The Effect of Teaching Supported by Encryption Activities on Student Achievement About the Concept of Relatively Prime Numbers

Abstract

1. INTRODUCTION

Mathematics, as a universal language with its formal structure, plays a crucial role in shaping and organizing human life. Despite this, many students perceive mathematics as abstract, disconnected from daily life, and difficult to understand, largely due to its inherent abstraction and the use of ineffective teaching methods. This perception underscores the importance of research about effective instructional approaches. Prior studies highlight that the use of activities, concrete materials, and technological tools in teaching positively influences students’ attitudes and achievement in mathematics. Activities are particularly valuable as they transform students from passive recipients into active participants, fostering higher-order thinking and problem-solving skills.

Interdisciplinary activities represent one promising approach to enriching mathematics instruction. Cryptology, the science of securing information through encryption, offers a unique opportunity to integrate mathematics with other disciplines. Researcher such as Koblitz (1997) have emphasized the potential of cryptology to enhance mathematics education by enabling students to discover mathematical concepts independently and engage in interdisciplinary learning. Cryptology encompasses cryptography, which develops secure systems, and cryptanalysis, which evaluates their reliability.

1.1. Polybius Encryption and Encryption Activities

Encryption transforms plaintext into ciphertext using systematic techniques, while decryption reverses the process. The Polybius encryption technique provides a historically significant and pedagogically adaptable method for designing instructional activities.The Polybius square, developed in ancient Greece, is one of the earliest encryption methods. Originally, the letters of the Greek or Roman alphabet were arranged in a table. To encrypt using this technique, each letter in the plaintext is replaced by a number representing its row and column position in the table. The version of Polybius square adapted to the Turkish alphabet is shown in Table 1.

Table 1

Polybius square adapted to the Turkish alphabet

1

2

3

4

5

6

1

A

B

C

Ç

D

E

2

F

G

Ğ

H

I

İ

3

J

K

L

M

N

O

4

Ö

P

R

S

Ş

T

5

U

Ü

V

Y

Z

Using Table 1 with Polybius encryption technique, for example, the Turkish plain text “OKU” can be encrypted as “36-32-51”. In this study, the Polybius technique was adapted to the Turkish alphabet and further modified to incorporate pairs of relatively prime and non-relatively prime numbers. This adaptation allowed students to engage with the concept of relatively prime numbers in a meaningful and applied context. Activities required students to encrypt and decrypt messages by analyzing number pairs, thereby reinforcing their understanding of divisibility, greatest common divisors, and the definition of relatively prime numbers.

1.2. Problem Statement

The concept of relatively prime numbers, introduced in the 8th-grade mathematics curriculum, is foundational yet challenging for students. Research indicates that learners often struggle with misconceptions, such as assuming that two numbers must both be prime to be relatively prime. To address these difficulties, instructional approaches that promote meaningful learning and correct misconceptions are essential. Encryption activities, by contextualizing abstract concepts and linking mathematics with cryptology, provide an innovative solution. However, no prior study has specifically examined the effect of encryption activities on the teaching of relatively prime numbers, which motivated the present research.

The study sought to answer the following questions:

1.Is there a significant difference in mathematics achievement levels between experimental and control group students?

2.Is there a significant difference in post-test scores between the two groups?

3.What are the experimental group students’ opinions regarding the encryption activities?

1. METHODOLOGY

A mixed-methods design was adopted to combine the strengths of quantitative and qualitative approaches. The quantitative component employed a posttest-only control-group  experimental design, while the qualitative component gathered student opinions through open-ended questions.

2.1. Study Group

The study group consisted of 80 students (39 experimental, 41 control) from two 8th-grade classes in a public middle school. Random assignment was used to designate one class as the experimental group and the other as the control group.

2.2. Data Instruments

As a posttest the “Relatively Prime Numbers Achievement Test” was used, that adapted from the Ministry of Education’s diagnostic test, comprised 12 multiple-choice items. The KR-20 reliability coefficient of the test was calculated as 0.56 for this study, indicating moderate reliability. An opinion form was used for qualitative part of the study, that developed based on prior studies. The form included five open-ended questions. Expert review and pilot testing ensured validity and clarity.

2.3. Encryption Activities

Two activities involving encryption and decryption processes were designed following activity design principles. In the first activity, students created their own ciphertext using a Polybius square adapted with relatively prime number pairs and exchanged these messages with their friends to decrypt. In the second activity, students encrypted a given plain text and decrypted a partially incomplete ciphertext, requiring the analysis of number pairs to determine relative primality.

2.4. Experimental Implementation and Data Collection

Both groups received identical instruction on the concept of relatively prime numbers. Subsequently, the experimental group engaged in encryption activities, while the control group solved additional practice problems. After the intervention, both groups completed the achievement test, and the experimental group provided written feedback on the activities.

2.5. Data Analysis

Normality tests indicated non-normal distributions; therefore, Mann–Whitney U tests were used to compare groups. Qualitative data were analyzed through descriptive coding, with inter-coder reliability calculated at 90%, exceeding the recommended threshold.

1. FINDINGS

Comparison of prior mathematics achievement revealed no significant differences between groups, confirming initial equivalence.

Post-test results showed that the experimental group achieved higher mean scores than the control group; however, the difference was not statistically significant (p > .05).

Students’ opinions highlighted several themes:

Interest and Engagement: Many students described the activities as enjoyable and motivating, noting that they made mathematics more appealing.

Conceptual Understanding: Students reported that encryption tasks helped them better grasp the meaning of relatively prime numbers by applying the concept in a practical context.

Challenge and Support: While some students found the tasks demanding, they appreciated the opportunity to collaborate and receive guidance from the teacher.

Interdisciplinary Connection: Several students expressed curiosity about cryptology and its relation to mathematics, recognizing the broader relevance of the activities.

1. DISCUSSION AND CONCLUSION

The findings suggest that encryption activities positively influence students’ attitudes and conceptual understanding, even though they did not produce statistically significant gains in achievement within the limited timeframe of the study. This aligns with prior research indicating that interdisciplinary and activity-based approaches enhance engagement and comprehension. The lack of significant quantitative differences may be attributed to the short duration of the intervention and the moderate reliability of the achievement test. Longer-term implementations with more robust assessment tools may yield clearer evidence of effectiveness.

Encryption activities, particularly those based on the Polybius technique, offer a promising instructional approach for teaching relatively prime numbers. They contextualize abstract mathematical concepts, foster problem-solving skills, and stimulate student interest through interdisciplinary connections. While the study did not find significant differences in achievement, qualitative evidence underscores the pedagogical value of such activities. Future research should extend the duration of interventions, employ larger samples, and explore additional mathematical topics to fully realize the potential of cryptology in mathematics education.