Course In Number Theory Research Articles

Implementation of Self-Evaluation Exit Tickets in Self-Organized Learning Environments (SOLEs) in Online Learning

Effectiveness is one of the main problems in the online learning process. This becomes a benchmark for success in achieving learning objectives. Therefore, educators must be careful in choosing creative and innovative strategies both in terms of learning strategies and assessment strategies. Like the Self-Evaluation Exit Ticket assessment strategy which offers various advantages for educators and students. This research aims to determine the effectiveness of applying the Self-Evaluation Exit Ticket technique as an assessment strategy in learning. The subjects in this research were eighteen Mathematics Education students at a private higher education institution in Indonesia who took Number Theory courses online. Where, the material that will be covered in this course is the basic concept of congruence, complete residue systems, reduced residue systems, Euler's concept, and Euler's theorem. This type of research is quantitative descriptive. Based on the research results, all subjects experienced improvement while implementing the Self-Evaluation Exit Ticket assessment strategy. This increase was seen at meeting 2, namely as many as 55% of students were able to determine their learning outcomes, whereas previously at meeting 1 only 33% of students were able to determine their learning outcomes. Subjects thought that learning implemented using Self-Evaluation Exit Tickets was more fun, interesting, and encouraged subjects to continue learning. This means that the Self-Evaluation Exit Ticket assessment strategy can be effective and possible to apply in online learning.

The Effect of teaching a Small Private Online Course (SPOC) on Elementary Number Theory course based on Constructivism Theory to Enhance Mathematics Logical Reasoning Ability for First Year Students of Xi’an University

Background and Aims: With the continuous advancement of new education curriculum reform, especially in China, the classroom roles of teachers and students are evolving. Logical reasoning is an important part of the core literacy of mathematics. The purposes of this research were: (1) Compare Mathematics Logical Reasoning Ability before and after the implementation of a Small Private Online Course (SPOC) on Elementary Number Theory course based on Constructivism theory. (2) Assess the student’s satisfaction with the Small Private Online Course (SPOC) on the Elementary Number Theory course. Methodology: The sample of this study was 40 first-year students in Xi'an University, Xi’an City, Shaanxi Province. They were selected by cluster random sampling. The research instruments were: 1) Seven lesson plans of a Small Private Online Course (SPOC) on Elementary Number Theory course based on Constructivism theory. 2) Mathematics Logical Reasoning Ability test paper. 3) Questionnaire for students’ satisfaction. The researcher used the Mathematics Logical Reasoning Ability test paper to conduct pre-tests and post-tests on the sample. Then the pre-test data and post-test data are analyzed. The content of data analysis includes: score mean, standard deviation, the correlation between pre-test data and post test data, t-test of paired dependent sample. Results: From the study, it was found that: (1) The mathematics Logical Reasoning Ability test paper data showed the mean score of post-test data were higher than pre-test scores at the.05 level of statistical significance (The mean of pretest was 19.48, the mean of post-test was 22.55, p <.05). (2) The mean scores of students’ satisfaction with the Small Private Online Course (SPOC) on the Elementary Number Theory course based on Constructivism theory were very high level. Conclusion: The students who learn through the Elementary Number Theory course will have a post-test score higher than the pretest score. The students’ satisfaction with a Small Private Online Course on Elementary Number Theory course based on Constructivism theory will be at a very high level.

Implementation of Inquiry Learning Model in Collaboration with PBL to Improve Student Understanding in Number Theory Course

The purpose of this study was to describe the presence or absence of the influence of the infusion learning model collaboration with Problem-Based Learning (PBL) to develop students' understanding of proof and mathematical argumentation in number theory courses. This research is an experimental study with a randomized control group pretest-posttest design, two groups namely the experimental group and the control group. The experimental group is the group that uses the infusion learning model in collaboration with PBL, while the control group is the group that uses conventional learning. The subjects of this study consisted of 40 students at a university in Jombang, Indonesia. Data collection techniques through observation sheets, proof understanding tests and observation sheets of students' mathematical argumentation abilities. The results of the research show that the significant difference between the average proof of understanding of students in the experimental group and the control group. The difference between the average proof of understanding in the experimental group and the average proof of understanding of students in the control group was 21.75. Furthermore, the significant difference between the average argumentation ability of students in the experimental group and the control group. The difference between the average argumentation ability of students in the experimental group and the average argumentation ability of students in the control group is 5.25. Therefore, the implementasion of infusion learning in collaboration with PBL is more effective than conventional learning models for developing students' ability to understand mathematical proof and argumentation. This learning model promotes the development of critical thinking skills, problem-solving, conceptual and different understanding needed to construct a formal proof, and strong and valid arguments.

