VHCC Course Description: MTH 128-129

Mathematical Modeling in the Sciences I-II

Course Number and Description:

Mth 128, Mathematical Modeling in the Sciences (4 CR): Introduces mathematical modeling analysis in conjunction with computer utilities and experimental observation of both computer models and the real world. Covers simple, mostly physical systems, and the accuracy of models of observed systems, the nature of the modeling process, the refinement of models, and nature of the functions, equations and matrices used in the modeling process. Prerequisites: Algebra I, Algebra II and Geometry or Equivalent. Lecture 3 hour. Laboratory 3 hours. Total 6 hours per week.

Mth 129: Studies complex and diverse physical and non-physical systems and rudimentary automata and statistical models. Models include statistical modes, cellular automata, and models from the fields of chemistry, biology, sociology and economics. Prerequisite: MTH 128. Lecture 3 hours. Laboratory 3 hours. Total 6 hours per week.

BROAD GOALS and GENERAL DESCRIPTION

In this course sequence, through a series of experiences and exercises, the student integrates experimental science, mathematical analysis, and computer modeling. The first semester will be concerned mainly with physical phenomena, though some biological systems and some second-semester topics involving relatively complex systems will be anticipated by introductory exercises. The typical exercise will be to become familiar, through computer simulations, experiment and speculation, with a phenomenon or system; to postulate a mathematical model, and study the behavior predicted by the model; to make carefully refined observations of that phenomenon and compare those observations to those predicted by a computer model based on the mathematical assumptions made; and to document and communicate the process clearly. The final activity will be to modify, if necessary, and codify the model symbolically.

The accuracy of measurement and the accuracy of computer approximations will be an important topic in most situations. There will be an emphasis on linear, quadratic, and exponential models, and on distinguishing these models by analysis of experimental and computer‑generated data, as well as on curve fitting. Second semester will employ polynomial and other functions, and will introduce models such as celular automata and other logic-based models which are not representable by traditional mathematical functions. Second semester will also expand to models from the fields of economics, sociology, chemistry, biology, and psychology, with attention to similarities and differences in mathematical behaviors exhibited by various systems. The concept and use of integrals, derivatives, the Fundamental Theorem of Calculus, matrices and differential equations will be developed progressively and used extensively through both semesters, though standard solution techniques will be left to Calculus courses; the only techniques used here will be very basic numerical and matrix-algebra techniques. Statistical models based on the Normal Curve and related distributions, and on the Central Limit Theorem, will also be included.

BASIC EXPERIMENTS

Several basic systems will be observed by all students, including experiments chosen from the following. Other experiments relevant to the process of mathematical modeling may be designed by students and/or instructors.

FIRST SEMESTER

1. Galileo’s Experiment: The speed and the position of an unimpeded glider down an air track whose slope is constant. The models here will exhibit linear and quadratic behavior. Timing will be done using simple pendulums in a way slightly different that Galileo’s.

2. Flow of water from a uniform vertical cylinder thru a uniform horizontal hole at a given height above a level surface. Three variables will be observed by different "teams", with each team observing one of the variables and predicting from that variable the behavior of the others. The three variables are: the depth of the water from surface to hole, the rate at which water is flowing from the hole, and the horizontal range of the stream. The related velocity of the stream and velocity of the water surface are directly implicit in the flow rate and the depth data, respectively. The chronicle of one quantity can be inferred from the chronicle of any other. Understanding the relationships among these quantities leads to a variety of computer models. These models constitute differential equations and their numerical solution. The functions governing these variables in time are all linear and quadratic, with minor (and difficult‑to‑observe) second‑order effects due to friction, viscosity, air resistance, and other factors.

3. The approach of the temperature of a cold or hot object (often an apple or a potato) to the temperature of a uniform room. The rate at which the temperature changes is proportional to the temperature difference between object and room, which gives rise to exponential behavior.

4. The weight of a sponge, originally dry, when one end is placed in a water reservoir. The graph looks more exponential than quadratic. The question of whether the graph is in fact exponential, and whether an exponential function can be created to fit the data, is central.

5. The amount of light transmitted from a constant‑intensity source as a function of the number of panes of glass through which it travels. The phenomenon is exponential in nature, and the data can be fit very well with an appropriate exponential function.

6. The amount of light transmitted through a pane of smoked glass as a function of the angle at which the light strikes the glass. This phenomenon involves trigonometric functions. Quadratic functions can be made to approximate the data fairly well, but ultimately fail in a consistent manner.

