For All Practical Purposes: Introduction to Contemporary Mathematics is a video instructional series on mathematics for college and high school classrooms and adult learners; 26 half-hour video programs.
This series goes beyond mathematics as a series of dry, problem-solving exercises to its utility in the areas of management science, statistics, social science, geometry, and computer science. See how mathematics influences everything from the success of savvy entrepreneurs to the fairness of voting practices. The modular construction of the series makes it useful across the curriculum.
Module 1 Management Science
Management science is the study of scheduling (people, workflow) in order to maximize efficiency and effectiveness. Algorithms are used to generate accurate and optimal solutions to problems involving street networks (garbage collection, postal deliveries), scheduling (airlines), and routing problems (salespeople, telephone relays). This overview conveys the scope and applicability of management science concepts.
2. Street Smarts
Many routing problems that involve traversing streets in a city can be solved by graphing. This program shows how to find efficient travel routes using graphs and Euler circuits. Students will learn how to apply the graph model and recognize Euler circuits in a graph. Curb inspections and other street-related jobs provide concrete illustrations of the central concepts.
3. Trains, Planes and Critical Paths
Nearest-neighbor and greedy algorithms are highlighted to show how they aid in solving complex routing problems. Critical path analysis is also featured in this program, as are order requirement directed graphs (digraphs). This program demonstrates how to find approximate solutions to the traveling salesman problem (TSP) and how to distinguish between Euler circuits and the TSP.
4. Juggling Machines
The scheduling requirements of airliners and police patrol cars illustrate just how crucial algorithms are to everyday life. List processing algorithms are used in simplified scheduling problems, constructed to provide insight into the behavior of scheduling processors. Bin packing problems and heuristic algorithms are also featured.
5. Juicy Problems
Economies depend on the optimal use of resources to produce goods and services at maximum profit. This program introduces linear programming as a powerful tool for determining the best combination of manpower and resource use. The corner principle, simplex method, and potentially faster linear programming methods are also discussed.
Module 2 Statistics
From baseball scores and roulette odds to national unemployment figures and quality control testing, statistics help us to understand information and make better decisions. This overview introduces the subject, featuring professionals in labor statistics and medicine who use statistical methods to determine probable outcomes in their fields.
7. Behind the Headlines
Data are collected for specific purposes by sampling or experimentation. Random sampling is employed to eliminate bias, and experiments are controlled so as to discover cause-and-effect relationships. Randomized comparative experiments are explained, as is the use of Latin square designs for statistical data gathering.
8. Picture This
Exploratory data analysis is the art of looking for unanticipated patterns in data. Uses of histograms, box plots, and scatterplots are explained, as are the meanings of mean, median, quartiles, and outliers in statistical parlance. The program illustrates concepts with examples relating to seismic analysis, Napoleon's march, and baseball.
9. Place Your Bets
Random events have unpredictable outcomes that over time follow a predictable pattern. This program travels to a casino to capture first-hand footage of this phenomenon in action. The topics covered include sampling distribution, normal curves, standard deviation, expected value, and the central limit theorem.
10. Confident Conclusions
Formal statistical inference, as opposed to exploratory data analysis, is based on calculations of probability. This program defines confidence intervals and demonstrates how to find a 95% confidence interval for a population proportion p. Application examples are drawn from a health study, manufacturing, and Gallup poll interviews.
Module 3 Social Choice
The dilemmas of modern life - making choices and taking chances - are highlighted. Issues surrounding individual choices are analyzed using game theory, one of the important developments of twentieth-century mathematics. Collective choice is analyzed using election theory, weighted voting, and apportionment. Real-life examples show the utility of the concepts.
12. The Impossible Dream
Five voting methods - plurality, plurality with runoff, Condorcet, Borda, and sequential runoff voting - are illustrated. Dramatic reenactments of a political convention and a news broadcast are presented to clarify concepts. An example of Nobelist Kenneth Arrow's theorem on voting theory is also featured.
