Cryptography: Linking Learning and Life

Project Director: Jamie Sloat

Cryptography: Linking Learning and Life

$2462

PROPOSAL SUMMARY

            The incessant question of “When will we ever use this” is heard in classrooms across America and illustrates the general disconnect students have between what they learn in the classroom and what they need to know in the real world.  Our project will help students form links between mathematics they learn in the classroom and problems of communication by focusing on the application of math in cryptography and coding theory. 

            We will help students to see the importance of math to technology, communication and secrecy by highlighting how math is used to keep information secret.  Students will see how math is used to make information difficult to decipher by anyone other than the intended recipient. 

            Since the need for secure encryption is evident through inspection of tools students use everyday (internet shopping, for example), we will help students discover the ways mathematics has been used to ensure higher levels of security.  This project will reinforce the need for students to be able to use what they learn in a real-world context and will help students link classroom life with its real world applications.

Description

This is an approximately seven-week unit employing cryptography as a means for linking together a number of key mathematical and computer concepts with real world application. It will all be brought together as students realize the implications for the community as a whole.

The unit begins with a description of cryptography and how it has been used in history. Introducing it in a historical perspective will allow students to see the potential impact that cryptography can have in today’s world and how it might be used in the future.  Additionally, a historical perspective will give the student’s concrete examples of real world applications rather than the hypothetical “you could use this in this way”. By connecting mathematics with historical events that the students are familiar with (such as World War II), the introduction will arouse further interest and prepare the students for applying cryptography today.

Next, students will be introduced to various forms of coding starting with the more basic shift and multiplier shift codes. Encoding and decoding help illustrate the notions of function and inverse function since all encoding is done with functions and decoding done by the inverse of the function used to encode. The next coding method the students will learn is what is referred to as two-step coding.  An example of two-step coding would be to take the number assigned to the letter, multiply it by two and add 3. The function equivalent of this coding process would be your basic linear function in point-slope form, c=2x+3, where c is the corresponding number to the encoded letter and x is the corresponding number to the original letter. As its name implies, there are two-steps in two-step coding. After students have mastered this technique of coding, we will move onto multi-step coding. Unlike two-step coding, multi-step coding provides a bigger challenge when searching for the inverse function that will decode your message. Students will have to learn the techniques of modeling multi-step codes with two-step codes and the best methods to do so. Since we will be teaching cryptography with linear functions, the functions will serve as the basis for lessons on the use of a graphing calculator. The coding aspect makes such exercises more compelling while the students begin to explore the uses of technology in modern day cryptography. The final section of the unit deals primarily with more advanced technological applications of coding by implementing codes in a computer programming context. This will help to give students experience with how the codes they have learned about can be applied quickly and on large scales.

The main goals of our project include:

·        Students will learn math.

o       They will learn about linear functions, inverses, graph theory, and modular arithmetic.

·        Students will learn that math applies to their lives in important ways.

o       An in depth understanding of the mathematical relationship between encoding and decoding.

o       Students will be able to design passwords and encryption methods that will not be easily broken.

·        Students will learn about the relevance of cryptography today.

o       Students will become aware of the responsibility to encrypt information wisely.

o       They will see that being able to protect electronic information is becoming more important in everyday life, e.g. in bill paying, investments, credit card accounts, and electronic communication.

RATIONALE

            For many students, there is a disconnect between math learned in the classroom and its application in the world outside of school.  Boaler^[1] observes this mental detachment of students from this vital subject in many of her conversations with students.  Students’ inability to make real-world connections between in-school and out-of-school learning has led to the outsourcing of technically advanced and complex jobs to countries where students are more prepared to use what they have learned in novel contexts.

            As the world becomes more technologically advanced and our students fall father behind in line with National Council of Teachers of Mathematics standards, it is imperative that we try to bridge the gap between ideas presented in school and their real world applications to ensure the future success of students.  During our project, we will help our students become better equipped to use knowledge and resources gained in the classroom to undertake problems they see in the community around them. 

            In our society, keeping sensitive information secret has been a vital component of how we communicate with each other and the world.  To that end, we will help our students understand the impact of mathematics on cryptography.  Students will connect the ideas they learn in the classroom to a real-world problem and see how math has changed the face of coding theory as well as how it is used outside of the classroom.

            By reinforcing the utility of ideas presented in the school setting to problems faced by corporations, we will show students the value of being able to bring together ideas they learn with problems they face, and thus help them to see how what they are learning is applicable to the world outside of school.

Potential Impact

This project will directly involve approximately 180 students during the first year of implementation. We hope to continue this program in the future as other students see the excitement of their peers and look forward to doing this project as well. The students will present their final projects at the end of the unit to a group faculty and peers.

In the next three years, we hope to have the project implemented into all Algebra 1 classes at our school.  This would directly impact the not only this years students, but every Algebra 1 student at our school over the next 3 years and beyond. As the teachers who are chosen to help with the implementation of this program experience the impact it will have on their students, they will hopefully be motivated to broaden their use of project based instruction throughout the school year. 

Evaluation Plan

The success of this unit will be evaluated individually as well as a group of students. Students will be learning about various encoding schemes in this unit. Along with each scheme, worksheets and homework assignments will be given to both assess the students’ progress in understanding the coding scheme as well as the underlying mathematical concepts.

            The unit has various lessons on how to crack codes. Students will have to not only crack different codes, but will have to write essays on the difficulties involved in such procedures. These essays can be used to ensure that the students understand the mathematical and coding principles involved.

            This unit also has a lesson where students will implement each encoding scheme on a computer. The implementations will demonstrate that the students understand not only how to use a code but show how codes are used in the real world.

The final evaluation of this project will be a scavenger hunt where students will have to decode a message to get to the items. This final evaluation will test the students abilities not only to decode particular schemes, but to determine which scheme is being used.

Calender

Budget