Conception of teaching Cryptography - TI89



Chapter 7

Analysis of the Instructional Design of

An Interactive Multimedia Based Learning Environment for Teaching Cryptography

Modern learning theory and existing paradigms of multimedia based learning environments have contributed to the design of the interactive multimedia based learning environment as an effective and stimulating learning tool. Therefore, I will first lay the groundwork by discussing the three most prominent learning theories: behaviorism, cognitivism and constructivism (section 7.1). I will then investigate their value for the tutorial design and explain why no single discussed learning theory is sufficient for the instructional design. However, I will explain why Dewey’s pragmatic approach lays the proper foundation for the design of the learning environment (section 7.2). Finally, accounting for specific learning conditions such as learning group, learning objectives and learner motivation, I will describe the design of the learning environment (in section 7.3).

7.1 Learning Theories

Learning is essentially making and maintaining connections. Biologically through neural networks. Mentally through concepts, ideas and meanings. Experientially through interaction between the mind and the environment. For thousands of years, learning theories have attempted to reflect such complex processes. The evolution of learning theories has brought us to a more complete understanding of learning. Many theories have been created; only few (i.e. behaviorism, cognitivism and constructivism) have been widely recognized and used in educational settings.

Learning theories are trendy in nature: they have provided adequate models for human learning at a given time and are usually replaced by more modern theories that enhance previous explanations. This cycle will very likely continue until we know which exact biological and cognitive processes are involved in human learning. Each learning theory seems to get us a step closer to this goal. I am mentioning this to caution that no learning theory fully reflects the complexity of the act of learning, however, each learning theory brings light to important aspects involved in learning.

As one of the first modern learning theories, Behaviorism became popular during the 1950's when B.F. Skinner proposed his content centered, behavioral, preprogrammed method of educating American children. I will describe the behaviorist view of learning in section 7.1.1. While the behaviorist view concentrated on investigating the observable behavior of humans and animals resulting from exposure to different stimuli such as reinforcement, punishment and conditioning, the Cognitivist School acknowledged that each learner possesses a brain and mental processes formed the primary object of study. Thus, the Cognitivist school explored the human brain and mental processes as factors that were irrelevant in Behaviorist models. I will describe the Cognitivist School in section 7.1.2. Constructivism is the youngest learning theory. In constructivism, the objective character of behaviorism and cognitivism is replaced by the subjectivity of the learner. The emphasis is placed on the individual that constructs knowledge or mental models through experiences in complex and authentic situations. Learning is viewed as the process of adjusting our mental models to accommodate new experiences. I will describe the constructivist view in section 7.1.3.

7.1.1 Behaviorism

Central to the theory of behaviorism is the study of behaviors that can be observed and measured (Good & Brophy, 1990). The mind is perceived as a "black box" that acts in a deterministic manner: response B will occur given stimulus A. Thus, to provoke a desired behavior, the correct stimulus has to be given. Thorndike’s law of exercise resulted from the observation that the more a stimulus response bond is practiced the stronger it will become. He also found that this bond strengthens even more when positive rewards are involved and weakens when negative responses are involved.

Behaviorist learning is viewed as a conditioned reflex that is acquired through interaction with the outer world. Teaching simply involves practicing the desired stimulus response relationships. The teacher knows what exactly the student has to learn and his efforts focus on how to teach. In his regard, behaviorism provides an authoritarian learning model: obedient student execution without reflection or critical thinking.

Key behaviorist researchers: I. Pavlov, E. Thorndike, J. B.Watson and B.F. Skinner.

Reflection: Behaviorism is concerned with regulation of learner’s behavior and not with cognitive processes inherent to humans. In that, behaviorism does not fully reflect human learning. Applying research on animal behavior (such as Pavlov’s dog experiments) to human behavior has to be limited.

Behaviorist teaching methods seem to work best in creating desired behaviors and physical skills. The learner shall be able to respond instinctively to provided stimuli. For example, World War II pilots were conditioned to react to silhouettes of enemy planes. Such responses were supposed to occur automatically.

A major limitation of behaviorist learning models is the inability to act in novel and unexpected situations that the learner is not trained for.

7.1.2 Cognitivism

Cognitivism followed behaviorism in an attempt to find explanations to the limitations of behaviorism. For example, children do not necessarily behave in a way that was reinforced; they may alter their behavior contrary to the reinforcements or rewards. Cognitivism recognizes that humans can process information mentally and don’t act necessarily in a predictable trained manner. Cognitivism emphasizes the inner cognitive processes occurring between stimulus and response. Although different classifications of cognitivism exist all agree that human thinking involves some information processing abilities. This explains why computers appear to be appropriate models when examining human thinking processes such as learning, memorizing or recalling.

