http://umastr1.math.umass.edu/~frankw/ccp/talks/IndianaMAA/index.htm
http://www.math.montana.edu/~frankw/ccp/talks/IndianaMAA/index.htm

An example and a metaphor

Almost every meeting of mathematicians has one or more talks on "The Impact of Technology on Mathematics" or "The Impact of Technology on Mathematics Education." The real news, however, is the impact of mathematics on technology or, even better, the synergy between mathematics and technology. Click here for a .pdf file that can be used as an example and a metaphor. This file produces six copies of the pattern shown at the right.

Make copies of this pattern on transparency film and also on ordinary paper. One pattern by itself represents either technology or mathematics. If you put one copy of the pattern on transparency film on top of another copy on plain paper and they don't line up exactly you will see completely new patterns -- the results of the synergy between mathematics and technology. This synergy is the subject of this talk.

• Algorithms and programming
• Images and animation
• Digital signal processing
• Simulations
• Bench-lab experiments
• Primary resources
• Tools
• Conclusions

• Mathematical algorithms
  • Numerical analysis
  • Parallel algorithms
  • Sorting algorithms
  • Fast Fourier Transform
  • Encryption, Code-Breaking, and Number Theory

• Logic
  • Programming languages
  • The limits of algorithms -- for example, Godel's Incompleteness Theorem, the Halting Problem and the Blum Speed-up Theorem
  • Program verification

• Shading
• Location and motion

• Fourier series, Fourier analysis, and wavelets
• Voice recognition
• Digital hearing aids

• Ripple Tank
  • Frank's ripple tank
  • WebPhysics ripple tank
  • An example and a metaphor
  • Computer algebra system ripple tank
• Probabilistic models
• Differential equation-based models
• Dynamic system modeling -- the Beer Game and others

Three questions

Suppose that for each birth the probability of having a male child is exactly 0.50. In these questions we ignore the possibility of twins. In country A each woman has two children. In country B each woman continues having children until she has her first son.

Will the percentage of sons in country A be lower, the same, or higher than in country B?

Will there be fewer, the same, or more children in country A than in country B?

Discussion.

Are you sure?

Discussion.

• Inexpensive equipment like the Texas Instruments' CBL (calculator-based laboratory), laser pointers for under $20, Tinkertoys, slides made with ordinary 35mm cameras, and videotapes made with consumer quality camcorders enable us and our students to do experts of our own design. For example, the TI-92 screen below shows a recording made with the CBL as sound from a casette recorder passed two microphones and the red image below is the diffraction pattern produced by a laser pointer using a slide printed on a consumer quality Postscript printer.

• The World Wide Web faciltates distributed collaborative experiments in which thousands of students collect and share data. Consider, for example, the problem of keeping a car cool when it is parked all day in the hot sun. People try all sorts of things. Some leave a window open a crack; others use a sunshade to block sun coming in the windshield; and others buy white cars.

  One possible experiment would be to have thousands of people each do an experiment recording the variation of temperature in a parked car over the course of a day along with other relevant variables. Because there are some many possible combinations of uncontrollable variables, a small number of individual recordings would be useless. But, the Web can enable us to collect enough data to be useful. Analyzing this data requires an understanding of statistics and is similar to analyzing other observational data in situations where there are many uncontrollable factors.

  In addition, the Web could enable us to share the results of a few controlled experiments performed, for example, by students whose parents ran car dealerships.

• Remotely accessible equipment. A growing number of Web sites provide access to scientific equipment that is either placebound, too expensive or too dangerous for every classroom. Examples include
  • An optical telescope at the Remote Access Astronomy Project at the University of California at Santa Barbara.

  • A radio telescope at the Haystack Observatory.

  • Liquid crystal optics at The ALCOM Science and Technology Center.

  • Online Experiments on Biological Timing at the Center for Biological Timing at the University of Virginia.

  • Scanning probe microscopy at INVSEE -- Interactive Nano-Visualization in Science and Engineering Education

• Massive and real time data
• Museum- and research-quality collections of specimens

All of these resources are useless if students are unable to formulate interesting questions; design experiments that can answer these questions; and analyze and interpret experimental results and other data. All this is the province of mathematics.

• Computer algebra systems
• Statistics packages
• Visualization
• Live mathematics on the Web

Conclusions

• Mathematical modeling is central to the intelligent use of technology.
• Data analysis is central to the intelligent use of massive and heterogeneous data.
• Probability underpins both probabilistic models and data analysis.
• Linear algebra for modeling, images, digital signal processing, and much more.
• Calculus and differential equations is essential to modeling.
• Three dimensional geometry, multivariable calculus for images and modeling.
• Logic for decision making and programming.

• Major conclusion -- Two year core!!!

Copyright c 1999
Frank Wattenberg
Educational and Productivity Solutions,
Texas Instruments, Inc.
7800 Banner Drive, MS 3908
Dallas, TX 75251
frankw@ti.com