Cryptography: Decoding Student Learning.

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ABSTRACT

PATTERSON, BLAIN ANTHONY. Cryptography: Decoding Student Learning. (Under the direction of Karen Keene.)

Cryptography is a content field of mathematics and computer science, developed to

keep information safe both when being sent between parties and stored. In addition to

this, cryptography can also be used as a powerful teaching tool, as it puts mathematics

in a dramatic and realistic setting. It also allows for a natural way to introduce topics

such as modular arithmetic, matrix operations, and elementary group theory. This thesis

reports the results of an analysis of students’ thinking and understanding of

cryptog-raphy using both the SOLO taxonomy and open coding as they participated in task

based interviews. Students interviewed were assessed on how they made connections to

various areas of mathematics through solving cryptography problems. Analyzing these

interviews showed that students have a strong foundation in number theoretic concepts

such as divisibility and modular arithmetic. Also, students were able to use

probabil-ity intuitively to make sense of and solve problems. Finally, participants’ SOLO levels

ranged from Uni-structural to Extended Abstract, with Multi-Structural level being the

most common (three participants). This gives evidence to suggest that students should

be given the opportunity to make mathematical connections early and often in their

academic careers. Results indicate that although cryptography should not be a required

course by mathematics majors, concepts from this field could be introduced in courses

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Cryptography: Decoding Student Learning

by

Blain Anthony Patterson

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Master of Science

Mathematics Education

Raleigh, North Carolina

2016

APPROVED BY:

Karen Hollebrands Molly Fenn

Karen Keene

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DEDICATION

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BIOGRAPHY

Blain Anthony Patterson was born on May 7, 1991 in Salem, Ohio. He grew up in the

small town of Wellsville, Ohio. Blain graduated from Youngstown State University with

a B.S. degree in Mathematics Education in 2014. Directly after graduating, he entered

the M.S. in Mathematics Education at North Carolina State University.

Blain originally intended to teach high school mathematics somewhere near his

home-town in Ohio. After meeting the love of his life, Sarah Elizabeth Ritchey, Blain had a

change of heart and decided that graduate school was in his future. After applying to

several graduate programs in both mathematics and mathematics education, he decided

that NCSU was a perfect fit.

Blain plans to enroll in the Ph.D. program in Mathematics Education at North

Car-olina State University for the spring 2016 semester. His interests focus on how

under-graduate students learn upper level mathematics, such as Linear and Abstract Algebra,

Number Theory, Probability, and Real Analysis.

Blain has always had a passion for teaching and mathematics. His long term goal

is to be a university professor and teach mathematics at a teaching-focused college. So

far, North Carolina State University has provided him with the skills to become a better

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ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Karen Keene, for her support from the time I

entered the program until completion of this project. Her guidance and expertise have

been an essential part of my success at North Carolina State University. I look forward

to working with her in the future. Also, I would like to thank Dr. Karen Hollebrands and

Dr. Molly Fenn for their suggestions and willingness to serve as committee members.

I would like to thank my father, Dorman Jeff Patterson II, for always believing in me

and pushing me to strive for excellence. I know without a doubt I would not be where

I am today without the constant love, patience, guidance, and support of my father. He

has always put my needs before his own and continues to do so to this day.

I would like to thank my best friend, Sarah Elizabeth Ritchey for making me a better

teacher, learner, and person. She has been by my side throughout my entire graduate

school journey. Sarah is the one that convinced me that graduate school was right for

me. She has guided me down a path of success and happiness. We are an unstoppable

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LIST OF TABLES

Table 2.1 SOLO Levels . . . 15

Table 2.2 Index of Coincidence . . . 28

Table 3.1 Student Demographics . . . 33

Table 3.2 SOLO Rubric . . . 40

Table 4.1 Student SOLO Classification . . . 83

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LIST OF FIGURES

Figure 1.1 Vigenere cipher with DOG as keyword. . . 4

Figure 1.2 Caesar cipher with a shift of 4. . . 4

Figure 1.3 Cipher wheel. . . 5

Figure 1.4 Public-key cryptography. . . 7

Figure 1.5 Elliptic curve. . . 8

Figure 1.6 Elliptic curve secant addition. . . 9

Figure 1.7 Elliptic curve tangent addition. . . 10

Figure 1.8 Elliptic curve vertical addition. . . 10

Figure 2.1 SOLO Taxonomy Pyramid. . . 17

Figure 2.2 Frequency distribution of the alphabet. . . 27

Figure 2.3 Euclidean algorithm for 1559 and 650. . . 30

Figure 4.1 Student A assigning numbers to letters. . . 42

Figure 4.2 Student A using an affine cipher with a= 10 and b= 0. . . 43

Figure 4.3 Student A using a Caesar cipher with 3 as the key. . . 43

Figure 4.4 Student A assigning numbers to letters. . . 44

Figure 4.5 Student A using a brute force method. . . 45

Figure 4.6 Student A using a factor tree. . . 46

Figure 4.7 Student B setting up a Hill cipher. . . 47

Figure 4.8 Student B multiplying matrices modulo 26. . . 48

Figure 4.9 Student B summarizes his work. . . 48

Figure 4.10 Student B using 1 as the key. . . 49

Figure 4.11 Student B displaying all possible keys. . . 50

Figure 4.12 Student B decrypting using various keys. . . 50

Figure 4.13 Student B reducing the total number of divisors. . . 51

Figure 4.14 Student B factors 8911 as 8911 = 7×1273. . . 51

Figure 4.15 Student B repeats his algorithm for 1273. . . 52

Figure 4.16 Student C assigning numbers to letters. . . 54

Figure 4.17 Student C encrypting use the one-time pad. . . 55

Figure 4.18 Student C displaying the ciphertext. . . 55

Figure 4.19 Student C using 25 as a key. . . 56

Figure 4.20 Student C listing possible two letter combinations. . . 57

Figure 4.21 Student C displaying the plaintext GOWOLFPACK. . . 57

Figure 4.22 Student C approximating √8911. . . 58

Figure 4.23 Student C dividing 8911 by 7. . . 59

Figure 4.24 Student C checking 7 and 11 as factors . . . 59

Figure 4.25 Student C displaying the prime factorization of 8911. . . 60

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Figure 4.27 Student D using frequency analysis. . . 62

Figure 4.28 Student D decrypting using a Caesar cipher. . . 63

Figure 4.29 Student D checking small prime divisors of 8911. . . 64

Figure 4.32 Student E using a Vigenere cipher with keyword GO. . . 66

Figure 4.33 Student E assigning numbers to letters. . . 67

Figure 4.34 Student E using 1 as they key. . . 67

Figure 4.35 Student E decrypting using various keys. . . 68

Figure 4.36 Student E using the Pollard Rho factoring algorithm with n = 8911. . 69

