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2004 NSF K-12 Math, Science, and Technology Curriculum Developers Conference

Cryptography: The Mathematics of Secret Codes

Janet Beissinger
Vera Pless
Cryptography: The Mathematics of Secret Codes

Our project is developing a module that teaches cryptography and the related mathematics to middle-grade students. This includes a book and interactive website, along with teacher materials. Cryptography is a motivating setting for teaching such topics as the addition and subtraction of positive and negative numbers; percents and decimals; division with remainder; modular arithmetic; prime factorization; greatest common factors; inverses; and exponents.

We have piloted our material in 15 classrooms in the Chicago area, 7 classrooms from around the country, 14 sites in a Chicago after-school program, and one museum in San Jose, CA. The schools represent diverse ethnic, language, and socio-economic populations and include both city and suburban schools. Grades range from fifth to eighth grade.

Two evaluators observed several pilot classrooms and conducted telephone interviews with teachers. The developers also visited classrooms. Teachers provided written feedback on the material and discussed their experiences with it at teachers’ meetings. Below is a summary of the feedback collected from pilot teachers:

• Teachers consistently commented that this material motivates and teaches students of all ability levels. Students who usually are low achievers found it non-threatening and enjoyable and therefore stayed on task. They were actively involved and were able to experience success. Working with the material gave them the opportunity to practice math skills. Stronger students were challenged to think about deeper mathematics that arises in this context.
• Teachers reported that working with this material improved students’ problem solving skills. They observed that their students persisted at solving the problems longer than usual. Students learned that they could attack the same problem with more than one strategy. (We intend to investigate more thoroughly the affects on problem solving ability.)
• The material appeals to girls, as well as boys. One teacher expressed surprise to see that her girls picked up concepts in cryptography faster that her boys, however we have not investigated whether this is true in most classrooms.
• Teachers said that students were interested in taking materials home and that they enjoyed showing parents how to encrypt and decrypt messages. They got their families involved.
• Teachers commented on the ability of their students to learn and apply some fairly advanced topics that come up naturally in cryptography, such as multiplicative inverses in modular arithmetic, relatively prime numbers, and modular congruency. Students were willing and able to work with these more general ideas because they are meaningful and useful in the cryptography setting. Teachers commented that learning the general ideas helped students to better understand the less general topics in the regular curriculum. For example, several teachers felt that working with the notion of inverse in the modular setting helped students to better understand what it means to be an inverse in regular arithmetic.

We are currently developing instruments to assess content knowledge, problem solving ability, and student attitudes and interest.

This project is supported by the National Science Foundation (Grant No. ESI-0352345). Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.