7. The pressure in a 2‑liter soft drink bottle as a function of time in an adiabatic expansion of the gas. The bottle is fitted with a stopper through which a valve has been inserted. Pressure is released slowly enough to permit reading a gauge inside the bottle, but quickly enough to prevent the exchange of a significant amount of heat. The data and the mathematical model are seen to exhibit behavior that cannot be modeled by any of the functions studied so far. This model is in fact solvable only by approximation; the differential equation cannot be integrated analytically. As a result the point can be made that it is impossible to write down the function which governs this system.

8. The terminal velocity of a spherical object attached by a thread to a glider on an air track, as a function of the net force when the system is at rest. This constitutes a measurement of the drag force on the object as a function of velocity. The function giving drag force as a function of velocity can be clearly seen to change from linear to quadratic, and finally to go beyond quadratic.

9. The activity of a yeast culture and its relationship to theories of population dynamics and microbiological and chemical processes.

10. The distribution of an individual's reaction times, as measured by keyboard responses to computer-generated "beeps", or by catching an object dropped without warning between the student's fingers. The average and standard deviations of individual drops are compared with the standard deviations of 16-drop means as an illustration of the Central Limit Theorem. Similar real and simulated experiments are conducted with situations involving binary probabilities.

11. Physiological responses (e.g., respiratory rate or heart rate) to various levels of physical activity as measured by work rate.

SECOND SEMESTER

First-semester phenomena will be observed and analyzed in greater depth. In addition, as time and interest permit, the following will be studied:

1. A variety of logic-defined cellular automata will be observed, with an emphasis on various graphical means of representing behavior and results.

2. Students will observe the aspect of their own learning behavior employed in the task of memorizing strings of letters. The exercise will be done at the computer, with data collected, stored, and analyzed by computer. Speculation on ways of explaining what is observed will lead to a variety of models to be compared with observed results.

3. Queuing behavior at traffic lights and in grocery stores will be observed and modeled.

4. Clock reactions in chemistry will be observed and modeled by a discrete proximity model.

5. Poisson distributions will be observed in the context of popping corn, raindrops and traffic distributions, with careful consideration of various influences that might tend to distort the distributions.

6. The distribution of darts thrown at a dartboard will be studied and modeled.

7. Models of interpersonal action/reaction cycles will be studied, both in a 2-person and multi-person context.

8. Population dynamics in various biological systems will be modeled and, within time constraints resulting from the duration of the course, observed.

9. Rudimentary economic models will be studied and observed in a role-playing context.

10. Physical models involving less tangible phenomena such as fields and thermodynamic processes will be observed and modeled.

Further experiments and models will emerge in response to these activities, and also in response to student interest. At least five such situations, chosen for relevance to course goals and feasibility, will be developed and studied each semester.

Experiments are often performed with the aid of a computer interface, though some are performed using cruder instruments in order to gain a better "feel" for certain situations. Analysis involves sufficient hand calculation to permit reflection on and mastery of the various processes influencing the various aspects of the situation, after which various computer utilities are employed to permit thorough analysis in a reasonable time. Pre‑programmed computer models are manipulated and observed. The observed behavior of the models is compared to that of the actual physical system. The basic model is analyzed for its plausibility, and the computational scheme is mastered.

SPECIFIC OBJECTIVES

The following specific objective will be implicit in each problem and each experiment:

The student will perform the experiment or solve the problem to the greatest depth permitted by time and ability, and will understand the problem or experiment within the context of other related problems and experiments.

MATHEMATICAL CONTENT

Mathematical techniques include numerical integration and differentiation, modeling diffusion and population dynamics by stochastic matrices, and solution of differential equations by iterative integration. These are presented at a level accessible to a highly capable student who has completed two units of College‑preparatory mathematics, or to a capable student with three units, and standard terminology is introduced only near the end of the course. The paradigm being the flow experiment, the somewhat cumbersome but meaningful terms "rate‑ from‑amount" and "amount‑from‑rate" are employed to describe integration and differentiation. These processes are to be understood directly, in terms of units, and in terms of graphs through the interpretation of individual trapezoids. Differential equations are solved numerically with the aid of flow charts specifying the various relationships involved in an iteration, with students expected to understand the meaning of each operation as it relates to the actual physical system, and to internalize the meaning of the solution process in such a way as to be able to modify the process for related but different situations. The approach of powers of a stochastic matrix to a limiting matrix is observed and measured, and related to common‑sense expectations related to the system observed, but for obvious reasons a deeper analysis is well beyond the scope of the course.