Figure 4.37 Student E factors 8911 as 8911 = 7×1273. . . 69

Figure 4.38 Student E using the Pollard Rho factoring algorithm with n = 1273. . 70

Figure 4.39 Student E factors 8911 as 8911 = 7×19×67. . . 70

Figure 4.40 Student F assigning numbers to letters. . . 72

Figure 4.41 Student F using an Affine cipher with a= 3 andb = 5. . . 72

Figure 4.42 Student F attempting to decrypt with 1 as a key. . . 73

Figure 4.43 Student F attempting to decrypt with various keys. . . 74

Figure 4.44 Student F testing prime divisors of 8911. . . 74

Figure 4.45 Student F factoring 8911 as 8911 = 7×1273. . . 75

Figure 4.46 Student F testing prime divisors of 1273. . . 76

Figure 4.47 Student F displaying the prime factorization of 8911. . . 77

Figure 4.48 Student G assigning numbers to letters. . . 79

Figure 4.49 Student G encrypting the message NCSU. . . 79

Figure 4.50 Student G summarizes her work. . . 79

Figure 4.51 Student G shifting the alphabet by 17. . . 80

Figure 4.52 Student G shifting the alphabet by 17. . . 80

Figure 4.53 Student G checking for small prime divisors of 8911. . . 81

Figure 4.54 Student G checking for small prime divisors of 1273. . . 81

Figure 4.55 Student E factors 8911 as 8911 = 7×19×67. . . 82

Figure D.1 Student A problem 6. . . 103

Figure D.4 Student B problem 6. . . 106

Figure D.7 Student C problem 6. . . 109

Figure D.10 Student D problem 6. . . 112

Figure D.11 Student D problem 7. . . 112

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Figure D.13 Student D problem 8 continued. . . 113

Figure D.14 Student E problem 6. . . 114

Figure D.17 Student F problem 6. . . 117

Figure D.20 Student G problem 6. . . 120

Figure D.21 Student G problem 7. . . 121

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Chapter 1 Introduction

In this chapter, I discuss the importance of cryptography for both security and

peda-gogical reasons and provide the background and significance for the study. I begin by

introducing some important concepts and definitions.

1.1 What is Cryptography?

Cryptography is often thought of as the practice and study of the techniques and

al-gorithms for secure communication. It is used on a daily basis, for keeping information

safe (Kahn, 1997). For example, anytime someone pays for some good or service online,

their credit card information is encrypted and decrypted using advanced algorithms that

are part of the cryptography. It can be described as the intersection of mathematics,

computer science, and electrical engineering (Singh, 1999).

Cryptography has a rich history. As a society, humans have been trying to keep

messages secret from the beginning of time. In fact, historians have discovered cryptic

hi-eroglyphics from around 2000 B.C (Kahn, 1997). Julius Caesar used a substitution cipher

to keep messages secret (Kahn, 1997). Eventually, complex machines, like the Enigma

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age had surfaced, more complex methods arose. With the abundance of history related

to cryptography available, an entire class could be devoted to studying this material.

1.2 Definitions and Examples

Encryption is the process of converting a message to something unintelligible (Kahn,

1997). The message we are converting is referred to as the plaintext and the message

that we get after converting and should be unintelligible is theciphertext (Kahn, 1997).

The inverse of encryption is referred to as decryption, where one converts ciphertext to

plaintext (Kahn, 1997). Acipher can be defined as a set of techniques and algorithms that

createencryption anddecryption methods (Kahn, 1997). Ciphers can bemono-alphabetic

orpoly-alphabetic.Mono-alphabetic can be thought of as a bijective function, where each

letter is uniquely mapped to another letter (Kahn, 1997). Note that a letter can be

mapped to itself.Poly-alphabetic ciphers may map the same letter to two different letters

(Kahn, 1997). In order to decrypt ciphertext, one needs to use a key. Cryptosystems can

be defined as the set of all possible plaintexts, ciphertexts, keys, and algorithms (Kahn,

1997). We define an algorithm as a well defined process with a finite number of steps.

Finally,cryptanalysis is the study of methods for cracking encryption algorithms without

a key (Kahn, 1997).

In addition to jargon associated with cyrptography, one must also consider notation.

The goal is to send messages securely. Keeping this in mind, the two parties who are

communicating should not make an intruder’s job any easier. This may be done by

removing all forms of punctuation, spaces, and capitalizing all letters (Kahn, 1997).

Removing punctuation simplifies the process, by not having to assign numbers to periods,

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there are only a small amount of words of a given size. Similarly, capitalizing all letters

will make breaking the cipher more difficult for the attacker. For example, sending the

message “I need help!” instead of INEEDHELP is not wise, since there are only a small

amount of words that are one letter in length.

What are some examples of ciphers? The Caesar cipher is one of the oldest ciphers

recorded. Julius Caesar used this particular cipher to encrypt messages of military

signifi-cance (Kahn, 1997). This cipher consists of shifting each letter in the alphabet to another

letter by first converting each letter to a number. A is assigned to 0, B is assigned to 1, C

is assigned to 2, and so on. Once each letter is represented by a number,x, the formula for encryption is Ek(x) =x+k mod 26 and decryption is Dk(x) =x−k mod 26. Modular

arithmetic uses 26 as the divisor, since there are 26 letters in the alphabet.

Another elementary cipher is the Vigenere cipher. Similar to the Caesar cipher, letters

are shifted by a key (number). However, instead of just a single number as the key,

messages are shifted by a key word. This makes the Vigenere cipher poly-alphabetic. For

example, supposed the key word is DOG. Note that D, O, and G are the fourth, fifteenth,

and seventh letters of the alphabet respectively. This implies we the first letter of the

plaintext will be shifted by 4, the second letter by 15, and the third letter by 7. The

plaintext will most likely be longer than 3 letters or the size of the key word in general.

If this is the case, the process repeats. In other words, the fourth letter will be shifted by

4, the fifth by 15, the sixth by 7 and so on.

An example will make this concept more concrete. Suppose MEETATDAWN is the

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Figure 1.1: Vigenere cipher with DOG as keyword.

This implies that M will be shifted by 4, the first E will be shifted by 15, the second

E will be shifted by 7, and so on. This will result in a ciphertext of PSKWOZGOCQ.

Decryption would happen in a similar manner, by using the same key.

These first two examples are called shift ciphers, since each letter in the plaintext is

shifted by a key to create the ciphertext. Although these methods are not used today in

the practical sense, they offer a great place to start. Modular arithmetic is an elementary

topic. In fact, students do modular arithmetic in grade school without realizing it.

Because of the elementary knowledge needed, shift ciphers are very accessible to

students. In particular, the Caesar cipher offers multiple representations. One can view

this cipher using Figure 1.2.

Figure 1.2: Caesar cipher with a shift of 4.

If a student is more algebraically inclined, they may favor the following formulas for

encryption and decryption.