Cognitivist learning does not focus on provoking the correct answer that follows a given stimulus. Rather, students shall learn rules, formulas and terminologies that form cognitive capacities and create inert knowledge. Learners possess transformation and adaptation capacities. Typically, complex problems are subdivided and posed as simplified stand-alone problems to the learner. Students develop problem-solving skills that help solve such predefined problems in their own way.

Key cognitivism researcher:

J. Piaget, E. Tolman, D. Ausubel, J. Bruner, L. Vygotsky.

Reflection: Cognitivism eliminates some of the limitations of behaviorist learning models. Learning also involves human cognitive capacities resulting in a variety of problem solving abilities. Learners apply their set of acquired rules to tackle even novel problems. The teacher-student is not authoritarian anymore, teachers act as advisors. They observe the learning process and possibly assist in solving the posed problem.

While the limitations of behaviorism can be seen in its narrow focus on physical behavior, the limitations of cognitivism are rooted in its overemphasis of mental information processing. Situations that involve instinctive (physical) behavior cannot be explained. Additionally, authentic problems that are complex in nature and embedded in their contextual settings cannot be solved using cognitivist strategies. These are typically acquired in given isolated predigested settings and not in ill-defined complex settings that require a different type of strategies.

7.1.3 Constructivism

As a philosophy of learning, constructivism can be traced at least to the eighteenth century and the work of the Neapolitan philosopher Giambattista Vico, who stated that humans can only clearly understand what they have themselves constructed. The fundamental difference to other learning theories is that constructivism denies the ability to objectively describe reality. Although reality may be objective in nature, we perceive it subjectively with our senses. Neurobiological evidence shows that we not only project reality but also interpret it at the same time (Anderson 1988).

Therefore, constructivists believe that "learners construct their own reality or at least interpret it based upon their perceptions of experiences, so an individual's knowledge is a function of one's prior experiences, mental structures, and beliefs that are used to interpret objects and events… What someone knows is grounded in perception of the physical and social experiences which are comprehended by the mind." (Jonasson, 1991).

Under the constructivist view, learning is seen as an active process in which humans construct knowledge in complex authentic situations. The fundamental basis of learning is discovery or reconstruction of knowledge. Learners will go through stages in which they develop, accept and, later on, reject their ideas. In the classroom, the teacher takes on the role as facilitator. He provides a classroom environment that allows learners to discover relationships and to develop ideas through activities that are of interest to them. Problems shall be situated in realistic settings and connect to learner’s previous experiences and knowledge.

The Assumptions of Constructivism (Merrill 1991)

• Knowledge is constructed from experience

• Learning is a personal interpretation of the world

• Learning is an active process in which meaning is developed on the basis of experience

• Conceptual growth comes from the negotiation of meaning, the sharing of multiple perspectives and the changing of our internal representations through collaborative learning

• Learning should be situated in realistic settings; testing should be integrated with the task and not a separate activity

Key Constructivism researchers: L. Vygotsky, J. Bruner, J. Dewey, J. Piaget, S. Papert, E. von Glaserfeld.

Reflection: The creation of knowledge is an individual process that can not be transmitted through a teacher. In that way, constructivism differs fundamentally from behaviorism and cognitivism. The focus is on learning as opposed to teaching; classes are student-centered and not teacher-centered. The teacher’s role shifts to that of a facilitator Since he may not know more than the learner, a solid amount of self confidence is needed. He considers how students learn, considers their beliefs and attitudes, thinks of learning as a process and supports cooperative learning.

7.1.4 Analysis of these Learning Theories

The following overview of these three most popular learning theories shall outline their key foci. Schuhman summarized the three prominent theories as follows:

Figure 1: Behaviorism, Cognitivism and Constructivism – An Overview (Schuhman, 1996)

When considering the key concepts of the three learning theories together with their particular strengths and weaknesses, none of them appears to be appropriate for all educational settings. Depending on specific learning conditions such as the learner, the learning situation and the learning objectives, specific learning theories appear to be more suiting than others. Schwier (1995) states that: We must allow circumstances surrounding the learning situation to help us decide which approach to learning is most appropriate. It is necessary to realize that some learning problems require highly prescriptive solutions, whereas others are more suited to learner control of the environment.

Any of the three learning theories may provide a meaningful and an appropriate model for an underlying learning situation. As long as their strengths and limitations are realized and their uses are newly assessed given the particular learning situation, their flexible combination can be very beneficial. If we want to our students to become self-driven autonomous learners that feel comfortable solving ill-defined problems in complex situations, constructivism should be our preferred learning model. Then, behaviorist and cognitivist models gain their significance by enabling the learner to reach that stage. It would be too farfetched to ask a learner to engage in the self-exploration of an unknown yet complex topic. Learning theories shall be related to educational content and expertise of the learner.

