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Acknowledgements
For All Practical Purposes: Introduction to Contemporary
Mathematics and the individual lesson descriptions are provided courtesty of
Annenberg/CPB.
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We are experiencing
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no control over the matter, but have notified the provider.
 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.
-- Annenberg/CPB
Select a
Topic
Allow about 30 minutes for viewing each presentation
- 1. Management Science—Overview
- 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.
- 6. Statistics—Overview
- 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.
- 11. Social Choice—Overview
- 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.
- 16. On Size and Shape—Overview
- 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.
- 21. Computer Science—Overview
- 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.
- 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.
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