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Students that are consider tactile learners may choose to use a cipher wheel shown in

Figure 1.3.

Figure 1.3: Cipher wheel.

If students have a sufficient background in linear algebra, the Hill cipher can be

discussed. Created by Lester S. Hill in 1929, the Hill cipher was the first poly-alphabetic

cipher which was practical to operate on more than three symbols at once (Singh, 1999).

In fact, the Hill cipher of dimension 6 was done mechanically using a system of gears and

chains (Kahn, 1997).

To encrypt using the Hill cipher, one must first convert letters to numbers using the

standard assignment (A is assigned to 0, B is assigned to 1, C is assigned to 2, and so

on). Once the plaintext is converted to numbers, the key matrix is chosen. In order for

decryption to be possible, the key matrix must be invertible modulo 26. In other words,

the determinant must be an integer that has an inverse modulo 26. The sender converts

the plaintext to n ×1 column vectors, where n is the dimension of the key matrix. To decrypt, one simply multiplies by the inverse of the key matrix.

Consider the following example. Suppose the plaintext is CAT, where C corresponds

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following. x=       2 0 19      

Now suppose the key matrix is below.

K =      

1 0 0

2 13 0

3 4 1

     

Notice that the determinant of K is 13, which has an inverse modulo 26. To encrypt, we compute the following.

Kx =      

1 0 0

2 13 0

3 4 1

            2 0 19       =       2 4 25       =y

Therefore the ciphertext is CEZ. To decrypt, one must compute K−1_y _{mod 26}_≡ _x_,

to recover the plaintext vector, x. Because matrix theory is a part of linear algebra, a strong foundation in that content area would be extremely useful for a student studying

cryptography to have.

Advancing to more recent work, modern cryptography methods are primarily

public-key systems. This means that the sender and receiver each have their own public-key, which are

related to a public key through computations (Kahn, 1997). The security of public-key

cryptography relies on the fact that the computations required to break the cipher is

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the cipher could not be broken in a reasonable amount of time. The general scheme of

public key cryptography can be seen in Figure 1.4.

Figure 1.4: Public-key cryptography.

The RSA algorithm is the most widely used public-key cryptosystem today. Created

by Ron Rivest, Adi Shamir and Leonard Adleman in 1978, this encryption method takes

advantage of the difficulty of factoring large composite integers (Kahn, 1997). With a

powerful computer, multiplying two large prime numbers is computationally feasible.

However, recovering the prime factors of a composite number takes the most powerful

computers a great deal of time (Kahn, 1997).

To encrypt a message with RSA, one must choose two “large” prime numbers p and

q and compute n =pq. Then choose a numbere, such that the greatest common divisor of eand (p−1)(q−1) is 1. The key (n, e) are made public, but pand q are kept private. In order for an adversary to break this cipher, this must factor n as n = pq, which is computationally infeasible.

Other examples of public-key cryptography include Diffie-Hellman, Cramer-Shoup,

ElGamal, and elliptic curve (Kahn, 1997). Although these methods are much more

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mathematics, including algebra, number theory, probability, and geometry.

One of the most accessible of these methods to students may be elliptic curve

cryp-tography. An elliptic curve is a curve in the plane of the form y2 ₌ _x3₊_Ax₊_B_{, with}

the condition that x3₊_AX₊_B _{has distinct roots (Abraham, Kapoor, and Singh, 2008);}

elliptic curves are also symmetric about thex-axis. Figure 1.5 shows a graph of an elliptic curve

Figure 1.5: Elliptic curve.

For computational purposes, these curves exists over finite fields, rather than all real

numbers. For example, letE be an elliptic curve over the integers mod 2. This implies that

A andB must be 1 or 0 and the only possible points on the curve are (0,0),(1,0),(0,1),

and (1,1). However, not all these points need be on the curve.

With each elliptic curve comes an algebraic structure, where two points can be added

in any order, each point has and inverse, there exists an identity point (the point at

infinity), and the associative property holds. Therefore, the points on an elliptic curve

form an abelian group (Abraham, Kapoor, and Singh, 2008). However to actually add

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In order to add two points, P and Q, on an elliptic curve one most construct the line between those two points. If line P Qgoes through the curve at a different point, R, one must construct the line perpendicular to the x-axis that goes through R. The point where this line intersects the curve isP +Q (Abraham, Kapoor, and Singh, 2008). This can be seen in Figure 1.6.

Figure 1.6: Elliptic curve secant addition.

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Figure 1.7: Elliptic curve tangent addition.

Finally, if two points are symmetric about thex-axis, the line the goes through these points will not intersect the curve at a third point. Since the points on the elliptic curve

form a group, the result must be another point on the curve. This point is called the point

at infinity, denoted byO. This point will be on any vertical line (Abraham, Kapoor, and Singh, 2008). This can be seen in Figure 1.8.

Figure 1.8: Elliptic curve vertical addition.

Clearly, to understand utilize elliptic curves for cryptography purposes, one must have

a firm grasp on fundamental geometric concepts. It may be the case that students have

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understand geometric concepts can give the researcher much need insight about their

ability to solve particular cryptography problems and vice versa.

1.3 Why Cryptography?

Cryptography is a field of mathematics associated with applications. Aside from security,

one application is teaching. Cryptography is a very engaging subject. It puts mathematics

in a dramatic setting, which can make it extremely engaging for students of all ages.

Children are fascinated by intrigue and adventure. “More is at stake than a grade on a

test: if you make a mistake, your agent will be betrayed” someone using cryptography to

send secret codes might say (Koblitz, 1997).

Additionally, cryptography enables students to discover mathematical concepts and

techniques on their own. The thrill of discovery can be highly motivating for students of

all ability levels. “After many hours the youngsters finally develop a method to break a

cryptosystem, then they will be more likely to appreciate the power and beauty of the

mathematics that they have uncovered” (Koblitz, 1997). Also, due to the uncertainty of

cryptography problems, students may be less likely to maintain the notion that every

problem in mathematics can be solve with a formula.

Finally, cryptography allows for interdisciplinary study (Koblitz, 1997). Mathematical

concepts like functions, inverses, modular arithmetic, and group theory are heavily used.

In addition to these, students can cross into other fields such as statistics and linguistics.

For example, a common way to break a code is to use the concept of frequency analysis.

This means that students would attempt to see if certain letters always represent other

letters based on their frequency (Koblitz, 1997). This strategy takes both statistical

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Cryptography can be used to teach mathematics to students of all ages. One can learn

how to send and receive secret messages at a very young age. Once the proper foundation

has put down, students can start to formalize encryption and decryption with algebraic

formulas. Finally, once students are exposed to mathematical theory, they can start to

analyze the strengths and weaknesses of various cryptosystems. Cryptography can be a

highly motivating topic for students. Students are engaged in problems where they must

pull information from various areas of mathematics, as well as their personal experiences.