Ertmer and Newby (1993) as described in Mergel (1998) match learning theories with the content to be learned as follows:

A behavioral approach can effectively facilitate mastery of the content of a profession (knowing what). Behavioral tasks requiring a low degree of processing (e.g., basic paired associations, discriminations, rote memorization) seem to be facilitated by strategies most frequently associated with a behavioral outlook (e.g., stimulus-response, contiguity of feedback/reinforcement).

Cognitive strategies are useful in teaching problem-solving strategies where defined facts and rules are applied in unfamiliar situations (knowing how). Cognitive tasks requiring an increased level of processing (e.g., classifications, rule or

procedural executions) are primarily associated with strategies

having a stronger cognitive emphasis (e.g., schematic organization, analogical

reasoning, algorithmic problem solving).

Constructivist strategies are especially suited to dealing with

ill-defined problems through reflection-in-action. Constructive

tasks demanding high levels of processing (e.g., heuristic problem solving,

personal selection and monitoring of cognitive strategies) are frequently easiest learned with strategies advanced by the constructivist perspective (e.g.,

situated learning, cognitive apprenticeships, social negotiation.)

Ertmer and Newby feel that the strategies promoted by different learning theories overlap (the same strategy for a different reason) and that learning theory strategies are concentrated along different points of a continuum depending on the learner’s task knowledge and the level of cognitive processing required.

 

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Ertmer and Newby's suggestion that theoretical strategies can complement the learner's level of task knowledge, allows the designer to make the best use of the strategies provided by the different learning theories. With this approach the designer is able to draw from a large number of strategies to meet a variety of learning situations.

7.2 A Pragmatic Approach

In designing instructional content, we shall not limit ourselves methodologically but rather be in a position that permits us to select the most suiting existing educational principles given the particular learning situation. Thus, we shall act pragmatically in a way that John Dewey proposed more than a century ago. John Dewey (1859-1952) was an American philosopher and educator who rejected authoritarian teaching methods. He was the founder of the “Experimental Laboratory School” and influential in the further development of constructivist learning models. He regarded education in a democracy as a tool to enable the citizen to integrate their culture and talents usefully. To accomplish those objectives, both curricula and pedagogical methods needed radical reform. Although not put forward by him, “learning by doing” within a dynamic social context describes Dewey's educational philosophy, called Pragmatism. Dewey's view of democracy as a primary ethical value permeated his educational theories.

As one of the principal figures in the “Progressive Education Movement” from the 1880s to 1904, Dewey set the tone for educational philosophy as well as concrete school reforms. His reactions to the prevailing theories and practices in education and his corrections made to these philosophies were vital for the development of educational thinking in the late nineteenth and early twentieth centuries. Dewey (1897) points out the major shortcoming of US schools in attempting to prepare the learner for later life:

“I believe that much of present education fails because it neglects this fundamental principle of the school as a form of community life. It conceives the school as a place where certain information is to be given, where certain lessons are to be learned, or where certain habits are to be formed. The value of these is conceived as lying largely in the remote future the child must do these things for the sake of something else he is to do; they are mere preparation. As a result they do not become a part of life experience of the child and so are not truly educative.”

Educational pragmatism does not add a new learning theory to the existing ones. It also does not value existing learning theories as good or bad, it rather classifies them as useful or not. As pragmatism (“Pragma”: greek for “Action” or “Practice”.) is guided by practical experience and observation rather than theory it asks “Does it work?" as opposed to "Is it right?"

Educational pragmatism is rooted in the social context the learner finds himself: “True education comes through the stimulation of the child’s power by the demands of the social situations in which he finds himself.” (Dewey 1897). Therefore, learning objectives vary with the particular learner and are thus not absolute. Dewey emphasizes the relational character of objectives, means and consequences in education in his works. In particular, this means, that different children may learn different things in the same educational setting depending on their individual capacities, interests and habits. Dewey (1897) believes that “Education must be conceived as a continuing reconstruction of experience and that the process and the goal of education are one and the same.” The best preparation for life is obtained by providing an environment which places the learner in authentic situations and lets him act on them. “To prepare him for the future life means to give him command of himself; it means so to train him that he will have the full and ready use of all his capacities; that his eye and ear and hand may be tools ready to command, that his judgment may be capable of grasping the conditions under which it has to work, and the executive forces be trained to act economically and efficiently”. (Dewey 1879) Examinations are of use if they test his fitness for social life and detect areas in which he shall be supported.

