Abstract Algebra Course Research Articles

Improving mathematical proof based on computational thinking components for prospective teachers in abstract algebra courses

Understanding and constructing mathematical proofs is fundamental for students in abstract algebra courses. The computational thinking approach can aid the process of compiling mathematical proofs. This study examined the impact of integrating computational thinking components in constructing mathematical proofs. The researcher employed a sequential explanatory approach to ascertain the enhancement of algebraic proof capability based on computational thinking through the Wilcoxon test. A total of 32 prospective teachers in mathematics education programs were provided with worksheets for seven meetings, which were combined with computational thinking stages. Quantitative data were collected from initial and subsequent test instruments. Moreover, three prospective teachers were examined through case studies to investigate their mathematical proof capability using computational thinking components, including decomposition, abstraction, pattern recognition, and algorithmic thinking. The study's findings indicated that CT intervention enhanced students' logical reasoning, proof-writing abilities, and overall engagement with abstract algebra concepts. The findings illustrate that integrating computational thinking into learning strategies can provide a framework for developing higher-order thinking skills, especially in proving, which are essential for studies in mathematics education programs.

The Challenges Facing Mathematics Teachers Pre-Service in the Abstract Algebra Course from their Point of View

تهدف هذه الدراسة إلى معرفة التحديات التي تواجه معلمي الرياضيات قبل الخدمة في مقرر الجبر المجرد من وجهة نظرهم، حيث اعتمدت الدراسة على المنهج الوصفي التحليلي، وقد أعدت الباحثة الاستبانة كأداة لتحقيق هدف الدراسة، واقتصرت عينة الدراسة على (31) طالبًا وطالبةً من طلبة المستوى الرابع، وتوصلت الدراسة إلى وجود عدد من التحديات تم تصنيفها في ثلاثة محاور، أهمها الفهم السطحي لمفردات الجبر المجرد، حيث يلجأ الطلبة إلى حفظ المعلومات لتحقيق النجاح فقط، وفي ضوء نتائج الدراسة أوصت الباحثة بالتركيز على الفهم المفاهيمي والإجرائي للطلبة عند تدريس المقرر، لأن المقرر يتطلب مستوى أعمق من البصيرة والتطور.

Utilizing a Poster Project as an Assessment in an Introductory Abstract Algebra Course

ABSTRACT Though posters have been used as a pedagogical tool in a variety of fields, the literature suggests that the use of posters as an educational assessment tool by the mathematics community has been limited. This is unfortunate given the numerous potential benefits of a poster, including the development of oral communication skills and the ability to examine a mathematical topic at a deeper level. We describe the use of a poster project in an introductory abstract algebra course with specific implementation guidance including a schedule of assignments and assessment criteria. We also include reflections on the success of this project over several semesters as well as advice for adapting the project to a remote course. Samples of student posters are available upon request to the authors.

Students’ Strategic Usage of Formative Quizzes in an Undergraduate Course in Abstract Algebra

Besides homework assignments, optional quizzes are a commonly used means for formative assessment in tertiary mathematics education. Instructors, for example, implement these to help students detect gaps in their understanding, or to foster a continuous and active engagement with the content. The extent to which these goals are reached, however, strongly depends on how students actually use such quizzes, which is currently an underexplored topic. We investigated this issue in an undergraduate abstract algebra course with a study using a mixed-methods design. Unlike previous research suggesting that students use optional quizzes in tertiary mathematics courses mainly for rehearsal or for a final self-check of their own performance, our study indicates that students also use such quizzes in many other ways, for example for planning their further learning, or for deepening their understanding of the content of the course. Furthermore, our study shows differences regarding students’ quiz usage during the semester and when preparing for the final exam. Finally, the data propose factors that influence the way students use optional quizzes, for example time constraints due to other obligations during the semester, the perceived difficulty of the questions, or the opportunity to discuss these with peers. This leads to several suggestions regarding the implementation of optional quizzes into tertiary mathematics courses.

How mathematicians assign homework problems in abstract algebra courses

While many aspects of the teaching and learning of advanced mathematics have been explored, the role, construction, and values of homework have been virtually ignored. This report draws on task-based interviews with six mathematicians to explore the relationship between an instructor’s learning goals and factors considered when selecting homework problems. All participants viewed homework as critical to student learning, and the majority of the participants’ claims focused on either the mathematics or how the problem would help students learn; no instructor gave primacy to evaluative reasons for homework. We highlight six themes used by participants to evaluate and select items for inclusion in homework. They are (1) knowing and recalling axioms and definitions, (2) developing an arsenal of examples, (3) developing new problem approaches, (4) remediating misconceptions, (5) making connections to prior and future material, and (6) valuing reading notes or text.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

We describe the design and implementation of lessons in undergraduate abstract algebra that integrate applications to teaching high school mathematics. Each lesson consists of a pre-activity, class activity, homework set, assessment questions and detailed instructor notes. The included applications intend to bridge advanced mathematics concepts to secondary school mathematics content for prospective secondary mathematics teachers, as well as provide instructors with a useful resource. Each lesson includes four guiding features: addressing meaningful content; making connections to mathematics teaching; emphasizing active learning; and providing comprehensive notes for instructors. Interviews with instructors and their students document how the instructors adapted the lessons to their classrooms and how the lessons influenced undergraduates' mathematical understanding.