Learning Obstacle to Ontogeny of Mathematics Teacher Candidates on Number Theory

Prospective mathematics teachers must have the characteristics of mathematical science, namely systematic, logical, and critical. This study aims to reveal the analysis of learning obstacles experienced by prospective mathematics teachers in number theory lectures with sub-material of division relations, FPB and KPK. This research uses a qualitative approach method in descriptive form. This research was conducted by collecting data through questionnaires and interviews and validated using the triangulation method. The study was limited to the analysis of ontogenic learning obstacles according to Brosseasu. The results of this study were that 4 out of 6 research subjects experienced the occurrence of ontogenic learning obstacles with details of 37.5% of prospective teacher students experiencing psychological ontogenic obstacles and instrumental ontogenic obstacles, then 25% experiencing didactic ontogenic obstacles. This indicates that students' readiness to take part in number theory courses is still weak. Students are weak in terms of psychological readiness, weak in readiness for complete mastery of related material previously studied and weak in terms of following the thought process needed in this course.

Students' mathematical argumentation ability when proving mathematical statements based on self-efficacy

Argumentation as an aspect of problem-solving has been studied in mathematics education. However, mathematical proof still needs to be resolved further. This study investigates students' mathematical argumentation skills when proving mathematical statements based on their self-efficacy. The research subjects were 43 mathematics education students at a university in Aceh Province who had taken a number theory course. The study used a qualitative approach with a case study design: students’ mathematical proving self-efficacy. Data was obtained using self-efficacy questionnaires and mathematical proof test instruments that experts have validated, while the data triangulation used was an in-depth interview. The results of this study reveal that students' mathematical argumentation skills in proving mathematical statements have differences based on their self-efficacy. The mathematical argumentation ability of students with high self-efficacy involves all aspects of argumentation well so that the construction of the proof is scientifically correct. Meanwhile, the argumentation ability of students with moderate or low self-efficacy still does not involve essential aspects of argumentation. So, the proof results are not scientifically correct because they have not arrived at the proper conclusion.

Development of Problem-Based Number Theory Learning Tools

Development of Problem-Based Number Theory Learning Tools" in the 2022 FMIPA Unimed Mathematics Education Lecture. This research is the development of learning tools that aim to describe the development process that produces valid, practical, and effective number theory learning tools. Problem-Based Number Theory Learning developed by following the Four D's: Define, Design, Develop, and Disseminate. The trial of device development was carried out in the S1 Mathematics Education Study program in the Number Theory lecture. In the development of number theory lecture tools produce: Textbooks, and 14 Lesson Plans (LP) that are valid, practical and effective. The implementation process is carried out by preparing components of teaching materials for the Number Theory course in the form of: textbooks, LP which are carried out in the Number Theory course in 2022. In the implementation of disseminate applying limited Number Theory teaching materials produced: LP as many as 14 valid, practical and effective meetings to the Lecturer Team in the Department of Mathematics FMIPA Unimed

Enhancement of Mathematical Conceptual Understanding in a Cloud Learning Environment for Undergraduate Students

Mathematical conceptual understanding is fundamental ability that students should have for learning practice in mathematics. This study presents a learning approach based on a mathematical understanding development strategy in the cloud learning environment to enhance undergraduate students’ comprehension of mathematical concepts. A quasi-experimental design with a pre-test and post-test control group was deployed in this study. The 56 undergraduate students, enrolled in the Number Theory Course, were involved in this study. The 28 students in the experimental group were taught using a mathematical understanding development strategy in a cloud learning environment and the 28 students in the control group were taught using a conventional approach. The research instruments were an instructional plan, and a mathematical conceptual understanding test. Data were analyzed by mean, standard deviation, and t-test. The results found that the mean score of mathematical conceptual understanding of students in the experimental group was higher than the 60 percent criterion. There were statistically significant differences between the mean scores of the students in the experimental and the control groups in their mathematical conceptual understanding in the post-test in favor of the experimental group at a significance level of .05.The results of this study inform the positive impact of learning approach based on a mathematical understanding development strategy in a cloud learning environment for undergraduate students.