Pragmatism provides a template for educational settings by placing the learner in the center of all educational activities. Based upon the learner’s capabilities and his social situation, the teacher provides appropriate learning experiences. Such experiences are educational once reflected upon and its diversity help him prepare for life. To provide these experiences, no single learning theory shall solely be used from a pragmatic point of view. Rather, didactic decisions are solely based on the specific learning situation as defined by the learner, the social setting and subject matter. Constructivist, cognitive and behaviorist principles may be used in combination based upon their utility for particular educational experiences.

In particular, this holds true for the design of multimedia based learning environments. Consequently, the selected media are a result of the desired learning situation to be created. Kerres (2002) states that “The situation determines the value of the used media, not the media themselves…Media for themselves are of no value, they obtain their value through their usage by humans in their specific contexts at specific times.

Teaching methods and media shall be employed in a “pragmatic” manner to produce the desired educational experiences. In particular, the learner shall be involved in activities that

• allow him to relate previous experiences to new ones.

• are authentic and of interest to him.

• provoke curiosity and the need to act.

• allow him to experience the consequences of his actions.

• require him to research and provoke original thoughts.

• don’t overwhelm the learner. He can manage the task or solve the problem.

• contribute to democratic and human structures in education.

• promote development of individual capacities.

I will now proceed to the design of the interactive multimedia based learning environment for teaching cryptography. In order to properly use Dewey’s pragmatic approach, I will analyze the particular learning conditions in the following section.

7.3 Design of the Multimedia Based Learning Environment

For a proper analysis of the particular learning conditions, I will use Kerres’ Guide for a media-didactic Conception (1999). This guide allows systematic analysis of the particular learning conditions as well as the reasons and functions of the used media.

7.3.1 Project

Teaching Cryptography in an American High School using An Interactive Multimedia Learning Environment.

7.3.1.1 Brief Description of Project

Cryptography is the art and science of encoding and decoding information. High School students at Antilles School in the U.S. Virgin Islands learn Cryptographic Methods and Applications through the use of an interactive Cryptography Tutorial.

7.3.1.2 Persons and Institutions involved

Dr. Michael Hortmann, University of Bremen, Germany

Salvatore Angilletta, University of Bremen, Germany.

7.3.1.3 Educational Setting

Antilles School is a college-preparatory K-12 school with an enrolment of 503 students of wide ethnic background. The spoken language is American English. The school has 3 computer labs with 20 workstations each. The course takes place in the School’s library that runs Windows2000 on each of their 400MHz Celeron / 64MB RAM workstations.

7.3.1.4 Characterization of Learning Group

Students that are part of this study are enrolled in the elective Advanced Placement Math classes. This learning group consists of 26 students whose ages range from 17 to 18 years. The students participating in this study are enrolled in Advanced Placement (AP) math classes in the school year 2002-2003. Advanced Placement math classes are elective courses that only students with a particular interest in Mathematics enroll. The study took place during the last two weeks of the school year after the students had completed their nation wide AP tests.

15 students are in 11th grade and are 17 years old, the other 11 students are 12th graders and are 18 years old. 13 students are male, 13 are female. Based upon a survey conducted before the study, the students using the tutorial had very little to no prior knowledge of cryptography. All students have experience in using the Internet, primarily to check email, to visit educational and recreational sites and to use Instant Messenger. The majority of students have limited experience in self-explorations and cooperative learning. Most students have been classmates for many years and have known each other very well. Moreover, all students appeared open-minded and ready to engage in the studies of cryptography.

Learner Motivation: None of the students has systematically studied cryptography before. I therefore consider them novices in the field of cryptography. Because students are exposed to critical privacy issues in today’s Internet usage such as secure online payments, email privacy or using PIN’s for automatic teller machines, they - generally speaking - quickly develop an interest for cryptography. Others are familiar with historic cryptographic events (i.e. Enigma machine during WW2) or may have used cryptographic or stenographic methods in a playful manner. As students of elective Advanced Placement Math classes, they usually have a strong mathematical interest and a solid background equal to 12 years of High School Mathematics.

7.3.1.5 Learning Content and Learning Goals

The security concerning online purchases (i.e. a book order at ), sending an encrypted Email or performing an ATM money withdrawal depends upon encryption method that is utilized. As these technological conveniences become increasingly important in our modern society, the question ultimately arises: How much security do the utilized encryption methods offer? In order to effectively evaluate the integrity of an encryption method, it is essential to understand the underlying mathematics of the encryption method (“Cipher”). Thus, the focus of modern cryptography and, in particular, of this tutorial is testing the security of existing encryption methods and exploring their underlying mathematical methods.

