Cryptography as a Teaching Tool

 

Is there a more effective way to teach Mathematics to high school students? When we was in high school the teacher would stand in front of the classroom, walk us through an example problem, then assign homework. That was our routine everyday for four years. We are not trying to declare this method ineffective, nor is Aaron. Rather, we both wonder if there is a more interesting, fun, and captivating way for students to learn math.

Throughout the four years of high school there were very few opportunities for us to apply the mathematics we learned in the real world. There were also few problems that even dealt with real life situations. When Aaron and I brainstormed, we bot h agreed that we would need to develop a unit plan that would enable high school students to apply the mathematics that they have been learning. Aaron and I both feel that application and analysis are intricate parts of learning, especially when it comes to math. We want to be able to teach math in such a way that the students would enjoy learning and see the need for it in life. Students are more inclined to really master the material if they can find ways to apply it to the real world and themselves and enjoy it in the process. To address desire, Aaron and I thought we should implement a unit that would introduce high school students to cryptography.

We feel cryptography is one of the most effective ways to approach the applications of math in today’s world. According to Webster, cryptography is defined as the writing and deciphering of messages in secret code. Cryptography is the study of co des. One might wonder how this relates to math. Most codes involve a great deal of math. Take credit card numbers for example, they are actually coded to prevent people from purchasing merchandise with false numbers (sixteen digits that could be chosen at random). The whole coding process is mathematically based. Let us look at a specific example.

8721 6608 7187 0319.

To find out if this is a valid credit card number, we must first double each of the odd digits. 8 becomes 16, 2 becomes 4, 6 becomes 12, and so forth. Our credit card number now looks like this:

16 7 4 1 12 6 0 8 14 1 16 7 0 3 2 9.

For any of the numbers that exceed 9, we must subtract 9 from them and get the following. We now have,

7741 3608 5177 0329.

If the sum of these numbers is divisible by 10, then the number is a valid credit card number. In this case the sum is 70, so it is indeed valid. Now as to whether or not this number is associated to an account, that is an entirely different matter a ltogether. This is one example of how math can be involved in coding.

We wanted to use cryptography because we knew that the possibilities for math lessons would be intriguing. Many different aspects of math are used for coding messages. A trouble we experienced was finding lessons where the level of math was appr opriate for high school math students. While substitute teaching in a local high school, Aaron had an opportunity to speak to students in upper level math courses about what concepts of math they had been introduced to. With that knowledge we were able to devise a unit with six lessons that we feel is appropriate and challenging for high school students. We feel that they could handle the lessons provided they were introduced to the following mathematical concepts: congruencies and modular arithmetic, factoring composite numbers into prime factors, linear equations, exponents and exponential equations, permutations and matrix multiplication.

Our reason for wanting to teach a unit like this is to assist students with their problem solving skills. If there is one thing that both Aaron and myself feel is very important to learning math, it is being able to solve problems. This feeds rig ht into the application part of the unit. We hope that by introducing a topic like codes, interest in the subject will increase and the students will actually want to solve the problems posed by coding and decoding. Then as a direct result the students will sharpen up on their problem solving skills. Another hope of ours is that this unit will promote critical thinking, which is a key element in developing keen mathematical skills. We do not expect the students to master mathematics. We expect studen ts to learn how to solve problems. The ability to think critically can only have positive affects even if the student loathes math.

We feel that cryptography promote problem-solving skills. One way cryptography encourages people to practice their problem solving skills is with the process of deciphering. Imagine, if you will, that you are in charge of deciphering an intercept ed coded message. You do not have the key either, how do you go about breaking it? That is what the British were faced with during World War II. The British were able to intercept the Germans coded messages, but before they solved the problem of not hav ing the key they were unable to read them. How they solved that problem is the subject for an entirely different paper. The fact that the coded messages provided a problem to be solved is all we are looking to present. Here is another problem, but this time it has nothing to do with warfare. Now imagine that you are in the business of selling computer software. You want to be able to take advantage of the Internet and allow people to purchase your software through telephone lines. Seeing how the tel ephone lines are not the safest place for people to let their credit card numbers float around, how will you, as a manufacturer, ensure the customers that it is safe to order over the Internet? You will need to use some sort of code system. Cryptography offers plenty of opportunities to practice problem-solving skills.

Teaching cryptography can be approached in many different ways. One advantage is that a unit like this can offer a teacher flexibility. This unit would be taught in the upper level math course. There would be fewer requirements that would have to be met by the teacher. As the teacher you would be able to organize the progression of this unit however you may wish. There is no text book from which to follow, nor are there any restrictions other than the level of math involved. There is only so mu ch math that high school students are exposed too. Another perk of the unit, for the teacher, is the fact that it is altogether different. It demands for a teacher to be creative. If teachers have the opportunity to try something new, something they cr eated, they are more likely than not to be a great deal more passionate about their work. If the teacher is passionate, then that attitude is likely to run off onto the students, and they may become passionate as well. When both the teacher and the stud ents are engrossed in the subject matter, then more learning takes place. This unit may be taught in different ways as well. The teacher can have the students work collaboratively, or instruct them in the "traditional" sense. By traditional we mean that the teacher stands in front of the class and lectures. This unit by design was intended for group work, but does not have to be taught in that manner.

