mu \mathrm{m}}=5080\kern0.5em \mathrm{cells}\kern0.5em \mathrm{thick}\\ {}\frac{\left(16\kern0.5em \mathrm{in}\right)\cdot \left(.0254\frac{\mathrm{m}}{\mathrm{in}}\right)}{50\kern0.5em \mu \mathrm{m}}=8128\kern0.5em \mathrm{cells}\kern0.5em \mathrm{wide}\end{array}} $$

Approximating the average adult male as a rectangular prism with these dimensions yields a volume of 1.47 × 10¹², roughly 1.5 trillion cells. Participants’ approaches included drawings, tree diagrams, tables, curve fitting, exponential growth, and linear growth.

• 3. There is an information desk on the street level in the Empire State Building. The two most frequently asked questions of the staff are (a) How long does the tourist elevator take to the top floor observatory? (b) If instead one decides to take the stairs, how long does this take? [Based on Torrhen’s solution]

• a. Height, h measured in feet, and speed r measured in miles per hour:

$$ \frac{\left(\frac{h}{5280}\right)}{r}\cdotp \frac{60\ \min }{1\ \mathrm{h}}\cdotp \frac{60\ \mathrm{s}}{1\ \min }=t $$

Assuming a rate of 20 mph and 1200 ft, it would take 40 s to ascend by elevator.

• b. Assuming that the stairs are constructed at a grade of 3 ft horizontal movement to 2 ft of vertical movement, total distance traveled for a 1200 ft climb is 1800 ft. With a walking pace of 2 mph, and using the same formula above, an ascension time of 613.6 s is obtained or 10.2 min.

Participants’ approaches included: drawings, empirical observations, linear relationships, and nonlinear relationships.

• 4. On November 20, 2011, Willie Harris, 42, a man living on the west side of Austin, TX died from injuries sustained after jumping from a second floor window to escape a fire at his home. What was his impact speed? [Based on one of Trystane’s solutions]

Assuming no air resistance, from kinematics,

$$ {v}_f^2-{v}_i^2=2 gd, $$

Where v_f is the final velocity, v_i is the initial velocity, g is the gravitational constant, and d is the distance fallen. Assuming v_i = 0 and a two-story fall of d = 25 ft, the impact velocity is 32 ft/s. Participants’ approaches included empirical observations, kinematics, energy, and differential equations.

• 5. To regulate the pH balance in a 300 L tropical fish tank, a buffering agent is dissolved in water and the solution is pumped into the tank. The strength of the buffering solution varies according to $1-{e}^{-\frac{t}{60}}$ g/L. The buffering solution enters the tank at a rate of 5 L/min. How much buffering agent is in the tank at any point in time? [Based on Orys’s solution]

$$ \frac{dQ(t)}{dt}+r\frac{Q(t)}{V}=r\left(1-{e}^{-\frac{t}{60}}\right) $$

Let Q(t) be the quantity of buffering agent in the tank at time t, V be the volume of the tank, r be the rate at which liquid enters and leaves the tank (assumed to be equal), and $c(t)=1-{e}^{-\frac{t}{60}}$ is the concentration of the buffering solution entering the tank. Assuming the liquid in the tank is well-mixed, the concentration of buffering agent leaving the tank is $\frac{Q(t)}{V}$.

Then the change in quantity of buffering agent in the tank over some time interval Δt is

$$ {\displaystyle \begin{array}{l}\Delta Q={Q}_{\mathrm{in}}-{Q}_{\mathrm{out}}\\ {}=\left( rc-r\left(\frac{Q}{V}\right)\right)\Delta t\end{array}} $$

Letting Δt → 0, we obtain the first-order, linear, and ordinary differential equation

$$ \frac{dQ(t)}{dt}=r\left(1-{e}^{-\frac{t}{60}}\right)-r\frac{Q(t)}{V} $$

or

$$ \frac{dQ(t)}{dt}=5\left(1-{e}^{-\frac{t}{60}}\right)-\frac{5}{300}Q(t) $$

The equation can be solved in a few ways to obtain

$$ Q(t)=300+A{e}^{\left(-\frac{t}{60}\right)}-5t{e}^{-\frac{t}{60}} $$

Where A is a parameter dependent upon the initial condition Q(0).

