Course Number and Description: 

Mth 170, Covers topics in the mathematics of social choice, management sciences, statistics, and growth. Uses physical demonstrations and modeling techniques to teach the power and utility of mathematics. Algebra I-II and Geometry or equivalent.

Lecture 3 hours per week.

In this course, 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 mathematical modeling of simple physical phenomena and the application of the related procedures and concepts to the statistics and rate phenomena required in management science.  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, polynomial, power-function and exponential models, and on distinguishing these models by analysis of experimental and computer‑generated data, as well as on curve fitting. 

Related models such as cellular automata and other logic-based models which are not representable by traditional mathematical functions will also be discussed. 

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.

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 the 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. 

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 most 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.