There is a strong emphasis on being able to perform complex calculation schemes in an organized manner and with reflection on the meaning of each result and process. Another strong emphasis is on the interpretation of problems involving varying rates, centered on the decision of when to use the "rate‑from‑amount" (differentiation) process and when to employ "amount‑from‑rate" (summing or integration). The numerical relationships between the graphs of a function and its derivative or an antiderivative are explored, and the effect of increment on the accuracy of approximation are examined.

The basic mathematics is to be learned by working through a set of computerized exercises and tutorials correlated with the experimental situation. The exercise sets employ similar mathematical techniques in a variety of contexts. The sets are generated with randomized numerical parameters and randomized selection of problem versions. There are approximately 100 exercises, each appearing in four different versions.

While the problems are not presented by levels, they can be categorized in four different levels. Most problems encountered will be at level II or III, where most students find their appropriate level of challenge. Level I problems are provided to allow the student to begin developing a context, and to build confidence. Lvel IV problems are provided to challenge the student. It has never occurred that a student achieves mastery of the full range of Level IV problems, which would require a rare degree of intelligence and diligence.

Level I is the easiest level. Any student who can do adequate work in college-preparatory mathematics courses and who devotes sufficient time can successfully complete these exercises. Some students will have minor difficulties with interpretation of some of the questions and situations, but these are easily resolved.

Level II builds on and generalizes the situations and techniques of Level I and provides a greater challenge. Nearly all students who devote an appropriate amount of time to the task can complete Level II, but many students require assistance and clarification at this level. The various exercises are strongly interrelated, and the interrelationships begin to appear at this level. Mastery of the basic exercises at Level II and an understanding of the fundamental interrelationships are basic requirements of the course.

Level III is designed to challenge a typical top-quartile student who has successfully completed at least 3 college-preparatory mathematics courses, or a 90th-percentile student who has completed at least two such courses. These students typically master about 50% of the material in the basic problem set at this level. Level III strongly emphasizes the interrelationships among various techniques, and provides a challenge to the student's ability to interpret problems of significant difficulty and complexity. A student with 50% mastery of this material will usually be eligble for an A, provided other work is consistent with this level of performance.

Level IV is very challenging to students with the typical preparation for this course. Success with a significant amount of Level IV work is rare. Though a 90th-percentile student with 3 college-preparatory courses would find much of it accessible after completing mastery of Level III, time does not usually permit this.

At all levels, students learn to devise and work through computation schemes to solve the various problems. Once basic schemes are mastered, students are asked to use the computer to facilitate their work, since the amount of calculation required to see certain patterns and results would be prohibitive. Programs for the basic calculation schemes are provided, though students with programming skills are encouraged to create their own programs.

COURSE REQUIREMENTS AND GRADING

To make a C for the course, students are required to be present and to be diligently on task during all classes and labs, and to demonstrate the ability to interpret the most rudimentary problems and perform the basic calculations.

To make at least a B:

Students are required to perform 10 basic experiments, many with corresponding simulations, to save their data to appropriately-named files accessible to other students, and to report their results in appropriate form.

Students are required to successfully complete a written test based on the problems at Levels I and II. This test demonstrates a basic knowledge of the mathematical techniques. The test will be given 3 times during the course, the first time approximately 1/3 of the way through the course (students desiring to make A's are strongly encouraged to successfully complete the test at this time), the second time about 2/3 of the way through the course (students who do not successfully complete the test by this time almost never make A's), and the third time at the end of the course.

To make an A:

Reports of the basic experiments must be of high quality and must demonstrate thorough understanding of the situation being investigated.

Adequate performance on a test of Level III knowledge is required.

Laboratory investigations of more complex situations than encountered in the 10 basic experiments are encouraged, and may substitute for some of the basic experiments, with prior approval by the instructor. Original experiments are encouraged.

Knowledge of optional problems, even at Level II, and Level IV knowledge of any problem, may substitute for Level III knowledge of the basic problems, as agreed upon between instructor and student.

INSTRUCTIONAL METHODOLIGIES AND MATERIALS

Course content will be delivered by interactive computer tutorials, by lecture, and by interaction of instructors with individual students and student teams, as well as by interactions among students. Computer network-based communication and access to thinking processes, and to representations of data, will be emphasized.

Testing will be by a combination of randomly generated tests, which may be taken and retaken until passed, and written tests designed to assess higher levels of learning than can be assessed by computer. The text will consist of the instructor's notes, distributed to students at the beginning of the course, and the interactive computer exercises.