The following topics are studied within the tutorial:

- Terminology used in Cryptography

- Distinction between 1 and 2 Key Cryptography

- Transposition Ciphers

- Substitution Ciphers

- Applications of Cryptography

- Caesar Cipher

- The One Time Pad

- Multiplication Cipher

- Linear Cipher

- Polyalphabetic Ciphers

- RSA Ciphers

- Cryptography Tools and Mathematical Tools on:

- Modular Arithmetic

- Euclidean Algorithm

- Extended Euclidean Algorithm

- Euler’s Phi Function

- Euler’s Theorem

- Letter Frequencies Counter

- Crypto Calculator

- Cipher Challenge

The tutorial was designed to motivate and require the learner to study encryption methods and their underlying mathematical concepts. The listed ciphers and the chosen sequential order were purposely selected to achieve this goal. Starting with the simplest cipher and ending with the most complex cipher, the study of each cipher teaches a new mathematical concept required to understand the following ciphers and ultimately the most complex and prominent cipher, the RSA cipher.

For example, one of the first ciphers to study is the Caesar Cipher that motivates learning the basics of modular arithmetic. Modular arithmetic itself is again the underlying mathematical concept of successive ciphers such as the RSA Cipher. These mathematical concepts are – in addition to the studied encryption methods - thoroughly explained in the “Crypto Tools” section of the tutorial.

Through the detailed study of the mathematical background of cryptography, the learner realizes that cryptography is an application of the mathematical discipline Number Theory – a theory that may be considered the last “pure” mathematical discipline as it had virtually no real-life applications until Cryptographers (which are mainly Mathematicians) started using Number Theory theorems to devise secure ciphers. Central concepts of number theory are i.e. Euler’s Theorem and the Euclidean Algorithm with its Extension.

Learning Goals

The learning goals reflect the above-mentioned issues of modern cryptography. Central here are the mechanism and the security issues revolving around the studied encryption methods as well as their mathematical foundation. The learning goals are described on each tutorial page and can be categorized into two classes: procedural and communicative. Students shall not only develop the skills necessary to understand and devise elementary Ciphers, they should also be able to communicate and intelligently discuss cryptography-related issues such as the security of a studied or devised cipher. The exercises provided on each page of the tutorial are designed to achieve the following learning goals:

The students shall

• develop an understanding of the primary uses and applications of cryptography.

• be able to distinguish between one- and two-key cryptography.

• be able to distinguish between transposition and substitution ciphers.

• be able to develop transposition and substitution ciphers.

• know how to cryptoanalyze transposition as well as the learned mono- and polyalphabetic ciphers.

• know how to use a letter frequency counter.

• develop an understanding of the mathematical basis of the RSA cipher and its underlying security.

• be proficient in modular arithmetic.

• conceptually understand the Euclidean algorithm and its extension.

• be able to communicate and intelligently discuss the learned cryptographic concepts.

7.3.2 Didactic Media

Cryptography is taught using “The Interactive Cryptography Tutorial” which provides a multimedia based learning environment designed by the author for this purpose. The tutorial is hosted on the school’s Internet server and thus is available to students from any campus computer as well as from home.

7.3.2.1 Reasons for using this Medium

There is a vast body of research that suggests that Interactive Multimedia based Learning Environments can be very beneficial and effective. Firstly, the interactive nature of the tutorial provides an authentic coding environment. Students can actively participate in encoding and decoding processes. Secondly, in addition to discovering and verifying encryption mechanisms, students are given the opportunity to create their own encryption methods as well as breaking existing ones. Moreover, the Interactive Multimedia based Learning Environment provides a platform for self directed investigations. A vast number of links to cryptography related Internet websites are provided that permit students to explore aspects of interest in cryptography on their own. Altogether, the tutorial provides a student centered learning environment that can be used to develop a thorough understanding and appreciation of cryptography and its use.

7.3.2.2 Costs and Benefits of Project

The time required to explain the usage and navigation through the tutorial is minimal. It does not pose any more difficult tasks than “surfing” the Internet. According to Kerres (2000), such skills are common place and may be considered as elementary culture techniques.

Benefits:

• closest way to simulate encoding and decoding in school settings.

• learning and reinforcing computer and Internet related skills

• important real life application of mathematics

• the hyperlinked learning environment provides immediate access to relevant Internet based cryptography information.

Costs:

• Creation of learning environment is very time consuming; it also requires advanced web design skills.

• More than rudimentary understanding of mathematics encountered.