13. More Equal Than Others
This program addresses the timely issue of fair representation. Fair division and apportionment problems are described, with the simple case of slicing a cake used to explain more complex cases in politics and voting. Reenactments also demonstrate how weighted voting and winning coalitions work.
14. Zero Sum Games
Game theory deals with strategies employed by parties with conflicting needs. Optimal strategies (pure and mixed) are described mathematically, and game matrices are explained. Expected value equations, a graphical interpretation, a restaurant illustrating the minimax technique, and an illegal parking example are presented.
15. Prisoner's Dilemma
This program explores social situations involving decision-making strategies in games of partial conflict. "Prisoner's dilemma" and games of "chicken" are explained in the broader context of corporate takeovers, national defense, politics, and labor relations.
Module 4 On Size and Shape
Geometry and its relationship to natural beauty and art are explored and analyzed. This program draws upon examples of geometric applications, from Leonardo da Vinci's "window" for recording proper linear perspective in art to symmetry-based classification systems in archaeology. The Fibonacci sequence fractals and their applications in many disciplines are also discussed and illustrated.
17. How Big Is Too Big
Problems dealing with geometric similarity and scale are examined. This program discusses tensile strength of building materials and their relationship to maximum size and proportion. Tiling patterns, two-dimensional Penrose tilings, and their importance to crystallography usage are also featured.
18. It Grows and Grows
Examples ranging from money in the bank to fish in the sea are used to explicate population growth. The mathematics of determining harvesting rates to maintain sustainable yields is explained. The program also emphasizes the importance of determining population size and related measures, concluding with an examination of demography and population pyramids.
19. Stand up Conic
The importance of understanding conic sections is explained using examples of their usage in telescopic lenses, airplane wing design, suspension bridges, and vehicle headlights. Hyperbola, parabola, and ellipse are defined, and Kepler's first law, reflective property, and laws of planetary motion are elucidated.
20. It Started in Greece
This program focuses on Euclidean geometry as a mathematical tool used to measure the world. The Great Pyramid, tunnel construction, and other examples are shown to illustrate congruent triangles, similarity, and the Pythagorean theorem. Students will also learn the parallel postulate and how to distinguish between Euclidean and non-Euclidean geometry.
Module 5 Computer Science
Math's essential role in the development of and relationship to the computer is best seen from a historical perspective. This program surveys the thought and contributions of Hilbert, Turing, and Neumann, revealing how their contributions have brought computers into the heart of modern mathematics. Contemporary mathematicians also explain how they employ the computer in exploring new horizons in their field.
22. Rules of the Game
Algorithms, complete directions for accomplishing tasks, are essential for solving computer-based problems - from playing chess to figuring out income tax. As the application of math to computing is still developing, new algorithms continue to be discovered. Sorting is a large part of what computers do, and this program explores two basic sorting algorithms: insertion sort and merge sort.
23. Counting by Twos
Computers store, process, and reproduce information - be it music or census numbers - in codes that represent data. This program demonstrates the difference between place value systems and two-symbol coding, the binary system and how it is used in computing, and how analog and digital information systems work.
24. Creating a Code
The possible ways of encoding computer information are infinite. Different codes have been devised for text, images, storage and transmission, encryption, and error correction. This program explains the utility of ASCII for text processing, use of the Laplacian pyramid, the advantages of the Hamming code, and the concept of pseudo-random number generation. Cable TV scrambling provides an application example.
Module 6 Summary
25. Moving Picture Show
Today's animation creates amazingly life-like images. But how is this done - and what role does mathematics play? Experts on site at Symbolics, Inc., a leader in computer-generated graphics, offer concise explanations of their handiwork. The creation of a three-dimensional model is demonstrated, followed by object animation, the rendering of a realistic simulation, and automated motion.
26. Summing Up
This concluding program stresses the purpose of understanding mathematics and the importance of mathematics to functioning in society. This message and the varied applications of mathematics in the world are reemphasized and illustrated through clips from the five topic clusters in the series.