Due to the engaging nature of cryptography, I want to investigate the following

re-search questions.

1. How does studying cryptography enable students to use their understanding of

different advanced mathematical content?

2. What different SOLO levels do students exhibit when they are solving cryptography

problems?

In this paper, I first synthesize literature on the SOLO taxonomy and student learning

of abstract algebra, linear algebra, number theory, probability, and algorithms. Then I

will discuss the research methods used including the setting, participants, recruitment

procedure, the interview protocol, and analysis. Next I will discuss the results of the

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Chapter 2 Literature Review

In this chapter, I discuss how students learn various areas of mathematics used in

cryp-tography. This includes abstract algebra, linear algebra, number theory, probability, and

algorithms. Also I present the SOLO Taxonomy, which will be used to assess student

understanding.

2.1 SOLO Taxonomy

One way I will study student understanding of advanced mathematics is through the

structure of observed learning outcomes (SOLO) taxonomy framework. Developed by

John B. Biggs and K. Collis in 1982, this framework provides a hierarchy for student

understanding of a particular subject. This particular framework was chosen because of

its simple, yet elegant way of assessing student understanding. According to Biggs and

Collis, (1982) “The SOLO taxonomy provides a simple and robust way of describing how

learning outcomes grow in complexity from surface to deep understanding.” The SOLO

taxonomy is made up of five modes (Sensori-motor, Ikonic, Concrete Symbolic, Formal,

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Extended Abstract).

Modes are related to various age groups. The Sensori-motor mode occurs soon after

birth, where a person reacts to the physical environment (Biggs and Collis, 1982). At

age two, one reaches the Ikonic mode, which is characterized by internalization of actions

in the form of images (Biggs and Collis, 1982). Concrete Symbolic occurs around age

six. Here a person would think through a symbol system such as language and number

systems (Biggs and Collis, 1982). In the Formal mode, which occurs around age 15,

one considers more abstract concepts (Biggs and Collis, 1982). Finally, Post Formal

is reached around age 22. Those in this mode are about to question or challenge the

fundamental structure of theories or disciplines (Biggs and Collis, 1982). For each given

mode, a person’s understanding can be categorized in one of the five stages of the SOLO

taxonomy.

Students functioning in Pre-structural are only requiring pieces of unconnected

in-formation, with little to no organization. Common behaviors associated with this stage

includes avoiding or repeating the question being asked. If the student does engage in

the problem, an incorrect process may be used, leading to irrelevant confusion (Chick,

1998). One can move into Uni-structural once a simple or obvious connection is made,

however this is the only focus of the student. Students functioning in this stage apply a

single process or concept, often resulting in an invalid conclusion (Chick, 1998). Once the

Multi-structural stage is reached, students have made a number of connections. However,

students may fail to synthesize information, which may indicate cognitive performance

below that required for a successful solution (Chick, 1998). If a student reaches the

Relational level, various aspects have become one and the student sees how the parts

combine to make a whole. In other words, students are starting to see the “big picture.”

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1998). Students functioning in this level have an adequate understanding of the given

subject area. Finally, in the Extended Abstract level, students can generalize and

trans-fer knowledge to a diftrans-ferent subject area or take knowledge from a diftrans-ferent area to solve

a problem. Extended Abstract responses are structurally similar to Relational responses,

but here students use concepts from outside the domain of assumed knowledge. “In this

taxonomy, the structure of the learned outcome occurs within each of Piaget’s stages of

cognitive development. More specifically, the three SOLO levels in the middle, namely,

Uni-structural, Multi-structural and Relational, fall within the same stage whereas the

extended abstract extends the level of abstraction into the next stage becoming the

Uni-structural level of that next stage (Jurdak and Mouhayar, 2013).” Table 2.1 summarizes

the SOLO levels.

Table 2.1: SOLO Levels

SOLO Level Description

Pre-structural No connections.

Uni-structural Single connection.

Multi-structural Multiple connections.

Relational Parts as a whole.

Extended Abstract Knowledge transfer.

It is also essential to describe how students transition from one stage to the next.

When transitioning from Pre-structural to Uni-Structural, a student may attempt to

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Multi-structural, one may attempt to handle multiple connections, but no significant progress

is made. During the transition from Multi-structural to Relational, a student may

rec-ognize several aspects of the problem, but fail to reconcile them. Finally, a transition

from Relational to Extended Abstract can be observed by seeing students make progress

towards a firm conclusion. Although participants in this study will be only be categorized

in the five main categories, these transitions allow for the researcher to better analyze

student understanding.

A key assumption made regarding this framework is that each level incorporates the

previous levels, then extends understanding. One can think of the SOLO taxonomy as

a pyramid, where each level provides a foundational support for the next. Since the

researcher often is asking questions that are open ended and have various entry points,

this allows for students to be categorized in any of the five stages. Figure 2.1 shows the

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Figure 2.1: SOLO Taxonomy Pyramid.

When posing questions in task-based interviews, one must keep in mind that

partic-ular questions may limit the SOLO level of possible solutions (Biggs and Collis, 1982).

For example, the question “What is public key cryptography?” would best require a

pre-structural response. However, posing tasks such as “Encrypt the plaintext WOLF using

an encryption method of your choice. Discuss the strengths and weaknesses of the chosen

encryption method,” may allow for various SOLO levels. This is because students could

do anything from not responding to using an encryption method that is outside the scope

of the course. The interviewer must create questions that allow for a large range of SOLO

level responses, which provides an efficient tool for measuring student understanding of

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2.2 Abstract Algebra

Abstract algebra is a foundational part of the field of cryptography. In fact, one can

even think of cryptography as “applied abstract algebra.” If this is the case, then

un-derstanding how students think about abstract algebra is essential to the analysis of

student understanding of cryptography. abstract algebra is often the first time students

are faced with high levels of abstraction. Research has been done to analyze how levels

of abstraction can be reduced while not sacrificing mathematical rigidity.

According to Hazzan (1999), abstraction level can be interpreted three different ways:

1. Abstraction level as the quality of the relationships between the object of thought

and the thinking person.

2. Abstraction level as a reflection of the process-object duality.

3. Abstraction level as the degree of complexity of the concept of thought.

The first interpretation of abstraction stems from the idea that whether something

is abstract or concrete is not inherent of the object, but the relationship that one has

with the object (Wilensky, 1991). This implies that abstraction can vary for each person

and object based on the previous connections between the two. The closer one is to an

object, the more connections one will have, hence the object will become less abstract

and more concrete (Hazzan, 1999). Using this perspective, one can correlate students’

mental processes with their tendency to make unfamiliar problems more familiar, i.e.

making the transition from abstract to concrete (Hazzan, 1999). This process is quite

common in Abstract Algebra when students use what they know about various numbers

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The abstraction level can also be view using process-object duality. To discuss this

duality, it is essential that one distinguishes between process conception and object

con-ception.“Process conception implies that one regards a mathematical object as a potential

rather than an actual entity, which comes into existence upon request in a sequence of

actions” (Sfard, 1991). A mathematical concept becomes an object when the concept is

conceived as one entity. Therefore, for a given mathematical idea, it is first conceived

as a process, which is less abstract, then becomes an object (Hazzan, 1999). Students

can reduce abstraction level by perceiving mathematical ideas as processes rather than

objects. The mental process that allows students to transition from process conception

to object conception was coined reflective abstraction by Piaget (Hazzan, 1999).