Reciprocal Sums and Counting Functions

Motivated by the gentle exploration of the distribution of prime numbers typical of an undergraduate number theory course, as well as by a recent breakthrough result in arithmetic combinatorics, we explore connections between the counting function, which counts the number of elements up to a threshold x, and the reciprocal sum function, which adds the reciprocals of the elements up to a threshold x, for subsets of the natural numbers.

EFEKTIVITAS E-MODUL INTERAKTIF BERBASIS PEMBELAJARAN PROJECT DITINJAU DARI KEMAMPUAN LITERASI MATEMATIS DAN KEPERCAYAAN DIRI MAHASISWA

The implementation of distance learning at every level of education, significantly higher education, has various obstacles. One of them was the lack of teaching materials that can support students to learn both guided and independent. Learning resources that were commonly used did not interactively support learning. This study aimed to determine the effectiveness of interactive e-modules based on project-based learning in mathematical literacy skills and student confidence in number theory courses. The type of research used was quasi-experimental with a quantitative approach. The research subjects were students of the 2021 mathematics education study program, which consisted of two classes with 32 experimental classes and 36 control classes. Technique sample used a census where each class was selected and determined treatment. Data collection techniques were used tests with literacy ability test instruments and self-confidence questionnaires. Data analysis with descriptive analysis and hypothesis testing using MANOVA and the effectiveness of learning with interactive e-modules were analyzed using a one-sample t-test. The results showed that the application of project-based learning-based interactive e-modules was effective in students' literacy skills, with a t-count result of 6.949 and a significance level of 0.000. The use of e-modules is more effective than the control class, with an average literacy ability of 81.25 in the experimental class and 62.91 for the control class. I am judging from the adequate self-confidence with a high category and the results of t count of 4.179 significant at 0.05 alpha. The results of the Manova analysis show that the application of interactive e-modules is more effective in terms of literacy skills and student self-confidence, with a significant Hotelling's trace of 0.000

Computational Thinking Process of Prospective Mathematics Teacher in Solving Diophantine Linear Equation Problems

<p style="text-align: justify;">Prospective teachers facing the 21st century are expected to have the ability to solve problems with a computer mindset. Problems in learning mathematics also require the concept of computational thinking (CT). However, many still find it challenging to solve this problem. The subjects in this study were twenty-one prospective mathematics teachers who took number theory courses, and then two research samples were selected using the purposive sampling technique. This study uses a qualitative descriptive method to describe the thinking process of prospective teachers in solving Diophantine linear equation problems. The results showed that the first subject's thought process was started by turning the problem into a mathematical symbol, looking for the Largest Common Factor (LCF) with the Euclidean algorithm, decomposition process, and evaluation. The second subject does not turn the problem into symbols and does not step back in the algorithm. The researcher found that teacher candidates who found solutions correctly in their thinking process solved mathematical problem used CT components, including reflective abstraction thinking, algorithmic thinking, decomposition, and evaluation. Further research is needed to develop the CT components from the findings of this study on other materials through learning with a CT approach.</p>

The Principle of Mathematical Induction: Applications in Physical Optics

The bare rudiments of the principle of mathematical induction as a method of proof date back to ancient times. In the contemporary university milieu, the demonstrative scheme is taught as part of a course in discrete mathematics, set theory, number theory, graph theory, group theory, game theory, linear algebra, logic, and combinatorics. In theoretical computer science, it bears the pivotal role of developing the appropriate cognitive skills necessary for the effective design and implementation of algorithms, assessing for both their correctness and complexity. Pure mathematics and computer science aside, the scope of its utility in the physical sciences remains limited. Following an outline of some elementary concepts from vector algebra and phasor analysis, the proofs by induction of a couple of salient results in multiple-slit interferometry are presented, viz., the fringe intensity distribution formula and the upper bound of the total fringe count. These specific optical instantiations serve to illustrate the versatility and power of the principle at tackling real-world problems. It thereby makes a welcome departure from the popular view of induction as a mere last resort for proving abstract mathematical statements.

