Teaching by using Mathematica notebooks

*To*: mathgroup at christensen.Cybernetics.NET*Subject*: [mg408] Teaching by using Mathematica notebooks*From*: herrmann at siam.math.tamu.edu (Joseph M. Herrmann)*Date*: Sun, 15 Jan 1995 11:36:41 -0600

For those interested in using Mathematica in teaching Mathematical modeling at Texas A&M University has been taught by having the students work through Mathematica notebooks which assign readings in the textbook "A First Course in Mathematical Modeling" by Frank R. Giordano and Maurice D. Weir, contain interactive examples and explanations, and assign exercises in which students create and analyze a mathematical model. Students complete an individual and group project during the class. The objective of the course is 1. Illustrate the broad range of problems which can be modeled mathematically. 2. Synthesize mathematical models from non-mathematical descriptions of problems. 3. Interpret the results of models and evaluate their implications. 4. Show the necessity of simplification and approximation in models and identify their effects. 5. Work cooperatively in groups. Currently there are 11 Mathematica notebooks. Eight Mathematica notebooks cover the first 9 topics of the syllabus below. The remaining 3 additional Mathematica notebooks create new Mathematica functions and illustrate their use with examples. 1. Simplex: Creates a tableau for a standard linear programming maximization problem. These tableaus are useful for solving linear programming problems by the standard maximization method, dual method, Big M method, or for achieving a sensitivity analysis. 2. Natural Spline: Determines the equations for the natural cubic spline functions which fit the data and includes examples of how to graph these functions automatically. 3. Clamped Spline: Determines the equations for the clamped cubic spline functions which fit the data for specified derivatives at the endpoints and includes examples of how to graph these functions automatically. Handouts: Theory of the Simplex Method Theory of the Dual Simplex Method Dimensional Analysis Syllabus: 1. Nuclear Arms Race Develop a simple probabilistic model Observe an equilibrium point Changing assumptions affects parameter values Sensitivity of the equilibrium point to change in parameters 2. The Modeling Process-identifying a problem Vehicular Stopping Distance Automobile Gas Mileage Elevator Service during the morning rush hour 3. Using Geometric Similarity in the Modeling Process Overall winner in weight lifting across weight classes Volume of lumber from the diameter of a tree at waist level Predicting Pulse rate from body weight Vehicular Stopping distance 4. Model Fitting--identifying the optimal parameters for a model Error criteria--Least square and Chebyshev Identifying Kepler's third law from observational data 5. Models requiring optimization--Linear programming/Critical points Linear programming economic models Inventory problem--minimize delivery and storage cost 6. Experimental Modeling Problems with using high order polynomials to interpolate data Splines 7. Project 1: Find the flow rate and water use of a small town from .5in height measurements of the water tank Numerical differentiation of data Fitting curves to transformed data Error analysis 8. Dimensional Analysis and Similitude Range of a cannonball Damped pendulum Terminal speed of a raindrop Turkey cooking times Model design for determining the drag force on a submarine 9. Simulation Modeling Monte Carlo simulation of area and volume Simulation of gas station delivery and storage cost for a stochastic demand Simulation of harbor waiting times for stochastic arrival and unloading times 10. Project 2: The projects for Spring 93 were A: Identifying a near optimal ratio of slurry, greens, and paper for composting. B: Identifying a near optimal work schedule for a coal tipple. 11. Differential equation models Population models Drug dosage Managing the fishing industry The notebooks on the Simplex and the splines are available from Mathsource. From Mosaic you could access it by http://www.wri.com/MathSource.html/ Simplex #0207-447 and Natural and Clamped Cubic spline coefficients #0207-436. If you are interested in additional information, discussion of the advantages or disadvantages of teaching in a format where students work in groups or independently at a computer while the teacher acts a coach and answers questions and stimulates inquiry, or any of the other materials, please contact me. Joseph Herrmann Joseph M. Herrmann Department of Mathematics, Texas A&M University College Station, Texas 77843-3368 (409) 845-1474 herrmann at math.tamu.edu