Finally, level of abstraction can be interpreted as the degree of complexity of the

concept of thought. This viewpoint hinges on the assumption that the more compound

an entity is, the more abstract it is (Hazzan, 1999). Students can reduce abstraction this

way by substituting a less complex, but related idea for a more complex one (Hazzan,

1999). For example, a student trying to decrypt a message may start by only examining

one letter at a time to determine what type of cipher was used. In an Algebra course,

a student may replacing an entire group with a single element to attempt to reduce the

abstraction level (Hazzan, 1999).

Knowing when students are attempting to reduce abstraction and how they are going

about this process can give one insight about which level of understanding the student is

at. Therefore, this enables a way to place students in a given stage based on their need

and ability to reduce levels of abstraction.

One can also consider instructional strategies for improving the understanding of

var-ious abstract concepts. According to Dubinsky et al. (1994), the following are strategies

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1. Going through the Action-Process-Object-Schema Steps

2. Computer Activities and Team Work to Clear Up Misconceptions

3. Meeting Prerequisites

4. Finding Alternatives to Linear Sequencing

Dubinsky et al. (1994) asks “How can we get students to take a specific step in the

development of a particular concept? In particular, what methods can be used to help

students to interiorize actions to construct processes, to reverse or coordinate processes to

construct other processes, to encapsulate processes, to construct objects, and to

thema-tize collections of processes and objects into schemas?” Dubinsky et al. have experienced

success in this regard by creating computer-based tasks to foster the constructing of

pro-cesses and objects and by having students work cooperatively. Regardless of the strategies

used, Dubinsky et al. suggest that students should reflect on their actions independently

or as a class.

Misconceptions may happen in any mathematics class. However, teachers must help

students clear up these misconceptions or at least make them aware that they have fallen

into them. The use of group work may be one solution (Dubinsky, Dautermann, Leron,

and Zazkis, 1994). Students tend to be more likely to seriously consider contradictions

presented by their classmates than ones presented by their teacher. When an instructor

claims something is or is not true, it is convention to simply accept what is said as truth

(Dubinsky, Dautermann, Leron, and Zazkis, 1994).

It is essential that students have a deep understanding of sets and functions before

entering an abstract algebra class (Dubinsky, Dautermann, Leron, and Zazkis, 1994). Sets

and functions play an integral role in the concepts of one-to-one, onto, and isomorphism.

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to have a deeper understanding of various algebraic structures (Dubinsky, Dautermann,

Leron, and Zazkis, 1994).

It is natural to present mathematics in a simple linear sequence. However, Jerome

Bruner prosed that a spiral curriculum is a suitable replacement (Dubinsky, Dautermann,

Leron, and Zazkis, 1994). This is done under the assumption that any mathematical topic

can be taught in a rigorous way at any age. Fundamental mathematical concepts can be

taught a young age, then revisited over the years, adding levels of sophistication and

rigor. Dubinsky et al. believes that instructors should use abstract algebra as a way to

revisit concepts such as sets and functions. This way, abstract algebra can be taught

through concepts students are familiar with, but have not quite mastered.

2.3 Linear Algebra

One of the first poly-alphabetic ciphers students can learn after the Vigenere cipher is

the Hill cipher. The Hill cipher encrypts messages by using an n ×n matrix that has an inverse modulo 26. In order for students to encrypt, decrypt, and codebreak the Hill

cipher, they must have a strong foundation in linear algebra. In this section, I provide

some background on student learning of linear algebra.

Although most mathematics majors take a linear algebra course, the courses often

focus on computation and procedural knowledge. Although these skills are important for

encryption and decryption, students often forget the procedures and algorithms (Dorier

and Sierpinska, 2001) so focusing on the concepts that support the procedures and

al-gorithms is important. For example, if a student forgets the formula for the inverse of a

2×2 matrix, if they had a conceptual understanding of where the formula came from,

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In addition to its use in classical cryptography, linear algebra tends to show up in

pure and applied mathematics, computer science, engineering, physics, and other

sci-ences. Clearly this is an important subject for many mathematics and science majors.

According to Dorier (2002) , the two main issues of teaching and learning linear algebra

are Epistemological Specificity and Cognitive Flexibility (Dorier, 2002).

Other difficulties which students are faced with when learning linear algebra is the

variety of languages, semiotic registers or representation, points of view, and settings

through which the objects of linear algebra can be represented (Dorier, 2002). According

to Hillel (2000), there are three basic languages associated with linear algebra.

1. Abstract Language

2. Algebraic Language

3. Geometric Language

This can be an issue if the instructor switches from one representation to the next

without any notice. Often students are confused the most when the language is switched

from the abstract to the algebraic. With each of these languages comes a corresponding

mode of thinking (Hillel, 2000).

1. Analytic-Structural

2. Analytic-Arithmetic

3. Synthetic-Geometric

It is essential that students can effectively use multiple representations of objects

when studying mathematics. In fact in any mathematical activity, representations are

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represented (Dorier, 2002). For example vectors can be represented graphically by arrows,

by rows or columns of coordinates, or symbolically as abstract elements of a vector space.

In order to reduce levels of abstraction some have considered a more geometric

ap-proach to linear algebra. The goal is to overcome some of the abstractions by giving more

concrete meaning to concepts through geometric figures. However a problem quickly

ar-rises when one relies too much on the geometry of linear algebra. Geometry is limited to

three dimensions, therefore various concepts have limited representations in a geometric

setting.

Also, if students learn linear algebra in a heavy geometric setting, they can have

a difficult time going back to more general and abstract cases. One example is that

students may struggle to imagine a linear transformation that would not be a geometric

transformation.

However, some students may use the geometric ideas to their advantage when

ap-propriate. In fact being able to decide when a geometric representation should be used

or not displays a high level of understanding of the subject. “It seems that the use of

geometrical representations or language is very likely to be a positive factor, but it has

to be controlled and used in a context where the connection is made explicit.”

Additionally, some have proposed a greater use of technology in the linear algebra

classroom. In fact, according to Dikovic (2007), every linear algebra instructor should

consider the following questions.

1. Why do some students learn more mathematics than other students in the same

class?

2. What can linear algebra teachers do to enrich or replace traditional lecturing in

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3. What contribution of technology could be in the fields of experimenting,

observa-tion, and discussing?