Pengembangan Media Pembelajaran Matematika Melalui Video Pembelajaran pada Mata Kuliah Teori Bilangan

Developing a video for learning mathematics in the Number Theory course as the purpose of this research. This needs to be done to help students understand the material during online learning. The development procedure in this R&D research is based on the ADDIE product development model (analyze, design, development, implementation, evaluation). The stages of this research are limited to the development and trial stages. Furthermore, at the trial stage, one class is carried out to take the Number Theory course in the odd semester for the Academic Year 2021-2022. The results of this study are (1) several learning videos have been formed in the Number Theory course based on sub-materials; (2) the learning video media is declared feasible based on the results of the assessment from a team of experts consisting of two validators, the results obtained from four aspects, namely the material aspect, language and image aspects, presentation aspects, and finally the display aspect with a final average score of 75 with appropriate criteria; and (3) the results of the trial obtained the results of the completeness test, the average score was 75.28 and the comparative test results obtained the average value. The mean of completeness of the experimental class and the control class were 75.28 and 60.28. The value of the experimental class is better than the control class.

Heuristic Trees as a Digital Tool to Foster Compression and Decompression in Problem-Solving

This design-based study addresses the issue of how to digitally support students’ problem-solving by providing heuristics, in the absence of the teacher. The problem is that, so far, digital tutoring systems lack the ability to diagnose students’ needs in open problem situations. Our approach is based on students’ ability to self-diagnose and find help. To this purpose, we introduce a new type of digital, interactive, help-seeking tool called a heuristic tree. Students’ use of this tool is supported by a help-seeking flowchart. The design of heuristic trees is based on our reinterpretation of the notion of heuristic in terms of terms of compression. Our research question is: How do heuristic trees and the help-seeking flowchart influence students’ problem-solving behaviour? This question was studied in the context of a number theory course for in-service mathematics teachers. During five weeks, fifty students worked on fifty-five problems supported by heuristic trees. Our data consists of video observations of two small groups of students, a teacher log, interviews with these two groups, and a survey filled in by twenty-three students. The main results are that the support by heuristic trees and the help-seeking flowchart allows students to work in the absence of a teacher and to engage strongly with problems, maintaining ownership of the solution methods. Moreover, as intended by the tree structure, students learned to focus not just on the small steps of the solutions, but also on the general heuristic techniques, theorems, and concepts that should be learned in the process of finding those solutions.

EFEKTIVITAS PROBLEM-BASED LEARNING SETTING STAD DITINJAU DARI HASIL BELAJAR DAN DISPOSISI MATEMATIS MAHASISWA

This study aims to describe the effectiveness of the STAD problem-based learning setting in terms of learning outcomes and mathematical dispositions. This research is an experimental research type of pre-experimental. This pre-experimental study used a one-group pretest-posttest design with a population of students in the second semester of the Mathematics Education Study Program at the University of Flores for the 2020/2021 academic year. The sample in this study was a class of second-semester students. The research instrument used was a test of learning outcomes in the form of a description of 6 questions and a non-test in the form of a mathematical disposition questionnaire in the form of a Likert scale with five answer choices. The data analysis technique used was the paired-sample t-test with the help of SPSS Windows Version 16. Based on the test results, obtained the value of sig. of 0.00 for learning outcomes and mathematical dispositions. This result shows that the problem-based learning setting of STAD is effective and good to be applied in the design of the learning process in a number theory course, in terms of learning outcomes and mathematical disposition abilities.

Developing Number Theory Textbook for Pre-Service Mathematics Teacher of International Program

The long-term objective of this research is to produce valid, practical, and effective textbooks for courses in international program of mathematics education of Universitas Ahmad Dahlan. For the short-term, this research aims to produce valid and practical textbook for number theory course. The validity was based on the relevancy towards the students’ expected competence, while the practicality was measured from the implementation of learning using the textbook. This research used design research with the type of development studies. It followed four steps, i.e. (1) preliminary research, (2) prototyping stage, (3) summative evaluation, and (4) systemic reflection and documentation. In the preliminary research, we analyzed the need of the textbook in the international program and found that the textbook of number theory course is very needed. In the prototyping stage, we wrote the textbook based on the need analysis. The prototype was then discussed with expert and revised in the stage of summative evaluation and applied to the lecture meetings. The result was the final product of the textbook used for the stage of systemic reflection and documentation when we wrote all the results according to the preset research framework. Finally, the developed number theory textbook is valid and practical.