Improving monoid concept understanding ability through problem posing learning model at abstract algebra course

This study aims to improve the ability to understand monoid concepts through Problem Posing learning. The method used is a quasi-experiment. The research design used was pre-test post-test group design. The research sample was 32 undergraduate students of the Mathematics Education Study Program FKIP Universitas Bengkulu at the Algebra course of even semester of 2018-2019 academic year. The instrument used was a 4 item test about the concept of monoid. Data were analysed qualitatively and quantitatively. The results of the study are as follows: 1) an increase in the ability to understand the monoid concept of students by 0.67, including the medium category; 2) the achievement of the ability to understand the monoid concept is 74.75 in the high category. The results of this study have implications for the importance of the role of questions in learning mathematics.

Analysis of Mathematical Proof Ability in Abstract Algebra Course

Mathematical proving is an important ability to learn abstract algebra. Many students, however, found difficulties in solving problems involving mathematical proof. This research aims to describe the students' mathematical proving ability and to find out the difference of the ability among students in private universities with three different levels of accreditation – A, B, and C. We used descriptive and comparative methods to reach the goals by involving mathematics education department students from A, B, and C-accredited private universities as its subjects. We used a test and interview to collect the data. The data of the students' mathematical proving ability were then statistically described and then compared among the three subject categories using the Kruskal-Wallis test and U Mann Whitney post hoc test. The results suggest that the students' mathematical proving ability from the A, B, and C-accredited universities respectively were 77.14 (high category), 39.32 (low category), and 36.78 (low category). Furthermore, the comparison results suggest that the significant differences only happen between universities with A and B accreditation level, and between the ones with A and C accreditation. Based on these findings, the mathematical proving ability of the students from B and C-accredited universities still needs to be improved by making the students accustomed to exercising with proof problems, motivating them to learn, and providing them learning materials that are easy to understand.

Examples of groups in abstract Algebra Course Books

This study has been conducted with the aim to examine the examples of Abelian and non-Abelian groups given in the abstract algebra course books in the university level. The non-examples of Abelian groups serve as examples of non-Abelian groups. Examples with solutions in the course books are trusted by the students and hence miscellaneous of those are required to clarify the subject in enough detail. The results of the current study show that the examples of Abelian groups are about the same among three course books, including number sets only with known operations. The examples of non-Abelian groups are rare in comparison and encapsulate the nonnumeric sets which are novel to students. The current study shows the mentioned examples are not sufficiently examined in the course books. Suggestions for the book writers are given in the study. Mainly it is suggested that more and various examples of Abelian and especially non-Abelian groups should be included in the course books.

Abstract Algebra to Secondary School Algebra: Building Bridges.

The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a three-semester project to help secondary mathe matics teacher candidates inform their teaching using abstract algebra.

The Implementation of Online Tutorial in Abstract Algebra Course and Pre-Service Teachers’ Perceptions toward Their Academic Achievement

In this 21st century education, technology plays a significant role. Blended learning, combining face-to-face class and e-learning, is being implemented in a Mathematics course at a teacher college located in Jakarta, Indonesia. Online tutorial is being used to discuss about topic in this course. The aim of this study is to explore pre-service teachers’ perception toward their achievement in learning Abstract Algebra with the implementation of Online Tutorial. The participants are pre-service teachers intake 2009 majoring Mathematics Education, enrolling in Abstract Algebra Course. Data are obtained from questionnaire, interview and focus group discussion about perceptions toward their academic achievement and interview and then analyzed qualitatively. Various responses are shown by students. There are positive response that support this Online Tutorial as a help for learning Abstract Algebra better, but others also think that the implementation need improvement to become more engaging and can cover all layer of students. Keywords - Achievement, Blended Learning, Higher Education, Mathematics, Online Tutorial, Perception, Pre-service Teachers

A Dynamical Systems Proof of Fermat's Little Theorem

(2003). A Dynamical Systems Proof of Fermat's Little Theorem. Mathematics Magazine: Vol. 76, No. 1, pp. 48-51.

An Attempt to Foster Students' Construction of Knowledge During a Semester Course in Abstract Algebra

An Attempt to Foster Students' Construction of Knowledge During a Semester Course in Abstract Algebra

A Historically Focused Course in Abstract Algebra

A Historically Focused Course in Abstract Algebra

A Historically Focused Course in Abstract Algebra

(1998). A Historically Focused Course in Abstract Algebra. Mathematics Magazine: Vol. 71, No. 2, pp. 105-111.

Using the Magma Computer Algebra System in Abstract Algebra Courses

Students meeting their first serious course in abstract algebra commonly experience difficulty understanding many of the basic concepts. This makes it very hard for them to interpret and generate proofs. The first step in reaching an understanding of such concepts is often best achieved by constructing and manipulating instances in particular algebraic structures. Computer algebra systems open up the possibility of students being able to experiment rapidly and conveniently with such concepts in a variety of structures having non-trivial size and complexity. The new Magma computer algebra system has a syntax and semantics based directly on fundamental algebraic notions, and consequently should provide an appropriate learning environment for those branches of mathematics that are heavily algebraic in nature. This paper describes the development of Magma-based exercises and a course methodology that utilizes Magma as a key learning tool in a Pass-level Rings and Fields course given at the University of Sydney.

Software as math

Discrete mathematics and abstract algebra courses are frequently required for computer science and engineering undergraduates. One could say these subjects define the mathematical basis for computing. Unfortunately, interest in the topics discussed in these courses frequently ends with the final exam. This paper presents a reason to continue looking and exploring.

Writing Assignments in an Abstract Algebra Course

Writing Assignments in an Abstract Algebra Course

A first course in abstract algebra (4th edition), by John B. Fraleigh. Pp 518. £17·95. 1989. ISBN 0-201-16847-2 (Addison-Wesley)

A first course in abstract algebra (4th edition), by John B. Fraleigh. Pp 518. £17·95. 1989. ISBN 0-201-16847-2 (Addison-Wesley)

A use for computing in an abstract algebra course

A use for computing in an abstract algebra course