4. How many reasons are there, for yes or no, for example for use of graphics calculators

for computing matrix inversion or for solving linear systems?

Dikovic (2007) suggests Maple, MATLAB, or Mathematica for their very powerful,

numerous functions. These functions include but are not limited to instantaneous numeric

and symbolic calculation, data collection and analysis, modeling, presenting two and three

dimensional graphics, and application development.

Hillel and et al. also suggest the use of Cabri, a dynamic geometry program, to

teach and learn linear algebra. Hillel and et al. believe that by using Cabri, students can

overcome the obstacle of formalism associated with vector spaces. Assuming the meaning

of various mathematical objects are well represented by the computer representations,

students can study vector spaces geometrically, rather than analytically (Hillel, Trgalova,

and Sierpinska, 1999). One must note that these geometric representations are limited to

two and three dimensions, so other representations must be considered (Dorier, 2002).

2.4 Number Theory

Number theory is often thought of as the purest form of mathematics (Cambell and

Zazkis, 2002). This is a classical subject that contains several beautiful and elegant proofs.

However, in recent years new light has been shed on number theory, specifically on

its application to modern cryptography. Public key systems, such as RSA, rely heavily

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RSA so effective. Additionally, foundational number theoretic concepts such as modular

arithmetic, divisibility, and primality are powerful tools to the field of cryptography.

What makes number theory even more important is its integration within

mathe-matics from elementary school through college (Cambell and Zazkis, 2002). It is very

common that elementary school students learn about different number systems, long

di-vision with and without remainders, and prime numbers (Cambell and Zazkis, 2002).

These topics are all rooted in number theory (Cambell and Zazkis, 2002). According to

The National Council of Teachers of Mathematics (NCTM), in pre-K through grade 2,

all students should be able to “develop a sense of whole numbers and represent and use

them in flexible ways, including relating, composing, and decomposing numbers, (NCTM,

2015).” These number theoretic concepts continue in grades 3 through 5, where NCTM

(2015) claims all students should be able to “recognize equivalent representations for the

same number and generate them by decomposing and composing numbers” and “describe

classes of numbers according to characteristics such as the nature of their factors.” In

grades 6 through 8 all students should be able to “use factors, multiples, prime

factoriza-tion, and relatively prime numbers to solve problems” and “develop meaning for integers

and represent and compare quantities with them,” according to NCTM (2015). NCTM

(2015) then explicitly states all students should be able to use “number-theory arguments

to justify relationships involving whole numbers” in grades 9 through 12.

Aside from these standards, number theory can help students make the transition

from arithmetic to algebra, by enabling students to develop better understandings of

the abstract conceptual structure of whole numbers and integers (Wagner, 2012).

Addi-tionally number theory has algebraic characteristics similar to variables (Wagner, 2012).

Therefore studying concepts such as number systems, division, and primality can develop

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Unfortunately, these concepts are forgotten after years of not engaging in them

(Cam-bell and Zazkis, 2002). Although high school students know what a remainder is and how

to find one, they most likely have not been introduced to the formal concept of modular

arithmetic. Cambell and Zazkis (2002) found that for preservice teachers, this poses a

serious problem. Their understanding of the concept of number is not where it should

be.

Additionally students that are required to take a number theory or modern

alge-bra course often struggle with elementary number theory concepts (Cambell and Zazkis,

2002). Students tend to solve problems in a procedural manner rather than conceptual. In

particular, students encounter difficulties when they are faced with problems that have a

wide range of strategies or representations (Cambell and Zazkis, 2002). For example,

stu-dents may face difficulties when attempting to link divisibility to factorization (Cambell

and Zazkis, 2002).

2.5 Probability

Studying cryptography requires keen intuitions about when particular methods are going

to work and when they are not. Instead of relying on intuition alone, probabilistic methods

are introduced. For example, suppose that students are asked to break a code, knowing

that it has been encrypted using a substitution cipher. Students can use the fact that

certain letters appear more often than others. Figure 2.2 shows the frequency distribution

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Figure 2.2: Frequency distribution of the alphabet.

With this information in mind, students can make predictions about the plaintext

based on the ciphertext. For example, if a student is faced with a string of ciphertext

where the letter J is most common, they may assume that J represents a vowel. This

is because in the English language, vowels appear more frequently than consonants,

with the exception of the letter T. However, frequency analysis is only effective when

a substitution cipher was used. How can one decide which type if cipher was used to

encrypt the plaintext?

The Index of Coincidence can be an efficient tool in determining whether a given

cipher is mono-alphabetic or poly-alphabetic (Kahn, 1997). Index of Coincidence refers

the the probability of choosing the same letter twice in a string of letters and is given by

IC =

Z P

i=A

fi(fi −1)

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where fi is the number of times the letter i appears for i = A, B, . . . , Z and N is

the number of letters in the ciphertext. Luckily, Index of Coincidence values have been

computed for various languages.

If all letters are equally likely, the Index of Coincidence would be approximately 0.038 (Singh, 1999). However, in the English language certain letters appear more frequently

such as the letter E, which appears approximately 13 percent of the time. This yields a

higher Index of Coincidence for English, approximately 0.067 (Singh, 1999). The Index of Coincidence for various languages can be seen in Table 2.2.

Table 2.2: Index of Coincidence

Language Index of Coincidence

English 0.067 Russian 0.068 Spanish 0.075 Portuguese 0.075 Italian 0.075 French 0.078 German 0.079 Random 0.038

When a cipher is mono-alphabetic, the Index of Coincidence does not change, since

the same letter frequencies exist. When the Index of Coincidence is closer to random,

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appropriate attack can be used.

The use of probability and statistics is an integral part of solving cryptography

prob-lems. However, many students have a lack of experience with probability, often only

studying this subject once. This begs the question, “How can we enable students to

un-derstand probability?” To answer this question, one must consider the following three

perspectives.

1. Building on the firm basis of students’ sound intuitions (Faulk, 1992).

2. Conventional teaching of probability does not establish enough connections between

the intuitions of the leaner and the mathematical theory (Borovcnik, 1991).

3. Intuitions are the product of personal experience (Fischbein, 1987).

Clearly, intuitions are important in learning probability. The learning of

probabil-ity should start with these intuitions, changing them as new knowledge is acquired by

the student. This can be done by developing secondary intuitions to create the link

be-tween the students’ preconceptions and the theory. However, it is essential to clear up

misconceptions as early as possible. Due to personal experiences, students may continue

to believe something is true, even when the mathematics say otherwise. For example, a

student may say that they understand that flipping a coin several times has the same

probability for heads each toss. However, after getting four tails in a row, students may

feel that that getting heads on the next toss is now more likely. According to Fischbein

(1987), “One of the fundamental tasks of mathematical education ... is to develop in

stu-dents the capacity to distinguish between intuitive beliefs, intuitive feelings and formally

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2.6 Algorithms

One can think of cryptography as the intersection of mathematics and computer science.