Mathematical Specialized Knowledge of a Mathematics Teacher Educator for Teaching Divisibility


 
 
 The knowledge of Mathematics teachers has been a very prominent focus of attention in the last decades. However, it leaves aside one of the dimensions involved in the development of this type of knowledge, specifically the knowledge of Mathematics teacher educators. In this paper, we discuss a mathematics teacher educator’s knowledge in the context of classes on Euclid’s division algorithm theorem in a Number Theory course for prospective secondary teachers. Some indicators of this specialized knowledge of mathematics teacher educators are presented and discussed.
 
 

Analysis of Self-Regulated Learning of Students in Mathematics Education Study Program in Number Theory Course

The number theory is the first mathematics content course in the UPH FIP Mathematics Education study program, which does not have a response class. The response class was conducted as a companion to four content subjects with close relations with school mathematics content, namely Calculus 1, Calculus 2, Geometry, and Linear Algebra. Based on the questionnaire that has been distributed, prospective mathematics teachers consider that the response class is beneficial for them in attending lectures and in exploring related mathematical concepts. However, on the other hand, students inevitably have to go back to learning to be determined with the response class. The implication is that students' self-regulated learning will not develop optimally. However, self-regulated learning is an essential aspect that prospective teachers need to have. This research was conducted to see how self-regulated learning of future mathematics teachers in number theory courses. The research instruments used are questionnaires, the scale of self-regulated learning, and interviews. This research is a qualitative descriptive study. The results of this study are that most students have a moderate level of learning independence. The aspects or indicators of learning independence that need special attention are setting learning goals/targets. Prospective teacher students need to continue to train and develop their learning independence to become professional teachers who continue to develop along with the times.

ANALISIS KREATIVTAS MAHASISWA DALAM MENYELESAIKAN PEMBUKTIAN LANGSUNG PADA MATA KULIAH TEORI BILANGAN

The way of thinking logically and reasoning in number theory courses can be seen by the number of proofs of the properties of numbers in some basic set of numbers. This qualitative research aims to analyze student creativity in solving proving questions directly. The research involved 37 third semester students taking lectures on number theory. The technique in this research is to provide a test of proving questions then ask students with their own desires to do in front of the class, analyze the result of students work and interview students about their work. Data analysis was performed by analyzing the creativity of the evidence carried out by students fulfilling which aspects of the aspects of creative thinking behavior and indicators of creative thinking abilities based on Mahmudi. As many as four students have beeb able to use different evidentiary techniques. The four students are creative with very low, low, medium, high creativity ability levels based on fulfillent of indicators of creative thiking abilities based Mahmudi.

Analysis of difficulties for pre-service mathematics teacher in problem solving of division and divisibility based on theory of action, process, object, and schemes

Action, process, object, and schemes is an acronym of APOS. The difficulty in the perspective of APOS theory is the inability of students to access specific schemes in all situations, thereby inhibit the construction of other concept. Knowing student’s difficulties in using the concept of division and divisibility is necessary to design genetic decomposition of that concept. This article is a report from qualitative research that discusses conceptual issues about the idea of division and divisibility given by prospective mathematics teacher students who use the APOS theory. The data for the study were collected from an interview. The researcher took three interview subjects from 30 pre-service mathematics teachers who have attended and completed a test on the division and divisibility in an elementary number theory course. These participants were selected based on their performance in the test. However, the interview conducted showed that there are five difficulties involved and they include solving the divisibility problem on integers, algebraic manipulation and recognize structures in algebraic expressions that correspond to the division algorithm or the definition of divisibility., distinguishing between variables and parameters on division problem presented in algebraic expression, associating between division algorithm and definition, and application of divisibility properties and theorems.

Description of The Number Theory Course Problem Solving Abilities Basen on Personality Type of The Math Student at Universitas Darussalam Ambon

This research is a qualitative research, which aims to describe students in solving problems based on each personality type. The results showed that: (1) understand the problems the guardian wrote qualified enough and need; (2) was making plans the subject could detect information to complete about; (3) in implementing the plan doing it in accordance with the subject but incorrect. (1) understand the problems the subject of idealist only record that asked; (2) was making plans the subject could detect formula; (3) carry out a plan the subject of using formulas were formerly, although one measure. (1) understand the problems the subject of type artisan wrote very clearly all information is; (2) was making plans the subject of determining. formula(3) implement plan, problem solving able to solve. (1) understand the problems the subject type rational not have inscribed requirements enough and the requirements necessary; (2) was making plans subject to determine the steps the; (3) implement plan, problem solving skilled in algorithms and accuracy.