7.3.2.3 Function of Didactic Medium

Introducing cryptography in form of a tutorial seemed most appropriate as it serves the following four important functions:

Firstly, cryptography is the art and science of encoding and decoding information such as words or texts. This process shall be made accessible to the learner in a realistic setting. The interactive components of the tutorial provide the means for students to immediately test, falsify and confirm own ideas immediately. In addition to discovering existing ciphers interactively, students are given the opportunity to create their own encryption methods as well as break existing ones.

Initial studies of cryptography shall consist of a high degree of interaction in order to develop a profound understanding of underlying cryptographic mechanisms. Ideally, by performing these cryptographic investigations, general encryption, decryption and crypto-analytic methods are explored by each learner.

Secondly, students have the option to do self-directed cryptography related studies on the Internet through the use of the tutorial. Each page contains hyperlinks that link to cryptography-related information on the Internet. In particular, the first section of the Tutorial (“Cryptography”) provides hyperlinks to a variety of historic, social and mathematical aspects of cryptography. Here, each learner has the opportunity to further explore uses, history or latest developments of cryptography that are of particular interest to him. I deem this section of the tutorial to be highly instrumental in allowing the learner to pursue that particular aspect of cryptography that is of most importance to him.

Cryptography is a highly interdisciplinary conglomerate of history, mathematics and social sciences among which students shall be able to gain access and interest to cryptography through their avenue of choice.

Moreover, performing self-directed cryptography related research on the Internet may be quite time consuming and not turn out to be successful as the Internet is itself an unorganized world. The tutorial provides order in the disorderly world of the Internet. Students are usually only one mouse-click away from finding the desired outside information so that time efficient learning and research can take place.

Thirdly, cryptography is intimately related to computer science. In practice, many aspects such as developing and testing ciphers as well as the art of breaking ciphers, called crypto-analysis, require the use of computers. Two examples are:

Breaking the mono-alphabetic ciphers which involves counting the number of letters in a text. Performing an RSA Cipher requires the computation of two large prime numbers. Naturally, just as any cryptographer uses powerful computer tools, learners can use the power of computers as well. A lot of precious time would be lost if such time consuming tasks would be performed by hand.

Fourthly, using interactive web pages to explore the mechanisms of Ciphers broadens the students’ abilities in making use of such digital media. Generally speaking, college-bound students are well-seasoned Internet surfers involving searching and navigating through the Internet, communicating interpersonally via Email and using Instant Messenger software. However, typically, such students have less experience in using interactive applications for the purpose of further exploring or discovering mathematical mechanisms. Commonly, the Internet is not used for inquiry-based tasks of a researcher. Such skills are formed and reinforced through the use of the tutorial.

7.3.3 Didactic Structure

When creating the tutorial, my main focus was to create a stimulating learning environment that allows the learner to easily, independently and joyfully study historic and modern encryption systems. In order to accomplish this goal, the most important question to answer is how to design the learning environment to produce the best learning results? Therefore, a crucial question has to be answered: Which learning theory serves the stated goals best? In the above discussion, I argue that no single learning theory possesses universal functionality. Instead – using a pragmatic approach - based upon the described learning group, the learning objectives and the learner motivation, suiting learn theoretic principles as laid out in section 7.1.4 shall be used.

As described before, in order to support self-driven autonomous learning that enables learners to solve ill-defined problems in complex authentic situations, constructivism provides my preferred learning model. Clearly, constructivist principles shall not be employed with novices of cryptography. Beginners will be frustrated when attempting to solve complex problems without a proper foundation. In general, behaviorist and cognitivist methods are more appropriate for beginners and intermediate learners to form a solid foundation.

Cognitivist methods are used with the introduction of all historic ciphers in the sections titled “Challenge”, “Substitution Ciphers”, “Transposition Ciphers”, “The Goldbug” and “The Adventure of the Dancing Men”. As their unknown underlying mechanisms are not too complex, students are given the opportunity to discover these ciphers by themselves using the interactive encryption tools. Students can take their time to figure the underlying mechanisms in a playful manner. No teacher pressure is involved, no risk of public embarrassment in case of failure, no grade pressure involved. A safe learning environment allows for placing the entire focus on the mechanisms of the cipher. This discovery based learning can be performed in cooperation with another learner or individually. Learners may also request hints from the teacher or other students. Explanations can be checked for correctness using the interactive encryption tools.

As learners proceed from one substation cipher to the next via the head menu, new cryptographic information is compared to the existing learner’s cryptographic knowledge structure. Substitution ciphers are combined, extended or altered to create new substitution ciphers. For example, the Linear Cipher is a combination of the Caesar Cipher and the Multiplication Cipher. Here, cognitivist principles are used as existing rules are modified and applied to form new rules.