Although the complex encryption methods require rigorous mathematical theory, heavy

computation must be done in real world problems. This requires both theory and

applica-tion of computer science. In particular, it is essential that one has a deep understanding

of algorithms, including representations, syntax, and time to run given algorithms.

An algorithm is defined to be a well defined process that requires a finite amount of

steps. One may view algorithms either graphically or using standard syntax. Figure 2.3

shows a graphical view of the Euclidean algorithm for 1599 and 650.

Figure 2.3: Euclidean algorithm for 1559 and 650.

This is an alternative to writing out the algorithm using symbolic notation, as seen

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1599 = 650×2 + 299

650 = 299×2 + 52

299 = 52×5 + 39

52 = 39×1 + 13

39 = 13×3 + 0

According to Byrne, Catrambone, and Stasko (1999), students benefit from seeing

both symbolic and graphical representations of algorithms. This is because “Conceptual

knowledge about the properties of an algorithm can help a learner to carry out the

algorithm’s steps. Similarly, being able to perform the step-by-step operations of an

algorithm may assist a learner in determining the veracity of a conceptual question about

it. (Byrne, Catrambone, and Stasko, 1999)” This will ultimately give students a deeper

understanding of algorithm and how they work. Consequently, students may gain insight

on effectiveness and running time.

Algorithms are an integral part of learning cryptography, since cryptanalysis is a

major focus. After presented with a given cryptosystem, students may be asked to analyze

its security. For example, the Caesar cipher is typically the first cipher discussed in a

cryptography course. A basic algorithm to break this cipher would be to try every key

from 1 to 25. Even if this was done by hand, the maximum amount of time to break

this cipher would be 25 steps. As more complex ciphers are discussed, the algorithms to

break them become more difficult and time consuming, allowing a perfect opportunity

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Chapter 3 Methods

In this chapter, I discuss the methods of data collection and analysis for this

qualita-tive study. This includes the setting of the interviews, participant demographics, and

a detailed description of the procedures used to collect and analyze the data from the

interviews.

3.1 Research Questions

1. How does studying cryptography enable students to use their understanding of

different advanced mathematical content?

2. What different SOLO levels do students exhibit when they are solving cryptography

problems?

3.2 Setting

This experiment was conducted at the end of the spring 2015 semester at a university

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interviewed in a one-to-one setting for approximately one hour. Those participating in

this study did so following the 700 level mathematics course “Applications of Algebra.”

The class consisted of approximately 20 students and the material was presented using

traditional lecture, twice a week for approximately 75 minutes All of the students in the

course were either mathematics (pure, applied, eduction) or computer science majors.

Although there are many topics that could be discussed in an applied algebra course,

the professor chose to focus solely on cryptography. Topics covered in this course include

shift ciphers, the Hill cipher, exponential ciphers, RSA, ElGamal, elliptic curve ciphers,

Pollard’s method, and quadratic sieves.

3.3 Participants

All participants were either Mathematics (Pure or Applied) or Computer Science majors.

Table 3.1 below summarizes the demographics of the participants in this research study.

Table 3.1: Student Demographics

Student Gender Age Major

A Male 22 Pure Mathematics

B Male 24 Pure Mathematics

C Female 22 Computer Science

D Female 23 Applied Mathematics

E Female 24 Pure Mathematics

F Female 21 Applied Mathematics

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A convenience sample was used for this project; the participants were selected based

on their willingness to participate. All students taking the course “Applications of

Alge-bra” were asked to participate by email and the first seven to respond were chosen.

3.4 Recruitment Procedure

Prior to being interviewed students were contacted through email and asked if they

would be willing to participate. Students were to sign an informed consent document,

where they agreed to the use of audio recordings and their written work in this research.

Participants’ work was kept and scanned for use in this research. All work can be found

in Chapter 4 and Appendix C.

3.4.1 Qualitative Studies

In this study, I am analyzing student interviews and work so it lends itself to the use of

qualitative research. Qualitative research emphasizes the importance of looking at

vari-ables in the natural setting in which they are found (Flick, 2009). Data can be gathered

through open ended questions and tasks that provide artifacts such as sample work and

direct quotations (Flick, 2009). Unlike in quantitative research, which attempts to

re-move the investigator from the investigation, the researcher is an integral part of the

investigation. The focus of qualitative research is to have a holistic view of what is being

studied, however there exist both advantages and disadvantages to doing qualitative

re-search. One advantage is that qualitative research provides more in depth, comprehensive

information (Flick, 2009). There is a cost, however, to gaining this information. Due to

subjectivity, establishing reliability and validity can be difficult. Additionally, it is very

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3.4.2 Task-Based Interviews

Task-based interviews are a particular form of clinical interviews, and date back to the

time of Piaget. “A clinical task-based interview can be seen as a situation where the

interviewer-interviewee interaction on a task is regulated by a system of explicit and

im-plicit norms, values, are rules” (Harel and Koichu, 2007). These interviews were originally

used to gain a deeper understanding of the cognitive development of children. In

math-ematics education, task-based interviews provide a means of gaining information about

a student or group of students’ mathematical inclination. In task-based interviews, the

interviewees interact with both the interviewer and a particular task environment. This

implies that one of the most important aspects of the task-based interviews is the task

itself. If done correctly, task-based interviews can be a very effective way of analyzing

mathematical behavior. Ericsson and Simon (1993) recommend the following monologue

to promote think-aloud taking.

Tell me everything you are thinking from the time you first see the question

until you give an answer. I would like you to talk aloud constantly from the

time I present each problem until you have given your final answer to the

question. I don’t want you to try to plan out what to say or try to explain to

me what you are saying. Just act as if you are alone in the room speaking to

yourself. It is most important that you keep talking. If you are silent for any

long period of time I will ask you to talk.

If tasks are to be used in various settings, by various interviewees, it is essential

that the procedure of interviewing can be repeated. This implies that the interview

protocol needs to be explicit, so that a different interviewee can use the same task in a

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many nonverbal events that effect the interview process. These could include nods, smiles,

and other use of body language. This issue makes completely standardizing task based

interviews a difficult feat.

Before the interview actually takes place, the expectations and rules must be discusses

between the interviewer and interviewee. This agreement between the two parties is

referred to as an experimental contract. These experimental contracts promote certain

types of social behavior, which in turn acts as a mediator between the subject and

knowledge of that subject. One implication of the experimental contract is that the

interviewee may try to offer the correct or preferred response, instead of their actually

thoughts. Therefore, it should be made clear that the interest is in the student’s thought

process, not a correct or incorrect answer. In the next section, I show the protocol I used

to conduct the interviews.

3.4.3 Interview Protocol

Hello, my name is Blain Patterson and I am conducting interviews for my Master’s Thesis.

