The materials below were developed for the use of students and faculty of the CUNY Teacher Academy and the Teacher Academy at York College, as well as other persons interested in Mathematics Education.

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This essay was used as a handout for the Teacher Academy at York sponsored celebration of "Pi Day" in 2007. It contains some information about properties of pi in other metrics than the Euclidean metric as well as web resources for students and future teachers to learn more about pi.

Pi Day

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### Modeling, Teaching using Contexts and Applications of Mathematics:

This essay outlines urban operations research problems (running errands, laying fiber optic cable, setting up study groups, locating a new fire house, etc.) that can be used as contexts for a variety of traditional middle school and high school skills at various grade levels.

Mathematical Modeling

A slightly modified version of these materials is available here.

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This note was developed to provide a context for teaching such 6th grade skills as graphing points on a line, learning to use absolute values, computing with signed numbers, and learning about distance for New York City 6th graders. It includes ideas about 1-dimensional and 2-dimensional facility location problems. (Where would be the best place to locate a mobile vaccination unit.) In the two-dimensional case it introduces the need in many urban situations for using taxicab distance rather than crow flies distance (Euclidean distance) as a way to measure how far apart two points may be. It also discusses how other distance functions such as Hamming distance can be used to get across the idea of how a spell check works (and to construct the genome of various species). Some spell checkers use Hamming distance and others use a phonetic based approach that involves the system known as Soundex, which is briefly described here. Soundex was developed as a tool for finding different variants of the way names can be written in latin characters based on how the name sounds (early immigrants to the US did not know how to spell their names in latin characters), and was developed for genealogical purposes.

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The concepts of mean, median, and mode are statistical tools for finding the "middle value" for a collection of data. When solving faciltity location problems (where should an ice cream truck park to make it convenient for customers to get to the truck) raises similar problems. The essay below suggests ways to connect up basic "measures of central tendency" in statistics with issues of facility location, and thus, can be used on a stand alone basis or for use with the notes about facility location above.

Statistics and Facility Location

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Society has benefited greatly from the telecommications revolution in the late 20th century, and rapidly continuing today. Fax machines, cell phones, wireless communications and HDTV are transforming our society. The following material consists of skeletal slides for a presentation about the ideas of error-correction and data compression that lie behind the telecommunication revolution. In particular, it is described how to send an imagine over a telephone or radio channel using binary codes.

The Mathematics Behind Your Cell Phone

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### Pedagogy:

Addition is a fundamental topic in K-12 education. However, there are a variety of subtle issues to address. When does finding an estimate make more sense then finding an exact answer? When does it make more sense to use a calculator rather than using a pencil and paper algorithm or solving the problem mentally? This brief note encourages future teachers to think about these issues.

How Should I Do This Addition Problem?

Future teachers need to be able to discuss in simple terms how mathematics differs from other areas of knowledge such as sociology, french literature, physics, geology, and chemistry. What makes mathematics special is that it derives theorems from simple axioms or rules. This note, which builds in part on the two papers above, illustrates two mathematical theorems where a proof can be provided that can be understood by elememntary, middle school and high school students. The proofs shown answer questions about how to design an efficient pot hole inspection route and locate a facilty along along a line by stating and proving theorems that shed light on these applied urban problems.

The Role of Proofs in Lower Grades

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### Resource Materials for Teachers:

A very skeletal bibliography of resources for teaching mathematics using contexts, mathematical modeling, and the use of applications for students who plan to become K-12 mathematics teacers.

Bibliography

A list of journals which publish articles dealing with research in Mathematics Education or provide good source materials for future and/or inservice mathematics teachers.

Bibliography

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Teacher Academy students may have some interest in mathematical modeling, which is taking on an increasing role in K-12 curriculum. One good source of modeling problems to think about can be found on the MathModels.org web site that is sponsored by COMAP, the Consortium for Mathematics and Its Applications:

Mathmodels.org

Here are some problems that I have contributed to MathModels.org:

Hotel Charges

School Quality and Class Size

Shaking Marbles

Heat Transfer Fan

Leaf Problem

Without A Trace

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Note: Thanks go to John Campanella (Queens High School for the Sciences) and Daniel Nizich (York Early College Academy) for feedback that helped improve these materials.

This work was supported in part by the Teacher Academy of York College. Specific funding was provided by: FIPSE (46274-07 01) and the Fund for PS (72042-07 01) to the Teacher Academy of CUNY.