Behaviorist methods are employed when ciphers were not successfully discovered by the learner. In that case, the following sections titled “How It Works” ask the learner to encode and decode in the explained manner. Answers can be verified using the encryption tools or the CTRL-A option. Immediate positive reinforcement shall strengthen the established stimulus response relationship and encourage the learner to continue with a positive attitude. Following the behaviorist ”law of exercise” with immediate feedback, the learner is asked to encode and decode a few times messages. According to Thorndike, the stimulus response bond becomes stronger with repetition and positive feedback.

Both cognitivist and behaviorist methods are employed on the pages that lay the mathematical foundation of the encryption systems (these are the pages titled “Modular Arithmetic” through “Euler’s Theorem” that are part of the head menu column titled “Crypto Tools”). Here, students can – in a limited manner – discover mathematical principles such as modular arithmetic (referred to as “clock arithmetic”). However, mostly explanations are provided that students are to follow. Following the explanations, students are asked to perform mathematical computation based on the given rules. It would be practically impossible to discover advanced mathematical theorems such as the Euler’s Theorem or the Euclidean Algorithm. These mathematical discoveries were made by mathematicians with a vast mathematical background and are impossibly rediscovered by students with a beginning knowledge in this field. This general instructional dilemma occurs in many areas of mathematics where students are simply introduced to underlying theorems and have to digest them. Usually, behaviorist strategies follow this initial exposure with a list of computational exercises to practice the newly introduced.

Cognitivist strategies are used in the sections titled “Master It”, “Decode It” and “Break It”. Here the ciphers are scrutinized, the underlying and previously established coding mechanism is the basis for the decoding and the crypto-analytic process. Existing coding rules are being manipulated. This is followed by behaviorist based decoding and crypto-analytic exercises that can be verified using the interactive encryption tools and the CTRL-A option.

Developing a thorough understanding of any cryptographic method requires investigating the three main aspects of cryptography: encryption, decryption and crypto-analysis. In the tutorial, the learner explores these 3 categories consecutively as shown in table1 below.

Table 1: The four tasks to investigate a cipher.

Constructivist strategies are effective once the learner has the ability to acquire advanced knowledge of cryptographic methods. Then, the learner can freely start to investigate aspects of cryptography that are of particular interest to him. He can use the provided link for self-directed cryptographic studies such as the historic or the practical security aspects of cryptography. The provided web links give the starting point to such investigations. A simple, user-friendly navigation system facilitates the tutorial navigation.

Motivation is the driving force of self-directed studies. The tutorial tries to stimulate intrinsic motivation as follows: Students are able to pursuit their preferred aspect of cryptography. I.e. historic, real-life applications or mathematical aspects can be pursued through the provided links to tutorial pages as well as to Internet pages. Real life situations are portrayed in the tutorial that show how even the learner uses and relies on cryptography. External motivation is provoked through the rewards that are given to cryptography experts that solve the provided Cipher Challenge.

Other assumptions of constructivist learning are fulfilled as well:

• The learner can freely choose the aspect(s) to be investigated and has the freedom of how to pursuit it.

• The selected topics are naturally embedded in their social setting and are not prepared or simplified by the teacher.

• Learners are encouraged to work cooperatively. The provided exercises initiate desired communication among students to exchange, create and clarify ideas regarding the mechanisms of ciphers. Group efforts can be very productive and rewarding i.e. when crypto-analyzing ciphers and when designing new ciphers. Such hybrid learning environments in which communication and cooperation are effective study elements have proven to be more attractive to the learner. Kerres (2000) found that the acceptance of individualized learning systems as rather limited whereas project-oriented exercises that support cooperative learning are more attractive to the learner.

• Learning is constructed through the free investigation of cryptographic rules. Learners add new knowledge to prior knowledge.

Sequential Structure of Learning Environment

The Cryptography tutorial is divided into seven sections that can be navigated through via the head menu located on top of each page. The first section (“Cryptography”) introduces basic concepts, terminology and applications of cryptography on 9 web pages. The next five sections (“Caesar Cipher”, “Multiplication Cipher”, “Linear Cipher”, “Polyalphabetic Ciphers” and “RSA Cipher”) are dedicated to the introduction of the mechanisms of such ciphers each using between 2 and 6 pages. The last section (“Crypto Tools”) is a reference section that is used throughout the study of the tutorial. It serves two purposes: it illuminates the underlying mathematical concepts of cryptography as well as provides necessary cryptography tools such as a crypto calculator and a letter frequency counter.