We also feel the student stand to gain greatly from a unit like this. If approached by design one of the benefits, although they may disagree at the time, is the experience of working with others. Group work is a touchy subject. Most teachers wo uld agree that group work is important, but some would not use it. The reason for that is the fact that with group work at the high school level, students will not work together like the teacher would desire. The more advanced students are more likely t han not to take on the entire project while the slower students would sit back and watch. Ideally, all students should be responsible to the group, but realistically it doesn’t always happen. We do see the argument being made, but Aaron and I both feel that group work, regardless, in the long run benefits the student. Not only that, if the students are in the level math course where they would see this lesson, they are most likely to be there because they want too. Another benefit for the students is the real life applications of cryptography. As we indicated before, if the students can apply what they learn, the chance of them really "owning" the material is much greater. There is also a sense of discovery with cryptography. New worlds o pen up for students when they discover new knowledge. Because of its problem solving and critical thinking nature, cryptography leaves a lot to be discovered. I addition to discovery, collaborative learning, and application, the students will also take with them a good foundation in math, due to the mathematical nature of cryptography.

The unit we designed serves multiple purposes. We want to answer the question we hear from students, ‘when are we ever going to use this stuff’. This unit uses math that our students are familiar with and applies to concepts that hopefully are intere sting and thought provoking. As we have stated before, we want our students to be able to think about the problems they see and be able to come up with complete creative solutions. This does not simply just happen. It has to be taught. This unit attem pts to teach the students how to think about mathematics and get meaning from the numbers.

In our first lesson, we simply want to introduce our students to the concept of coding. We talk about the history of coding, the effect that it has had on our history, our world. For instance, If the Polish, British and Americans did not break En igma, who knows exactly how for Germany could have gone. Because of math and it’s application, one could argue that math played an enormous part in ending World War II. This is the thinking we want our students to have. We want them to see just how imp ortant mathematics are and how powerful math can be. We begin our coding by looking at the Caesar Cipher and the Vingenere Cipher and have our students get somewhat aquatinted with coding.

Our second lesson deals with many different codes. We want to show our students some of the variety of coding and we also in this lesson show them some of the current ways that coding is used to help our society. Postal codes, Bank Identification codes are two of the codes we show them that deal with patterns in numbers that make up coding systems. Matrices are what we use in our third lesson to do our coding. We simply use 2X2 matrices to keep the math easy enough so that they do not get lost in it. We liked this method because it used a different mathematical method to code.

The fourth lesson revolves around Enigma, how it works, and all of the definitions and terms that go with it. We also introduce our students to Finite State Transducers. We feel that through the FST, the students would have an easier time relating to Enigma. This is a lesson that could easily take two or three class days.

The fifth lesson is the other major focus of the unit, along with Enigma. The project for the unit comes out of this lesson. We take a look at linear equations and how they can be used in coding. Modular arithmetic can also be used in accommodat ion of the linear equations. Through this project, composition of functions is another spin off that can be taken. It is a project that can become quite complicated. We think that a few days would also be spent on this lesson. It is interesting becaus e the aspect of a plugboard, reflector and states can be implemented using linear equations. Using the form mx + b allows a person to assign values to m and b and make the assignment somewhat systematic. By assigning different values to m and b us ing more than one equation, the aspect of rotors can also be added to this project. The other advantage to beginning with linear equations is the fact that you can deal with other equations as well. Exponentials can work if you limit your domain so that the function is still one to one for the part that the student is using. As can be seen, depending on the class and the teacher, this lesson can be taken in many different directions and we believe that adds to the effectiveness of this unit.

Finally, we look at RSA coding, which is a current type of coding system that deals with prime numbers, modular arithmetic and exponintials. We keep the numbers to a level that the students can deal with them, but also show how easily this can bec ome very difficult. We also want the students to reflect on some of the current issues of cryptography. We planned on having the student’s do some research and give us their ideas on security and other issues. We thought this would make a nice ending t o our unit.

Cryptography is an area that seems to be growing in popularity. We feel privileged to work on this and put something constructive together. At New Mexico State University, they offer a class to high school students on Cryptography and next weeken d, April 18; there is actually a workshop for secondary teachers at the university on Cryptography and high school mathematics. We feel that the teaching of math is moving in the direction of more application and that our unit addresses this need.

We struggled with this project. We both learned a great deal, but there really is not much information at all out there about the teaching of Cryptography. It is definitely a subject that deserves so much more attention than it is receiving at th e present moment. With both of us going into education, we learned a great deal of what each of us teach in the classroom and at some point, that will include Cryptography.


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