Participants’ approaches included linear relationships, difference equations, and differential equations.

• 6. How many piano tuners are there in the city of New York? [Based on Orys’s solution]

$$ {\displaystyle \begin{array}{l}8,000,000\frac{\mathrm{people}}{\mathrm{New}\ \mathrm{York}}\cdot \frac{1\ \mathrm{piano}\mathrm{s}}{16\ \mathrm{person}}\cdot \frac{1\ \mathrm{tuning}}{\mathrm{piano}\cdot \mathrm{year}}\cdot \frac{\$100}{\mathrm{tuning}}\cdot \frac{1\ \mathrm{piano}\ \mathrm{tuner}}{\frac{\$60,000}{\mathrm{year}}}\\ {}=200\ \mathrm{piano}\ \mathrm{tuner}\mathrm{s}\ \left(\mathrm{per}\ \mathrm{New}\ \mathrm{York}\right)\ \end{array}} $$

The relevant parameters are that there are 8 million people in New York City, that the ratio of pianos to people is 1:16 (based on 1 piano per four four-person families), a piano needing to be tuned once per year, costing $100 per visit, and that the piano tuner must make $60,000 per year in order for it to be a viable job. Participants’ approaches included empirical observations, rates, and proportions.

Appendix B: Using the MMC to analyze modeling processes

The modeler’s process in the falling body problem can be mapped against the MMC with the stages and transitions indicated as letters [a]–[f] and transitions [1]–[6], respectively. In what follows, I show how the MMC might predict a modeler would solve the falling body problem. The given solution was generated by a study participant, but I have written it as an epistemic narrative rather than as a series of direct quotes.

A body is falling from a second-story window [a]. The modeler recognizes that it will impact the ground and wishes to know how quickly it is moving when it hits [1]. By thinking about the body falling and wondering about its impact speed, the modeler has created the situation model [b], an internal mental representation of the real situation. The modeler then thinks about variables, parameters, and constraints that may influence the quantity of interest. The modeler may simplify/structure [2] the problem by making assumptions like the body is subject only to gravity (neglect air resistance) or by noticing that quantities such as mass of the body, height of fall, duration of fall, initial velocity may influence its final velocity and therefore need to be included as variables.

The simplifying/structuring activity [2] produces the real model [c], which is an idealized version of the real situation. In the case of the falling body, the modeler may assume that initial velocity, duration of the fall, and acceleration due to gravity will influence its impact speed. She may sketch a free-body diagram and determine that impact speed does not depend on the mass of the falling body and that the object falls from rest. Next, the modeler will mathematize [3] the real model to produce the mathematical presentation [d], v_f = v_i + at.

Mathematical analysis [4] in this case involves inputting known, estimated, or assumed values for v_i, a,and t and evaluating the expression for v_f. Depending on the conditions from the real situation [a], the modeler may assume that the initial velocity is 0 m/s (dropped from rest), acceleration is roughly 10 m/s², and that the object fell in roughly 1 s [2]. In other scenarios, the modeler may need to use explicit algebraic or other mathematical techniques to obtain a result (e.g., solving for time when velocity is given). The outcome is the mathematical result [e], in this case the number 10. The modeler then interprets [5] the mathematical result in terms of the problem’s context to obtain the real result [f] 10 m/s. Finally, the modeler will validate [6] the real result by comparing it to empirical measurements, a known answer, or real-world constraints.

Upon validating, the modeler decides whether the model is “good enough,” whether it needs revision or should be rejected. If it needs revision, the modeler will re-enter the MMC at the situation model [b] and repeat the cycle. For example, the modeler may decide, based on measurements, that 10 m/s is an overestimate. He or she may choose to revise the model by incorporating air resistance or perhaps estimating the parameter values more precisely.