I am interested in how you make connections to various areas of mathematics through

learning cryptography. I am going to ask you a series of questions. I am interested how

you think about each problem, so I would like for you to talk to me while you work. Also,

I am going to keep your work, take notes, and record audio. I am provided you with a

consent form that you must sign before the interview begins. Do you have any questions

before we start? Please answer the following questions (with time in between).

Introductory Questions

1. Why did you decide to take this course?

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3. What was your favorite part of the course and why?

4. What was your least favorite part of the course and why?

5. Explain the difference between private and public key cryptography. Discuss

ad-vantages and disadad-vantages of each.

Tasks Used

6. Consider the following plaintext: NCSU.

a. Encrypt this message using a cipher of your choice.

b. Discuss the strengths and weaknesses of the cipher your choose.

7. Consider the following ciphertext: XFNFCWGRTB.

a. Decrypt this message.

b. Discuss the strengths and weaknesses of your decryption method.

8. Consider the number 8911.

a. Is it prime? Why or why not? (If student answers yes, move on to parts b and

c. Otherwise, move on to question 9.)

b. How did you determine primality?

c. Factor this number.

Closing Questions

9. What areas of mathematics did you use in this course and how were they used?

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3.4.4 Rationale for Protocol Questions

The first four questions were meant to learn about the participants’ background, which

can provide insight to their problem solving abilities. Questions five through eight are

meant to directly categorize student understand into one of the five levels of the SOLO

taxonomy. Question five asks students to encrypt using any cipher of their choice and

discuss its effectiveness. This is an open ended question that allows for a wide range of

SOLO levels, since students are to choose methods using as much or as little mathematics

as needed. In a similar way, question six allows students to use any strategies they choose,

however is more limited in the possible strategies that can be implemented. Note that

students were given the opportunity to use a calculator for number eight, but were still

asked to show as much work as possible. Although they were asked to factor this specific

number, the factoring algorithm they use will vary. All three task questions allow for a

wide range of mathematics used and SOLO classifications. Once the task portion of the

interview is complete, students were asked to summarize their experience in questions

nine and ten. Audio clips were recorded and student worked was kept for the analysis.

Additionally, the researcher recorded notes while the students were answering questions

and working through problems.

3.5 Analysis

Recall that the SOLO taxonomy is being used to classify student understanding of

cryp-tography. For classification purposes, I am assuming that all students are functioning

at the Formal mode of the SOLO taxonomy. This is due to the fact that all students

participating in this study are college students.

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identified when a participant used something they learned in “Applications of Algebra,”

something they learned in another course, or something they learned while studying

independently. This was done by keeping a list of concepts covered in the course and

marking when a participant used one of these concepts to solve a problem. If a concept

used was not covered in the course, I asked the student to clarify where they learned that

concept and how they used it to solve a given problem.

If a student refused to engage in the task, they were placed in the Pre-structural SOLO

level. Students who used a single relevant aspect of cryptography to solve the problems

were placed in the Uni-structural SOLO level, whereas students who used multiple

rele-vant aspects of cryptography to solve the problems were placed in the Multi-structural

SOLO level, and students who made multiple connections were placed in the

Multi-structural level. If a student used multiple aspects of cryptography together to solve

a problem, they were placed in the Relational SOLO level. Finally, students who used

knowledge outside of the scope of cryptography were placed in the Extended Abstract

SOLO level, since the Extended Abstract level of the SOLO taxonomy is associated with

making connections both in and outside a given subject area. Table 3.2 summarizes how

the understanding of cryptography for participants A through G will be classified using

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Table 3.2: SOLO Rubric

SOLO Level Description

Pre-structural Inability or refusal to engage in the task.

Uni-structural Uses one relevant aspect of cryptography.

Multi-structural Uses several relevant aspects of cryptography.

Relational Uses multiple relevant aspects of cryptography together.

Extended Abstract Uses knowledge outside of the scope of cryptography.

Student work is described in Chapter 4 along with the classification of each student

in one of the five SOLO levels. Using student work, notes taken during the interview,

and audio recordings, detailed descriptions of each interview have been compiled. This

was done by transcribing each of the seven interviews and written work. I then used

a list of concepts (encryption methods, decryption methods, algorithms) discussed in

“Applications of Algebra” and marked when a participant used the various concepts. I

also made note of when a participant used a single concept, multiple concepts disjointly,

multiple concepts together, or concepts outside of cryptography. Then using the rubric

in Table 3.2, the understanding of cryptography for students A through G was placed in

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Chapter 4 Results

In this chapter, results are presented including student background information (previous

coursework and interest in cryptography as shared in the interviews), a description of

the students’ responses and some related work samples, and SOLO level classification.

Students A through G have been labeled based on the order in which the interviews took

place. Each student description includes their responses to the introductory questions,

solutions to the tasks, and closing questions.

4.1 Student A

4.1.1 Introduction

Student A decided to take this this course because he felt very confident in his algebraic

thinking. “I did really well in linear and abstract algebra, so I thought it would be

inter-esting to see some applications.” Student A has taken linear algebra, abstract algebra,

and real analysis. He has yet to take course in number theory, but claims he has worked

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favorite part of the course was working with error correcting codes, because he liked the

application of vector space theory. His least favorite part of the course was the elementary

ciphers, such as the Caesar and Vigenere. He felt that these could have all been covered

in the first day, so that there was more time to focus on public key cryptography.

4.1.2 Problem Solving

Student A started encrypting NCSU by assigning letters to numbers. However, Figure

4.1 shows that instead of starting at 0 and counting to 25, he started at 1 and counted

to 26

Figure 4.1: Student A assigning numbers to letters.

This came as a surprise, since the student had been using the former method the

entire semester. When asked about his method, student A commented “It really doesn’t

matter if I start at 1 instead of 0, I just need to make sure I do everything modulo 27.”

This lead the student to convert NCSU to 14,3,19,21.

Student A was originally going to encrypt the plaintext NCSU using an affine cipher

Cryptography: Decoding Student Learning.

ABSTRACT

DEDICATION

BIOGRAPHY

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

Chapter 1

Introduction

1.1

What is Cryptography?

1.2

Definitions and Examples

1.3

Why Cryptography?

Chapter 2

Literature Review

2.1

SOLO Taxonomy

2.2

Abstract Algebra

2.3

Linear Algebra

2.4

Number Theory

2.5

Probability

2.6

Algorithms

Chapter 3

Methods

3.1

Research Questions

3.2

Setting

3.3

Participants

3.4

Recruitment Procedure

3.4.1

Qualitative Studies

3.4.2

Task-Based Interviews

3.4.3

Interview Protocol

3.4.4

Rationale for Protocol Questions

3.5

Analysis

Chapter 4

Results

4.1

Student A

4.1.1

Introduction

4.1.2

Problem Solving