The tutorial ends with the “Cipher Challenge” which is the ultimate test of the Cryptography Tutorial. It covers 8 of the most prominent ciphers which are investigated in the tutorial. The ability to cryptoanalyze all 8 ciphers requires profound knowledge of the learned ciphers. Students that already have a thorough knowledge of cryptography may choose to begin the tutorial with the Cipher Challenge. The Cipher Challenge contains hyperlinks that provide the learner with quick and easy access to all the necessary information concerning the ciphers. Although the purpose of the cipher challenge is ultimately to test the learner’s knowledge of the material, its successful completion is very rewarding. In the cryptography class, students that master the Cipher Challenge are encouraged to email the solution to me and are awarded an appropriate prize.

The tutorial’s sequential design supports the natural learning process i.e. “from simple to difficult”. Through gradually learning more complex ciphers, the learner may confidently progress through the lessons. For example, to master the first basic ciphers the learner is solely required to have mastered addition and to know the chronology of the letters in the alphabet. With each cipher the learner learns a particular mathematical concept that is required to master the subsequent ciphers. The final cipher to study, the RSA Cipher, is a difficult cipher to comprehend and requires the knowledge of all mathematical concepts studied previously.

Further Didactic Aspects of Learning Environment

Immediate Feedback: A learner shall be able to keep track of his learning progress. This can be achieved through answering the posed questions and checking the answers using the highlighting function of the key combination CTRL-A which displays the hidden answers right next to the question.

The Cipher Challenge was created as stimulation for users that like to challenge themselves by taking on the job of a cryptoanalyst. Here, the user is referred to the relevant sections of the tutorial for assistance. Final solutions to the challenge may be emailed to me resulting in immediate feedback.

Navigation System: I designed a menu driven tutorial. The head menu gives the user an immediate overview of the full extent of the tutorial. Sequential tutorial navigation is made simple: from left to right, top to bottom. Any page can be accessed at any time allowing for quick look ups, references, reviews, etc. “Back”, “Next” and “Home” links are provided on the top and “Back” and “Next” buttons at the bottom of every page enabling quick access to neighboring pages and to the starting page.

The provided links to pertinent pages are usually provided in the right column of each page. Different tutorial pages contain different links with some links used more than once. I decided to leave the link column on each page unchanged. A central link page which contains the links of all other tutorial’s web pages appears to be useful. It would have eliminated the right columns on each page. However, I decided not to create this page as a user is now two as opposed to one mouse click away from reading the desired pertinent information. This does not seem to be a significant factor when accessing a few outside websites only. However, significantly more time is needed when a great number of outside web sites are visited. I also find that the right column is not very wide and does not take away a significant amount of space from the remaining page. However, it does provide the links where they seem most pertinent.

Page Length: It was my intention to keep the page length as small as possible. This goal was realized on most pages of the learning environment. However, the pages that contain a lot of text are pages that are usually used as reference pages or contain detailed encryption explanations that the user may skip without a loss of the overall picture. Additionally, pages that contain a lot of text often contain links to other pages or interactive encryption tools that allow the user to verify the written information and by doing so continues the interactive format and reduces any monotony.

7.3.4 Learning Organization

The two-week long cryptography course took place at the end of the school year 2002-2003. The 26 students took their nationwide Advanced Placement Math exams. Teacher and students met for 50 minutes five times a week. In the first class, the students are introduced to the structure of the tutorial and its usage. The teacher acts as a facilitator who answers questions, assists in Internet research or any other needed area. The teacher was accessible inside and outside the classroom and via email.

The tutorial is hosted on the school’s Internet server. During class periods, students accessed the tutorial from their workstation in the school’s library. However, students also accessed the tutorial from home. The Internet-accessible tutorial enabled students to utilize the tutorial outside of their regular class period and, thus, is a preferable solution to a tutorial that is exclusively accessible through the school’s network. The ability to access the tutorial outside of class should not be underestimated as it enables the students to review lessons at home or to engage in self-directed research.

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Behaviorism: Based on observable changes in behavior. Behaviorism focuses on a new behavioral pattern being repeated until it becomes automatic. The mind is perceived as a black box, stimulus A produces response B.

Cognitivism: Based on the thought process behind the behavior. Changes in behavior are observed, and used as indicators as to what is happening inside the learner's mind.

Constructivism: Based on the premise that we all construct our own perspective of the world, through individual experiences and schema. Constructivism focuses on preparing the learner to problem solve in ambiguous situations.

“Challenge” - permits students to discover the encryption and decryption method of the investigated cipher from encrypted sample texts.

“How it works and Master” - develops and verifies students’ understanding of the encoding method of the cipher.

“Decode It” - develops and verifies the students’ understanding of the decoding method of the cipher.

“Break It” - develops and verifies the students’ proper understanding of how to cryptoanalyze the cipher.

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